# Assistant conversation WC6ZWF - Canonical URL: - Resource IRI: - Type: assistant conversation - Mode: chat ## Transcript ### 1. User Import the attached PDF documents. [Uploaded PDFs] - Toward_a_Realistic_Science_of_Environments.pdf: Sheaf file #JAZ9RM - True_Grid.pdf: Sheaf file #JCG48M - Naive_Physics.pdf: Sheaf file #3X4CUE - More_Things_in_Heaven_and_Earth.pdf: Sheaf file #FDTKYK - New_Foundations_for_Qualitative_Physics.pdf: Sheaf file #ERXDR8 - Vague_Reference_and_Approximating_Judgements.pdf: Sheaf file #6GQCXJ - Categories_in_Top_Level_Ontologies.pdf: Sheaf file #D3DKEW - Ontologies_of_Common_Sense_Physics_and_Mathematics.pdf: Sheaf file #DP29LL - On_Credentials.pdf: Sheaf file #2WJXNZ - Truth_and_the_Visual_Field.pdf: Sheaf file #DK95UN ### 2. Assistant tool calls #### Tool call: `document_import` ```json { "action": "stage", "file_ids": [ "JAZ9RM", "JCG48M", "3X4CUE", "FDTKYK", "ERXDR8", "6GQCXJ", "D3DKEW", "DP29LL", "2WJXNZ", "DK95UN" ], "name": "Import 10 attached PDFs" } ``` ### 3. Tool result: document_import { "status": { "counts": { "pending": 10 }, "files": [ { "error": null, "status": "pending", "file_id": "2WJXNZ", "execution_id": null }, { "error": null, "status": "pending", "file_id": "3X4CUE", "execution_id": null }, { "error": null, "status": "pending", "file_id": "6GQCXJ", "execution_id": null }, { "error": null, "status": "pending", "file_id": "D3DKEW", "execution_id": null }, { "error": null, "status": "pending", "file_id": "DK95UN", "execution_id": null }, { "error": null, "status": "pending", "file_id": "DP29LL", "execution_id": null }, { "error": null, "status": "pending", "file_id": "ERXDR8", "execution_id": null }, { "error": null, "status": "pending", "file_id": "FDTKYK", "execution_id": null }, { "error": null, "status": "pending", "file_id": "JAZ9RM", "execution_id": null }, { "error": null, "status": "pending", "file_id": "JCG48M", "execution_id": null } ] }, "sources": [], "action": "stage", "run_id": "K6HSTJ", "file_ids": [ "JAZ9RM", "JCG48M", "3X4CUE", "FDTKYK", "ERXDR8", "6GQCXJ", "D3DKEW", "DP29LL", "2WJXNZ", "DK95UN" ], "run_iri": "https://m.sheaf.less.rest/K6HSTJ" } ### 4. Assistant tool calls #### Tool call: `document_import` ```json { "action": "extract", "run_id": "K6HSTJ" } ``` ### 5. Tool result: document_import { "status": { "counts": { "completed": 10 }, "files": [ { "error": null, "status": "completed", "file_id": "2WJXNZ", "execution_id": "pex_XQBUjIhGzaK6" }, { "error": null, "status": "completed", "file_id": "3X4CUE", "execution_id": "pex_wQnU6ZDZzw2d" }, { "error": null, "status": "completed", "file_id": "6GQCXJ", "execution_id": "pex_SUfAJAP8YIKZ" }, { "error": null, "status": "completed", "file_id": "D3DKEW", "execution_id": "pex_Joppj68TFLEh" }, { "error": null, "status": "completed", "file_id": "DK95UN", "execution_id": "pex_QQnjm6YXbZuE" }, { "error": null, "status": "completed", "file_id": "DP29LL", "execution_id": "pex_w9sHaOticPAP" }, { "error": null, "status": "completed", "file_id": "ERXDR8", "execution_id": "pex_58AmF-BaDkRi" }, { "error": null, "status": "completed", "file_id": "FDTKYK", "execution_id": "pex_QDtN230jlQI_" }, { "error": null, "status": "completed", "file_id": "JAZ9RM", "execution_id": "pex_WQSHI1v31dsz" }, { "error": null, "status": "completed", "file_id": "JCG48M", "execution_id": "pex_24m2Jl7AeYuY" } ] }, "action": "extract", "completed": 1, "run_id": "K6HSTJ", "submitted": 10 } ### 6. Assistant tool calls #### Tool call: `document_import` ```json { "action": "inspect", "run_id": "K6HSTJ" } ``` ### 7. Tool result: document_import { "status": { "counts": { "completed": 10 }, "files": [ { "error": null, "status": "completed", "file_id": "2WJXNZ", "execution_id": "pex_XQBUjIhGzaK6" }, { "error": null, "status": "completed", "file_id": "3X4CUE", "execution_id": "pex_wQnU6ZDZzw2d" }, { "error": null, "status": "completed", "file_id": "6GQCXJ", "execution_id": "pex_SUfAJAP8YIKZ" }, { "error": null, "status": "completed", "file_id": "D3DKEW", "execution_id": "pex_Joppj68TFLEh" }, { "error": null, "status": "completed", "file_id": "DK95UN", "execution_id": "pex_QQnjm6YXbZuE" }, { "error": null, "status": "completed", "file_id": "DP29LL", "execution_id": "pex_w9sHaOticPAP" }, { "error": null, "status": "completed", "file_id": "ERXDR8", "execution_id": "pex_58AmF-BaDkRi" }, { "error": null, "status": "completed", "file_id": "FDTKYK", "execution_id": "pex_QDtN230jlQI_" }, { "error": null, "status": "completed", "file_id": "JAZ9RM", "execution_id": "pex_WQSHI1v31dsz" }, { "error": null, "status": "completed", "file_id": "JCG48M", "execution_id": "pex_24m2Jl7AeYuY" } ] }, "action": "inspect", "files": [ { "status": "completed", "file_id": "2WJXNZ", "quality": { "blocks": 193, "pages": 21, "math_expressions": 0, "page_continuations": 3, "block_types": { "Caption": 2, "Footnote": 41, "ListGroup": 4, "PageFooter": 1, "PageHeader": 42, "Picture": 2, "SectionHeader": 21, "Table": 1, "Text": 79 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/2WJXNZ.datalab.json" }, { "status": "completed", "file_id": "3X4CUE", "quality": { "blocks": 233, "pages": 22, "math_expressions": 0, "page_continuations": 7, "block_types": { "ListGroup": 3, "Picture": 1, "SectionHeader": 22, "Text": 207 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/3X4CUE.datalab.json" }, { "status": "completed", "file_id": "6GQCXJ", "quality": { "blocks": 195, "pages": 20, "math_expressions": 286, "page_continuations": 9, "block_types": { "Caption": 6, "Equation": 28, "Figure": 2, "ListGroup": 3, "PageFooter": 1, "PageHeader": 20, "Picture": 5, "SectionHeader": 23, "Text": 107 }, "empty_equation_blocks": 0, "equation_blocks": 28, "pages_with_math": 13 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/6GQCXJ.datalab.json" }, { "status": "completed", "file_id": "D3DKEW", "quality": { "blocks": 302, "pages": 31, "math_expressions": 15, "page_continuations": 3, "block_types": { "Caption": 4, "Figure": 2, "Footnote": 25, "ListGroup": 14, "PageFooter": 32, "PageHeader": 93, "SectionHeader": 24, "Table": 2, "Text": 106 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 3 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/D3DKEW.datalab.json" }, { "status": "completed", "file_id": "DK95UN", "quality": { "blocks": 79, "pages": 8, "math_expressions": 0, "page_continuations": 5, "block_types": { "Caption": 1, "ListGroup": 1, "PageFooter": 1, "PageHeader": 13, "Picture": 1, "SectionHeader": 10, "Text": 52 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/DK95UN.datalab.json" }, { "status": "completed", "file_id": "DP29LL", "quality": { "blocks": 313, "pages": 32, "math_expressions": 255, "page_continuations": 2, "block_types": { "Caption": 10, "Equation": 12, "Figure": 10, "Footnote": 59, "ListGroup": 6, "PageFooter": 32, "PageHeader": 1, "SectionHeader": 27, "Text": 156 }, "empty_equation_blocks": 0, "equation_blocks": 12, "pages_with_math": 22 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/DP29LL.datalab.json" }, { "status": "completed", "file_id": "ERXDR8", "quality": { "blocks": 121, "pages": 13, "math_expressions": 47, "page_continuations": 6, "block_types": { "Footnote": 15, "PageFooter": 13, "PageHeader": 26, "SectionHeader": 7, "Text": 60 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 3 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/ERXDR8.datalab.json" }, { "status": "completed", "file_id": "FDTKYK", "quality": { "blocks": 81, "pages": 15, "math_expressions": 0, "page_continuations": 1, "block_types": { "Caption": 2, "Footnote": 8, "ListGroup": 6, "PageHeader": 15, "Picture": 2, "SectionHeader": 6, "Text": 42 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/FDTKYK.datalab.json" }, { "status": "completed", "file_id": "JAZ9RM", "quality": { "blocks": 77, "pages": 11, "math_expressions": 0, "page_continuations": 4, "block_types": { "Footnote": 2, "ListGroup": 1, "PageFooter": 11, "SectionHeader": 9, "Text": 54 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/JAZ9RM.datalab.json" }, { "status": "completed", "file_id": "JCG48M", "quality": { "blocks": 111, "pages": 17, "math_expressions": 0, "page_continuations": 6, "block_types": { "Caption": 4, "Footnote": 8, "ListGroup": 5, "PageHeader": 1, "Picture": 3, "SectionHeader": 13, "Table": 1, "Text": 76 }, "empty_equation_blocks": 0, "equation_blocks": 0, "pages_with_math": 0 }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/K6HSTJ/JCG48M.datalab.json" } ], "run_id": "K6HSTJ" } ### 8. Assistant tool calls #### Tool call: `document_import` ```json { "action": "import", "run_id": "K6HSTJ" } ``` ### 9. Tool result: document_import { "next": "Resolve metadata, rebuild indexes, and validate reader pages.", "action": "import", "documents": [ { "status": "imported", "title": null, "file_id": "2WJXNZ", "document_id": "KYQGNH", "document_iri": "https://m.sheaf.less.rest/KYQGNH" }, { "status": "imported", "title": null, "file_id": "3X4CUE", "document_id": "KY3Y9U", "document_iri": "https://m.sheaf.less.rest/KY3Y9U" }, { "status": "imported", "title": null, "file_id": "6GQCXJ", "document_id": "XZX6PE", "document_iri": "https://m.sheaf.less.rest/XZX6PE" }, { "status": "imported", "title": null, "file_id": "D3DKEW", "document_id": "88BVY3", "document_iri": "https://m.sheaf.less.rest/88BVY3" }, { "status": "imported", "title": null, "file_id": "DK95UN", "document_id": "PHAFYA", "document_iri": "https://m.sheaf.less.rest/PHAFYA" }, { "status": "already_imported", "file_id": "DP29LL", "document_id": "GN66WW", "document_iri": "https://m.sheaf.less.rest/GN66WW" }, { "status": "imported", "title": null, "file_id": "ERXDR8", "document_id": "TQPVBD", "document_iri": "https://m.sheaf.less.rest/TQPVBD" }, { "status": "imported", "title": null, "file_id": "FDTKYK", "document_id": "FJ5KCA", "document_iri": "https://m.sheaf.less.rest/FJ5KCA" }, { "status": "imported", "title": null, "file_id": "JAZ9RM", "document_id": "2F8T3H", "document_iri": "https://m.sheaf.less.rest/2F8T3H" }, { "status": "imported", "title": null, "file_id": "JCG48M", "document_id": "DT9Y7X", "document_iri": "https://m.sheaf.less.rest/DT9Y7X" } ], "run_id": "K6HSTJ" } ### 10. Assistant tool calls #### Tool call: `document_import` ```json { "action": "metadata", "run_id": "K6HSTJ" } ``` ### 11. Tool result: document_import { "action": "metadata", "documents": [ { "status": "already_resolved", "document_id": "GN66WW" }, { "match": { "reason": "title match accepted for Crossref type journal-article", "source": "doi", "identifier": "10.1515/jso-2019-0034", "doi": "10.1515/jso-2019-0034", "score": 1.0, "accept?": true, "crossref_title": "On Credentials", "crossref_type": "journal-article" }, "metadata": { "title": "On Credentials", "usage": { "cost": { "total": 0.001052, "tokens": 0.001052, "tools": 0.0, "storage": 0.0, "images": 0.0, "input_cost": 7.9e-4, "output_cost": 2.62e-4, "reasoning_cost": 0.0, "line_items": [ { "count": 3161, "id": "token.input", "kind": "tokens", "component": "token.input", "cost": 7.9e-4, "quantity": 3161 }, { "count": 175, "id": "token.output", "kind": "tokens", "component": "token.output", "cost": 2.62e-4, "quantity": 175 } ] }, "output_tokens": 175, "input_tokens": 3161, "total_cost": 0.001052, "reasoning_tokens": 0, "input_cost": 7.9e-4, "output_cost": 2.62e-4, "reasoning_cost": 0.0, "cached_tokens": 0, "total_tokens": 3336, "tool_usage": {}, "image_usage": {}, "cache_creation_tokens": 0 }, "year": "2020", "model": "google:gemini-3.1-flash-lite-preview", "notes": "Published online August 7, 2020. The journal title is inferred from the DOI prefix '10.1515/jso', which corresponds to the Journal of Social Ontology.", "issue": null, "pages": null, "authors": [ "Barry Smith", "Olimpia Giuliana Loddo", "Giuseppe Lorini" ], "doi": "10.1515/jso-2019-0034", "volume": null, "isbn": null, "source_filename": null, "publication": "Journal of Social Ontology", "confidence": "high" }, "document_id": "KYQGNH", "wrote": true }, { "match": { "reason": "title match score below 0.62", "source": "doi", "identifier": "10.1080/09515089408573121", "doi": "10.1080/09515089408573121", "score": 0.5, "accept?": false, "crossref_title": "Naive physics", "crossref_type": "journal-article" }, "metadata": { "title": "Naïve Physics: An Essay in Ontology", "usage": { "cost": { "total": 0.001117, "tokens": 0.001117, "tools": 0.0, "storage": 0.0, "images": 0.0, "input_cost": 8.67e-4, "output_cost": 2.5e-4, "reasoning_cost": 0.0, "line_items": [ { "count": 3468, "id": "token.input", "kind": "tokens", "component": "token.input", "cost": 8.67e-4, "quantity": 3468 }, { "count": 167, "id": "token.output", "kind": "tokens", "component": "token.output", "cost": 2.5e-4, "quantity": 167 } ] }, "output_tokens": 167, "input_tokens": 3468, "total_cost": 0.001117, "reasoning_tokens": 0, "input_cost": 8.67e-4, "output_cost": 2.5e-4, "reasoning_cost": 0.0, "cached_tokens": 0, "total_tokens": 3635, "tool_usage": {}, "image_usage": {}, "cache_creation_tokens": 0 }, "year": "1994", "model": "google:gemini-3.1-flash-lite-preview", "notes": "The document also contains a repository reference (ijn_01627178) from HAL.", "issue": "2", "pages": "225-244", "authors": [ "Barry Smith", "Roberto Casati" ], "doi": "10.1080/09515089408573121", "volume": "7", "isbn": null, "source_filename": null, "publication": "Philosophical Psychology", "confidence": "high" }, "document_id": "KY3Y9U", "wrote": false }, { "match": { "reason": "no DOI or ISBN found", "source": "none", "score": 0.0, "accept?": false }, "metadata": { "title": "Vague Reference and Approximating Judgments", "usage": { "cost": { "total": 6.43e-4, "tokens": 6.43e-4, "tools": 0.0, "storage": 0.0, "images": 0.0, "input_cost": 4.35e-4, "output_cost": 2.08e-4, "reasoning_cost": 0.0, "line_items": [ { "count": 1741, "id": "token.input", "kind": "tokens", "component": "token.input", "cost": 4.35e-4, "quantity": 1741 }, { "count": 139, "id": "token.output", "kind": "tokens", "component": "token.output", "cost": 2.08e-4, "quantity": 139 } ] }, "output_tokens": 139, "input_tokens": 1741, "total_cost": 6.43e-4, "reasoning_tokens": 0, "input_cost": 4.35e-4, "output_cost": 2.08e-4, "reasoning_cost": 0.0, "cached_tokens": 0, "total_tokens": 1880, "tool_usage": {}, "image_usage": {}, "cache_creation_tokens": 0 }, "year": "2003", "model": "google:gemini-3.1-flash-lite-preview", "notes": "Article title and authors identified from the first page; 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Assistant tool calls #### Tool call: `get_document` ```json { "id": "88BVY3" } ``` ### 15. Tool result: get_document DOCUMENT #88BVY3 Title: Untitled thesis Outline: - 1 #DY7KEM Categories in Top-Level Ontologies: Revisiting the Aristotelian Background - 1.1 #Y994UA 1. What are Categories? - 1.1.1 #HC4LC5 1.1 The word “category” - 1.1.2 #5JW3Z5 1.2 The Interpretation of Aristotle's Categories - 1.1.3 #CM3E3X 1.3 Aristotle’s Ten Categories - 1.2 #9UCDSE 2. Universals and Particulars - 1.3 #FHJA5L 3. Dependent and Independent Entities - 1.3.1 #9FLUAK 3.1 The Priority of Particular Substances - 1.3.2 #94L6NZ 3.2 The Relation of Dependence - 1.3.3 #T27VSK 3.3 Generic Dependence - 1.4 #T97WK5 4. Continuants and Occurrents - 1.4.1 #SC7ZP5 4.1 Time and Existence - 1.4.2 #HNHSLF 4.2 SNAP and SPAN - 1.5 #L5HS8A 5. Putting It All Together - 1.5.1 #C8ATRQ 5.1 The Ontological Square - 1.5.2 #F997NM 5.2 Adding Processes: The Ontological Sextet - 1.5.3 #CTHD63 5.3 Ontological relations - 1.6 #K2VRML 6. Complex Entities - 1.6.1 #7AVA8H 6.1 States of Affairs - 1.6.2 #78RVGJ 6.2 Sets - 1.6.3 #5QFVKE 6.3 Mereology: Wholes and Their Parts - 1.6.4 #CCUHN7 6.4 Classes - 1.7 #48D2BK 7. Top-Level Ontologies Today - 1.8 #NHHNAK References ### 16. Assistant tool calls #### Tool call: `read` ```json { "blocks": [ "DY7KEM" ], "expand": true } ``` ### 17. Tool result: read DOCUMENT #88BVY3 Untitled thesis SECTION #DY7KEM Categories in Top-Level Ontologies: Revisiting the Aristotelian Background EXCERPT #ZJY3G9 p. 0 Ludger Jansen 1 , Barry Smith 2 EXCERPT #PWGMZ6 p. 0 Abstract. In the field of applied ontology, it is now commonplace to refer to a top-level ontology, and the Basic Formal Ontology (BFO) has even been recognized as an ISO standard. Like other contemporary top-level ontologies, BFO makes use of distinctions that have been developed in philosophy, many of which are already to be found in Aristotele. In this essay, we revisit Aristotele's metaphysics and discuss the similarities and differences with BFO. EXCERPT #3VQT6C p. 0 Traditionally, the task of ontology has been to represent reality, for example in terms of a division between different modes of being. More recently, ontologies are being used to support not only philosophers but also scientists and others in their representation of reality. Indeed, with the advent of computers there has arisen a new discipline of applied ontology, whose task is to support the integration and discoverability of data deriving from different sources, nowadays including financial, industrial, governmental and other organizations, by providing logically supported classification systems (Munn 2008). 3 EXCERPT #R6E2X4 p. 0 An important instrument for all these purposes is the technique of classification. But, in any classification we have to select what will be the classes or kinds we place at the very top. What should the top level of an ontology, or indeed of any classification, look like? What are the most general classes of all classifications? It is upon questions such as this that we shall focus in what follows. EXCERPT #J2KENE p. 0 1 Cusanus Professor for Philosophy at the PTH College Brixen in Bressanone, Italy. Supernumery Professor at the University of Rostock. Email: ludger.jansen@pthsta.it. Affiliation: PTH Brixen College; University of Rostock. ORCID: https://orcid.org/0000-0002-0097-6359 . EXCERPT #PWSLWU p. 0 2 Distinguished Professor of Philosophy, Professor of Biomedical Informatics. Professor of Computer Science and Engineering. Director of the National Center for Ontological Research. Email: phismith@buffalo.edu. Affiliation: University at Buffalo, NY, USA. ORCID: https://orcid.org/0000-0003-1384-116X . EXCERPT #X82DHV p. 0 3 This essay is an updated and extended version of Jansen, Ludger. "Categories: The top-level ontology". Applied Ontology. An Introduction, eds. K Munn, B. Smith, Frankfurt: Ontos 2008. The text has been heavily revised, and several old sections have been deleted. Introduction and Sect. 7 are totally new. To a fair extend, references originate from this first version, though they have been supplemented by pointers to the more recent literature. EXCERPT #4MS9GN p. 0 EXCERPT #SSNL3H p. 1 EXCERPT #K42RSM p. 1 EXCERPT #X8HUM6 p. 1 EXCERPT #YS4B7A p. 1 Authors in the fields of informatics and knowledge representation have suggested various answers to these questions. An early candidate for the role was SUMO, short for ‘Suggested Upper Merged Ontology’, 4 which was developed from an open-source project bringing together freely available, non-commercial ontologies into a common system. Nowadays Basic Formal Ontology (BFO) is a popular candidate top-level ontology, with over 600 domain ontologies defined in its terms (Otte 2022). 5 BFO has been recognized by the International Standards Organization (ISO) and the International Electrotechnical Commission (IEC) as an ISO standard (ISO/IEC: 21838-2), confirming that it satisfies the requirements for being a top-level ontology set forth in standard ISO/IEC:21838-1 (2021). In addition, use of BFO as top-level ontology has been mandated by the Joint Enterprise Standards Committee of the US Department of Defense and Intelligence Community. BFO also provides the architecture for the agency wide data repository of the US Department of Homeland Security. EXCERPT #83YWML p. 1 The first thinkers to address the idea of a standard ontology were, in fact, philosophers, most notably Aristotle in his short treatise on the Categories . This classical text can be read as addressing exactly those questions that matter to applied ontologists today when they think about top-level ontologies (Jansen 2007). From the point of view of traditional philosophy, the question of a top-level ontology is tantamount to the question of the most basic categories of being, and our strategy for addressing this question is to examine Aristotle’s theory of categories and describing some of the ways in which this theory has influenced current work in the discipline of applied ontology. EXCERPT #Q7G3D9 p. 1 There were three principal influences which helped to shape the earliest versions of BFO: Aristotle’s Categories ; the literature of geographic information science (cf. Smith & Mark 2001); and the realist phenomenological approach to ontology pioneered by Edmund Husserl, Adolf Reinach, Roman Ingarden and other members of the so-called Munich-Göttingen school (cf. Smith 1996). We will here address only the first of these influences by presenting the ontology laid out by Aristotle in his Categories and showing how it maps to the philosophical foundations underlying BFO. We first clarify what a category is (section 1). From there, we continue to introduce three ontological dichotomies that are at the root of both Aristotle’s ontology in the Categories and of BFO: the distinction between universals and particulars (section 2), the distinction between dependent and independent entities (section 3), and the distinction between continuants and occurrents (section 4). We then present two diagrams that synthesize various ontological views that can be generated along the lines of these three distinctions: the EXCERPT #PYN5H7 p. 1 4 See Jansen 2008 for a discussion of other early suggestions, such as OpenCyc and the Sowa Diamond. EXCERPT #74VTPZ p. 1 5 For a regularly updated list, cf. http://basic-formal-ontology.org/users.html . EXCERPT #TD6RVA p. 1 EXCERPT #WQ4KLB p. 2 EXCERPT #GFH6NE p. 2 EXCERPT #R9A8PF p. 2 EXCERPT #DR73QZ p. 2 ontological square, inspired by Aristotle's Categories , and the ontological sextet, which is, as we claim, a more adequate representation of the ontological structure of reality, and represents the basic structure of BFO (section 5). We conclude by pointing out some points where we need to go beyond the simple picture of Aristotle's Categories in order to represent adequately our world as described, for example, by biology and medicine (section 6). SECTION #Y994UA 1. What are Categories? SECTION #HC4LC5 1.1 The word “category” EXCERPT #6DC88W p. 2 As far as we know, Aristotle was the first to use the Greek word kategoria as a technical term in the context of philosophy. Originally, the noun kategoria and its corresponding verb, katēgorein , belonged to legal discourse. There, kategoria means the accusation in front of the judge, and katēgorein means to accuse someone . Probably because an accusation asserts something of someone, the verb can also mean to make known or to assert , and it was used in this way by Plato. 6 EXCERPT #Q6EFRY p. 2 Aristotle uses the active verb phrase katēgorein ti tinos in the sense of: to assert something about something, but even more often he uses the passive katēgoreisthai ti tinos or katēgoreisthai ti kata tinos in the sense of: is said of something. The noun kategoria is used by him in a variety of ways, including using the plural of the noun in the sortal sense to mean ‘kinds of predicates’ or ‘kinds of predication’. Such kinds of predicates are for example quantitative or qualitative predicates such as ‘3 foot long’ or ‘wet’. It is in this sense that the Greek word kategoria can be translated into English as category (Jansen, 2006). EXCERPT #UBFQDR p. 2 We have evidence that Aristotle's conception of the categories developed in three phases. First, as in Topics I 9, the distinction of different categories was meant exclusively as a classification of predicates. In this first phase, the categories served as aids for finding arguments and for avoiding or discovering false inferences, and it was in this way that talk of categories found a place in the theory of argumentation, which is what the Topics are about. EXCERPT #ZRPF6K p. 2 The second phase is represented in Aristotle's Categories . Here the division of categories encompasses, not only predicate terms such as ‘is tall’ or ‘is hungry’, but also subject terms, including proper names such as ‘Socrates’ or ‘Plato’, which function in sentences only as the subject of predication but never as predicates in their own right EXCERPT #6NGW77 p. 2 6 See for example Theaetetus 208b; Phaedrus 73b. Theaetetus 167a links the two meanings. EXCERPT #ATZMRT p. 2 EXCERPT #3DKJYE p. 3 EXCERPT #FEHGNJ p. 3 EXCERPT #A7SZG8 p. 3 EXCERPT #QPD8KG p. 3 ( Categories 5, 3a 36-37). This represents a step in the direction of ontology and away from concerns with the theory of argumentation. EXCERPT #8UXKWJ p. 3 The third phase finds its expression in the Metaphysics . There we find Aristotle's famous observation that 'to be' and 'a being' are used in as many different ways as there are categories ( Metaphysics V 7, 1017a 22-23). Here, the division into separate categories becomes a full-fledged part of one of the most important of Aristotle's ontological works. SECTION #5JW3Z5 1.2 The Interpretation of Aristotle's Categories EXCERPT #5PWHAP p. 3 Aristotle's theory of categories was the subject of much dispute in antiquity, and it has been interpreted in a variety of ways in the history of philosophy ever since. Partly, this has to do with the fact that category theory had many different facets even in the works of Aristotle himself. This came about because Aristotle repeatedly subjected his ideas to further development and highlighted different aspects when presenting his theory. But we can distinguish four prototypical conceptions of what categories are (which often appear in combination), according to whether they classify: EXCERPT #9SQTMY p. 3 (1) subject and predicate terms and their associated meanings , (2) mental or extra-mental concepts , (3) meanings of the copula 'is', or (4) beings or entities . 7 EXCERPT #MPUNV9 p. 3 Here, we draw on the last of these, which was certainly the main conception of the late Aristotle, namely that categories are the highest species of beings. EXCERPT #T8S5AC p. 3 One reason why the question of the highest species of beings is important turns on the way in which definitions are conceived by Aristotle and his successors. Aristotle himself was interested in real definitions , that is, in definitions of things. A definition is then a phrase indicating a thing's essence ( ti esti , what it is). Only substances have essences, and so only substances are definable – an outcome that is confusingly bolstered by the fact that the same word ousia is used to mean both 'substance' and 'essence' (Jansen 2017). The essence of a thing is what is expressed in its definition. EXCERPT #KJY27Q p. 3 One standard technique for creating definitions has its roots in Aristotle's thinking on this topic, and thus in some circles of contemporary applied ontology its results are referred to as 'Aristotelian definitions' (Rosse, Mejino & Jose 2003). Such Aristotelian definitions are constructed by joining a genus term with a specific difference, following the template: EXCERPT #WD8PWD p. 3 7 For these four options cf. Bonitz 1853, Ebert 1985, Kahn 1978, Oehler 1986. EXCERPT #444ZK8 p. 3 EXCERPT #T6Y7GC p. 4 EXCERPT #P5NJAH p. 4 EXCERPT #ZVWSYB p. 4 EXCERPT #9LNDRD p. 4 An S is a G which Ds EXCERPT #3R2NDN p. 4 Here, ‘ S ’ stands for the species to be defined, ‘ G ’ for the genus term, and ‘ D ’ for the differentia , or in other words for the specific difference which picks out all and only those instances of G that are also instances of S , as for example, in the classical definition of human beings as rational animals, where “animal” is the genus term and rationality the specific difference. EXCERPT #XBVZR4 p. 4 A problem arises for such an approach to definitions, however, since it works only where the term to be defined has some more general term which can be used as starting point in creating its definition. Hence the need for a top-level ontology comprising ‘top-level general terms’, 8 which are primitive in the sense that they cannot be defined, though they can in various ways be elucidated , for example by specification of necessary conditions for instantiation, and by provision of examples. Aristotle’s ideas on categories represent the first attempt to create a top-level ontology so conceived. SECTION #CM3E3X 1.3 Aristotle’s Ten Categories EXCERPT #QDNE4G p. 4 There are many lists of categories in the extant works of Aristotle, of different length and content. 9 In Topics I 9, Aristotle says explicitly that there are ten categories, which he then proceeds to delineate. A list of ten categories can also be found in the Categories (see Table 1). The ontology thereby envisaged is nicely summarized in the following passage: EXCERPT #CY4UMJ p. 4 Expressions which are in no way composite signify substance, quantity, quality, relation, place, time, position, state, action, or affection. To sketch my meaning roughly, examples of substance are ‘man’ or ‘the horse’, of quantity, such terms as ‘two cubits long’ or ‘three cubits long’, of quality, such attributes as ‘white’, ‘grammatical’. ‘Double’, ‘half’, ‘greater’, fall under the category of relation; ‘in the market place’, ‘in the Lyceum’, under that of place; ‘yesterday’, ‘last year’, under that of time. ‘Lying’, ‘sitting’, are terms indicating position, ‘shod’, ‘armed’, state; ‘to lance’, ‘to cauterize’, action; ‘to be lanced’, ‘to be cauterized’, affection. (Aristotle, Categories 4, 1b25–2a4, transl. Edghill) EXCERPT #DS3UMG p. 4 8 As Bonaventure describes the problem in his Itinerarium mentis in Deum c. 3, 3: “The function of the intellectual faculty consists in understanding the meaning of terms [...]. Now, the intellect grasps the meanings of terms when it comprehends in a definition what a thing is. But definitions are constructed by using more universal terms; and these are defined by more universal terms until we come to the highest and most universal. Consequently, unless these latter are known, the less universal cannot be grasped in a definition.” EXCERPT #PFHPZK p. 4 9 A synopsis of these lists can be found in the appendix of Oehler 1986. EXCERPT #CSLK77 p. 4 EXCERPT #SB56WF p. 5 EXCERPT #39URVQ p. 5 EXCERPT #LV8X64 p. 5 EXCERPT #VYMPU5 p. 5 Table 1 lists the Greek terms Aristotle uses for his ten categories, together with their verbal translations, Latin equivalents, and some modern terms now in use. Aristotle presents this system of categories by adverting to how each category is represented in ordinary language. More particularly, Aristotle would sometimes name categories not directly, but rather by using questions whose answers would make reference to entities in the respective categories. Many of the names we currently use for these categories then have their origins in the corresponding Latin interrogative expressions (see Figure 3). EXCERPT #562MGJ p. 5 Table 1: Different Terms for Aristotle's Categories EXCERPT #BC8YN6 p. 5 Aristotle's Term English Translation Latin Term Modern Terms ti esti ousia What is it? Essence quod est, quiditas, essentia Essence, Substance poson How much? quantum, quantitas Quantum, Quantity poion How is it? quale, qualitas Quality pros ti Related to what? relativum Relative, Relation pou Where? ubi Place pote When? quando Time keisthein Lying, Being situated situs Position, Posture echein Having habitus poiein Doing agere Action paschein Suffering pati Passion EXCERPT #9W29QS p. 5 Kant accused Aristotle of choosing his categories in a 'rhapsodic manner', without any guiding principles. It is for this reason, or so Kant argues, that Aristotle could never be certain that his list of categories was complete ( Critique of Pure Reason , A 81 = B 106- EXCERPT #9TB2DR p. 5 EXCERPT #U7YJ7V p. 6 EXCERPT #C7AWW4 p. 6 EXCERPT #A857VN p. 6 EXCERPT #FHNAWL p. 6 107). Later Aristotelians, such as Thomas Aquinas 10 or Franz Brentano (1862) 11 , undertook the task of reconstructing a system that yields the Aristotelian categories in the precise order in which they are named and discussed in the Categories . 12 EXCERPT #Z2S7P7 p. 6 To do justice to Aristotle, however, we should note that the works by him that survived are mainly lecture notes and not polished works ready for publication. This helps to explain the disparities between Aristotle's various lists, disparities which draw attention also to the fact that the elements in his lists are not all of the same standing. EXCERPT #MMNMKM p. 6 There are three important dichotomies in terms of which we can understand how Aristotle's categories are organized: EXCERPT #NL95G9 p. 6 • they make room for both universals (kinds, types) and the particulars which are the instances of these universals (section 2 below); • they encompass dependent as well as independent entities (section 3); • they divide reality into continuants and occurents (section 4). EXCERPT #6S36FX p. 6 Taken together, these dichotomies help to systematise Aristotle's list of categories, as can be seen in Figure 1. EXCERPT #QAZCZE p. 6 10 See Aquinas, In Physicorum Aristotelis expositio III, lectio 5, Nr. 322 [15] and In Metaphysicorum Aristotelis expositio V, lectio 9, Nr. 891-892. EXCERPT #PVXEK3 p. 6 11 On Brentano's idea see Smith 1987. EXCERPT #5QTYMC p. 6 See also Simons 1992 and Jansen 2007 for new proposals for the hierarchical organization along the lines suggested in this communication. EXCERPT #CQ2DBG p. 6 12 See Jansen 2007 for a new suggestion of a hierarchy of Aristotle's categories along the lines suggested here. EXCERPT #U4EK9D p. 6 EXCERPT #4R7HC7 p. 7 EXCERPT #YFDCHE p. 7 EXCERPT #FUN5Y2 p. 7 EXCERPT #HSZQU3 p. 7 graph TD Particular[Particular] --> Independent[Independent] Particular --> Dependent[Dependent] Independent --> Substance[Substance] Dependent --> Occurrent[Occurrent] Dependent --> Continuant[Continuant] Occurrent --> WithoutChange[Without change] Occurrent --> InvolvingChange[Involving change] Occurrent --> WithinWhichChange[Within which change] Continuant --> InASingleBearer[In a single bearer] Continuant --> InAPlurality[In a plurality of bearers] Continuant --> WhereinBearers[Wherein bearers can be] WithoutChange --> Position[Position] InvolvingChange --> Action[Action] InvolvingChange --> Passion[Passion] WithinWhichChange --> Time[Time] InASingleBearer --> Quality[Quality] InASingleBearer --> Quantity[Quantity] InAPlurality --> Relation[Relation] Relation --> Having[Having] WhereinBearers --> Place[Place] A hierarchical tree diagram representing Aristotle's categories. The root is 'Particular', which branches into 'Independent' and 'Dependent'. 'Independent' leads to 'Substance'. 'Dependent' branches into 'Occurrent' and 'Continuant'. 'Occurrent' branches into 'Without change', 'Involving change', and 'Within which change'. 'Without change' leads to 'Position'. 'Involving change' branches into 'Action' and 'Passion'. 'Within which change' leads to 'Time'. 'Continuant' branches into 'In a single bearer', 'In a plurality of bearers', and 'Wherein bearers can be'. 'In a single bearer' branches into 'Quality' and 'Quantity'. 'In a plurality of bearers' leads to 'Relation', which then leads to 'Having'. 'Wherein bearers can be' leads to 'Place'. EXCERPT #J2G55G p. 7 Figure 1: A schematic representation of Aristotle's categories. EXCERPT #PMBZCT p. 7 Source: Smith 2022, modified after Jansen 2007. SECTION #9UCDSE 2. Universals and Particulars EXCERPT #CDAA5E p. 7 Concerning the first of these dichotomies, we note that universal and particular are not themselves categories in Aristotle's sense, and nor do they divide the categories into distinct groups. Rather, both universals and particulars can be classified according to the categories; hence the distinction between universals and particulars is orthogonal to Aristotle's categorial distinctions. We can call it 'transcategorical' (Lowe, 2006, 21). EXCERPT #8BRF4X p. 7 This dichotomy is given systematic treatment in the second chapter of the Categories , where Aristotle distinguishes between what can and what cannot be predicated of another entity. Predication requires an aspect of generality. Particulars, such as Socrates or my height, cannot be predicated of other entities. Sentences that contain as predicates expressions such as 'is Cicero' or 'is my height' are not predications in the technical sense at work here. Rather, they are identity claims comparable with 'Tully is Cicero' or 'My height is 5 feet'. A general expression such as 'human' can, in contrast, appear both as the subject and as the predicate of predicating assertions, as in 'A human is a vertebrate', and 'Cicero is a human'. EXCERPT #84SX4H p. 7 EXCERPT #FXKPUA p. 7 EXCERPT #JCTBJ2 p. 8 EXCERPT #X8M5EZ p. 8 EXCERPT #V3HHFL p. 8 SECTION #FHJA5L 3. Dependent and Independent Entities SECTION #9FLUAK 3.1 The Priority of Particular Substances EXCERPT #YSZF8P p. 8 Aristotle is quite clear that his ten categories are not to be viewed as equals. Pride of place is given to the so-called substances, most notably material objects like organisms. They are called 'substances' because their existence is in a sense basic – substances allow entities of all other categories to exist: qualities are always qualities of substances, relations are ultimately relations between substances, and so on. Qualities and relations are, thus, ontologically dependent on substances. From Aristotle's perspective, this dependence on substances is even that which guarantees the unity of ontology ( Metaphysics IV 2). EXCERPT #9G8EKR p. 8 Customarily, the dependent categories are called accidents and are placed in opposition to substances. Examples of accidents are being hot , being hungry , being seated , or being asleep . A traditional criterion for the opposition of substances and accidents can be found in the second chapter of the Categories : qualities and quantities are in a substance , while substances are not in but are, rather, identical with a substance. But it is not entirely clear how this 'being in something else' is to be understood (Smith & Mulligan 1982). A heart is in a body and a tapeworm is in its host, but these are not cases of the 'being in something else' relation that Aristotle had in mind. Thus he explicitly excludes 'being-in' in the sense in which a part is in a whole (as the heart is in the body). And a parasite such as a tapeworm is not even a part of its host, any more than a foetus is a part of its mother, or a tub of yogurt is a part of a refrigerator. EXCERPT #S3UG6W p. 8 The criterion of ontological dependence helps to solve this problem. The tapeworm could leave its host and move into another host. Both tapeworm and host are independent entities. A headache or an instance of colour, in contrast, cannot leave its bearer in this way and continue to exist. It is not possible for the Cheshire Cat to disappear and leave its grin behind. 13 The headache and the colour are dependent for their existence upon a specific bearer, a substance which has this headache, or this colour, among its properties. These properties cannot migrate from one substance to another. EXCERPT #YHEPTD p. 8 In addition, Aristotle also sees a similar dependence between universals and particulars. For him, universals are ontologically dependent on particulars that instantiate EXCERPT #3W5QMB p. 8 13 We discuss this topic further under the heading of 'tropism in subsection 5.1 below. EXCERPT #3VTH9K p. 8 EXCERPT #GGKHXC p. 9 EXCERPT #6N3PJ8 p. 9 EXCERPT #V2T7F5 p. 9 EXCERPT #8E8W8H p. 9 them. In the Categories , Aristotle distinguishes between primary substance ( protē ousia ), that is, a particular substance, such as an organism, and secondary substance ( deutera ousia ), that is, a kind of substance, like the kind Man or the kind Bed . Of these two, Aristotle accords special ontological status to the particular substances. Every universal is then predicated either of some particular substance, or of some particular accident that is in some particular substance in the sense explained. 14 SECTION #94L6NZ 3.2 The Relation of Dependence EXCERPT #DL4QYK p. 9 Substances do not require entities of the other categories further down the list in order to exist. But entities of these other categories do require some entity in the category of substance to serve as ground or fundament for their existence. It is in this sense that substances are said to be ontologically independent entities, where accidents are ontologically dependent . More precisely: substances are ontologically independent of accidents, while accidents are ontologically dependent upon substances. The notion of ontological dependence can be formally captured through a counterfactual criterion. The rough idea is that an entity x is ontologically dependent upon an entity y if x could not exist if y did not exist. EXCERPT #Y6NAHH p. 9 To elaborate further on this relation, we examine how it applies within the categorial framework laid down by Aristotle. First, we note that substances are independent entities. That is to say, they do not rely upon anything else in order to exist. This relation is an ontological one – it is a constraint which operates only in the plane of existence (Ingarden 1965, II/1; Koslicki 2012). Thus our usage of ‘dependence’ and ‘independence’ here is distinct from what we find in other domains, for example when we talk of dependent children or of someone’s being drug dependent. EXCERPT #J3VZ7H p. 9 Such dependence relations hold, now, for all instances of accident categories. More precisely, it holds that, if s is a substance and a is one of s ’s accidents, then a cannot exist unless s exists. This means that a is ontologically dependent on s , or, in a more traditional formulation used already by Aristotle, that a is ontological ‘prior’ to s . In the case of substances and their accidents it is also often said that accidents ‘inhere’ in their substance. This specific form of dependence is a matter of the ways a and s exist – they are different sorts of beings in reality, or, as ontologists express the matter, they belong to different ontological categories. 15 EXCERPT #AE49X7 p. 9 14 Categories 5. 2a 34-35; 2b 3-5; 2b 15-17. In later texts, Aristotle affirms the centrality of particular substances and their special importance with respect to the other categories, which he then also calls ‘affections of the substances’. Metaphysics IV 2, 1003b6: ousiai – pathē ousias ; see also Metaphysics XIV 2, 1089 b 23: ousiai – pathē – pros ti . EXCERPT #YE4XL8 p. 9 15 Aquinas refers in this connection to different degrees of being, with God at the highest degree. EXCERPT #Y9JR9H p. 9 EXCERPT #ESWENC p. 10 EXCERPT #X96ZBN p. 10 EXCERPT #382DK5 p. 10 EXCERPT #5A5P4D p. 10 Quantities and qualities specifically depend on the one substance they inhere in; they cannot switch their bearers. This many-one pattern (many accidents in one substance) can be modified in various ways, as we will see in the next subsection. EXCERPT #U8LL5E p. 10 Second, there are relational entities, which are ontologically dependent on two or more bearers, possibly at the same time (see Smith et al. , 2006). Relational accidents – such as being owner of or being in the marketplace – also show that not all things that are ontologically dependent on a certain entity do in fact inhere in that entity. Relational processes, such as kisses or hits involving two persons, are ontologically dependent upon each of their relata taken singly, but they inhere not in their relata taken singly, but in the totality which these relata form. It is also possible for two or more entities to be mutually ontologically dependent. There can only be a timbre of a musical tone, for example, if there is also a pitch and a loudness (Smith 1997). This also applies for reciprocal relational accidents. There can only be a husband if there is a wife, there can only be an employer if there are employees, and in each case vice versa . SECTION #T27VSK 3.3 Generic Dependence EXCERPT #G65AZM p. 10 Accidents are in every case dependent on the specific substance or substances in which they inhere. This variety of dependence is often called ‘specific dependence’. Not all dependence relations are of this kind. Remember that, according to Aristotle, universals depend on their instances. They do not, however, depend on any one specific instance. It suffices for the universal human to exist that any instance of human exists, and in fact there is no instance of this species that exists now and already existed in Aristotle’s time. EXCERPT #WFPQWG p. 10 Similarly, being a doctor is not dependent upon the existence of any particular individual patient; any patient at all would be sufficient. By the same token, the existence of a patient does not end when there is no longer any doctor treating him. Only if there are no more doctors at all would there be no more patients. And while being an employer requires having at least one employee, the employees can vary over time. This variety of dependence is normally called ‘generic dependence’. Universals are thus generically dependent on their instances. Doctors are generically dependent on their patients, employers on their employees, and so on. This relation can be viewed also at the level of universals, so that, for two universals F and G , being F is generically dependent upon being G if and only if nothing can be F unless something is G . EXCERPT #9T7UEK p. 10 Taken together with Aristotle’s idea that the existence of a universal depends on the existence of at least one instance thereof, we can then carry this idea over to the instances of universals, and define: EXCERPT #92X9SC p. 10 EXCERPT #X5BFAA p. 11 EXCERPT #AK79MH p. 11 EXCERPT #P7YG6U p. 11 EXCERPT #JXWUUH p. 11 An instance of the universal F is generally dependent on the universal G =def. an instance of F cannot exist unless there exists some instance of G . EXCERPT #PE4PQA p. 11 This kind of dependence is important for a wide range of phenomena. Organisms are generically (but not specifically) dependent on their constituent cells and molecules. Literary texts cannot exist without their exemplars. A poem can be written or printed on paper, stored on a computer, learned by heart, or stored as the recording of a recitation. The poem exists as soon and as long as one of these ‘concretizations’ of it exists, but it does not need to be the same concretization for the entire duration of its existence (Arp, Smith & Spear 2015, 105–107, following Ingarden 1974). SECTION #T97WK5 4. Continuants and Occurrents SECTION #SC7ZP5 4.1 Time and Existence EXCERPT #D5JD5J p. 11 There is another way in which Aristotle’s list of categories can be divided into two sub-groups. Note, first, that, where a substance such as a bacterium, a quantity such as a length of 20 meters, or a quality such as an instance of redness, exists in toto at every point in time at which it exists at all, actions and passions are as it were spread out over the course of some time interval. Whenever we encounter a bacterium, we encounter the whole bacterium – and this is so at each point in time over the course of the bacterium’s life. The process (action) by which a bacterium reproduces, by contrast, or by which it moves through a medium, takes place in time and is manifested over a time span. The process of bacterial reproduction has a beginning and an end; it is composed of various phases that follow one another in time. Instances of process universals like reproduction and movement have temporal parts. By contrast, the bacterium itself has spatial parts – for example, a nucleus, a membrane, a cytoplasm – each of which exists as one and the same entity through time, even while potentially gaining and losing qualities, and even while gaining and losing parts. EXCERPT #X6Z9MC p. 11 Hence, we see that there are two kinds of entities. First, there are for example organisms, which continue to exist through time, and for this reason they are called continuants . Next to an organism, however, there is its life or history . It and its successive phases occur in time , and for this reason they are called occurrents . The organism itself is present as a whole at every time at which it exists. For the organism’s life, in contrast, there is no time it is wholly present. Rather, it unfolds itself in successive phases. EXCERPT #54HV8M p. 11 The words ‘continuant’ and ‘occurrent’ can be traced back to the Cambridge logician William Johnson. Johnson defines ‘continuant’ as ‘that which continues to exist while its states or relations may be changing’ (1921, 199). More recently, David Lewis (1986, 202) drew a similar distinction between endurers and perdurers : EXCERPT #BGL97J p. 11 EXCERPT #ZH7SW4 p. 12 EXCERPT #3BX74Z p. 12 EXCERPT #L8SWZA p. 12 EXCERPT #X2U7GH p. 12 Something perdures iff it persists by having different temporal parts, or stages, at different times, though no one part of it is wholly present at more than one time; whereas it endures iff it persists by being wholly present at more than one time. EXCERPT #2YBAJ6 p. 12 Distinguishing between these two modes of existence is often seen as marking a distinction between two competing theories of the ontology of reality, referred to as endurantism and perdurantism, respectively (Donnelly 2011). Lewis, for example, is a perdurantist. He held that all entities are four-dimensional occurrents, 16 a view which is accordingly often referred to as four-dimensionalism, reflecting the fact that occurrents are seen as occupying four-dimensional chunks of spacetime. There is no David Lewis, on the perdurantist view, but rather a process of davidlewis filling out a certain spacetime region. EXCERPT #CRQXDG p. 12 Other philosophers, in contrast, follow Aristotle – and common sense – in embracing a view according to which there are two very different modes of existence exemplified by the bacterium on the one hand and its movement on the other (McCall & Lowe 2009). We, too, hold a view of this sort – namely that we need to acknowledge both continuants and occurrents in order to achieve an accurate representation of reality. EXCERPT #TA62SS p. 12 However, the opposition between entities which continue to exist over time and entities whose existence is spread across successive regions of time does not present an exhaustive classification. This is because it captures only those entities whose existence is, in fact, extended in one or other way over multiple points in time. But there are in addition also what we can think of as temporal boundary entities, including instants of time, on the one hand, and also instantaneously existing qualities and quantities (see Johansson, 2005). If, for example, the temperature of a body is increasing continuously from 2° to 3°C across a certain stretch of time, then it has at just one time point somewhere in the middle the instantaneous quality of exactly 2.5°C. If a tumour grows continuously during its growth process and maintains constant density, then there are no two points at which the tumour has the same weight. If a surface changes its colour continuously from, say, blue to red, then there are no two points in time at which this surface has the same colour. EXCERPT #SJUD4G p. 12 Temporal instants and the temporal boundaries of processes are limit cases of occurrent and are therefore included in the occurrent class. And we must similarly define ‘continuant’ in such a way as to comprehend also instantaneous existents in the realm of specifically dependent continuants, for example the instantaneous quality of 2.5°C in the body temperature example above. EXCERPT #6BXF5W p. 12 16 For an overview of this discussion, see for example Lowe, 2002, 49–58. EXCERPT #43AZ4H p. 12 EXCERPT #4YZ58X p. 13 EXCERPT #B3EFUM p. 13 EXCERPT #KMK7GF p. 13 SECTION #HNHSLF 4.2 SNAP and SPAN EXCERPT #QQXLXP p. 13 If we picture the world at any single point in time, we will discover in our picture planets, people, animals, artifacts, colours, sizes, and relations. But changes, processes, events and happenings that are taking place at that point in time will not be visible in the picture. In order to represent the latter, we need a sequence of pictures; we need something like a film. In order to obtain an all-inclusive picture of our ever-changing world, we thus need two kinds of representation. On the one hand, in order to capture the continuants, we need snapshots of the world at particular points in time. We might call such snapshots SNAP ontologies (following Grenon and Smith, 2004). Included among SNAP entities are substances, quantities, qualities, relations and positions. It will include also the boundaries of substances, collections of substances, spatial regions such as points, lines, surfaces, and spatial volumes, as well as places such as niches and holes, and also the environments in which substances are to be found (Smith 2001). Over and above the traditional category of continuants, SNAP ontologies comprise also the merely instantaneously existing instances of qualities and quantities which would otherwise be ontologically homeless. EXCERPT #78PQP9 p. 13 On the other hand, we need a representation of change, something like a film which represents entire time spans. Grenon and Smith (2004) called these representations SPAN ontologies . Included among SPAN entities are processes and temporal regions, as well as time instants and instantaneous process boundaries which serve as their boundaries. Spatiotemporal regions and the boundaries of such regions are also included, since they too exist along the temporal dimension. Boundaries of processes include for example the beginning and ending of a race, the beginning and ending of a millennium, or the beginning and ending of your life as a 2-year old. SECTION #L5HS8A 5. Putting It All Together SECTION #C8ATRQ 5.1 The Ontological Square EXCERPT #K6EMKY p. 13 Joining together the universal–particular dichotomy and the dichotomy between inhering and non-inhering entities yields a fourfold distinction of entities represented by the so-called ontological square (Figure 4), which captures the core structure of Aristotle’s early ontology. 17 We can think of both Aristotle’s list of categories and the ontological square as transparent partitions of reality (Bittner & Smith 2001). That is, if we look through (as it were) the respective cells in these partitions, then it is as if we can see the corresponding entities in each cell. EXCERPT #3ERY6E p. 13 17 See Smith, 2003a. On the history of such diagrams see Angelelli, 1967, 12; see also Wachter 2000, 149. An alternative interpretation, drawing on Categories 2 is given in Jansen 2014/15, where the ontological square is seen as combining the universal–particular dichotomy with the concrete–abstract dichotomy. EXCERPT #HVUFXA p. 13 EXCERPT #7HJUBL p. 14 EXCERPT #S6TGH4 p. 14 EXCERPT #VYDYS2 p. 14 EXCERPT #EGMVDA p. 14 substantial (not in a subject) accidental, non-substantial (in a subject) universal (predicated of a subject) III. substance universals human being horse IV. accident universals being white knowing particular (not predicated of a subject) I. individual substances this human being this horse II. individual accidents this individual whiteness this individual knowing EXCERPT #P8FD9M p. 14 Figure 4: Aristotle's Ontological Square EXCERPT #5Y394C p. 14 One of the most important contemporary exponents of the idea of a four-category ontology is E. J. Lowe (2006), whose account in his The Four-Category Ontology provides an overhaul of the ontological square designed to match the needs of contemporary philosophers but still very much in Aristotle's spirit. On the other hand, many contemporary philosophers reject some of the fields recognized by proponents of the ontological square. These rejections take a number of different forms, including: EXCERPT #A896Z3 p. 14 • Nominalist philosophers accept only particulars, i.e., only entities from the two lower fields, I and II. Some nominalist philosophers even try to make do with only one of these two categories. For example, the so-called tropists accept only the existence of particular accidents in field II, which they call 'tropes'. This would be a view close to one in which the world consists exclusively of accident instances, a view under which individual substances such as you and me are viewed as more or less loosely connected bundles of such tropes. This view has been defended, for example, by Donald C. Williams (1953, 2018) and Keith Campbell (1990). • Plato, in contrast, ascribed real being only to universals, i.e., to the entities in the two upper fields, III. And IV. A modern defender of such a position was Bertrand Russell, EXCERPT #3C6N6M p. 14 EXCERPT #V5AYDM p. 15 EXCERPT #V99UJ2 p. 15 EXCERPT #BRLE3B p. 15 EXCERPT #DVT8RW p. 15 who wanted to eliminate the level of individuals, 18 most likely under the influence of Leibniz's theory of individual concepts. 19 EXCERPT #A4KK9J p. 15 • Very many twentieth-century philosophers embraced a view amounting to the acceptance of only cells I. and IV. This is because they see the language of First-Order Logic as their tool for understanding reality, as though they regard its syntax as providing a mirror of reality. The particulars in cell I. correspond, on this account, to the individual constants (' a ', ' b ', ' c ' ...), and the property universals in cell IV. to the predicate variables (' F ', ' G ', ' R ' ...). The idea that the formula ' F(a) ' is the key to ontology has been dubbed fantology by Smith (2005a). Representatives of fantology include the Wittgenstein of the Tractatus (Wittgenstein 1960) and an elaborated version of this view, traces of which can be found in the works of almost all of the principal figures of 20th-century analytic philosophy, in the work of David Armstrong, who accepts only particular substances and what he calls property universals (Armstrong, 1978 and 1997). • As concerns the ways in which analytic philosophers treat entities in what we are calling cell IV., different accounts exist as to whether terms such as 'horse' or 'human being' refer (i) to genuine entities (e.g., Bigelow and Leckey 2022), or (2) to one or other logical or set-theoretic constructions or to concepts in people's minds (e.g., Goodman and Leonhard 1940), or (3) such terms are merely façons de parler and so lacking in referents of any sort (e.g., Quine 1964). EXCERPT #JHQPST p. 15 Ontologists who want to eliminate one or more of the fields of the ontological square represent one or other kind of reductionist position. They are required to produce an alternative explanation for why we suppose in our everyday understanding that these things – people, their lives, the species mus musculus , the colour red, the number 2, the redness of this apple – exist. They do this mainly through explaining our reference to entities in these fields as merely a roundabout way of talking about entities in other, philosophically more highly favoured, fields. SECTION #F997NM 5.2 Adding Processes: The Ontological Sextet EXCERPT #TN4GYR p. 15 But can Aristotle truly fit all his categories into the ontological square? In particular, does he deal adequately with the role of processes in his category system? He does list 'doing' EXCERPT #EJDJFG p. 15 18 See for example Russell, 1940, ch. 6; and 1948, Part II, ch. 3 und Part IV ch. 8; 1959, ch. 9. For a similar position see Hochberg 1965, 1966, and 1969. EXCERPT #7CT7EG p. 15 19 Russell (1948) attributes this conception explicitly to Leibniz. See also Armstrong, 1978, I 89: "while the influence of Leibniz on Russell is clear, it is less clear that Leibniz held this theory of the nature of particulars." EXCERPT #ZQA884 p. 15 EXCERPT #EPM68W p. 16 EXCERPT #E5ZQF3 p. 16 EXCERPT #PZ7E3Z p. 16 EXCERPT #DQGJR7 p. 16 (action) and ‘suffering’ (passion) among his ten categories, and seems to understand them as the active and passive sides of a change ( kinesis , De anima III 2, 426a2, cf. Physics III 3, 202a13 –21). Moreover, he refers elsewhere to processes that are not associated with change, as for example in Metaphysics IX 6, where he calls these non-change processes energeiai . It is difficult to say where processes such as seeing or living are to be put in the list of ten categories. It seems, indeed, that Aristotle resists the idea that processes should be treated as fundamental entities. EXCERPT #EBUSET p. 16 On the side of mainstream analytic philosophy, Donald Davidson appears against this background as something of a hero, thanks to his argument in favour of the need for what he called an ‘ontology of events’ (Davidson 1970, 1980). He starts with a problem we face in understanding the semantics of a sentence such as ‘John buttered the toast slowly’. Here the ‘slowly’ requires an entity with which the corresponding adverbial characteristic could be associated. This is why Davidson analyses the sentence as predicating something of a certain event, namely, a buttering event that is slow. Davidson hereby breaks out from the radically simplifying fantologically reduced ontological square by admitting a cell devoted to events as particulars. EXCERPT #2JSE2L p. 16 A picture of the world which did not provide a special place for occurrents would indeed be incomplete. We need to account for the dynamic, processual character of the world, and for this reason we need to add occurrents, including not only processes but also temporal intervals in our ontology. There are also of course important relations that obtain between occurrents and continuants, for example individual substances participate in individual processes. There are also important kinds of processes, for example nuclear decay or apoptosis, which have all of the features of universals in the continuant sphere. All of which suggests that we expand the ontological square to an ontological sextet , as illustrated in Figure 5 (Smith, 2005a). EXCERPT #L9N8YC p. 16 EXCERPT #69DW37 p. 17 EXCERPT #T89QF5 p. 17 EXCERPT #4Q5RZ6 p. 17 EXCERPT #K3K3RN p. 17 graph TD SU[Substance Universal] PU[Property Universal] PUS[Process Universal] SP[Substance Particular] IP[Individual Property] IPr[Individual Process] PU -- "predicated of" --> SU SU -- "instantiates" --> SP SP -- "exemplifies" --> PU PU -- "instantiates" --> IP IP -- "inheres in" --> SP PUS -- "instantiates" --> IPr IPr -- "participates in" --> SP Figure 5: The Ontological Sextet and Associated Formal-ontological Relations. A diagram showing six entities in a 2x3 grid. Top row: Substance Universal, Property Universal, Process Universal. Bottom row: Substance Particular, Individual Property, Individual Process. Relations: 'predicated of' from Property Universal to Substance Universal; 'instantiates' from Substance Universal to Substance Particular; 'exemplifies' from Substance Particular to Property Universal; 'instantiates' from Property Universal to Individual Property; 'inheres in' from Individual Property to Substance Particular; 'instantiates' from Process Universal to Individual Process; 'participates in' from Individual Process to Substance Particular. EXCERPT #3URGS7 p. 17 Figure 5: The Ontological Sextet and Associated Formal-ontological Relations SECTION #CTHD63 5.3 Ontological relations EXCERPT #9SKJDB p. 17 The discussion of ontological dependence, the ontological square and the ontological sextet already revealed that there is a number of relations that are of utmost relevance for ontology. First and foremost, these are the basic relations that obtain among entities in the six fields of Figure 5, namely: EXCERPT #2JRGVP p. 17 • individual accidents inhere in individual substances. • accident universals are predicated of substance universals. • individual substances instantiate substance universals. • individual accidents instantiate accident universals. • individual substances exemplify accident universals. EXCERPT #7SP7SE p. 17 The relations of inheritance, exemplification, instantiation, and participation govern the relations among the entities in the four fields of the ontological square. If we look at the ontological sextet, we can add relations to deal with processes ('events' in Davidson's terminology) drawing on the participates_in relation, as follows: EXCERPT #PUJ8AL p. 17 • individual substances participate in processes. EXCERPT #BMT6NV p. 17 EXCERPT #HUUEQ5 p. 18 EXCERPT #6FJVE3 p. 18 EXCERPT #E6DL2H p. 18 EXCERPT #CFU5J6 p. 18 • individual processes instantiate process universals. EXCERPT #ZMV78W p. 18 One important feature of ontological relations is their generality. Regardless of which area of reality we want to represent, we must take relations like inherence and instantiation into account. In all domains that display changes over time, the relation of participation will be needed. EXCERPT #9CDU9Y p. 18 A second feature of ontological relations is their ‘formal’ nature. This means that these relations hold in a way that does not involve any additional ‘matter’ in the world. Rather, they hold simply because of the very ontological nature of their relata. For more familiar ‘material’ relations something different holds. If Mary is in love with Peter, this is not simply because there are Mary and Peter, but because there is this third thing, namely, Mary’s love for Peter. Similarly, if Peter and Mary are married, this is because there was this processual entity, their marriage, that made them a married couple. It is because of their generality that ontological relations are so important for applied ontology. Because they apply in virtually all domains of reality, much work has been done in recent decades to formalize and standardize the representation of such relations, which are being re-used in myriad ontology applications. SECTION #K2VRML 6. Complex Entities EXCERPT #XJTQWL p. 18 In addition to the categories we have discussed thus far, contemporary ontologists have considered other candidate categories, such as states of affairs, sets and classes, and mereological sums. As they are regularly referred to in writings on applied ontology, we introduce them briefly here. SECTION #7AVA8H 6.1 States of Affairs EXCERPT #2NTMAA p. 18 The idea of states of affairs (in Latin status rerum , German Sachverhalte ) has a long history, starting out from its usage in clinical trials (Smith 1992), but it reached the apogee of its influence in the era of Lotze (Milkov 2023), and then of Lotze’s student Carl Stumpf (Chrudzimski 2015), who, in turn, inspired Husserl and through him Adolf Reinach. Husserl’s and Reinach’s work on states of affairs then runs in (partial) parallel with Wittgenstein’s thinking on this topic as laid down in his Tractatus (Smith 1978). Since then, interest in the idea has waned, having been pushed aside, by set-theoretic approaches to semantics on the one hand, and by truthmaker approaches on the other (see for example Faroldi & Van De Putte 2023). EXCERPT #WMHZTR p. 18 States of affairs are complex entities that can be represented in normal language by means of ‘that’ clauses such as that the ball is green or that the cat is on the mat . The state of affairs that John is sick is a complex entity composed of a substance (this person), and a certain quality or disposition (the sickness). The state of affairs that a certain molecule is attached to a receptor is composed of a substance (the molecule), a part of a substance (the receptor), and the two-place relation of being attached. EXCERPT #H84UBT p. 18 EXCERPT #ZXFCHD p. 19 EXCERPT #NQSYEB p. 19 EXCERPT #R36NPZ p. 19 EXCERPT #6GKE8E p. 19 One problem with attempts to specify the ontology of states of affairs turns on an apparent redundancy which arises from the fact that, if we are representing (for example) the aforementioned ball, then we are already representing something that is green. If we now represent the state of affairs that the ball is green , and if we assume that this state of affairs includes all that exists on the side of reality that is salient to (part of the truthmaker for) our representation, then the ball's greenness quality – in contrast to its other qualities of being round, weighing 4 ounces, being made of plastic – would seem to figure twice in the state of affairs that we are representing. It figures once as it were coiled up inside the object alongside all its other qualities, and then again as somehow uncoiled, or made in some sense explicit, by our representation (Ingarden 1965, II/1 §§ 39–42; Smith 1978). This seems, however, to add an unfortunate epistemological or cognitive or language-dependent dimension 20 to the idea of a state of affairs, which makes such entities not strictly ontological. Part of the background behind the idea of 'truthmakers' (Mulligan, Simons & Smith 1984) was the aim of providing the means to solve this problem. EXCERPT #6ZNJRT p. 19 On the other hand there are states of affairs where an epistemological dimension is ontologically in order – where the epistemological dimension is part of the objective reality which makes the corresponding assertion true. Consider the state of affairs that the doctor believes that her patient has the flu . This is composed of the doctor and the intentional relation of believing, and (if the doctor has a true belief) of the further state of affairs that the patient has the flu . EXCERPT #TJTDPW p. 19 From the perspective of applied ontology we can now identify at least part of the reason for the contemporary disregard of states of affairs as lying in the difficulty we face if we attempt to construct logically coherent taxonomies of states of affairs in the face of the sorts of combinatorial explosions we encounter in taking account of intentional relations such as this. SECTION #78RVGJ 6.2 Sets EXCERPT #PFB668 p. 19 Sets are well known from mathematics, where the term (in German ' Menge ') was introduced by Cantor in 1883 to refer to 'a collection M of definite, well-differentiated EXCERPT #JS9CYT p. 19 20 Compare Strawson (1950): "If you prise the statements off the world you prise the facts off it too; but the world would be none the poorer." EXCERPT #57E39J p. 19 EXCERPT #H8TSJH p. 20 EXCERPT #YPKCHM p. 20 EXCERPT #CY9NZV p. 20 EXCERPT #GTUUKT p. 20 objects m [...] into a whole'. 21 Mathematical views of sets proceed axiomatically, and a number of different axiomatic set theories have been devised. We are interested here, however, in how set theory in general might help us in understanding the sorts of classification or categorization of real-world entities in which ontologists are interested. Does it make sense, for example, to try to understand Aristotle's list of categories, or the ontological square, in set-theoretic terms? Does it make sense to understand real-world phenomena such as a horse race or a ride on the L Train in terms of set theory? Can even a stamp collection be correctly described as a set of stamps? EXCERPT #UWZP25 p. 20 To answer this question, we note that sets can be represented either by simply listing their elements or by pointing to a common feature of these elements. As examples of the first, we can write '{2, 3, 5, 7}' or '{Aristotle, 2, my stethoscope}', reflecting the fact that sets can be built out of unrelated elements. As an example of the second, we can write 'set of prime numbers less than 10'. Here we represent a set by specifying certain necessary and sufficient conditions for membership. Further examples would be 'the set of all patients in Leipzig at noon on November 1, 2008', or 'the set of all such patients with a fever'. EXCERPT #7N3TX8 p. 20 Sets are identical if and only if they contain the same elements. From this it follows that sets are in a certain sense timeless; hence: sets can include elements which exist at different times and at no times. They are also outside space (if the elements of a set move about in space the set is not affected in any way). Sets are in all of these respects distinguished from other collections – for example, my stamp collection or your collection of cancer tissue samples – in that its members are fixed, though this feature that is not clearly captured in Cantor's definition). This means that sets are something abstract; they live outside the world of what happens and is the case. EXCERPT #LS9K5S p. 20 In his "Against Set Theory", Peter Simons quotes a number of authorities in set theory who have, as he says, pulled the wool over the eyes of philosophers and others by presenting sets as something wholly natural and uncontroversial: EXCERPT #M6MYCA p. 20 The technique, usually applied on or about page 1 of a textbook of set theory, is to claim that we are already familiar with sets under some other names or guises, and then trade on this supposed familiarity to sell us a bill of fare which is ontologically far from neutral and far from benign. EXCERPT #2ADQMM p. 20 Simons quotes several authorities from mathematics and logics: EXCERPT #58A5BZ p. 20 21 Translation taken from Oliver & Smiley 2018. The German original (Cantor 1895, 481 and 1932, 282) reads: "Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterscheidbaren Objecten m [...] zu einem Ganzen." EXCERPT #N85QXV p. 20 EXCERPT #UCM7YG p. 21 EXCERPT #Z9BZYY p. 21 EXCERPT #9M2QA4 p. 21 EXCERPT #MK6QXZ p. 21 Consider a collection of concrete objects, for instance of the apples, oranges etc. in a fruit shop. We may call it a set of fruit, the individual apples etc. being the members (or elements) of the set. Conceiving the collection as a new single concept is an elementary intellectual act. (Fraenkel 1953, 4) EXCERPT #GLWD4D p. 21 Or: EXCERPT #CP8GAW p. 21 In our examples, sets consisted of concrete and familiar objects, but once we have sets, we can form sets of sets, for example the set of all football teams. (Van Dalen, Doets and de Swart 1978, 1) EXCERPT #22A7LJ p. 21 So, as Simons goes on, “one can buy a set from a fruiterer, or we can buy a set called ‘Manchester United’ or ‘Juventus’” (Simons 2005). Because they exist outside space and time, however, sets are peculiar entities, in ways which would forestall their use for the sorts of ontological purposes pursued by Aristotle or Lowe or by contemporary applied ontologists. SECTION #5QFVKE 6.3 Mereology: Wholes and Their Parts EXCERPT #L2NEMN p. 21 The mentioned problems have induced some logicians and philosophers to develop an alternative to the set-theoretic approach under the heading of ‘mereology’ 22 , and like set theory, mereology has been subjected to a number of different axiomatizations (Simons 1992; Varzi & Cotnoir 2021). EXCERPT #6RWWF7 p. 21 In the world of sets, every element is just what it is, and has the granularity that it has – whether this be of molecules, of cells, of whole organisms or of entire populations. Mereological sums, in contrast, are concrete entities that can be partitioned on various levels of granularity. Each human being is an organism, and a mereological sum of cells, and a mereological sum of molecules – and all of these are at any given time identical. Moreover, in mereology – as contrasted with set theory – there is no requirement that an ultimate bottom layer of mereological simples needs to be specified or accepted (Smith & Brogaard 2002). EXCERPT #SPJP8W p. 21 My stomach, my sandwich, and the Midwestern US states can each comprise such a mereological sum. Just as with sets, there is, on some formalizations, virtually no limitation to the building of mereological sums. And just as with sets, many mereological sums (such as the sum of Napoleon and the pebbles in this bowl) have an artificial character, though some have sought axiomatic ways of restricting mereology in such a EXCERPT #HR4H8Y p. 21 22 Simons, 1987; Ridder, 2002; see also Husserl’s third Logical Investigation , “On the theory of wholes and parts” (Husserl 1970). EXCERPT #QPSG6R p. 21 EXCERPT #RG9MBL p. 22 EXCERPT #W4KRRF p. 22 EXCERPT #25X3PS p. 22 EXCERPT #4MNMTE p. 22 way as to allow credence only to what we can think of as ‘natural wholes’ (see Simons 2006). EXCERPT #4424YK p. 22 While sets are abstract entities even when composed of concrete elements, mereological sums composed of concrete elements are themselves concrete. Mereological sums exist in space and time, but only – on most formalizations – for so long as all of their parts exist. Like membership in set theory, parthood is temporally rigid in classical mereology: A mereological sum does not survive the loss or destruction of even one of its parts. Gaining or losing a part will result in another mereological sum. EXCERPT #HY5HYB p. 22 In many ontologies, part-whole relations are used as formal-ontological relations. The theory of granular partitions (Bittner & Smith 2001, 2003) introduces an approach which attempts to blaze a third trail between set theory and mereology by linking the concreteness of mereological sums with the hierarchical nature of the element-of relation. SECTION #CCUHN7 6.4 Classes EXCERPT #JKMKQ2 p. 22 Although the words ‘set’ and ‘class’ are often used as synonyms, we will here use them to signify different things. In many standard mathematical treatments, sets can be composed arbitrarily by placing singular terms between curly brackets as in: {Aristotle, the year 1969, New York}. But there are also sets that are defined by means of a uniform property, or a conjunction of such properties, as in: like the class of all things that are red, or the class of all humans, or the class of all electric charges. We will reserve the term ‘class’ for collectives of the latter sort. EXCERPT #9GTMDP p. 22 This is the approach followed by Smith and Ceusters (2006, 60) for whom ‘class’ signifies ‘a collection of all and only the particulars to which a given general term applies’. When the general term connected to a class represents a universal, we can speak of a natural class , which is the totality of instances of a universal. Where sets may be constructed by enumeration of their members, natural classes require that there be universals of which they are the extension. Two natural classes are identical if their defining general term represents the same universal. Because not all general expressions correspond to universals, not all classes are natural classes. The non-natural classes are called ‘defined classes’, as for example: the class of diabetics in Paris on a certain day, or the class of Italian restaurants in San Diego. EXCERPT #9GN24P p. 22 Where sets can have members of arbitrarily different sorts, ‘class’ on this reading refers to collections of members which are in some sense constrained, as for example in: the class of mammals, the class of red things, the class of electrons. The account of classes suggested here thus attempts to mitigate the arbitrary features of set construction. 23 EXCERPT #4CSHA9 p. 22 EXCERPT #XV9D9N p. 23 EXCERPT #HW3QYB p. 23 EXCERPT #VG39BQ p. 23 EXCERPT #SMKM8W p. 23 Unlike what is the case in set theory, class theory does not require us to know what things there are in the world in order to say, for example, that the class of red things and the class of round things are different from one another. EXCERPT #SFNX9H p. 23 In addition, natural classes, but not sets, can survive the destruction or coming into existence of new instances; for sets are individuated by their elements, where natural classes are individuated by a (possible) universal which stays the same even as it has different instances at different times. 24 EXCERPT #JYSAX8 p. 23 The result of dividing entities into classes is called a classification . Instead of speaking of a class we sometimes speak of a taxon , which is derived from the Greek tattein meaning: to place in order, which form in turn the nodes of a taxonomy . A taxonomy must be distinguished from a partonomy . While a classification or a taxonomy divides a universal into species or kinds, a partonomy divides a whole into its parts. SECTION #48D2BK 7. Top-Level Ontologies Today EXCERPT #LUNL5W p. 23 What should an ontology look like at the highest level? In this essay, we used Aristotle's Categories as a guideline for our understanding the development of crucial ontological distinctions that underlie many modern top-level ontologies, including BFO. We already pointed out that Aristotle focusses mainly on continuants and does not really develop the analysis of occurrents. BFO addresses this problem by enriching the ontological square to form the ontological sextet. EXCERPT #YRWGR9 p. 23 There is another respect in which Aristotle's analysis should be supplemented. As presented in the foregoing, Aristotle's theory of categories conforms to a high degree with our common-sense understanding of reality. It shares with common sense above all a view of the world as of a single granularity – the granularity of organisms and their qualities – where our contemporary scientific understanding requires a multi-granular approach, incorporating cells and cell components, molecules, atoms, electrons, and so forth, as well as planets, stars, galaxies, black holes, and other entities dealt with by cosmology. EXCERPT #A5U9YD p. 23 23 There are earlier attempts to link intensional elements with set theory; for example, in Feibleman, 1974. The remarks presented here draw on Johansson 2006. See also Smith 2005b and Smith et al. 2005. EXCERPT #MG2EAP p. 23 24 We here leave open the question as to how one might deal with the natural class corresponding to the universal dodo . EXCERPT #2Z79HP p. 23 EXCERPT #GFCNL5 p. 24 EXCERPT #84RU88 p. 24 EXCERPT #9JEQUW p. 24 EXCERPT #P3ERPC p. 24 Reality, on this approach, appears as a complex hierarchy of levels that are nested within each other. Molecules are embedded in the interior of cells, cells in leaves, leaves in trees, trees in forests and so on (Smith 2001). As our everyday perceptions and actions are tuned to the entities appearing on the level of the common-sense world, so various sciences are tuned to entities on other levels within this complex hierarchy. For example, there is not only macroscopic anatomy, with offshoots such as clinical, surgical and radiological anatomy, but also microscopic anatomy, with sub-disciplines such as histology, cytology and secondary branches such as anatomical embryology, anatomical genomics, neuroanatomy, and so on. Astronomy, similarly, incorporates multiple disciplines such as planetary science, stellar astronomy, galactic astronomy, radio astronomy, and infrared astronomy, which focus on different aspects of the cosmos at different levels of granularity. EXCERPT #9A66DM p. 24 Basic Formal Ontology shares many of the features of the Aristotelian ontology set forth above, but as is made clear throughout the BFO handbook (Arp, Smith and Spear 2015), BFO embraces the principle of perspectivalism, whereby the BFO ontology can be implemented in association with domain ontologies at many different levels of granularity. For example, the BFO class object might comprehend cells and cell components in one implementation, and planets and their satellites in another. In this way BFO can serve as an ontology that simultaneously supports both common-sense and scientific realism, doctrines which are otherwise seen as being incompatible. EXCERPT #7UQEDS p. 24 BFO and its many users thereby provide support for the case that the progress of science is not a step away from Aristotle towards something better, any more than quantum physics is a step away from classical Newtonian physics. 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EXCERPT #BMNPAS p. 27 EXCERPT #6PNYUP p. 28 EXCERPT #C3795F p. 28 EXCERPT #DSSFQL p. 28 EXCERPT #RRJXSN p. 28 Macdonald, Cynthia. "Tropes and Other Things." Contemporary Readings in the Foundations of Metaphysics. eds. Stephen Laurence and Cynthia Macdonald. Oxford: Basil Blackwell, 1998. 329–350. McCall, Storrs and E.J. Lowe. "The Definition of Endurance, Analysis 69 (2009): 277–280. Milkov, Nikolay, Hermann Lotze's Influence on Twentieth Century Philosophy, Berlin: de Gruyter 2023 Mulligan, Kevin, Peter Simons & Barry Smith, "Truth-Makers", Philosophy and Phenomenological Research 44 (3):287–321 (1984) Munn, Katherine, "Introduction: What is Ontology for", in K. Munn and B. Smith (eds.), Applied Ontology: An Introduction, Berlin: de Gruyter, 2008, 7–19. Oliver, Alex and Timothy Smiley, "Cantorian set theory", Bulletin of Symbolic Logic 24 (4):393–451 (2018). Ontology portal, as of August 8, 2006, http://www.ontologyportal.org/ (as of August 8, 2006). Plato. "Phaedrus." Plato: The Collected Dialogues Including the Letters. ed. Edith Hamilton and Huntington Cairns. Princeton: Bollingen Series LXXI, Princeton University Press, 1961. Plato. "Theaetetus." Plato: The Collected Dialogues Including the Letters. ed. Edith Hamilton and Huntington Cairns. Princeton: Bollingen Series LXXI, Princeton University Press, 1961. Quine, Willard Van Orman, "On What There Is", in: From a Logical Point of View, 2nd edition, revised, Cambridge, Massachusetts: Harvard University Press, 1964, 1–19. Rosse, Cornelius and Mejino, Jose L V. "A Reference Ontology for Bioinformatics: The Foundational Model of Anatomy." Journal of Biomedical Informatics 36 (2003), 478–500. Russell, Bertrand. Human Knowledge: Its Scope and Limits. London: Allen & Unwin, 1948. Russell, Bertrand. An Inquiry into Meaning and Truth. New York: Norton 1940. Simons, Peter. Parts. A Study in Ontology. Oxford: Clarendon Press, 1987. Simons, Peter. "Categories and Ways of Being." Philosophy and Logic in Central Europe from Bolzano to Tarski. Selected Essays. ed. Peter Simons. Dordrecht/Boston/London: 1992. 377–394. Simons, Peter. "Against Set Theory." Erfahrung und Analyse. eds. M. E. Reicher and J. C. Marek. Vienna: HPT&ÖBV, 2005, 143–152, 145. EXCERPT #PXJUZB p. 28 EXCERPT #JHEBM5 p. 29 EXCERPT #YSUP6W p. 29 EXCERPT #5PUSWS p. 29 EXCERPT #3HA9RA p. 29 Simons, Peter. "Real wholes, real parts: Mereology without algebra." The Journal of Philosophy 103.12 (2006): 597–613. Smith, Barry. "An Essay in Formal Ontology", Grazer Philosophische Studien 6 (1):39–62 (1978). Smith, Barry. "The substance of Brentano's ontology", Topoi 6:1 (1987) 39–49. Smith, Barry. "Sachverhalt", Historisches Wörterbuch der Philosophie , Volume 8. Basel: Schwabe, (1992), 1102–1113. Smith, Barry. "Realistic Phenomenology", in: Lester Embree (ed.), Encyclopedia of Phenomenology . Kluwer Academic Publishers 1996, pp. 586–590. Smith, Barry & David M. Mark. "Geographical categories: an ontological investigation", International Journal of Geographical Information Science 15.7 (2001): 591–612. Smith, Barry. "On Substances, Accidents and Universals. In Defence of a Constituent Ontology", Philosophical Papers , 27 (1997), 105–127. Smith, Barry. "Objects and their environments: From Aristotle to ecological ontology". In: Andrew U. Frank, et al. (eds.), The Life and Motion of Socio-Economic Units . London: Taylor & Francis, 2001. 79–97. Smith, Barry. "Aristoteles 2002." In: Kann man heute noch etwas anfangen mit Aristoteles? eds. Thomas Buchheim, Hellmut Flashar and Richard A.H. King. Darmstadt: Meiner Felix Verlag, 2003. Smith, Barry. "Against Fantology." Experience and Analysis . eds. M. E. Reicher and J. C. Marek. Vienna: HPT&ÖBV, 2005 (cited as Smith 2005a). Smith, Barry. "The Logic of Biological Classification and the Foundations of Biomedical Ontology." Logic, Methodology and Philosophy of Science. Proceedings of the 12th International Conference . eds. Peter Hájek et al. London: King's College Publications, 2005. 505–520 (cited as Smith 2005b). Smith, Barry. "Biomedical Ontologies." In: P. L. Elkin (ed.), Terminology, Ontology and their Implementations , Springer 2022, 125–169. Smith, Barry & Berit O. Brogaard, "Quantum mereotopology", Annals of Mathematics and Artificial Intelligence 2002, 36 (1):153–175) Smith, Barry & Werner Ceusters. "Ontology as the Core Discipline of Biomedical Informatics. Legacies of the Past and Recommendations for the Future Direction of Research." Computing, Philosophy, and Cognitive Science , eds. G. D. Crnkovic and S. Stuart. Cambridge: Cambridge Scholars Press, 2006. EXCERPT #5VCQDT p. 29 EXCERPT #M78VX6 p. 30 EXCERPT #9N7KDV p. 30 EXCERPT #9QBK7A p. 30 EXCERPT #URSPLL p. 30 Smith, Barry, et al. “Relations in Biomedical Ontologies.” Genome Biology 6 (2005): R46. Smith, Barry, et al. (2006). Towards a Reference Terminology for Ontology Research and Development in the Biomedical Domain. In Proceedings of KR-MED, CEUR , vol. 222. pp. 57–65. Smith B, et al. (2007). “The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration.” Nature Biotechnology 25 (11): 1251–5. Van Dalen, Dirk, Doets, H. C., and de Swart, H. 1978 Sets: Naïve, Axiomatic and Applied , Oxford: Pergamon. Varzi, Achille C; Cotnoir, A. J. Mereology . Oxford: Oxford University Press 2021. Wachter, Daniel von. Dinge und Eigenschaften , Dettelbach: Verlag J. H. Röhl 2000. Williams, D.C. “The Elements of Being.” Review of Metaphysics 7 (1953): 3–18 and 171–192. Williams, Donald C. The Elements and Patterns of Being: Essays in Metaphysics . Oxford University Press 2018. Wittgenstein, L. Tractatus logico-philosophicus. Tagebücher 1914–1916. Philosophische Untersuchungen. Schriften , vol. 1. Frankfurt am Main: Suhrkamp 1960. EXCERPT #53KFLD p. 30 ### 18. Assistant tool calls #### Tool call: `update_document_metadata` ```json { "authors": [ "Barry Smith", "Roberto Casati" ], "document_id": "KY3Y9U", "doi": "10.1080/09515089408573121", "kind": "journal_article", "title": "Naïve Physics: An Essay in Ontology", "venue": "Philosophical Psychology", "year": "1994" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Thomas Bittner", "Barry Smith" ], "document_id": "XZX6PE", "kind": "journal_article", "title": "Vague Reference and Approximating Judgments", "venue": "Spatial Cognition and Computation", "year": "2003" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Ludger Jansen", "Barry Smith" ], "document_id": "88BVY3", "title": "Categories in Top-Level Ontologies: Revisiting the Aristotelian Background" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Barry Smith" ], "document_id": "PHAFYA", "kind": "book_chapter", "title": "Truth and the Visual Field", "venue": "Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science", "year": "1997" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Jean Petitot", "Barry Smith" ], "document_id": "TQPVBD", "kind": "book_chapter", "title": "New Foundations for Qualitative Physics", "venue": "Evolving Knowledge in Natural Science and Artificial Intelligence", "year": "1990" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Barry Smith" ], "document_id": "FJ5KCA", "kind": "journal_article", "title": "More Things in Heaven and Earth", "venue": "Grazer Philosophische Studien", "year": "1995" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Barry Smith" ], "document_id": "2F8T3H", "kind": "journal_article", "title": "Toward a Realistic Science of Environments", "venue": "Ecological Psychology", "year": "2009" } ``` #### Tool call: `update_document_metadata` ```json { "authors": [ "Barry Smith" ], "document_id": "DT9Y7X", "kind": "book_chapter", "title": "True Grid", "venue": "Spatial Information Theory: Foundations of Geographic Information Science", "year": "2002" } ``` ### 19. Tool result: update_document_metadata Updated metadata for #KY3Y9U: kind, title, authors, year, venue, doi ### 20. Tool result: update_document_metadata Updated metadata for #XZX6PE: kind, title, authors, year, venue ### 21. Tool result: update_document_metadata Updated metadata for #88BVY3: title, authors ### 22. Tool result: update_document_metadata Updated metadata for #PHAFYA: kind, title, authors, year, venue ### 23. Tool result: update_document_metadata Updated metadata for #TQPVBD: kind, title, authors, year, venue ### 24. Tool result: update_document_metadata Updated metadata for #FJ5KCA: kind, title, authors, year, venue ### 25. Tool result: update_document_metadata Updated metadata for #2F8T3H: kind, title, authors, year, venue ### 26. Tool result: update_document_metadata Updated metadata for #DT9Y7X: kind, title, authors, year, venue ### 27. Assistant tool calls #### Tool call: `document_import` ```json { "action": "validate", "run_id": "K6HSTJ" } ``` ### 28. Tool result: document_import { "action": "validate", "documents": [ { "title": "Ontologies of Common Sense, Physics and Mathematics", "valid": true, "document_id": "GN66WW", "readable_chunks": 278, "reader_path": "/GN66WW", "source_pages": 32 }, { "title": "On Credentials", "valid": true, "document_id": "KYQGNH", "readable_chunks": 147, "reader_path": "/KYQGNH", "source_pages": 21 }, { "title": "Naïve Physics: An Essay in Ontology", "valid": true, "document_id": "KY3Y9U", "readable_chunks": 226, "reader_path": "/KY3Y9U", "source_pages": 22 }, { "title": "Vague Reference and Approximating Judgments", "valid": true, "document_id": "XZX6PE", "readable_chunks": 165, "reader_path": "/XZX6PE", "source_pages": 20 }, { "title": "Categories in Top-Level Ontologies: Revisiting the Aristotelian Background", "valid": true, "document_id": "88BVY3", "readable_chunks": 174, "reader_path": "/88BVY3", "source_pages": 31 }, { "title": "Truth and the Visual Field", "valid": true, "document_id": "PHAFYA", "readable_chunks": 60, "reader_path": "/PHAFYA", "source_pages": 8 }, { "title": "New Foundations for Qualitative Physics", "valid": true, "document_id": "TQPVBD", "readable_chunks": 76, "reader_path": "/TQPVBD", "source_pages": 13 }, { "title": "More Things in Heaven and Earth", "valid": true, "document_id": "FJ5KCA", "readable_chunks": 65, "reader_path": "/FJ5KCA", "source_pages": 15 }, { "title": "Toward a Realistic Science of Environments", "valid": true, "document_id": "2F8T3H", "readable_chunks": 62, "reader_path": "/2F8T3H", "source_pages": 11 }, { "title": "True Grid", "valid": true, "document_id": "DT9Y7X", "readable_chunks": 104, "reader_path": "/DT9Y7X", "source_pages": 17 } ], "run_id": "K6HSTJ", "embedding_status": "completed", "embedding_errors": 0, "search_rows": 13212 } ### 29. Assistant Imported and validated all 10 PDFs: - Toward a Realistic Science of Environments — #2F8T3H - True Grid — #DT9Y7X - Naïve Physics: An Essay in Ontology — #KY3Y9U - More Things in Heaven and Earth — #FJ5KCA - New Foundations for Qualitative Physics — #TQPVBD - Vague Reference and Approximating Judgments — #XZX6PE - Categories in Top-Level Ontologies: Revisiting the Aristotelian Background — #88BVY3 - Ontologies of Common Sense, Physics and Mathematics — #GN66WW - On Credentials — #KYQGNH - Truth and the Visual Field — #PHAFYA All documents are readable, indexed, and have no embedding errors. #GN66WW was already present, so Sheaf reused the existing document rather than creating a duplicate. ### 30. User great, let's put them in the ontology folder, read them and write good microabstracts for them ### 31. Assistant tool calls #### Tool call: `list_documents` ```json {} ``` ### 32. Tool result: list_documents FOLDERS - Accounting Algebra & Property Theory - Living Structure, Wholeness & Beauty - Living Structure, Wholeness & Beauty - ontology - Procedural Generation & Co-Creation - Roads, Trails & Movement - Scenario-Based & Behavioral Programming - Technology, Scale & Conviviality - Terrain, Hydrology & Erosion - Water Simulation & Rendering Accounting Algebra & Property Theory (4) - #GANFP9 A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology - 2024 | David Ellerman | 53 pp. Micro abstract: Traces an overlooked subset–partition duality—from logic and category theory through entropy and quantum indefiniteness to selectionist and generative mechanisms in biology. - #NBH3BE Algebraic Models for Accounting Systems - 2010 | Derek J. S. Robinson, José García Pérez, Robert A. Nehmer, Salvador Cruz Rambaud | World Scientific | 255 pp. Micro abstract: Develops accounting theory algebraically: balance vectors as modules, transactions as a group, and whole accounting systems as automata with quotients, homomorphisms, and decision algorithms for audit and control. - #7ESDBJ Economics, Accounting, and Property Theory - 1982 | David P. Ellerman | Lexington Books | 110 pp. Micro abstract: Ellerman's vector-accounting monograph: double entry generalized to property vectors ("accounting without valuation"), grounding a property-theoretic account of appropriation, the firm, and goodwill. - #C8FHDZ On implication and negation in partition logic - 2025 | , David Ellerman | Open Journal of Mathematical Sciences | 9 pp. | doi:10.30538/oms2025.0250 Micro abstract: Develops implication as a refinement-sensitive operation on set partitions, showing how relative negation yields local Boolean cores within the non-distributive algebra of partitions. Living Structure, Wholeness & Beauty (9) - #MH5J8D Beautimeter: Harnessing GPT for Assessing Architectural and Urban Beauty Based on the 15 Properties of Living Structure - 2025 | Bin Jiang | AI | 12 pp. | doi:10.3390/ai6040074 Micro abstract: Presents Beautimeter, a GPT-based tool that scores buildings and urban scenes against Christopher Alexander’s 15 properties of living structure to assess their coherence and beauty. - #XW22YY Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods - 2005 | Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt | Center for Environmental Structure | 21 pp. Micro abstract: Argues that living neighborhoods arise from generative codes: ordered, participatory steps that let buildings and public spaces unfold from local people, land, and context. - #SKRF4C Geography as a Science of the Earth’s Surface Founded on the Third View of Space - 2022 | Bin Jiang | Annals of GIS | 14 pp. | doi:10.1080/19475683.2021.1966502 Micro abstract: Recasts geography around an organismic view of space, using scaling and spatial dependence to understand—and deliberately create—places with greater living structure. - #PXG56P Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole - 2009 | Christopher Alexander | Unpublished manuscript | 66 pp. Micro abstract: Proposes harmony-seeking computation as a creative process that repeatedly strengthens latent centers in a configuration while preserving and deepening the larger whole. - #MJKTBB Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space - 2023 | Bin Jiang, Chris de Rijke | Annals of the American Association of Geographers | 19 pp. | doi:10.1080/24694452.2023.2178376 Micro abstract: Measures an image’s structural beauty by recursively extracting its nested substructures, revealing a compact hierarchy that also captures visual saliency. - #3XSLTA Structural Beauty: A Structure-Based Computational Approach to Quantifying the Beauty of an Image - 2021 | Bin Jiang, Chris de Rijke | Journal of Imaging | 15 pp. | doi:10.3390/jimaging7050078 Micro abstract: Proposes a quantitative measure of structural beauty based on how many substructures an image contains and how strongly they form a hierarchy across scales. - #ZU8GZV Structure-Preserving Transformations - 2002 | Christopher Alexander | The Nature of Order, Book Two: The Process of Creating Life | 4 pp. | doi:10.2307/j.ctv27ftw6c.5 Micro abstract: Explains structure-preserving transformations: incremental changes that extend the centers and relationships already present in a place rather than weakening its wholeness. - #AULNWD The Nature of Poetic Order - 1998 | Richard P. Gabriel | Warren Wilson Alumni Conference, Mount Holyoke | 99 pp. Micro abstract: Gabriel's slide essay relating poetry's formal order to Christopher Alexander's ideas of generative structure, exploring how constraint and pattern produce living order in creative work. - #BYG3BQ Wholeness as a Hierarchical Graph to Capture the Nature of Space - 2015 | Bin Jiang | International Journal of Geographical Information Science | 14 pp. | doi:10.1080/13658816.2015.1038542 Micro abstract: Models spatial wholeness as a hierarchical graph of mutually reinforcing centers, using PageRank and scaling depth to quantify the life of parts and wholes. ontology (21) - #FQCWKV A Theory of Granular Partitions - 2003 | Barry Smith, Thomas Bittner | Foundations of Geographic Information Science | 33 pp. Micro abstract: Formalizes granular partitions as hierarchical cell systems projected onto reality, combining cognitive selectivity with mereological structure for naming, classifying, mapping, and representation. - #SF7KYZ About the Unreal - 2025 | Barry Smith, Jim Logan, John Beverley | Proceedings of the Joint Ontology Workshops (JOWO), Episode XI | 14 pp. Micro abstract: Models fiction, blueprints, simulations, and other information about unreal entities through logical combinations of actual classes, avoiding commitments to nonexistent dummy instances. - #CGE2NC Against Fantology - 2005 | Barry Smith | Experience and Analysis | 22 pp. Micro abstract: Critiques the idea that first-order logic reveals reality’s ontology, tracing its atomism, timelessness, Booleanism, and reductionism before proposing a six-category ontology and an enhanced Davidsonian formal language. - #JZG4PM Against Fantology Again - 2016 | Ingvar Johansson | The Theory and Practice of Ontology | 12 pp. Micro abstract: Extends the critique of fantology through default ontologization, arguing that Quine’s canonical notation is incoherent about classes and excludes intentional phenomena and distinct modes of existence. - #E5CLFY Agglomerations - 1999 | Barry Smith | Spatial Information Theory: Cognitive and Computational Foundations of Geographic Information Science | 16 pp. | doi:10.1007/3-540-48384-5_18 Micro abstract: Defines agglomerations as geographically dispersed yet unified aggregates—populations, cultures, organizations, and diasporas—and develops a realist mereotopology for their boundaries, identity, and change. - #3CCZ4A Bodily Systems and the Spatial-Functional Structure of the Human Body - 2004 | Barry Smith, Igor Papakin, Katherine Munn | Ontologies in Medicine | 26 pp. | doi:10.3233/978-1-60750-945-5-39 Micro abstract: Integrates anatomy and physiology by modeling the body as a nested spatial-functional hierarchy whose parts are demarcated as system elements through the functions they bear and realize. - #7YZU95 Boundaries: An Essay in Mereotopology - 1997 | Barry Smith | The Philosophy of Roderick Chisholm | 32 pp. Micro abstract: Reconstructs and extends the Brentano–Chisholm mereotopology in which dependent, coincident boundaries account for contact and the continuum across points, lines, surfaces, and bodies. - #3TZK66 Capabilities: An Ontology - 2024 | Barry Smith, David Limbaugh, Eric Merrell, John Beverley, Peter M. Koch | Proceedings of the Joint Ontology Workshops (JOWO), Episode X | 14 pp. Micro abstract: Defines a capability as a disposition in whose realization an organism or group has or had an interest, placing capabilities between dispositions and functions in Basic Formal Ontology. - #XYERFR Carving Up Reality - 2004 | Barry Smith | Categories: Historical and Systematic Essays | 14 pp. Micro abstract: Explains how context-sensitive, coarse-grained partitions guide reference and perception while preserving transitive parthood and distinguishing fiat demarcations from boundaries grounded in reality. - #9G4F42 CLASSIFYING PROCESSES: AN ESSAY IN APPLIED ONTOLOGY - 2012 | Barry Smith | Ratio | 21 pp. | doi:10.1111/j.1467-9329.2012.00557.x Micro abstract: Extends Basic Formal Ontology to scientific process data through process profiles—quality, rate, and cyclical aspects that ground measurements, time-series graphs, and representations of dynamic systems. - #GSLMP8 Diagrams, Documents, and the Meshing of Plans - 2013 | Barry Smith | Visual Learning, vol. 3: How to Do Things with Pictures: Skill, Practice, Performance | 14 pp. Micro abstract: Shows how diagrams and evolving networks of documents mesh plans, obligations, and specialized labor to enable coordinated collective action beyond the limits of linear text. - #M8BQ3S Do Mountains Exist? Towards an Ontology of Landforms - 2003 | Barry Smith, David M. Mark | Environment and Planning B: Planning and Design | 22 pp. Micro abstract: Argues that mountains are object-like in everyday thought but elevation fields in environmental science, motivating a geospatial ontology that supports both perspectives. - #KSESR8 Drawing Boundaries - 2019 | Barry Smith | The Philosophy of GIS | 26 pp. | doi:10.1007/978-3-030-16829-2_7 Micro abstract: Updates the distinction between human-demarcated fiat boundaries and physically grounded bona fide boundaries, tracing its uses in geography, property, ecology, and Basic Formal Ontology. - #56MWAA Environmental Metaphysics - 2001 | Achille C. Varzi, Barry Smith | Metaphysics in the Post-Metaphysical Age: Proceedings of the 22nd International Wittgenstein Symposium | 12 pp. Micro abstract: Develops an ontology of token niches as tenant–medium–retainer structures, using physical and fiat boundaries to explain environmental fit, protection, movement, and niche construction. - #KG5TBB Making space: the natural, cultural, cognitive and social niches of human activity - 2021 | Barry Smith | Cognitive Processing | 11 pp. | doi:10.1007/s10339-021-01049-y Micro abstract: Shows how legal decisions, plans, historical reasoning, and language create fiat spatial and spatiotemporal entities, then draws limits and practical lessons for ontology-supported AI. - #B98HVX Objects and Their Environments: From Aristotle to Ecological Ontology - 2001 | Barry Smith | The Life and Motion of Socio-Economic Units | 26 pp. | doi:10.1201/9781482268096-14 Micro abstract: Extends Aristotelian substance–accident ontology into a realist theory of behavioral settings and ecological niches as nested, bounded wholes in which organisms, objects, and activities mutually fit. - #BV47YZ On Drawing Lines on a Map - 1995 | Barry Smith | Spatial Information Theory: A Theoretical Basis for GIS | 10 pp. | doi:10.1007/3-540-60392-1_31 Micro abstract: Builds a typology of spatial boundaries around the fiat–bona fide distinction, applying it to maps, political and property divisions, scattered objects, linguistic framing, and truthmakers. - #D8LRQM Ontological Foundations for Geographic Information Science - 2004 | Barry Smith, David M. Mark, Max J. Egenhofer, Stephen C. Hirtle | A Research Agenda for Geographic Information Science | 8 pp. | doi:10.1201/9781420038330.ch12 Micro abstract: Sets a research agenda for geospatial ontology, linking formal accounts of geographic objects, processes, scale, and vagueness to human concepts, interoperable data, and ontology-driven GIS. - #GN66WW Ontologies of Common Sense, Physics and Mathematics - 2023 | Barry Smith, Jobst Landgrebe | arXiv | 32 pp. | doi:10.48550/arXiv.2305.01560 Micro abstract: Separates upper ontologies for common sense, physics, and mathematics, arguing that classical models connect measurable universals to reality while modern physics depends on mathematical entities without commonsense counterparts. - #WYP3G6 Ontology and Geographic Kinds - 1998 | Barry Smith, David M. Mark | Proceedings of the 8th International Symposium on Spatial Data Handling (SDH ’98) | 7 pp. Micro abstract: Argues that geographic kinds are intrinsically spatial and boundary-centered, requiring mereology and topology to connect physical reality, cultural categorization, cognition, and GIS representation. - #9YMD2E SNAP and SPAN: Towards Dynamic Spatial Ontology - 2004 | Barry Smith, Pierre Grenon | Spatial Cognition & Computation | 35 pp. | doi:10.1207/S15427633SCC0401_5 Micro abstract: BFO's bicategorial framework: SNAP snapshot ontologies of continuants and a SPAN ontology of processes in spacetime, linked by trans-ontological relations to capture change — demonstrated on the ontology of geodynamics. Procedural Generation & Co-Creation (14) - #ABD2B8 Between Tech and Art: The Vegetation of Horizon Zero Dawn - 2018 | Gilbert Sanders, Guerrilla Games | Game Developers Conference (GDC) 2018 | 87 pp. Micro abstract: A production breakdown of Horizon Zero Dawn’s vegetation pipeline, covering global wind simulation, layered foliage motion, coverage-preserving alpha mipmaps, shading, asset LODs, placement, and cascaded shadows. - #4TH488 Explainable AI for Designers: A Human-Centered Perspective on Mixed-Initiative Co-Creation - 2018 | Antonios Liapis, G. Michael Youngblood, Jichen Zhu, Rafael Bidarra, Sebastian Risi | 2018 IEEE Conference on Computational Intelligence and Games (CIG) | 8 pp. | doi:10.1109/CIG.2018.8490433 Micro abstract: Defines explainable AI for game designers, mapping co-creative systems by their explainability, initiative, and domain overlap so explanations serve concrete design tasks. - #9NQ94D Extracting Physics from Blended Platformer Game Levels - 2020 | Adam Summerville, Anurag Sarkar, Joseph C. Osborn, Sam Snodgrass | Joint Proceedings of the AIIDE 2020 Workshops (CEUR Workshop Proceedings, Vol. 2862) | 7 pp. Micro abstract: Infers playable jump physics from generated platformer levels, including hybrid physics models for levels that blend the geometry and style of multiple games. - #66Q3W3 Ghost of Tsushima: Procedural Grass - 2021 | Eric Wohllaib, Sucker Punch Productions | Game Developers Conference (GDC) 2021 | 55 pp. Micro abstract: Explains Ghost of Tsushima’s compute-driven grass pipeline, from tiled placement and culling to indirect drawing, cubic Bézier blade geometry, variable LOD, wind animation, and material shading. - #QHMFH2 Improved Alpha Testing Using Hashed Sampling - 2019 | Chris Wyman, Morgan McGuire | IEEE Transactions on Visualization and Computer Graphics | 12 pp. | doi:10.1109/TVCG.2017.2739149 Micro abstract: Develops hashed alpha testing, a stable quasi-random thresholding method that preserves distant alpha-mapped foliage and hair while controlling flicker, anisotropy, and interactions with TAA and alpha-to-coverage. - #7GR3AQ Procedural Content Generation through Quality Diversity - 2019 | Ahmed Khalifa, Antonios Liapis, Daniele Gravina, Georgios N. Yannakakis, Julian Togelius | 2019 IEEE Conference on Games (CoG) | 8 pp. | doi:10.1109/CIG.2019.8848053 Micro abstract: Argues for quality-diversity algorithms in procedural generation, producing broad collections of varied, playable content while exposing the design space for exploration and co-creation. - #CQBDX4 Procedural Content Generation via Machine Learning (PCGML) - 2018 | Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass | IEEE Transactions on Games | 15 pp. | doi:10.1109/TG.2018.2846639 Micro abstract: Defines and surveys PCGML: generating functional game content directly from models trained on existing examples, with uses spanning creation, completion, repair, critique, and compression. - #EARFEK Procedural Generation of Villages on Arbitrary Terrains - 2012 | Adrien Bernhardt, Adrien Peytavie, Arnaud Emilien, Eric Galin, Marie-Paule Cani | The Visual Computer | 10 pp. | doi:10.1007/s00371-012-0699-7 Micro abstract: Presents a three-stage procedural model that grows terrain-responsive village roads and settlements, partitions land into plausible parcels, and generates slope-adapted buildings with open shape grammars. - #EDURTK Real-Time GPU Tree Generation - 2025 | Bastian Kuth, Carsten Faber, Dominik Baumeister, Max Oberberger, Pirmin Pfeifer, Quirin Meyer, Seyedmasih Tabaei | High-Performance Graphics – Symposium Papers | 10 pp. | doi:10.2312/hpg.20251168 Micro abstract: Introduces a GPU work-graph pipeline that generates, animates, edits, and continuously LODs detailed seasonal trees every frame, replacing gigabytes of baked geometry with kilobytes of parameters. - #GBXEP3 Realistic Modeling and Rendering of Plant Ecosystems - 1998 | Bernd Lintermann, Matt Pharr, Oliver Deussen, Pat Hanrahan, Przemyslaw Prusinkiewicz, Radomír Měch | Proceedings of SIGGRAPH ’98 | 12 pp. | doi:10.1145/280814.280898 Micro abstract: Presents a foundational pipeline for authoring plant ecosystems through terrain design, ecological simulation, procedural plant models, approximate instancing, and efficient rendering of billion-primitive scenes. - #BDBBL6 Real‐time Realistic Rendering and Lighting of Forests - 2012 | Eric Bruneton, Fabrice Neyret | Computer Graphics Forum | 11 pp. | doi:10.1111/j.1467-8659.2012.03016.x Micro abstract: Combines detailed z-field trees with terrain shader-maps to render immense forests in real time, preserving sun, sky, canopy, and ground-lighting effects through seamless, scale-consistent transitions. - #PQ68ZH Responsive Real-Time Grass Rendering for General 3D Scenes - 2017 | Klemens Jahrmann, Michael Wimmer | Proceedings of the 2017 Symposium on Interactive 3D Graphics and Games (I3D ’17) | 10 pp. | doi:10.1145/3023368.3023380 Micro abstract: Renders every grass blade as responsive tessellated geometry on arbitrary 3D surfaces, with per-blade wind, gravity, and collision physics plus aggressive culling that retains dense fields in real time. - #WZ8DHP Runtime Evaluation of Procedural Content Generation in an Endless Runner Game Using Autonomous Agents - 2026 | Rishabh Kar | arXiv | 25 pp. | doi:10.48550/arXiv.2605.01783 Micro abstract: Integrates procedural generation and validation in an endless runner, using aerial and ground agents to detect blocked or unnavigable content before the player reaches it. - #NRBMD5 Towards Friendly Mixed Initiative Procedural Content Generation: Three Pillars of Industry - 2020 | Frederic Fol Leymarie, Gorm Lai, William Latham | Proceedings of the International Conference on the Foundations of Digital Games (FDG '20) | 4 pp. | doi:10.1145/3402942.3402946 Micro abstract: Distills three requirements for industry-friendly co-creative PCG tools: preserve designer control, keep feedback loops short, and fit into existing production pipelines. Roads, Trails & Movement (8) - #G3TBNG A Sequential Two-Step Algorithm for Fast Generation of Vehicle Racing Trajectories - 2016 | J. Christian Gerdes, John Subosits, Nitin R. Kapania | Journal of Dynamic Systems, Measurement, and Control | 12 pp. | doi:10.1115/1.4033311 Micro abstract: Generates near-optimal racing trajectories quickly by alternating between a minimum-time speed profile and a convex path update that reduces curvature. - #B6P8L4 Active walker model for the formation of human and animal trail systems - 1997 | Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár | Physical Review E | 34 pp. | doi:10.1103/physreve.56.2527 Micro abstract: Models trail systems as self-organization: walkers reinforce attractive routes while unused traces fade, producing dendritic ant trails and low-detour pedestrian networks. - #V4TQYB Interactive procedural street modeling - 2008 | Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka | ACM Transactions on Graphics | 10 pp. | doi:10.1145/1360612.1360702 Micro abstract: Lets designers generate and edit large street networks through tensor fields, combining procedural speed with brush-like global and local control over street patterns. - #UYLTYJ Modelling the Evolution of Human Trail Systems - 1997 | Dirk Helbing, Joachim Keltsch, Péter Molnár | Nature | 11 pp. | doi:10.1038/40353 Micro abstract: Shows how pedestrian trails emerge through feedback between destination-seeking walkers, existing paths, and vegetation recovery, yielding a compromise between directness and shared infrastructure. - #GY93FG Mountain Trail Formation and the Active Walker Model - 2009 | J. P. Hague, S. J. Gilks | International Journal of Modern Physics C | 22 pp. | doi:10.1142/S0129183109014059 Micro abstract: Extends the active-walker model to steep terrain, explaining zigzag mountain trails through slope avoidance, directional persistence, and mutual reinforcement by ascending and descending walkers. - #LXV9AT Principles of Trail Layout and Design - 2019 | California State Parks | California State Parks Trails Handbook | 64 pp. Micro abstract: A field-oriented guide to durable trail design, emphasizing curvilinear alignment, natural drainage, sustainable grades, control points, and close reading of landform and soils. - #XDEFZS Procedural Generation of Roads - 2010 | A. Peytavie, E. Galin, E. Guérin, N. Maréchal | Computer Graphics Forum | 10 pp. | doi:10.1111/j.1467-8659.2009.01612.x Micro abstract: Automatically routes and constructs roads with an anisotropic shortest-path method that weighs slope and obstacles while treating surface segments, bridges, and tunnels consistently. - #ARP5U7 The Topography of Minoan Peak Sanctuaries - 1983 | A. A. D. Peatfield | The Annual of the British School at Athens | 8 pp. | doi:10.1017/s0068245400019729 Micro abstract: Argues that Minoan peak sanctuaries were chosen for visibility and proximity to local settlements, forming a beacon-like sacred network whose contraction tracked settlement abandonment rather than cultic collapse. Scenario-Based & Behavioral Programming (8) - #P2W4J5 Adaptive Behavioral Programming - 2011 | David Harel, Nir Eitan | 8 pp. | doi:10.1109/ictai.2011.109 Micro abstract: Adds reinforcements to live sequence charts and BPJ so scenario-based programs can learn from their environment, specifying goals to pursue and scenarios to avoid, with modular learning decompositions. - #XQ5NKX Challenges in Modeling and Unmodeling Emergence, Rule Composition, and Networked Interactions in Complex Reactive Systems - 2023 | Assaf Marron, David Harel, Guy Frankel, Irun Cohen, Smadar Szekely | 8 pp. | doi:10.5220/0011728900003402 Micro abstract: Position paper on modeling emergence, rule composition, and networked interactions in complex reactive systems, introducing "unmodeling"—explicitly excluding entities and behaviors from model execution. - #D4VB7S Distributing Scenario-Based Models: A Replicate-and-Project Approach - 2017 | Assaf Marron, Daniel Gritzner, David Harel, Guy Katz, Joel Greenyer, Shlomi Steinberg | MODELSWARD 2017 | 16 pp. | doi:10.5220/0006271301820195 Micro abstract: Distributes scenario-based models by replicating the full specification on every component and projecting it per component, mimicking centralized behavior while sharply reducing synchronization. - #CSJARA Enhancing Scenario-Based Modeling Using Large Language Models - 2026 | Assaf Marron, David Harel, Guy Katz, Smadar Szekely | Communications in Computer and Information Science | Springer Nature Switzerland | pp. 43-68 | 26 pp. | doi:10.1007/978-3-031-96841-9_3 Micro abstract: Extended methodology for combining LLM chatbots with scenario-based modeling: iterative generation of stand-alone scenarios checked by analysis and human review, framed as a step toward Wise Computing. - #3JCRAD On Augmenting Scenario-Based Modeling with Generative AI - 2024 | Assaf Marron, David Harel, Guy Katz, Smadar Szekely | MODELSWARD 2024 | 12 pp. | doi:10.5220/0012427100003645 Micro abstract: Outlines a structured method for using generative-AI chatbots in modeling: iteratively generate scenario-based model fragments, then analyze and inspect them to converge on an accurate system model. - #QV3BWZ On tracing reactive systems - 2011 | David Harel, Shahar Maoz | Software & Systems Modeling | 22 pp. | doi:10.1007/s10270-010-0151-2 Micro abstract: Introduces model-based trace visualization and exploration for reactive systems, using scenario-based (LSC) abstractions and the Tracer prototype, demonstrated on a PacMan game. - #TDS4H2 Relaxing Synchronization Constraints in Behavioral Programs - 2013 | Amir Kantor, David Harel, Guy Katz | LPAR 2013 (Logic for Programming, Artificial Intelligence, and Reasoning) | 17 pp. | doi:10.1007/978-3-642-45221-5_25 Micro abstract: Proposes eager execution for behavioral programs: fast b-threads run ahead when synchronization outcomes are predictable, improving performance, modularity, and distributability, shown in a C++ BP framework. - #M5788P Towards Behavioral Programming in Distributed Architectures - 2015 | Amir Kantor, Assaf Marron, David Harel, Gera Weiss, Guy Katz, Guy Wiener | Science of Computer Programming | 58 pp. | doi:10.1016/j.scico.2014.03.003 Micro abstract: Extends behavioral programming to distributed architectures: b-threads as Erlang processes, eager execution to relax synchronization, and modular distributed execution, demonstrated on simulations and a quadrotor. Technology, Scale & Conviviality (2) - #WYH36B The City as Convivial Centre - 1974 | Leopold Kohr | Tract, no. 12 (Gryphon Press) | 18 pp. Micro abstract: Kohr's essay arguing that cities exist for convivial life rather than economic function, and that human-scale size is what lets a city serve as a centre of leisure, culture, and encounter. - #67REFX The Question Concerning Technology - 1977 | Martin Heidegger | The Question Concerning Technology and Other Essays (Harper & Row) | 23 pp. Micro abstract: Heidegger's essay on the essence of technology as Enframing (Gestell), a mode of revealing that reduces the world to standing-reserve, and on art as a possible saving power. Terrain, Hydrology & Erosion (8) - #NV2YRW FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation - 2024 | Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain | Computer Graphics Forum | 13 pp. | doi:10.1111/cgf.15243 Micro abstract: A GPU framework for routing surface flow through terrain and its depressions fast enough to make erosion, river, lake, and ecosystem simulations interactive. - #2284QZ From features to fingerprints: A general diagnostic framework for anthropogenic geomorphology - 2019 | Damian Evans, Erle C Ellis, Giulia Sofia, Paolo Tarolli, Wenfang Cao | Progress in Physical Geography: Earth and Environment | 34 pp. | doi:10.1177/0309133318825284 Micro abstract: Integrates geomorphology, archaeology, and high-resolution remote sensing into a framework for reading anthropogenic landforms as landscape-scale sociocultural fingerprints. - #96ZMGK Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion - 2016 | Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin | Computer Graphics Forum | 11 pp. | doi:10.1111/cgf.12820 Micro abstract: Generates large, controllable mountain terrains by coupling user-painted tectonic uplift with fluvial erosion, then turning the resulting stream graph into detailed landforms. - #K82AS7 Legacy sediment: Definitions and processes of episodically produced anthropogenic sediment - 2013 | L. Allan James | Anthropocene | 11 pp. | doi:10.1016/j.ancene.2013.04.001 Micro abstract: Broadens legacy sediment to episodically produced anthropogenic alluvium and colluvium, and explains its deposition, storage, and remobilization through sediment delivery–transport capacity dynamics. - #DWXKYQ Physically-based analytical erosion for fast terrain generation - 2024 | Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer | Computer Graphics Forum | 14 pp. | doi:10.1111/cgf.15033 Micro abstract: Turns the stream power law into an interactive terrain tool, replacing thousands of erosion time steps with analytical solutions and a direct control for landscape age. - #MTDKDE Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models - 2014 | Clarence Lehman, David Mulla, Richard Barnes | Computers & Geosciences | 17 pp. | doi:10.1016/j.cageo.2013.04.024 Micro abstract: Introduces Priority-Flood, a simple, optimal algorithm that removes drainage-blocking depressions from elevation models and can also derive watersheds and flow directions. - #AK7NGE Procedural Riverscapes - 2019 | A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial | Computer Graphics Forum | 12 pp. | doi:10.1111/cgf.13814 Micro abstract: Builds editable, animated riverscapes from bare terrain by carving hydrologically plausible channels and blending real-time procedural water primitives instead of simulating fluids. - #DMTA8Y Terrain Generation Using Procedural Models Based on Hydrology - 2013 | Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin | ACM Transactions on Graphics | 10 pp. | doi:10.1145/2461912.2461996 Micro abstract: Generates controllable, multiscale terrain from a sketched drainage network, representing rivers and landforms as an editable hierarchy of continuous procedural primitives. Water Simulation & Rendering (12) - #RBS5K6 A Layered Particle-Based Fluid Model for Real-Time Rendering of Water - 2010 | Daniel Scherzer, Florian Bagar, Michael Wimmer | Computer Graphics Forum | 7 pp. | doi:10.1111/j.1467-8659.2010.01734.x Micro abstract: Renders particle-based water and volumetric foam in real time using perspective-aware surface smoothing, physically guided foam formation, and layered depth compositing. - #C4AY2M A Survey of Ocean Simulation and Rendering Techniques in Computer Graphics - 2011 | B. Crespin, D. Ghazanfarpour, E. Darles, J.-C. Gonzato | Computer Graphics Forum | 17 pp. | doi:10.1111/j.1467-8659.2010.01828.x Micro abstract: Surveys ocean graphics from spectral deep-water models to near-shore fluid simulation, then covers the foam, spray, and light transport needed for convincing rendering. - #WZMZGY Advected river textures - 2009 | Dirk Arnold, Stephen Brooks, Tim Burrell | Computer Animation and Virtual Worlds | 11 pp. | doi:10.1002/cav.288 Micro abstract: Combines a 2D Navier–Stokes solver, hydrostatic pressure columns, and advected procedural textures to render detailed, terrain-responsive rivers at real-time frame rates. - #92XRH7 Lagrangian Texture Advection: Preserving both Spectrum and Velocity Field - 2011 | Qizhi Yu, E. Bruneton, F. Neyret, N. Holzschuch | IEEE Transactions on Visualization and Computer Graphics | 13 pp. | doi:10.1109/tvcg.2010.263 Micro abstract: Advects fluid textures with deformable particle grids, preserving both the input texture’s visual spectrum and exact motion along the velocity field without cumulative stretching. - #8SERGP Real-time Breaking Waves for Shallow Water Simulations - 2007 | Markus Gross, Matthias Müller-Fischer, Nils Thürey, Simon Schirm | 15th Pacific Conference on Computer Graphics and Applications (Pacific Graphics 2007) | 8 pp. | doi:10.1109/PG.2007.33 Micro abstract: Adds real-time overturning waves to shallow-water heightfields by detecting steep fronts and spawning connected particle sheets that collapse into splashes and foam. - #CWC7H9 Real-time Rendering of Enhanced Shallow Water Fluid Simulations - 2013 | Antonio Susín, Jesús Ojeda | Computers & Graphics | 9 pp. Micro abstract: Builds a real-time rendering pipeline for shallow-water simulations, adding fine surface detail, advected foam, photon-based caustics, and screen-space reflection and refraction. - #MVUJ8Z Real-time Rendering of River Networks - 2010 | Quintijn Hendrickx, Rafael Bidarra, Ruben M. Smelik | Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games | 1 pp. Micro abstract: Renders branching river networks efficiently with quadratic Bézier curves, GPU distance fields, and streaming normal maps instead of dense geometry or particle simulation. - #5MGCZ5 Real-time River Representation by Dynamic Control of Data on Waves - 2008 | Makoto Kosugi, Nobuhiko Mukai, Yasuhiro Kato | 4 pp. | doi:10.3169/itej.62.2063 Micro abstract: Dynamically switches river-wave models by viewing distance, preserving nearby reflection and wave detail while retaining wind-driven motion across the full landscape. - #XDESU9 Scalable real‐time animation of rivers - 2009 | Eric Bruneton, Fabrice Neyret, Nicolas Holzschuch, Qizhi Yu | Computer Graphics Forum | 11 pp. | doi:10.1111/j.1467-8659.2009.01363.x Micro abstract: Animates rivers across vast terrains by computing steady flow locally and advecting fine surface detail only where visible, at a screen-space sampling density. - #869NHK Very Fast Real-Time Ocean Wave Foam Rendering Using Halftoning - 2011 | Ian Parberry, Jennifer R. Alford, Mary Yingst | Proceedings of the 6th International North American Conference on Intelligent Games and Simulation (GAMEON-NA) | 7 pp. Micro abstract: Uses a spectrally controlled halftone mask to make texture-based ocean foam dissipate as clumped bubbles that pop, rather than merely fading transparently. - #A2QB8L Water Flow in Portal 2 - 2010 | Alex Vlachos | ACM SIGGRAPH 2010 Course: Advances in Real-Time Rendering in 3D Graphics and Games | 54 pp. Micro abstract: Explains Portal 2’s production-friendly water shader: artist-painted flow maps advect layered normal maps, while offsets and noise suppress repetition and pulsing on limited hardware. - #PBM2TC Water surface wavelets - 2018 | Chris Wojtan, Matthias Müller-Fischer, Miles Macklin, Nuttapong Chentanez, Stefan Jeschke, Tomáš Skřivan | ACM Transactions on Graphics | 13 pp. | doi:10.1145/3197517.3201336 Micro abstract: Represents water waves as spatially varying directional amplitudes, enabling vast interactive simulations with fine detail, moving obstacles, solid coupling, and direct artistic control. Unfiled (10) - #88BVY3 Categories in Top-Level Ontologies: Revisiting the Aristotelian Background - Barry Smith, Ludger Jansen | 31 pp. - #3CV92Z moppe - #FJ5KCA More Things in Heaven and Earth - 1995 | Barry Smith | Grazer Philosophische Studien | 15 pp. - #KY3Y9U Naïve Physics: An Essay in Ontology - 1994 | Barry Smith, Roberto Casati | Philosophical Psychology | 22 pp. | doi:10.1080/09515089408573121 - #TQPVBD New Foundations for Qualitative Physics - 1990 | Barry Smith, Jean Petitot | Evolving Knowledge in Natural Science and Artificial Intelligence | 13 pp. - #KYQGNH On Credentials - 2020 | Barry Smith, Giuseppe Lorini, Olimpia Giuliana Loddo | Journal of Social Ontology | 21 pp. | doi:10.1515/jso-2019-0034 - #2F8T3H Toward a Realistic Science of Environments - 2009 | Barry Smith | Ecological Psychology | 11 pp. - #DT9Y7X True Grid - 2002 | Barry Smith | Spatial Information Theory: Foundations of Geographic Information Science | 17 pp. - #PHAFYA Truth and the Visual Field - 1997 | Barry Smith | Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science | 8 pp. - #XZX6PE Vague Reference and Approximating Judgments - 2003 | Barry Smith, Thomas Bittner | Spatial Cognition and Computation | 20 pp. ### 33. Assistant tool calls #### Tool call: `get_document` ```json { "id": "2F8T3H" } ``` #### Tool call: `get_document` ```json { "id": "DT9Y7X" } ``` #### Tool call: `get_document` ```json { "id": "KY3Y9U" } ``` #### Tool call: `get_document` ```json { "id": "FJ5KCA" } ``` #### Tool call: `get_document` ```json { "id": "TQPVBD" } ``` #### Tool call: `get_document` ```json { "id": "XZX6PE" } ``` #### Tool call: `get_document` ```json { "id": "88BVY3" } ``` #### Tool call: `get_document` ```json { "id": "GN66WW" } ``` #### Tool call: `get_document` ```json { "id": "KYQGNH" } ``` #### Tool call: `get_document` ```json { "id": "PHAFYA" } ``` ### 34. Tool result: get_document DOCUMENT #2F8T3H Title: Toward a Realistic Science of Environments Outline: - 1 #HXSPJA Toward a Realistic Science of Environments - 1.1 #PGEWCQ The Ride on Lake Constance - 1.2 #UGZPUT The Problem of the 'Two Worlds' - 1.3 #BKDBX6 Scheler's Doctrine of the Milieu: An Illustration of How Things Go Wrong - 1.4 #EK457V Ecological Realism - 1.5 #DFNFN3 The Ontology of the Niche - 1.6 #2SNEM6 Is Gibson a Realist? - 1.7 #ZNLB8P Reasons for Representationalism - 1.8 #2ZCXW7 References ### 35. Tool result: get_document DOCUMENT #DT9Y7X Title: True Grid Outline: - 1 #FFC7EB True Grid - 1.1 #CA2U4M 1 Through a Glass Clearly - 1.2 #UEYZZC 2 Theatrum Orbis Terrarum - 1.3 #AXLW7Z 3 Fiat Lux - 1.4 #LBDZZH 4 Qualitative Geometry - 1.5 #NJMY36 5 Rays of Marvelous Subtlety - 1.6 #UNR4PK 6 How to Tell the Truth with Maps - 1.7 #DFMTJ5 7 Semantic Projection - 1.8 #XKWVDP 8 Directions of Fit - 1.9 #EVX3VY 9 Windowless Monads - 1.10 #KHBJYM 10 Fit Happens - 1.11 #DCWT2Q Acknowledgements - 1.12 #38GLQJ References ### 36. Tool result: get_document DOCUMENT #KY3Y9U Title: Naïve Physics: An Essay in Ontology Outline: - 1 #984VXE Naïve Physics: An Essay in Ontology - 1.1 #LERGJV ► To cite this version: - 2 #6AAQJL Naive Physics: An Essay in Ontology(1) - 2.1 #N99NM2 Introduction - 2.2 #3M8H6K I. GESTALT THEORY AND THE HISTORY OF NAIVE PHYSICS - 2.2.1 #KLB9F2 Avenarius and Mach - 2.2.2 #3QP57R Köhler, Lipmann, Bogen - 2.2.3 #AN4X5A Gibson - 2.2.4 #JH2FDB The Austro-Italian School of Gestalt Theory - 2.2.5 #ZSU3C6 Phenomenology - 2.3 #Z2JSWR II. NAIVE PHYSICS AND ARTIFICIAL INTELLIGENCE - 2.3.1 #YCP6DT Historical Antecedents - 2.3.2 #QKV6B5 Critique of Artificially Intelligent Naive Physics - 2.4 #BQHK5F III. BRANCHES OF NAIVE PHYSICS - 2.4.1 #CYFWW4 1. Objects, Natural Units and Natural Kinds - 2.4.2 #LZAZAB 2. Events, Processes and Causality - 2.4.3 #272Z8G 3. Stuffs, States of Matter, Qualities - 2.4.4 #5ZPALD 4. Surfaces, Limits, Boundaries, Media - 2.4.5 #AX6HDN 5. Motivation, Requiredness, Value - 2.5 #LS7S69 Conclusion - 2.6 #NUXU89 References - 2.7 #DA9YQJ Endnotes ### 37. Tool result: get_document DOCUMENT #FJ5KCA Title: More Things in Heaven and Earth Outline: - 1 #DVWAMH MORE THINGS IN HEAVEN AND EARTH - 1.1 #WSZ4H4 1. Heaven - 1.2 #9XN2MV 2. Earth - 1.3 #MHPRY4 3. Performative Maps - 1.4 #L3379F 4. Conclusion: Impossible and Incomplete Spatial Objects - 2 #4UBCX4 REFERENCES ### 38. Tool result: get_document DOCUMENT #TQPVBD Title: New Foundations for Qualitative Physics Outline: - 1 #CYJHV6 New Foundations for Qualitative Physics - 1.1 #BWMMSE Preamble - 1.2 #CDWRVD Manifestations of Matter. I: Spatial Movement - 1.3 #5WEURB Manifestations of Matter. II: Spectral Rays - 1.4 #4WU4DE Manifestations of Matter. III: Qualitative Discontinuities - 1.5 #YNPWYT A Theory of the Commonsense World - 1.6 #DBCBET Qualitative Ontology and the Science of Cognition ### 39. Tool result: get_document DOCUMENT #XZX6PE Title: Vague Reference and Approximating Judgments Outline: - 1 #UR75WH Vague Reference and Approximating Judgments - 1.1 #L35HYH Crisp Granular Partitions - 1.1.1 #U33NC7 Language - 1.1.2 #A7KYQ7 Theory A - 1.1.3 #58M7S8 Theory B - 1.2 #JM424D Vague Granular Partitions - 1.2.1 #Y9BDQJ The Theory - 1.2.2 #UHNNJV Equivalence of Candidate Referents - 1.2.3 #FA6PAW Semantic partition - 1.3 #F7CTDR Approximating Judgments - 1.3.1 #RAK36R Approximation in Egg-yolk Partitions - 1.3.2 #PHAMTM Egg-yolk Partitions vs. Semantic partition - 1.3.3 #GJDGHT Approximation in Complex Partitions - 1.3.4 #UCKBZM Approximation and Judgments - 1.3.5 #EUKWZB Partition Theory and Approximation - 1.3.6 #H67PPK Constraining Approximation - 1.3.7 #B4N4D4 Well-Formed Approximations - 1.3.8 #V7FNPA Precise Approximation - 1.3.9 #HDWEPC Constraining Approximation - 1.4 #PT3REE Properties of Reference Partitions - 1.5 #4TYZWW Conclusions - 1.6 #W4CE7E Acknowledgments - 1.7 #TG3Z68 References ### 40. Tool result: get_document DOCUMENT #88BVY3 Title: Categories in Top-Level Ontologies: Revisiting the Aristotelian Background Outline: - 1 #DY7KEM Categories in Top-Level Ontologies: Revisiting the Aristotelian Background - 1.1 #Y994UA 1. What are Categories? - 1.1.1 #HC4LC5 1.1 The word “category” - 1.1.2 #5JW3Z5 1.2 The Interpretation of Aristotle's Categories - 1.1.3 #CM3E3X 1.3 Aristotle’s Ten Categories - 1.2 #9UCDSE 2. Universals and Particulars - 1.3 #FHJA5L 3. Dependent and Independent Entities - 1.3.1 #9FLUAK 3.1 The Priority of Particular Substances - 1.3.2 #94L6NZ 3.2 The Relation of Dependence - 1.3.3 #T27VSK 3.3 Generic Dependence - 1.4 #T97WK5 4. Continuants and Occurrents - 1.4.1 #SC7ZP5 4.1 Time and Existence - 1.4.2 #HNHSLF 4.2 SNAP and SPAN - 1.5 #L5HS8A 5. Putting It All Together - 1.5.1 #C8ATRQ 5.1 The Ontological Square - 1.5.2 #F997NM 5.2 Adding Processes: The Ontological Sextet - 1.5.3 #CTHD63 5.3 Ontological relations - 1.6 #K2VRML 6. Complex Entities - 1.6.1 #7AVA8H 6.1 States of Affairs - 1.6.2 #78RVGJ 6.2 Sets - 1.6.3 #5QFVKE 6.3 Mereology: Wholes and Their Parts - 1.6.4 #CCUHN7 6.4 Classes - 1.7 #48D2BK 7. Top-Level Ontologies Today - 1.8 #NHHNAK References ### 41. Tool result: get_document DOCUMENT #GN66WW Title: Ontologies of Common Sense, Physics and Mathematics Outline: - 1 #6KMTK8 Ontologies of common sense, physics and mathematics - 1.1 #MFV3NU 1 Ontology of physics - 1.1.1 #JJQFJY 1.1 The nature of theories and models in physics - 1.1.1.1 #ZUFVAD 1.1.1 Classical and modern physics - 1.1.1.2 #ZHEERU 1.1.2 Laws as models - 1.1.1.3 #CAHVQY 1.1.3 The scope of physics - 1.1.1.4 #FR2Q9H 1.1.4 The need for approximations - 1.2 #9BTD4U 2 Physics upper ontology - 1.2.1 #NYRFHH 2.1 System elements - 1.2.2 #XLSYCZ 2.2 Magnitudes - 1.2.2.1 #MSMT2E 2.2.1 Quantification via constants - 1.2.2.2 #KTYY4Y 2.2.2 Quantities and units of measurement - 1.2.3 #NEM8TT 2.3 Models - 1.3 #JGRA8D 3 A mathematics upper ontology - 1.3.1 #E3Q2KL 3.1 The nature of mathematical entities - 1.3.2 #YFJ8WQ 3.2 Mathematics upper ontology - 1.4 #KHZPSR 4 Examples from classical and quantum physics - 1.4.1 #AWHTTL 4.1 The harmonic oscillator - 1.4.1.1 #J9BHWH 4.1.1 Mathematical ontology of the oscillator model - 1.4.1.2 #RTGZW9 4.1.2 Magnitudes in the harmonic oscillator model - 1.4.2 #FJ2UE9 4.2 Entangled photons model - 1.4.2.1 #2LXWR4 4.2.1 Mathematical ontology of the entangled photon model - 1.4.2.2 #MDGYDQ 4.2.2 Magnitudes in the entangled photons model - 1.4.2.3 #W7M2FJ 4.2.3 The meaning of the model - 1.5 #74LR6W 5 The ontological relation of common sense, physics and mathematics - 1.6 #VPLR2G 6 Discussion - 1.7 #86KNF3 References ### 42. Tool result: get_document DOCUMENT #KYQGNH Title: On Credentials Outline: - 1 #EXDAJ5 Article - 2 #XBYXRY Barry Smith, Olimpia Giuliana Loddo and Giuseppe Lorini* On Credentials - 3 #XS833B 1 Introduction - 4 #WKUZRV 2 What are Credentials? - 4.1 #K9YZET 2.1 Credentials as Status Indicators - 4.2 #83D5QY 2.2 Credentials as Institutional Objects - 4.3 #29CHSE 2.3 Counterfeiting Credentials - 4.4 #BPKCKS 2.4 Credentials as Bearer-dependent, Portable, Inspectable Documents - 5 #89CNFV 3 Toward a Typology of Credentials - 5.1 #2ACGZB 3.1 Bearer Credentials versus Identifying Credentials - 5.2 #M88RCU 3.2 Credential Templates versus Credentials Proper - 5.3 #Q73TWT 3.3 Credentials with Displayed Text versus Credentials without Displayed Text - 6 #8Y4CDY 4 The Functions of Credentials - 6.1 #QXC782 4.1 Performing Document Acts with Credentials - 6.2 #M9UNER 4.2 Deontic Functions and Causal Functions of Credentials - 7 #PX6RZD 5 History of Credentials - 7.1 #KVRFJU 5.1 Pre-History of Credentials - 7.2 #U7CQF7 5.2 The Way of Saint James - 7.3 #G6DVYZ 5.3 Present-day Pilgrim Credentials - 8 #G6S8LB 6 Conclusions - 9 #K76VLJ References ### 43. Tool result: get_document DOCUMENT #PHAFYA Title: Truth and the Visual Field Outline: - 1 #WQR5VN Contributors - 2 #GYV9QJ NATURALIZING PHENOMENOLOGY - 2.1 #JMETCH Truth and the Visual Field - 2.2 #BZ6CLE PREAMBLE: GIBSON AND PHENOMENOLOGY - 2.2.1 #D6U7A6 1. TYPES OF BOUNDARIES - 2.2.2 #6AFHK7 2. FIAT OBJECTS - 2.2.3 #EMC9A9 3. FIAT BOUNDARIES AS CREATED ENTITIES - 2.2.4 #ADGTJ6 4. FIAT OBJECTS IN PERCEPTION - 2.2.5 #NGNLAC 5. LANGUAGE-GENERATED FIAT OBJECTS - 2.3 #6LAAZD 6. TRUTH ### 44. 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Tool result: read DOCUMENT #2F8T3H Toward a Realistic Science of Environments SECTION #HXSPJA Toward a Realistic Science of Environments EXCERPT #JHJND5 p. 0 Barry Smith phismith@acsu.buffalo.edu EXCERPT #KZQMG9 p. 0 Preprint version of paper in Ecological Psychology 21 (2), April-June 2009, 121-130 SECTION #PGEWCQ The Ride on Lake Constance EXCERPT #38FZTQ p. 0 If mind is a creature of adaptation, then our standard representationalist theories of mind are in need of revision. For such theories are inspired by Cartesian ways of thinking. They thus conceive the subject of mental experience in isolation from any surrounding physico-biological environment and do not grapple with the interconnections between the world of human thought, feeling and action and the environment of human behavior as this is described by physics and evolutionary biology. Mind is an all-or-nothing affair, that is not coherently integrated with the causal-energetic world of what happens and is the case. EXCERPT #FEW6KE p. 0 One group of more holistically inclined thinkers, forming what is commonly referred to as the Berlin School of Gestalt psychology, offer the beginnings of a more adequate approach. The members of this school, especially Max Wertheimer, Wolfgang Köhler, Kurt Koffka, and Kurt Lewin, sought to understand the relations between mental acts and external objects as participants in a larger complex of interactions between subjects and objects in a common physical and biological environment. Koffka and Lewin in their turn influenced the American psychologist J. J. Gibson, and it is against this background that Gibson's ecological psychology was born. SECTION #UGZPUT The Problem of the 'Two Worlds' EXCERPT #89MZUW p. 0 The Gestalt psychologists had no qualms in accepting the reality of the world described in physical theories, and they were among the first to investigate the relations between mental experiences and associated processes in the brain. When turning to the external environment of human behavior and perception, however, they still embraced a Cartesian approach and saw this environment as something like a manifest image constructed by the human subject. Hence they were left with the problem of explaining the relation between this constructed environment and the world of physics. EXCERPT #EYGQUZ p. 0 EXCERPT #FAK6T9 p. 1 To see the nature of the problem, it will be useful to quote an important passage from Koffka's Principles of Gestalt Psychology in which a fateful distinction between two environments – the psychological (or 'behavioral') and the physical (or 'geographic') – is introduced: EXCERPT #DCP9CZ p. 1 On a winter evening amidst a driving snowstorm a man on horseback arrived at an inn, happy to have reached shelter after hours of riding over the wind-swept plain on which the blanket of snow had covered all paths and landmarks. The landlord who came to the door viewed the stranger with surprise and asked him whence he came. The man pointed in the direction straight away from the inn, whereupon the landlord, in a tone of awe and wonder, said: 'Do you know that you have ridden across the Lake of Constance?' At which the rider dropped stone dead at his feet. EXCERPT #8G74TY p. 1 In what environment, Koffka asks, did the behavior of the stranger take place? EXCERPT #FP5R8F p. 1 The Lake of Constance. Certainly [... and it is] interesting for the geographer that this behaviour took place in this particular locality. But not for the psychologist as the student of behaviour. EXCERPT #2DVEX7 p. 1 The latter, Koffka concludes, will have to say that there is a second sense to the word 'environment,' according to which EXCERPT #WMJ25Q p. 1 our horseman did not ride across the lake at all, but across an ordinary snow-swept plain. His behaviour was a riding-over-a-plain, but not a riding-over-a-lake. (Koffka 1935, pp. 27f.) EXCERPT #JE5UN6 p. 1 How, then, are we to understand the relationship between the physical and the psychological environment? The tale of the ride across Lake Constance tells us that we cannot conceive them as identical in every case. But we cannot say, either, that they are always distinct (and thus embrace a 'two-world' hypothesis). Certainly we would then gain the advantage of a uniform domain for psychological science; but we would also face the problem of explaining how this psychological domain (and hence our psychological experience) might ever come into contact with the domain of physics. Perhaps, then, we can defend the view according to which the two environments are connected via some sort of partial identity relation. Identity obtains in those cases where there is a match between experience and objects, but fails in cases of mismatch of the sort described by Koffka. The problem with this view, however, is that it would imply a peculiar ontological heterogeneity of psychological experiences (which from the perspective of the experiencing subject would yet appear homogeneous). For it would imply that we would somehow, in the course of our mental experience, be crossing back and forth between genuine interaction with physical reality on the one hand and quasi-solipsistic concern with our own psychological creations on the other. It would imply also a corresponding disunity of the domain of psychological science. EXCERPT #GPFAFK p. 1 EXCERPT #NFR3YP p. 2 The Berlin Gestaltists embraced the first of these two alternatives; that is, they embraced the two-world (two-environment) thesis. They were consequently not able to come to a coherent account of the relationship between the environment of psychologically experienced objects and the world of physical things. This is true even of the most sophisticated theorists of the psychological environment such as Kurt Lewin and Fritz Heider. For them, too, the psychological environment is something dependent upon the ego (something that is present even in dreams: Heider 1959), and thus not something that could be accepted as a part of physical reality. This view survives today in philosophical psychology where it is described under the heading 'methodological solipsism' (Fodor 1980). The issue of the relationship between psychological and physical phenomena is bracketed in order to ensure an ontologically uniform domain for psychological science within which both true and false beliefs and both veridical and non-veridical perceptual phenomena can enjoy equal rights in a single homogeneous stream of representations. SECTION #BKDBX6 Scheler's Doctrine of the Milieu: An Illustration of How Things Go Wrong EXCERPT #7B2MX2 p. 2 To see the nature of the problem from another perspective it is useful to look at the thinking of the phenomenologist Max Scheler on what he called the 'milieu' of practical life, which anticipates the later, and more influential theory of the 'life world' developed by Husserl (1954). The things which are relevant to our acting, Scheler tells us, in describing his own version of the Gestalt's psychological environment, EXCERPT #3R8972 p. 2 have of course not the slightest to do either with Kant's 'thing in itself' or with the objects conceived by science (through the supposition of which science 'explains' natural facts). The sun of the milieu of human beings is not the sun of astronomy. The meat that is stolen, bought, or what have you, is not a sum of cells and tissues with the chemicophysical processes which take place within them. The sun of the milieu is different at the EXCERPT #WWM4WD p. 2 EXCERPT #6TY5Z7 p. 3 North Pole, in moderate zones, and at the equator, and its beams are felt as different beams. (Scheler 1954, p. 158f., Eng. trans., p. 139) EXCERPT #AF9VVS p. 3 The problem with this passage is clear. As schoolboys with microscopes know, meat that is stolen and bought does most certainly possess cells and tissues which undergo chemicophysical processes. It simply cannot be the case that the things in our practical, commonsensical environment have ‘not the slightest’ to do with the objects conceived by science. EXCERPT #WNBYF6 p. 3 Electrical and magnetic currents, Scheler wants to hold, may affect me ‘objectively;’ but they do not belong to my milieu, which comprehends ‘only that which I effectively experience.’ But is it really possible to mark out a ‘world’ – a psychological environment – of what is effectively experienced without at the same time letting in all manner of cells, tissues, fields, currents, and the chemicophysical properties relating thereto? EXCERPT #VG3QUQ p. 3 One option pursued by Scheler is to see milieu-things as being in some sense intermediate between persons and the ‘objective’ reality that is studied by physics, as belonging to an intermediate realm ‘lying between our perceptual content and its objects on the one hand and those objectively thought objects on the other.’ (Scheler 1954, p. 159, Eng. trans. p. 140). But how, then, to provide a satisfactory account of the relationship between such milieu-things and their physical counterparts? EXCERPT #WTK5W3 p. 3 For Scheler the milieu is something holistic: it is not the sum of things I perceive or take an interest in or pay attention to. Rather, I can be intentionally related only to what already belongs to my milieu. The milieu is a fund of objects, comprising all of that of which I have the ability to take account in my practical day-to-day dealings with the world. It can include not only food, utensils, people, buildings, the words which I read on the page, but also the laws which I obey or disobey, the value-qualities which make things attractive or repulsive, and other features which seem to belong entirely outside the realm of material things. Thus one and the same landscape presents different milieus to one who recognizes the authorities which prevail within it, to a criminal, and to one who is an enemy in time of war. The same forest is likewise a different milieu to a forester, a hunter, a hiker, as also to a deer or a lizard. ( Op. cit. , p. 161f., Eng. trans. p. 142f.) SECTION #EK457V Ecological Realism EXCERPT #X5Y226 p. 3 With the work of Gibson we meet a new approach to this problem, in which the external, physical, geographical environment is at last given its due. To a much greater degree than is manifest in even the most radical Gestaltist writings, EXCERPT #QGQT58 p. 3 EXCERPT #LJ7ETN p. 4 Gibson emphasizes that normal psychological experience is to be understood not in terms of a succession of relations between interior mental acts on the one hand and objects in some special ‘realm’ on the other, but rather in terms of something like a topological nesting , whereby the sentient organism is housed or situated within a surrounding environment of which it serves as interior boundary. 1 EXCERPT #LN2NQD p. 4 Perceptions and actions are to be understood, from this perspective, as mere dependent features of an encompassing organism-environment relation, whereby both organism and the environment are to be conceived as falling within the realm of physics. In perception, as in action, from the Gibsonian point of view, we are caught up with the very things themselves in the surrounding world, and not with ‘sense data’ or ‘representations’. Perception is not a matter of the processing of sensations. Rather, it is part of that direct linkage between the perceiving organism and its environment which grows out of the fact that, in its active looking, touching, tasting, feeling, the organism as purposeful creature is bound up with those very objects—the ripened fruit, the crumpled shirt, the empty glass, the broken spear—which are relevant to its life and to its tasks of the moment. EXCERPT #WNN5PC p. 4 Gibson thus embraces a radically externalistic view of mind and action. We have not a Cartesian mind or soul, with its interior theater of ‘contents’ and the consequent problem of explaining how this mind or soul and its psychological environment can succeed in grasping physical objects external to itself. Rather, we have a perceiving, acting organism, whose perceptions and actions are always already inextricably intertwined with the parts and moments, the things and surfaces, of its external environment. SECTION #DFNFN3 The Ontology of the Niche EXCERPT #F7W4HW p. 4 For Gibson, reality is a complex hierarchy of inter-nested levels of parts and sub-parts: molecules are nested within cells, cells are nested within leaves, leaves are nested within trees, trees are nested within forests, forests are nested within Special Federal Forest Protection Zones, and so on. (Gibson 1986, p. 101.) Each type of organism is tuned in its perception and action to objects on specific levels within this complex hierarchy – to objects (‘affordances’) which are the environmental correlates of innate and learned traits on the side of the organism. EXCERPT #88MKRH p. 4 1 Important elements of this approach, especially as concerns the way in which individual human beings within socially determined physical-behavioral settings (for example in schools, factories, churches) were developed by Roger Barker, a contemporary of Gibson’s who applied ecological psychology to the world of human behavior. See Heft 2001, Schoggen 1989, and Smith 2001. Schoggen describes physical-behavioral settings as consisting of ‘highly structured, improbable arrangements of objects and events that coerce behavior in accordance with their own dynamic patterning.’ (1989, p. 4) On a formal theory of organism-environment relations that is rooted in Gibsons’s and Barker’s ideas, see Smith and Varzi 1999 and 2002. EXCERPT #2AD9T9 p. 4 EXCERPT #9RJEF2 p. 5 These environmental correlates together form what Gibson calls the organism's 'ecological niche'. A niche is that into which an animal fits (as a trained hand fits into a well-fitting glove, as trained eyes fit well onto a page of words). The niche is that in relation to which the animal is habituated in its behavior. (Gibson 1986, p. 129.) It embraces not only things of different sorts, but also shapes, textures, boundaries (surfaces, edges), all of which are organized in such a way as to enjoy affordance-character for the animal in question in the sense that they are relevant to its survival. The given features motivate the organism; they are such as to intrude upon its life, to stimulate the organism in a range of different ways. EXCERPT #MXGF69 p. 5 The perceptions and actions of human beings are tuned to the characteristic shapes and qualities and patterns of behavior which form their environments. The scope of this attunement is in our case extended further via artifacts, such as microscopes and telescopes, and via cultural phenomena such as languages and institutions of law and politics. To learn a language is in part to extend the range of objects in relation to which we are able spontaneously to adjust our behavior and thus it is to extend radically the types of niche or setting into which we can spontaneously fit. SECTION #2SNEM6 Is Gibson a Realist? EXCERPT #DYPLME p. 5 Human environments, like the environments of all other animals, are parts of physical reality and this is so however far they may be extended through artifacts of different sorts. Yet still, a science of human environments will look very different from any branch of physics. This is not because environments form a separate psychological reality of subjective constructs. Rather, it is because what belongs and what does not belong to the environment of each given organism is dependent in non-rule-governed ways on that organism's location, background, traits, needs, and so forth. The delineation between the environmental and the extra-environmental is on the one hand spatial – no part of Minsk is part of my environment of the moment when I am asleep in Buffalo. But it is also a matter of granularity: the molecules in my bathroom mirror are, unless I am engaging in specific molecule-related activities, are not part of the environment of my morning ablutions. 2 EXCERPT #AT3CPL p. 5 As Gibson saw, we face a challenge to develop a realist science of environments which will be ' consistent with physics, mechanics, optics, acoustics, and chemistry' by taking seriously the idea that ecological facts are 'facts of higher EXCERPT #KMBWAH p. 5 2 On the puzzles raised here – where x is part of my environment but not the parts of x – see Smith 2003. On the underlying formal theory of granular partitions see Bittner and Smith 2003. EXCERPT #E2ZCYC p. 5 EXCERPT #5J6TVN p. 6 order that have never been made explicit by these sciences and have gone unrecognized.’ (Gibson 1979, p. 17) He uses the term ‘ecology’ precisely in order to designate the discipline that should encompass these higher-order facts; it is ‘a blend of physics, geology, biology, archeology, and anthropology, but with an attempt at unification’ on the basis of the question: what can stimulate the organism? (Gibson 1966, p. 21) EXCERPT #5VG6PE p. 6 Gibson thus stands out from the bulk of contemporary psychologists in rejecting representationalism in favor of what he calls ‘direct realism’, a position according to which we are, as a result of adaptation, bound up directly and spontaneously in our normal psychological experience with the objects themselves in the physical world – because we ourselves form part of the physical environment. EXCERPT #YPKVZP p. 6 There is a puzzle, however. For Gibson’s ecological perspective is in other respects very close to phenomenological theories of the environment like the Schelerian theory referred to above, which have been held to dictate precisely a representationalist reading. In an important paper entitled “Is Gibson a Relativist?” Stuart Katz helps us to understand how this apparent conflict could have arisen by drawing attention to passages in Gibson’s work which seem to negate a realist interpretation of his views and thus draw him closer to the phenomenologists. Katz points in particular to the following characteristic statements from Gibson’s Ecological Approach to Visual Perception : EXCERPT #H4J7FW p. 6 animal and environment make an inseparable pair. Each term implies the other. No animal could exist without an environment surrounding it. Equally, although not so obvious, an environment implies an animal (or at least an organism) to be surrounded. (1979, p. 8) EXCERPT #E6TZRS p. 6 The affordances of the environment are what it offers the animal, what it provides or furnishes , whether for good or ill. - I mean by [affordance] something that refers to both the environment and the animal in a way that no existing term does. It implies the complementarity of the animal and the environment. (1979, p. 127) EXCERPT #LE3NG2 p. 6 ... an affordance is neither an objective property nor a subjective property; or it is both if you like. An affordance cuts across the dichotomy of subjective-objective. - It is both physical and psychical, yet neither (1979, p. 129). EXCERPT #AHJ9G6 p. 6 EXCERPT #GXW2ZR p. 7 These passages dictate, according to Katz, a reading of Gibson according to which different species live in different worlds . Water is for you and me a substance; for fish it is a medium which substances inhabit. Hence the question arises: EXCERPT #CJFWG4 p. 7 Do terrestrial animals perceive water correctly and aquatic species incorrectly, or vice versa? Gibson as relativist tells us no. Each lives in a different world and, complementarily, each perceives differently. Water is a substance in one world and a medium in another; it is not absolutely substance, nor is it absolutely medium. 'The animal and its environment, remember, are reciprocal terms.' One could never say what water is, without saying for whom it is, and conversely. (Katz 1987, p. 120) SECTION #ZNLB8P Reasons for Representationalism EXCERPT #8PG8LX p. 7 Whether valid or not, Katz's argument is significant. If it is valid, then this implies that phenomenologists such as Husserl or Scheler can claim a hitherto unrecognized ally among experimental psychologists. If, on the other hand, the argument is flawed, then in coming to understand why this is so, we will discover which modifications of standard Husserlian views must be made if we are to bring them into harmony with Gibsonian realism. EXCERPT #SQH4ZQ p. 7 To see whether Katz's argument is valid or not, we note that there are two principal motivations for representationalist views of perception: (1) the problem of perceptual error , and (2) the problem of apparent global incompatibilities between different systems of representations. EXCERPT #3LBR3S p. 7 The existence of perceptual error, according to familiar arguments (involving bent sticks and like phenomena), reveals that perception itself cannot be solely a product of sensory inputs. It tells us that, on occasion at least, for example in cases of hallucination, perceptual objects are in some sense or to some degree created or constituted by or with the help of the perceiver. A representationalist, again, is one who holds that the objects that are given in perception are always constructed or constituted in this sense (hence they belong to a special world, a world of representations). The representationalist is able to do justice to the fact of perceptual error without abandoning the goal of a unified theory of perception, but only at the price of cutting off her theory from any roots in the real world of mind-independent objects. The realist solution to the problem of error, on the other hand, denies that what is phenomenologically experienced as the unitary stream of phenomena of perception is in fact a unitary matter. Rather, it distinguishes two types of perceptual setup, and correspondingly two distinct tasks for the theory of perception. On the one hand is the task of providing a theory of perception in the strict sense – a theory of successful, veridical, world-embrangled perception of the normal sort. On the other hand is the quite different task of giving an account of perceptual error (of the different types of shortfall from this standard, veridical case). There is a theory of smooth running in the realm of perception, and a supplementary account of accidents and breakdowns. EXCERPT #6VYME2 p. 7 EXCERPT #5XE6MP p. 8 A second motivation for representationalism might be formulated as follows: our common-sense perceptual space has a Euclidean structure (or a structure closely related thereto); the space of the physicist has another, quite different structure; and it may well be that the perceptual spaces of mice, of spiders, of clams, have other structures again. Not all of these structures can be true of space as it is in itself. Hence, the argument goes, our (and the mouse's, and the spider's) perceptual spaces are mere 'representations'. And what goes for space holds for other features of the manifold environments of perception, too – so that it is as if each species lives in its own special world. EXCERPT #2PYAHL p. 8 It is a constructivist, relativist, representationalist, projectivist, methodologically solipsist, Kantianist, Uexküllist conclusion of this sort which Katz attributes to Gibson. But, to remain with Katz's own example, space (as we may here assume) is a continuum. Like all continua it can be partitioned in a range of mutually incompatible ways (as a cheese can be sliced in such a way as to produce either triangular or rectangular or disk-shaped segments, but not all of these at once). In principle, therefore, all members of a family of mutually conflicting 'perceptual spaces' may well turn out to be compatible after all, if they can be interpreted as expressing distinct partitions , for example partitions on different levels of granularity, of one and the same reality. (Bittner and Smith 2003) In this way the second motive for representationalism may be resisted, too, and therewith also for a representationalist reading of Gibson. In full conformity with the realist perspective, different languages, different theories, and different families of organisms are able to generate their own precisely fitting partitions of one single reality. The various animal behavior-systems generate partitions of reality into their own ecological niches. But these are not separate worlds. Rather, they are partitions of one and the same world, effected for different purposes and at different levels of granularity. EXCERPT #KM4AR2 p. 8 The ultimate common ontology governing these environments will be scientific, but it will have room not only for physics but also for mesoscopic structures built up on the basis of physics, including structures of the type to which human behaviour and perception are tuned. But to do justice to this complex edifice, the realistic science of organisms, biology, will need as its counterpart a realistic science of environments. EXCERPT #42VZUC p. 8 SECTION #2ZCXW7 References EXCERPT #KXXWMG p. 9 Bittner, Thomas and Smith, Barry 2003 “A Theory of Granular Partitions”, Foundations of Geographic Information Science , Matthew Duckham, Michael F. Goodchild and Michael F. Worboys (eds.), London: Taylor & Francis, 2003, 117–151. Fodor, Jerry 1980 “Methodological Solipsism Considered as a Research Strategy in Cognitive Psychology,” Behavioral and Brain Sciences , 3, 63-73. Gibson, J. J. 1966 The Senses Considered as Perceptual Systems , London: George Allen and Unwin. Gibson, J. J. 1979 The Ecological Approach to Visual Perception , Boston: Houghton-Mifflin, repr. 1986, Hillsdale, NJ: Lawrence Erlbaum. Heft, Harry 2002 Ecological Psychology in Context: James Gibson, Roger Barker, and the Legacy of William James's Radical Empiricism , Mahwah, NJ: Lawrence Erlbaum. Heider, Fritz 1959 “The Description of the Psychological Environment in the Work of Marcel Proust,” in Heider, On Perception and Event Structure, and the Psychological Environment, Selected Papers (Psychological Issues , Vol. 1, No. 3), New York: International Universities Press, 85-107. Husserl, Edmund 1954 Die Krisis der europäischen Wissenschaften und die transzendente Phänomenologie. Eine Einleitung in die phänomenologische Philosophie , W. Biemel, ed. (Hua VI), The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy , translated by David Carr, Evanston IL: Northwestern University Press, 1970. Katz, Stuart 1987 “Is Gibson a Relativist?”, in A. Costall and A. Still, Cognitive Psychology in Question , Brighton: Harvester, 115-127. Koffka, Kurt 1935 Principles of Gestalt Psychology , London: Routledge and Kegan Paul. EXCERPT #LXRP2L p. 9 EXCERPT #P7778N p. 10 Scheler, Max 1954 Der Formalismus in der Ethik und die materiale Wertethik , 4th edition, Bern: Francke, English translation by M. S. Frings and Roger L. Funk as Formalism in Ethics and Non-Formal Ethics of Value , Evanston: Northwestern University Press, 1973. EXCERPT #MPPMLJ p. 10 Schoggen, P. 1989 Behavior Settings. A Revision and Extension of Roger G. Barker's Ecological Psychology , Stanford: Stanford University Press. EXCERPT #PCEZR2 p. 10 Smith, Barry 2001 "Objects and Their Environments: From Aristotle to Ecological Psychology", in Andrew Frank, Jonathan Raper and Jean-Paul Cheylan (eds.), The Life and Motion of Socio-Economic Units , London: Taylor and Francis, 2001, 79–97. EXCERPT #JWD7YN p. 10 Smith, Barry 2004 "Carving Up Reality", in M. Gorman and J. Sanford (eds.), Categories: Historical and Systematic Essays , Washington: Catholic University of America Press, 225–237. EXCERPT #GQQU7S p. 10 Smith, Barry and Varzi, Achille 1999 "The Niche", Noûs , 33: 2, 198–222. EXCERPT #TJJ9UX p. 10 Smith, Barry and Varzi, Achille 2002 "Surrounding Space: The Ontology of Organism-Environment Relations", Theory in Biosciences , 121, 139–162. EXCERPT #NVDLXP p. 10 Uexküll, Jacob von 1928 Theoretische Biologie , Berlin: J. Springer. EXCERPT #JC8L97 p. 10 DOCUMENT #2F8T3H Toward a Realistic Science of Environments SECTION #EK457V Ecological Realism EXCERPT #X5Y226 p. 3 With the work of Gibson we meet a new approach to this problem, in which the external, physical, geographical environment is at last given its due. To a much greater degree than is manifest in even the most radical Gestaltist writings, EXCERPT #QGQT58 p. 3 EXCERPT #LJ7ETN p. 4 Gibson emphasizes that normal psychological experience is to be understood not in terms of a succession of relations between interior mental acts on the one hand and objects in some special ‘realm’ on the other, but rather in terms of something like a topological nesting , whereby the sentient organism is housed or situated within a surrounding environment of which it serves as interior boundary. 1 EXCERPT #LN2NQD p. 4 Perceptions and actions are to be understood, from this perspective, as mere dependent features of an encompassing organism-environment relation, whereby both organism and the environment are to be conceived as falling within the realm of physics. In perception, as in action, from the Gibsonian point of view, we are caught up with the very things themselves in the surrounding world, and not with ‘sense data’ or ‘representations’. Perception is not a matter of the processing of sensations. Rather, it is part of that direct linkage between the perceiving organism and its environment which grows out of the fact that, in its active looking, touching, tasting, feeling, the organism as purposeful creature is bound up with those very objects—the ripened fruit, the crumpled shirt, the empty glass, the broken spear—which are relevant to its life and to its tasks of the moment. EXCERPT #WNN5PC p. 4 Gibson thus embraces a radically externalistic view of mind and action. We have not a Cartesian mind or soul, with its interior theater of ‘contents’ and the consequent problem of explaining how this mind or soul and its psychological environment can succeed in grasping physical objects external to itself. Rather, we have a perceiving, acting organism, whose perceptions and actions are always already inextricably intertwined with the parts and moments, the things and surfaces, of its external environment. DOCUMENT #2F8T3H Toward a Realistic Science of Environments SECTION #DFNFN3 The Ontology of the Niche EXCERPT #F7W4HW p. 4 For Gibson, reality is a complex hierarchy of inter-nested levels of parts and sub-parts: molecules are nested within cells, cells are nested within leaves, leaves are nested within trees, trees are nested within forests, forests are nested within Special Federal Forest Protection Zones, and so on. (Gibson 1986, p. 101.) Each type of organism is tuned in its perception and action to objects on specific levels within this complex hierarchy – to objects (‘affordances’) which are the environmental correlates of innate and learned traits on the side of the organism. EXCERPT #88MKRH p. 4 1 Important elements of this approach, especially as concerns the way in which individual human beings within socially determined physical-behavioral settings (for example in schools, factories, churches) were developed by Roger Barker, a contemporary of Gibson’s who applied ecological psychology to the world of human behavior. See Heft 2001, Schoggen 1989, and Smith 2001. Schoggen describes physical-behavioral settings as consisting of ‘highly structured, improbable arrangements of objects and events that coerce behavior in accordance with their own dynamic patterning.’ (1989, p. 4) On a formal theory of organism-environment relations that is rooted in Gibsons’s and Barker’s ideas, see Smith and Varzi 1999 and 2002. EXCERPT #2AD9T9 p. 4 EXCERPT #9RJEF2 p. 5 These environmental correlates together form what Gibson calls the organism's 'ecological niche'. A niche is that into which an animal fits (as a trained hand fits into a well-fitting glove, as trained eyes fit well onto a page of words). The niche is that in relation to which the animal is habituated in its behavior. (Gibson 1986, p. 129.) It embraces not only things of different sorts, but also shapes, textures, boundaries (surfaces, edges), all of which are organized in such a way as to enjoy affordance-character for the animal in question in the sense that they are relevant to its survival. The given features motivate the organism; they are such as to intrude upon its life, to stimulate the organism in a range of different ways. EXCERPT #MXGF69 p. 5 The perceptions and actions of human beings are tuned to the characteristic shapes and qualities and patterns of behavior which form their environments. The scope of this attunement is in our case extended further via artifacts, such as microscopes and telescopes, and via cultural phenomena such as languages and institutions of law and politics. To learn a language is in part to extend the range of objects in relation to which we are able spontaneously to adjust our behavior and thus it is to extend radically the types of niche or setting into which we can spontaneously fit. DOCUMENT #2F8T3H Toward a Realistic Science of Environments SECTION #2SNEM6 Is Gibson a Realist? EXCERPT #DYPLME p. 5 Human environments, like the environments of all other animals, are parts of physical reality and this is so however far they may be extended through artifacts of different sorts. Yet still, a science of human environments will look very different from any branch of physics. This is not because environments form a separate psychological reality of subjective constructs. Rather, it is because what belongs and what does not belong to the environment of each given organism is dependent in non-rule-governed ways on that organism's location, background, traits, needs, and so forth. The delineation between the environmental and the extra-environmental is on the one hand spatial – no part of Minsk is part of my environment of the moment when I am asleep in Buffalo. But it is also a matter of granularity: the molecules in my bathroom mirror are, unless I am engaging in specific molecule-related activities, are not part of the environment of my morning ablutions. 2 EXCERPT #AT3CPL p. 5 As Gibson saw, we face a challenge to develop a realist science of environments which will be ' consistent with physics, mechanics, optics, acoustics, and chemistry' by taking seriously the idea that ecological facts are 'facts of higher EXCERPT #KMBWAH p. 5 2 On the puzzles raised here – where x is part of my environment but not the parts of x – see Smith 2003. On the underlying formal theory of granular partitions see Bittner and Smith 2003. EXCERPT #E2ZCYC p. 5 EXCERPT #5J6TVN p. 6 order that have never been made explicit by these sciences and have gone unrecognized.’ (Gibson 1979, p. 17) He uses the term ‘ecology’ precisely in order to designate the discipline that should encompass these higher-order facts; it is ‘a blend of physics, geology, biology, archeology, and anthropology, but with an attempt at unification’ on the basis of the question: what can stimulate the organism? (Gibson 1966, p. 21) EXCERPT #5VG6PE p. 6 Gibson thus stands out from the bulk of contemporary psychologists in rejecting representationalism in favor of what he calls ‘direct realism’, a position according to which we are, as a result of adaptation, bound up directly and spontaneously in our normal psychological experience with the objects themselves in the physical world – because we ourselves form part of the physical environment. EXCERPT #YPKVZP p. 6 There is a puzzle, however. For Gibson’s ecological perspective is in other respects very close to phenomenological theories of the environment like the Schelerian theory referred to above, which have been held to dictate precisely a representationalist reading. In an important paper entitled “Is Gibson a Relativist?” Stuart Katz helps us to understand how this apparent conflict could have arisen by drawing attention to passages in Gibson’s work which seem to negate a realist interpretation of his views and thus draw him closer to the phenomenologists. Katz points in particular to the following characteristic statements from Gibson’s Ecological Approach to Visual Perception : EXCERPT #H4J7FW p. 6 animal and environment make an inseparable pair. Each term implies the other. No animal could exist without an environment surrounding it. Equally, although not so obvious, an environment implies an animal (or at least an organism) to be surrounded. (1979, p. 8) EXCERPT #E6TZRS p. 6 The affordances of the environment are what it offers the animal, what it provides or furnishes , whether for good or ill. - I mean by [affordance] something that refers to both the environment and the animal in a way that no existing term does. It implies the complementarity of the animal and the environment. (1979, p. 127) EXCERPT #LE3NG2 p. 6 ... an affordance is neither an objective property nor a subjective property; or it is both if you like. An affordance cuts across the dichotomy of subjective-objective. - It is both physical and psychical, yet neither (1979, p. 129). EXCERPT #AHJ9G6 p. 6 EXCERPT #GXW2ZR p. 7 These passages dictate, according to Katz, a reading of Gibson according to which different species live in different worlds . Water is for you and me a substance; for fish it is a medium which substances inhabit. Hence the question arises: EXCERPT #CJFWG4 p. 7 Do terrestrial animals perceive water correctly and aquatic species incorrectly, or vice versa? Gibson as relativist tells us no. Each lives in a different world and, complementarily, each perceives differently. Water is a substance in one world and a medium in another; it is not absolutely substance, nor is it absolutely medium. 'The animal and its environment, remember, are reciprocal terms.' One could never say what water is, without saying for whom it is, and conversely. (Katz 1987, p. 120) ### 46. Tool result: read DOCUMENT #DT9Y7X True Grid SECTION #FFC7EB True Grid EXCERPT #F7BK8K p. 0 Barry Smith EXCERPT #R4MUT5 p. 0 Department of Philosophy, Center for Cognitive Science and NCGIA University at Buffalo, NY 14260, USA phsmith@buffalo.edu EXCERPT #N2XMFJ p. 0 Abstract. The Renaissance architect, moral philosopher, cryptographer, mathematician, Papal adviser, painter, city planner and land surveyor Leon Battista Alberti provided the theoretical foundations of modern perspective geometry. Alberti's work on perspective exerted a powerful influence on painters of the stature of Albrecht Dürer, Leonardo da Vinci and Piero della Francesca. But his Della pittura of 1435–36 contains also a hitherto unrecognized ontology of pictorial projection. We sketch this ontology, and show how it can be generalized to apply to representative devices in general, including maps and spatial and non-spatial databases. EXCERPT #QAJ7Y4 p. 0 An engraving by Albrecht Dürer titled 'The Draftsman's Net'. It depicts a man lying on a bed, looking out a window at a landscape. In the foreground, a draftsman is seated at a table, working on a large sheet of paper. The scene is set in a room with a window that looks out onto a landscape with a tree and a building. The engraving is signed 'DÜRER' in the bottom right corner. Albrecht Dürer's engraving 'The Draftsman's Net' showing a man lying on a bed looking out a window at a landscape, with a draftsman sitting at a table in the foreground. EXCERPT #RWGZLF p. 0 Fig. 1 Albrecht Dürer's interpretation of 'The Draftsman's Net' SECTION #CA2U4M 1 Through a Glass Clearly EXCERPT #TZP77Q p. 0 The Della pittura of the Renaissance artist and art theorist Leon Battista Alberti, dating from 1435–36, is the first modern treatise on painting. It defends a view according to which the proper goal of the artist is to produce a picture that will represent the visible world as if the observer of the picture were looking through a window . EXCERPT #KT9QW9 p. 0 This open window conception reflects a time when painting is still an adjunct of architecture: paintings are designed to enhance one's home. The aesthetic experience of a building's interior and the aesthetic experience of the paintings on its walls are meant to be fused into one: the picture must be so painted that a spectator's imagination is drawn towards the wall-plane, not away from it. This is why Renaissance painters, acting as interior decorators, revived and elaborated the system of perspective already used by interior decorators at Pompeii and elsewhere in the ancient world. (Collingwood 1938, p. 153) EXCERPT #XWPPJP p. 1 Alberti's conception of the painting was extremely influential. Indeed the art historian Erwin Panofsky argues that, while there are elements of perspectival foreshortening in earlier works of art, one can properly speak of a perspectival intuition of space only where a 'whole picture is as it were transformed into a "window" through which we should then believe ourselves to be looking into the space'. (Panofsky 1927) To rub the same point home, Dürer, in his woodcuts, always depicts the process of perspective projection in such a way that this process is situated in a room in which a section of window clearly appears. ('Perspective' means, roughly, 'seeing through' or 'seeing clearly'.) EXCERPT #2AAVX6 p. 1 A black and white woodcut illustration of Alberti's 'Reticolato' perspective device. It shows a table with a gridded window frame (reticolato) and a gridded notepad. Lines of perspective radiate from a point on the table, passing through the grid and extending to a distant cityscape visible through the window. The scene is set in a room with a tiled floor and a wall. A black and white woodcut illustration of Alberti's 'Reticolato' perspective device. It shows a table with a gridded window frame (reticolato) and a gridded notepad. Lines of perspective radiate from a point on the table, passing through the grid and extending to a distant cityscape visible through the window. The scene is set in a room with a tiled floor and a wall. EXCERPT #33JCRU p. 1 Fig. 2 Alberti's Reticolato EXCERPT #JLQUSD p. 1 Alberti presented his ideas on perspective in terms of his so-called ' reticolato ', also known as Alberti's 'grid' or 'grill' ( graticola ), a mechanical aid to painters in the execution of the fenestra aperta technique, which involved creating a grid across an actual window in order to enable the artist to transfer the scene visible through the window to a correspondingly gridded canvas. Because parallax is here so strong, it is unlikely that such devices were ever in fact used by painters. Even the slightest movement on the painter's part will have a dramatic effect upon the scene perceived. We might think of the reticolato , rather, as a pedagogical device, designed to help the artist understand how perspective works. EXCERPT #5E8UQC p. 1 Figure 2 depicts rays extending from the abstractly represented (single) eye of the artist, passing out through the cells of the artist's grid and forming a visual pyramid along their way to their final destination: an array of planes in the background of the figure. To the right of the grid is a correspondingly gridded notepad to which the artist is supposed to transfer the contents of each successive cell, contents that have been 'measured' by the rays, which reach out like feelers to touch the corresponding portions of reality. In this way the artist can apprehend in systematic and accurate fashion the visual qualities in the scene before him. Dürer's treatise on measurement, his Underweysung der Messung of 1527, illustrates a range of similar machines by which an artist might 'scientifically' depict people and objects along these same lines. The machines employ a glass plate or frame divided into small squares by a net or veil of black thread. This allows the imagined artist to locate marks within the space of the painting in such a way that their shapes, sizes and relative positions conform to what we would see if we were observing corresponding objects in reality. At the same time the grids encourage a new way of seeing, through which a portion of the visible world is organized into a geometric composition. SECTION #UEYZZC 2 Theatrum Orbis Terrarum EXCERPT #LQTNJT p. 2 The practical problem of projecting an array of objects existing in three-dimensional space onto a two-dimensional plane was solved at around the time of Brunelleschi, who is held to have created the first painting – of the Baptistery of St. John in Florence – in 'true perspective', sometime between 1415 and 1425. The problem was solved theoretically by Alberti in Book I of Della pittura , which presents the mathematical theory of the way in which a plane intersects a visual pyramid in exactly the way that is captured intuitively in images of the reticolato . Over the next century and a half Brunelleschi's and Alberti's work, and that of their contemporaries and successors, including not only Dürer but also Piero della Francesca and Leonardo da Vinci (all of whom were influenced by Alberti), transformed painting in a way which enabled European art for the first time to free itself from the inhibiting burden of those earlier traditions of visual representation which had remained unaware of perspective. EXCERPT #QJXLF8 p. 2 The theoretical solution of the problem of perspective put forward in Della pittura was a scientific discovery of the first importance, and it ranks with the later contribution of Desargues in launching our contemporary understanding of projective geometry. But why did mankind have to wait until the fifteenth century, 1700 years after Euclid's Elements and Optics , to take what Panofsky calls 'the apparently small step of intersecting the visual pyramid with a plane'? How can this be lag explained, if perspective had been present in all seeing from the very start? Samuel Edgerton, in his The Renaissance Rebirth of Linear Perspective , presents a two-part solution to this problem, holding 1. that there arose among a certain group of citizens of Florence in the early years of the fifteenth century a new way of apprehending visual space as a structure ordered by an abstract uniform system of linear coordinates, and 2. that the decisive impetus towards this new way of seeing was inspired by developments in cartography, and specifically by the rediscovery of Ptolemy's Geographia , a work dating from around 140 A.D., which arrived in Florence in 1400 to great acclaim. EXCERPT #R6GW6J p. 2 In more traditional metaphysical systems, such as were employed, for example, by Aristotle, a distinction had been drawn between the realm of astronomy, which is subject to precise, intelligible mathematical laws, and the sublunar world of change and decay, which is only partially intelligible to mortals such as ourselves. The principal achievement of Ptolemy's Geographia turned on its demonstration of the possibility of using a regular mathematical grid system to map the entire known world. Ptolemy thereby showed how the earth below could be comprehended in a uniform way in terms of a single mathematical system. Essential to this achievement was the idea that the grid not only have the mathematical properties of an exhaustive tessellation, but also that it be transparent . Ptolemy's grid is not a part of any of reconstruction of some abstract mathematical realm. It is designed, rather, to help us to grasp this world, the world of sensate matter, as it really is. EXCERPT #XK6NMJ p. 3 The impact of Ptolemy's transparent grid system was so great that already by 1424 Florence has acquired the reputation of a center of cartographic and geographic study, and its influence may have extended, through commentaries on Florentine versions of the Geographia , to Christopher Columbus. Ptolemy's grid system also began to be taken up as a basis for territorial boundary-demarcation. Certainly grid systems had been used for surveying purposes since much earlier times, above all by the agrimensores who had introduced centuriation into many parts of the Roman Empire. But like the grids used in the seventeenth century in dividing up the Dutch polders Roman centuriation applied always to the demarcation of intraterritorial lines. During the wars of 1420, however, a longitudinal line was proposed as the boundary between the two states of Milan and Florence. Edgerton (1975, p. 115) conjectures that this may have been the first occasion when an imaginary mathematical line – a fiat boundary – was recognized as a political-territorial limit. EXCERPT #W4PBN8 p. 3 As Veltman (1977) points out, there are a number of problems with the details of Edgerton's account. Yet the similarities between Ptolemy's method of projecting arcs of circles visible on a globe onto a planar map and the method of perspective painting encapsulated in Alberti's reticolato are strikingly close, and the hypothesis that Alberti recognized the significance of Ptolemy's cartographic projection method for painting is supported further by Alberti's own claims on behalf of his reticolato , for example that it 'affords the greatest assistance in executing your pictures, since you can see any object that is round and in relief, represented on the flat surface of the veil.' (Alberti 1435/36, pp. 68 f.) EXCERPT #B3GQGG p. 3 In his introduction to the English translation of Della pittura , Spencer conjectures (1956) that Alberti may have arrived at his solution to the problem of perspective also through his own experiences in the domain of surveying. 1 Between 1431 and 1434, which is to say just before the completion of Della pittura , Alberti composed a small work entitled Descriptio urbis Romae in which he sets forth both a method for surveying and a table of sightings obtained in applying this method to yield what he calls a 'picture' of Rome. 2 A surveyor needs some means to determine the proportionate distance between any two quantities. There can be no doubt that Alberti understood such a method, 3 and one which did not make use of trigonometry, which had not yet been invented. Further evidence that surveying is a source of Alberti's construction is provided by the privileged role awarded by Alberti to the measure of a staff held at arm's length. Spencer points out further that Piero della Francesca gives an account of a perspective construction – based on EXCERPT #9AD3RP p. 3 1 The association between optics and surveying has a long tradition, as is shown not least by the inclusion of four theorems on surveying in Euclid's Optics . The philosopher Al-Farabi could write of optics in the tenth century that it 'makes it possible for one to know the measurement of that which is far distant, for example, the height of tall trees and walls, the width of valleys and rivers, the height of mountains and the depth of valleys, rivers' (from Veltman 1999). EXCERPT #UUJFVZ p. 3 2 An account of the instrument which Alberti invented for this purpose is given by Spencer as follows: EXCERPT #RNB9HW p. 3 a bronze disc [is] mounted parallel to the surface of the earth and divided on the circumference into 48 degrees. At the centre a metal or wooden ruler, divided into 50 degrees, is pivoted. ... When the ruler is placed at right angles to the line of sight, it becomes possible to compute the distance of the object – given its width – or its actual width – given the distance – by means of the similarity of triangles. (Spencer 1956, pp. 113 f.) EXCERPT #WJATWE p. 3 3 In his Ludi mathematici composed for Borso d'Este about 1450 he demonstrates the well known operation of determining the width of a stream by means of a staff and the similarity of triangles. (Spencer, 1956, pp. 114 f.) EXCERPT #ZNSHUR p. 4 plan and elevation drawings connected by lines from a point of sight and cut by a perpendicular – which is essentially Alberti's surveying method from the Descriptio urbis Romae moved indoors to the drawing board. SECTION #AXLW7Z 3 Fiat Lux EXCERPT #RA9ENQ p. 4 Alberti's contribution to the history of cartography has been noted by others. Our purpose here is to show that Della pittura contains also a contribution to our understanding of the ontology of pictures which can be generalized to projective devices in general. There are, according to Alberti, two kinds of matter with which the painter must be concerned. On the one hand is the three-dimensional matter of the observable world, which exists in space and light. On the other hand is the two-dimensional matter of the painting, a simulacrum of reality that is produced by the painter, who 'must find a means of controlling the matter of the macrocosm if he is to represent it in his microcosm.' (Spencer 1956, p. 19) The first kind of matter is composed of surfaces in three-dimensional reality, the second of marks the artist makes on the flat plane of the canvas. (Compare Gibson 1980) This second kind of matter exists, if the artist is successful, in the form of a visual story ( istoria ) that is constructed out of points, lines and planes (marks) on a panel or canvas. The latter are grouped together to form (for example) limbs, bodies and groups of bodies related together in a way that is analogous, as Alberti sees it, to the way in which words, phrases, sentences and paragraphs are related together in natural language. Alberti develops rules for manipulating these various elements in an istoria , based on the four principles of dignità, varietà, modestia and verisimilitudo . Together with geometry, these principles constitute the basis of a rational art or indeed of a science of painting. EXCERPT #7A9U4X p. 4 No painter can paint well who does not know geometry. 4 The observed scene, the scene that is visible and that is to be represented by the artist, is made of finite surfaces out there in the world. The painter's job is to find the appropriate shapes, sizes and positions for the counterparts of these surfaces within the microcosm of the painting in such a way as to constitute an istoria . EXCERPT #E379WJ p. 4 The totality of surfaces in the macrocosm exists objectively, though it changes from moment to moment with changes in the ambient light. (It is as if the sun, by a sort of divine fiat, makes a selection of which surfaces shall belong to this totality from moment to moment.) In addition to this global selection, however, each observer effects his own local selection from this totality in such a way as to yield a framed arrangement of observed surfaces of the sort which we see when we look through an open window. The array which results through this local selection is dependent upon the observer's position and on the scope and direction of his gaze. Moreover, some surfaces in the observed array are foreshortened because they have parts which are obstructed for this observer or are such as to fall outside the scope of what the artist will choose to represent. In this way there is created out of an in principle infinite totality a selection of a sort that can be comprehended by a finite mind. EXCERPT #ZXCTVC p. 4 4 Alberti 1435–36, p. 89. Compare Leonardo's Non mi legga chi non e matematico . ('Let no one read me who is not a mathematician.') SECTION #LBDZZH 4 Qualitative Geometry EXCERPT #QR7BYC p. 5 For all of this, however, the results of this act of selection are, because they fall within the first kind of matter, still something entirely objective: they belong to the world of space and light out there. Compare the ontological status of the events which take place on the stage in a theater. Certainly the latter constitute a play only because of the way they are perceived and understood (and separated off by fiat from the events around them); but they exist nonetheless objectively, as movements of bodies and props. These movements are however subject to a further series of effects because of the ways the spectators in the theater react towards them. They find one movement threatening, another welcoming, and so on. And so also in our present case: the objective array of surfaces is subjected, when viewed by an observer, to effects of an analogous sort. Some surfaces will appear to be larger or of a different color or shape, some figures will dominate, others will recede into the background. EXCERPT #5HU5EF p. 5 The artist's job, according to Alberti, is to project the objective array of surfaces into the microcosm of the painting in such a way as to achieve a maximally beneficial (moral) effect. The buildings, too, in which the painting is to be displayed, should likewise be designed on the basis of a combination of geometrical and moral principles, and the same applies also (Alberti was a pioneer of urban planning) to the city in which these buildings are to be arranged. (Westfall 1974) EXCERPT #XKC8E8 p. 5 Alberti is sometimes described as the first universal genius, and his work, whether on painting, on architecture, on town planning, or on the morality of the family, always transcends the purely theoretical sphere. This is no less true in the domain of mathematics, where Alberti was the first to present the geometrical principles of linear perspective. For even here his concerns point always in the direction of practical implications. As he himself expresses it: mathematicians examine the form of things as separated from their matter. Those, however, who wish the object to be seen 'will use a more sensate wisdom'. (1435–36, p. 42) Alberti's interest is accordingly not in form separated from matter, but rather in form as it is visible, which means: in the matter that is located in space and that is affected by ambient light. EXCERPT #QF3LPC p. 5 He thus develops a version of Euclid's geometry not in terms of abstractions but in terms of concrete visible 'signs' or 'marks' (recall that the term used by Euclid himself for what we call 'point' is ' semeion ' or 'sign'): EXCERPT #9XDNVL p. 5 The first thing to know is that a point is a sign [ signum ] which one might say is not divisible into parts. I call a sign anything which exists on a surface so that it is visible to the eye. ... Points joined together continuously in a row constitute a line. So for us a line will be a sign whose length can be divided into parts, but it will be so slender in width that it cannot be split ... If many lines are joined closely together like threads in a cloth, they will create a surface. (Alberti 1435–36, p. 42) EXCERPT #T79342 p. 5 It is in the same vein that Alberti proposes for the outer edge by which a surface is bounded the terminology of 'brim' [ ora ] or 'fringe' [ fimbria ], terms connoting the edge of a piece of cloth or garment. In a separate tract entitled De punctis et lineis apud pictures Alberti writes: 'Points and lines among painters are not as among mathematicians, [who think that] in a line there fall infinite points.' (Edgerton 1975, p. 81) Alberti hereby anticipates contemporary work on so-called qualitative geometries (Bennett et al. , 2000), which means: geometries based, not on abstract mathematical points, but rather on finite regions. Both his theory of perspective and his theory of the organization of marks or signs to form an istoria are formulated in qualitative-geometrical terms. SECTION #NJMY36 5 Rays of Marvelous Subtlety EXCERPT #NAC5UE p. 6 The surfaces in the objective array and their qualities of color, shape and size are, Alberti tells us, 'measured with sight'. What he means by this he explains by referring to 'the maxims of the philosophers' for whom there are rays that serve the sight 'which carry the form of the thing seen to the sense.' These visual rays, which are depicted in Figure 2 as extending between the single, fixed eye and the array of surfaces seen in the background, constitute what we have called a 'visual pyramid'. They are such that 'by a certain marvelous subtlety' they penetrate the air and 'all thin and clear objects' until EXCERPT #Q8VKV6 p. 6 they strike against something dense and opaque, where they strike with a point and adhere to the mark they make. Among the ancients there was no little dispute whether these rays come from the eye or the plane. This dispute is very difficult and is quite useless for us. It will not be considered. We can imagine those rays to be like the finest hairs of the head, or like a bundle, tightly bound within the eye where the sense of sight has its seat. The rays, gathered together within the eye, are like a stalk; the eye is like a bud which extends its shoots rapidly and in a straight line on the plane opposite. (Alberti 1435–36, pp. 44 f.) EXCERPT #EQ5UKM p. 6 Alberti's reference in this passage to 'the ancients' relates to the disputes among philosophers between so-called intromissionist and extromissionist views of visual perception. For the intromissionists, vision is to be explained in terms of light passing from the object and into the eye. For the extromissionists, vision is an active process involving 'visual rays', which move from the eye and out into the world of surfaces beyond. When Euclid, in his Optics , demonstrates theorems about visual angles, then it is in terms of an extromissionist theory of visual rays that these theorems are formulated. (The validity of the laws of geometrical optics work is not affected by the direction of the visual rays.) EXCERPT #R3YRUW p. 6 For Euclid visual rays are homogeneous. For Ptolemy, on the other hand, another extromissionist, the centermost visual ray, which flows directly from the eye and strikes at right angles the surface of what is seen, is privileged as contrasted with 'median' rays on the fringes of the cone of rays emanating from the eye. When Galen isolated the eye's crystalline lens as the seat of visual power, he sees the lens, still in extromissionist terms, not as a receiver but rather as a transmitter of visual force. EXCERPT #VNCR8N p. 6 Why, we might reasonably ask, did Euclid and Ptolemy and Galen, and many other prominent thinkers of the ancient world, defend what must nowadays seem so counterintuitive a view of visual perception? One reason was the supposed power of cats and other nocturnal animals to see in the dark. The primary argument for extromissionism however turned on the atomism embraced by many ancient thinkers. The extromissionists pointed out that it would be impossible that every point on a large visual surface should be transmitted simultaneously to a single point via atoms of light. The 'effluxes of things so large as, say, a camel or a mountain could not very well pass through the tiny pupil of the eye'. (Edgerton 1975, p. 67) EXCERPT #CX3DG8 p. 6 It was the Arab thinker Alhazen who, by solving this 'large efflux' problem, established the viability of intromissionist optics by showing how refraction can filter out excess information in the light. Alhazen showed how it was in fact possible even in atomistic terms for every point on the surface of an object seen in nature 'to convey its form to the seat of vision within the eye – in an exact one-for-one, place-for-place proportionate way.' (Edgerton 1975, p. 74) Alhazen's theory of refraction explained also the privileged status of rays close to the axis of sight: they travel unrefracted through to the optic nerve. EXCERPT #NYE87G p. 7 In his commentary on Aristotle's De sensu , Albertus Magnus distinguishes four positions on this dispute: EXCERPT #P7S8A8 p. 7 a. extromission of visual rays (for example Empedocles); b. intromission of atoms (corporeal images: Democritus); c. intromission of forms (spiritual images: Aristotle); d. simultaneous extromission and intromission of rays (Plato); EXCERPT #YZWPN4 p. 7 Extromissionism lives on in the thinking of Augustine and Anselm, but the success of the intromissionist theory is given a powerful boost when it is taken up by three thirteenth-century English scholars: Robert Grosseteste, Roger Bacon and John Pecham, who saw in the new optical theories of the transmission of light the model of how God spreads the light of grace to his subjects in the world. As Veltman points out (1999), however, Leonardo could still defend a combined extromissionist-intromissionist view (and in presenting the arguments for an extromissionist component in the visual process Leonardo refers to the power which maidens have in their eyes 'to attract the love of men.'). Even by the time of Kepler the debate was still not conclusively settled. EXCERPT #VA873L p. 7 Indeed, extromissionism still lives on today. And this is so even in spite of all subsequent developments in our understanding of the physics of light and of the physiology of the eye. It lives on in contemporary philosophy and cognitive science in the context of discussions of what is nowadays called 'intentionality'. 5 And it is in these terms, I suggest, that the visual rays of Alberti – and perhaps even of Euclid 6 – are to be understood. EXCERPT #W4CLTM p. 7 'Intentionality' is the term employed especially in the tradition of Brentano, Husserl and Ingarden to refer to the directedness of the mind towards its objects. Husserlians sometimes speak of the arrow of intentionality, and Husserl himself (1970) uses the terminology of 'mental rays' for example when he distinguishes between single-rayed and many-rayed acts of perception. All cognitive directedness towards objects, for Husserl, depends on perceptual intentionality and all perceptual intentionality depends on action. (Mulligan 1995) Language, including maps and diagrams, Husserl sees as a vehicle by which intentional directedness is leveraged beyond the realm of objects given in direct perception. EXCERPT #A29HT3 p. 7 Gibson's ecological psychology, too, can be understood in these terms. It represents a mixed intromissionist-extromissionist view, according to which each organism, in each given context, is tuned to certain specific types of invariants within its surrounding ocean of energy. The organism picks up the information available in the environment that is relevant to its actions in a spontane- EXCERPT #MSFTLN p. 7 5 Compare also the extromissionist theory of vision propounded in Pylyshyn (1989), which presents a view of the visual system as employing a limited number of visual indexes that go out into the world and adhere to whatever it is that the visual system wants to interrogate. These indexes are hypothesized to allow access to the objects that they individuate, and because they are sticky, they are able to track objects and to be updated automatically as the object moves about in the environment. EXCERPT #Z2E4T7 p. 7 6 On this reading, the visual rays which Euclid conceives as projecting from the eye are not to be conceived in physical terms at all. For the theory of perspective belongs not to physical but rather to geometric optics, which is what results when we adopt simplifying assumptions to the effect that the wavelength of light is zero and that rays propagate through homogeneously refractive media along straight lines. Euclid's visual rays would then be analogous to the abstractly conceived (fiat) lines of his own geometry, rather than to rays of light in the proper, physical sense – or indeed to X-rays, or to any other physical manifestations of 'marvelous subtlety'. EXCERPT #E27LTS p. 8 ous process which involves, not inference or other cognitive processes, but rather (in our terminology) something like a transparent grid into the cells of which the affordances of the environment exactly fit. EXCERPT #BGWENQ p. 8 The views presented here can now be seen as a generalization of the Husserlian and Gibsonian accounts of organism-environment interactions to apply also to the projection onto reality that is involved in our uses of maps, pictures, databases, catalogues and taxonomies of various sorts. When we use a proper name to refer to an object, then a relation of projection comes into existence: the name projects out towards the object, be it present or absent, in virtue of an intentional ray. When we use a photograph to refer to an object, then a relation of projection likewise comes into existence: the photograph projects out towards the object in virtue of a whole pattern of intentional rays. And similarly, when using maps or spreadsheets we employ labelled grids to project in multi-rayed fashion onto corresponding objects in reality. EXCERPT #X9YNYN p. 8 H 1 He 2 Li 3 Be 4 B 5 C 6 N 7 O 8 F 9 Ne 10 Na 11 Mg 12 Al 13 Si 14 P 15 S 16 Cl 17 Ar 18 K 19 Ca 20 Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30 Ga 31 Ge 32 As 33 Se 34 Br 35 Kr 36 Rb 37 Sr 38 Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48 In 49 Sn 50 Sb 51 Te 52 I 53 Xe 54 Cs 55 Ba 56 La 57 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80 Tl 81 Pb 82 Bi 83 Po 84 At 85 Rn 86 Fr 87 Ra 88 Ac 89 Rf 104 Ha 105 ?? 106 Lanthanide Series Ce 58 Pr 59 Nd 60 Pm 61 Sm 62 Eu 63 Gd 64 Tb 65 Dy 66 Ho 67 Er 68 Tm 69 Yb 70 Lu 71 Actinide Series Th 90 Pa 91 U 92 Np 93 Pu 94 Am 95 Cm 96 Bk 97 Cf 98 Es 99 Fm 100 Md 101 No 102 Lr 103 EXCERPT #XL8WNL p. 8 Fig. 3 The Periodic Table of the Chemical Elements SECTION #UNR4PK 6 How to Tell the Truth with Maps EXCERPT #8NZETC p. 8 A good map casts a transparent net over the surface of the earth in just the way Alberti's reticolato casts a transparent net over some portion of objective reality. As the painter's grid casts into relief a certain visual scene, so the grid of the map casts into relief a certain spatial region. There is a deep-seated analogy here; but it is an analogy that has nothing to do with perspective – for it obtains even in relation to maps and plans of strictly two-dimensional planar arrays. It has to do, rather, with the highly general concept of a transparent grid and with the associated highly general notion of projection , both of them notions which (as Figure 3 makes clear) can be applied even to types of organization which are entirely non-spatial. (Bittner and Smith, in this volume.) EXCERPT #PZG2L9 p. 9 To see what all of these cases have in common we need to boil Alberti's reticolato down to its essential elements, which we can list as follows: EXCERPT #ALTNLS p. 9 1. the eye (or point of projection), 2. projective rays, 3. the artist's grid, 4. the constituent cells of the artist's grid, 5. the totality of objective visible surfaces, 6. the target grid: the artist's grid as projected onto the objective visible surfaces, 7. the constituent cells of the target grid EXCERPT #VH67DQ p. 9 Extending from the point of projection (1.), projective rays (2.) bring about a one-one correspondence between the two arrays of cells (4. and 7.), within the artist's grid (3.) and the target grid (6.), respectively. We shall call a structure of this sort a true grid . A true grid is transparent to the corresponding objects in reality. EXCERPT #M3LQQH p. 9 Almost all our customary maps are true grids in the sense defined. In this case the term for the eye or station-point (1.) is replaced by that of the user of the map. The projective rays (2.) are replaced by relations of rigid designation (to be discussed below). The counterpart of the artist's grid (3.) is constituted not just by the rectilinear grid of the map but more generally by whatever is the pattern of contour and border lines and cartographic icons (4.) to be found on the map. The counterpart of (5.) is the corresponding portion of the earth's surface, and the counterparts of (6.) and (7.) are the results of projecting the grid of the map onto this more or less planar region. EXCERPT #5TT3SV p. 9 The image is a black and white diagram illustrating the concept of cartographic projection. On the left side, there is a map of the British Isles, including Great Britain and Ireland, overlaid with a grid of irregular cells representing the map's grid. On the right side, there is a grayscale image of a portion of the Earth's surface, showing land and water. A grid of cells is also visible on the Earth's surface, representing the target grid. Several straight lines, representing projective rays, originate from a point on the map's grid and extend towards the Earth's surface, showing how the map's grid is projected onto the real-world surface. A diagram illustrating cartographic projection. It shows a map of the British Isles on the left, with a grid of cells. On the right, a portion of the Earth's surface is shown, also with a grid. Projective rays originate from a point on the map and extend towards the Earth's surface, demonstrating the mapping process. EXCERPT #X3N6YB p. 9 Fig. 4 Cartographic Projection EXCERPT #PR2R7Q p. 9 As in the reticolato , so also here we need to distinguish in the ontological structure of the map between two distinct grids. On the one hand is the grid of the map itself, which is in the simplest case a system of regular or irregular cells, each cell enjoying a certain intrinsic position within the grid and thus also standing in certain determinate relations to its neighboring cells. On the other hand is an isomorphic grid on the side of the target portion of the surface of the earth (in Figure 4 a grid of English counties). Regular cells in such grids will standardly be identified by their coordinates within the grid itself; irregular cells will standardly be assigned proper or common noun labels such as 'Berkshire' or 'lateral geniculate nucleus'. EXCERPT #FBAZDY p. 10 In the case of the reticolato , the projective rays can be made to point in whichever direction the user might desire. In the case of a map, on the other hand, the projective rays tie the cells of the map rigidly to corresponding portions of reality. This means that when a user buys a map, he buys not any simple piece of paper but rather a complex cognitive device from out of which, as soon as he begins to use the map, there will project invisible arrows – rays of marvelous subtlety – which tie its constituent cells rigidly to corresponding features on the ground. 7 SECTION #DFMTJ5 7 Semantic Projection EXCERPT #5QETQ9 p. 10 Such lines of projection are at the basis, too, of the so-called picture theory of meaning defended by Wittgenstein in the Tractatus : EXCERPT #Q2ZKRW p. 10 The pictorial relationship consists of the assignments of the picture's elements to the things. These correlations are, as it were, the feelers of the picture's elements, with which the picture touches reality . (2.1514 f., italics added) EXCERPT #JVXZAW p. 10 A true proposition, for Wittgenstein, is a picture or map of a state of affairs in reality. It is a propositional sign in its projective relation to the world . (3.12) Each (atomic) proposition in the Tractarian framework consists of simple signs (names), which stand in a projection relation to corresponding simple objects. If the proposition is true, then these simple signs stand to each other in the propositional picture as the corresponding objects stand to each other in the world. Here the counterpart of the artist's grid, in Alberti's terminology, is the propositional sign, a complex of names arranged in a certain order, the names serving as the equivalent of the constituent cells. The counterpart of the target grid is a state of affairs in the world. EXCERPT #9TGCJ7 p. 10 7 We can now see our way to resolving a thorny problem highlighted by Ernst Gombrich, which turns on the apparent irreversibility of the projective function involved in maps and pictures. The theory of perspective representation, as Gombrich notes, EXCERPT #ED6SWG p. 10 was treated as if it were a mapping procedure. It was claimed that it enabled the artist to represent what has been called 'measurable space'. Yet ... it is clear from the theory of central projection that you cannot reverse the process: while we can work out what the projection of a three-dimensional object will be like on a given plane, the projection itself does not give us adequate information about the object concerned, since not one but an infinite number of related configurations would result in the same image, just as not one but an infinite number of related objects would cast the same shadow if placed in the beam emanating from a one-point source. (1974, pp. 190 f.) EXCERPT #RBPADL p. 10 Gombrich extends this indeterminacy of projection also to maps: the answer to the question 'how can we ever know whether a picture or a map represents a particular building?' is, he says, simple: we cannot . (Op. cit., p. 175) But surely something has gone wrong with Gombrich's thinking here. Millions of people are, after all, using maps perfectly successfully every day to find particular token buildings. To do justice to this fact we need to recognize that the lines of projection emanating from maps, as also from representational paintings, and photographs and other similar projective devices are not purely geometrical. Rather, they involve also a combination of semantic, perceptual and other projective elements and some of their constituent cells are analogous to proper names, which tie their users rigidly to specific object tokens. EXCERPT #NS3K67 p. 11 It is a complex question how far Wittgenstein's picture theory of meaning can be extended to language in general. Where it does unproblematically apply is in relation to simple lists, which constitute true grids in the sense here intended – provided only that the items listed do indeed exist in reality. Even a single name, for example 'Mama', constitutes a true grid under the obvious ('Mama' – Mama) projective relation. EXCERPT #ALZU85 p. 11 Moreover, as Figure 3 once more reveals, a system of concepts, too, can form a true grid in the sense defined. The idea of a projective relation from concepts to corresponding categories on the side of target objects in reality is at work in the following passage from Millikan: EXCERPT #DMED94 p. 11 The membership of the category 'cat,' like that of 'Mama,' is a natural unit in nature, to which the concept cat does something like pointing , and continues to point despite large changes in the properties the thinker represents the unit as having. ... The difficulty is to cash in the metaphor of 'pointing' in this context. (Millikan 1998) EXCERPT #S4GD5V p. 11 The generalized reticolato and the associated notion of projection can, I suggest, help to cash in Millikan's pointing metaphor in precisely the way required. EXCERPT #2SJXAV p. 11 The generalization from Alberti's reticolato to maps is in one sense simple. Both involve grids which are recognizable as such; both involve relations between spatial neighborhoods which can easily be defined in topological terms; both types of grid can also be subject to the same types of geometrical transformations. It is at first sight difficult to see how we can generalize beyond these sorts of cases to talk of semantic or conceptual grids. In light of recent advances in mereotopology and in the study of so-called 'conceptual neighborhoods' or 'continuity networks', and also in light of the work on granular partitions outlined in the paper by Bittner and Smith (in this volume), we can more readily understand what such generalized grids involve. They all share in common – in the ideal case – the presence of a domain (the user's grid) and a co-domain (the target grid), with systems of mereotopological relations defined on each, and with a notion of correspondence or mapping connecting the two of a sort which – in the case of a true grid – preserves mereotopology. EXCERPT #YBAGQV p. 11 But not all grids are true. For while the examples dealt with so far have involved an isomorphism between cells in the grid and corresponding objects, grids may fall short of such perfection by involving some mismatch between user's grid and target grid. Such a mismatch can come about either because the projective relation is not well-defined (cells in the grid are putatively projected onto objects where there are no such objects) or because the cells of the grid do not stand to each other in relations isomorphic to the relations between the corresponding target objects. EXCERPT #7NJEH5 p. 11 Even grids which satisfy both of these requirements may still fall short of the sort of perfection that is manifested in the examples of optical, cartographic and conceptual projection referred to above. The grids of the latter satisfy a requirement to the effect that the cells within the target grid fit exactly to the corresponding cells of the user's grid. This condition can in various ways be weakened. Bittner and Stell (1998) offer an approach to spatial grids otherwise similar to the one advanced here but within which the restriction on cell-object fit is relaxed through the notion of 'rough' location. Smith and Brogaard (2001, 2001a) show how the theory of true grids can be used to develop a new version of the supervaluationist theory of vagueness. Their work turns on the idea that grids are always such as to involve a certain coarse-grainedness, which implies that their cells trace over parts or features of reality which fall beneath a certain size. This in turn means that the latter can vary while the user's grid – which represents our cognitive access to the relevant objects – remains the same. The phenomenon of vagueness, from this perspective, is just the other side of the coin from the phenomenon of granularity. It arises because of the possibility of a variation that falls, in a given context, beneath the threshold of salience. SECTION #XKWVDP 8 Directions of Fit EXCERPT #9UWM7H p. 12 Each true grid – be it optical, cartographic, or conceptual – effects a tiling of the portion of reality towards which it is directed. In some cases, as in the case of a gridded map or an Albertian grill, this imputed tiling – a system of fiat boundaries in reality to which the grid directly corresponds – is of no intrinsic significance. Even a purely arbitrary imputed tiling may, however, acquire significance if its fiat cells are put to specific practical purposes by colonial administrators or postal authorities. (Smith 2001) In some cases, however, the grid of a map reflects prior independently existing boundaries on the side of the objects themselves. This is so, for example, of the irregular grid depicted in Figure 4 above, which reflects not only the fiat division of England into counties but also the bona fide division between England and the sea, which (partially) surrounds it. In yet other cases – this is so above all in the case of cadastral maps – the grid of a map stands to its target grid in a symbiotic relationship. As the objects change (because the fiat boundaries of land parcels are redrawn or re-measured, or because the land itself has been subject to erosion), so corresponding changes are made in the grid of the cadastre; and as administratively motivated changes in the grid of the cadastre are effected, so this may bring about changes in the objects (land parcels) on the ground. EXCERPT #WQJ4BG p. 12 We can thus distinguish, for true maps, three different sorts of cases: EXCERPT #ALS7KK p. 12 1. the target grid depends exclusively on the grid of the map (a map-to-world direction of fit) 2. map grid and target grid are mutually dependent upon each other (as in the case of a cadastre) 3. the grid on the map reflects a pre-existing grid in reality (a world-to-map direction of fit). 3. can be divided into further sub-cases, according to whether the grid of the map reflects 3a. bona fide boundaries on the side of the target objects themselves 3b. pre-existing fiat boundaries on the side of the target objects 3c. some combination of bona fide and pre-existing fiat boundaries. EXCERPT #3UXFEL p. 12 The same family of cases can be distinguished also in the domain of conceptual projection. Corresponding to 1. is the case where the distinctions in the world are mere reflections of our concepts (for example when the baseball coach divides up his team by assigning positions to his players at the beginning of the game). An example under 2., the symbiotic case, might be the set of prize categories used by dog shows, which both reflects the divisions on the side of the sample domain and also, over time, may itself bring about adjustments to these divisions. Corresponding to 3., finally, is the case where concepts reflect pre-existing distinctions among objects in reality, whether bona fide (3a.), for example the distinctions between the six different sorts of quark; or fiat (3b.), for example distinctions between different tax brackets; or mixed (3c.), for example the distinctions among bird species or among languages and dialects. SECTION #EVX3VY 9 Windowless Monads EXCERPT #QCK2SE p. 13 Epistemological skepticism is a view to the effect that conceptual classifications of type 3a. are forever beyond our reach. Such epistemological skepticism goes hand in hand with the views of many artists in recent times, who have been pleased to ignore perspective geometry, just as they have ignored Alberti's four principles of dignità, varietà, modestia and verisimilitudo . From at least the time of Duchamp, the visual arts have been freed from their connection to everyday life (and to beauty and harmony) and they have been recontextualized in the museum. The function of painting, if it has one, is not at all that of providing a window on the world, but rather that of drawing attention to itself. A painting is no longer conceived as a transparent device enabling the perceiver to grasp the reality beyond. Rather, it is an object in its own right, and the viewer is called upon to relish its materiality and its quality of opaqueness. EXCERPT #RWGZVK p. 13 Talk of a 'correct' perspectival representation, with its implication to the effect that there is some single detached master point of view, has at the same time come to be disparaged as a remnant of outmoded phallogocentric thinking. How can one or another method of painting be 'true' or 'correct', when there is no single notion of reality against which its results could be matched? As Henri Lefebvre puts it in his Production of Space : EXCERPT #QNJRKP p. 13 The fact is that around 1910 a certain space was shattered. It was a space of common sense, of knowledge ( savoir ), of social practice, or political power ... a space, too, of classical perspective and geometry, developed from the Renaissance onwards on the basis of the Greek tradition (Euclid, logic) and bodied forth in Western art and philosophy, as in the form of the city and town. (1974, p. 25) EXCERPT #GFZAXH p. 13 There is a simple argument from the realist side against all such nonsense. It is the same argument which can be used against all attempts to see reality as somehow dependent upon people's beliefs, and against all attempts to identify scientific truths with mere conventions of time or culture. In our present case the argument would run as follows: if perspective geometry is not inherent in the world – a structure waiting to be discovered – but rather a convention, which had to be invented , then this has the consequence that Renaissance men were living in a different world from the world of their medieval predecessors. It is this consequence which makes the skeptical and anti-realist positions seem so glamorous and exciting. But it is evidently a consequence no less absurd than the thesis that, with the dawning of the realization that the earth is round, the earth itself acquired a new geometry. To this, characteristically, the defender of the anti-realist view will respond that of course he is not wishing to be taken literally when he says that Renaissance men were living in a different world , or that they were ' producing ' or ' shattering ' a certain space. Such remarks are, he will say, mere metaphors. But then surely the interesting questions pertain, not to what metaphors have been fashionable at different points in human history, but rather to what the true structure of reality is – and this is a question which makes sense only from the realist perspective. EXCERPT #8AVFUG p. 13 As the physiologist M. H. Pirenne (1952) shows, even granting the simplifying assumptions of geometrical optics, perspective paintings correspond to the way we see the world around us with a very high degree of approximation. The best explanation of this correspondence lies in the thesis that the mathematical forms captured in the geometry of perspective are – modulo certain well-understood and in normal circumstances negligible simplifications – out there in the world, waiting for us to apprehend them through abstraction. They thus serve as truthmakers for the theory of perspective; and as Pirenne nicely puts it, the strange fascination which perspective had for the Renaissance mind was precisely 'the fascination of truth.' EXCERPT #CHGN9V p. 14 Certainly our understanding of perspective has developed over time. For whatever reason, it took a long time before people were ready to perform the abstraction of these mathematical forms. We can hazard that part of this reason turns on the need, before this step could be taken, for a certain detachment from the world of objects through the cultivation of the standpoint of the neutral, scientific observer, a standpoint which Renaissance thinkers, like some of their Greek predecessors, enjoyed, but which medieval thinkers lacked. Renaissance thinkers such as Alberti were able to grasp the world as an abstract, mathematical ‘container’ – as a stage upon which men move, and have their exits and entrances. It is indeed in the Renaissance that the theatrical audience is for the first time separated from and forced to adopt a particular point of view (or as we might also say, a particular perspective) in relation to the spectacle on the stage. SECTION #KHBJYM 10 Fit Happens EXCERPT #HXBEYN p. 14 There is nothing subjective in Alberti’s reticolato . As Pirenne makes clear, the geometrical relationship between an object and its projection on the picture plane obtains quite independently of whether there is an eye at the vanishing point. As the technology of laser-guided missiles reveals, the laws of perspective hold independently of the existence of subjects, observers, artists or cultures: they are laws governing the way light, space and the surfaces of objects are related together. The laws of perspective are laws of geometrical optics; they have nothing to do with neurology or psychology. Correspondingly, the picture drawn in perspective aims not at representing anything like the retinal image or any pattern of nervous stimulation on the side of an observer. Rather it aims to send to the eye the same distribution of light as that which the object itself would send. EXCERPT #868TWE p. 14 This corresponds to the theory of picture perception sketched by the great theorist of realism J. J. Gibson and encapsulated by his student Kennedy (1974) in the form of a definition of a picture as: ‘a surface treated so that it yields light to a particular station point, usually on a normal to the picture surface, which could have come from a scene in the real world.’ (Compare Gibson 1978.) EXCERPT #K446N4 p. 14 Gibson naturally recognizes that there are other sorts of pictures (including maps), some of which involve conventional elements (symbols, icons), which have nothing to do with the conveyance of light to the eye in a way which simulates the light that is projected from surfaces in three-dimensional space. Even these pictures can, however, be interpreted in realist fashion on the basis of the general theory of projection sketched above. There are of course also many cases of pictorial images in which perspectival or other features of the represented scene are distorted in one or other way. As the Gibsonian realist can insist, however, the fact that pictures are sometimes made, for whatever reason, in such a way as to embody such distortions does not imply that all pictures are lacking in the sort of transparency for which the followers of Alberti strove. EXCERPT #MHVS9U p. 14 All maps must be of a certain scale or combination of scales, just as every grid must have a certain resolution or granularity of cells. And since reality itself (as Gibson 1979 emphasizes) contains entities accessible at many different scales, it follows that no single grid can be complete. Rather, as scientific practice shows, we need grids of many different resolutions if we are to do justice to reality in its many aspects. This implies, as the enemies of realism are fond of pointing out, that there is no ‘God’s eye perspective’ or ‘view from nowhere’. This does not, however, mean that we are justified in drawing the conclusion that every single one of the myriad perspectives which we have at our disposal embodies a false view of reality. The inference from partiality to falsehood might indeed be valid, but only in a world without windows – a world in which no single one of our grids enjoys the condition of transparency. EXCERPT #Q8JTQD p. 15 The fact that there are maps which deviate, for whatever reason, from the strictly veridical representation of reality does not take away from the fact that – leaving aside any small errors which may have been made in the application of the relevant projection system – almost all maps are true of the corresponding portion of reality. This applies to Mercator's map, and it even applies to Saul Steinberg's View of the World from Ninth Avenue . Maps must of course embody some projection system in representing three dimensions on a planar surface. Yet those who see in this an argument to the effect that all maps must necessarily involve some form of systematic distortion are simply revealing their own misunderstanding of the nature of projection. They are like those who, on noticing that the Circle Line is represented on maps of the London Underground as a yellow band, complain of 'distortion' because yellow, rather than some other color, has been used. SECTION #DCWT2Q Acknowledgements EXCERPT #FLUNF9 p. 15 Support from the American Philosophical Society, and from the NSF (Research Grant BCS-9975557: "Geographic Categories: An Ontological Investigation") is gratefully acknowledged. SECTION #38GLQJ References EXCERPT #355TQ2 p. 15 Alberti, Leon Battista ca. 1435–1436 De pictura praestantissima , original (Latin) edition: Basel 1540 (reprinted Portland, Oregon 1972), Italian translation: Della pittura , Venice 1547. Cited according to the English translation: On Painting , 1956 (complete text available at http://www.noteaccess.com ). Bennett, B., Cohn, A. G., Torrini, P. and Hazarika, S. M. 2000 "A Foundation for Region-Based Qualitative Geometry", Proceedings of ECAI 2000 , Berlin, 204–208. Bittner, Thomas and Stell, John G. 1998 "A Boundary-Sensitive Approach to Qualitative Location", Annals of Mathematics and Artificial Intelligence , 24, 93–114. Bittner, Thomas and Smith, Barry (in this volume) "A Taxonomy of Granular Partitions". Collingwood, R. G. 1938 The Principles of Art , Oxford: Oxford University Press. Edgerton, Samuel Y. 1975 The Renaissance Rebirth of Linear Perspective , New York: Basic Books. Gibson, James J. 1978 "The Ecological Approach to Visual Perception in Pictures", Leonardo , 11:3, 227–235. Gibson, James J. 1979 The Ecological Approach to Visual Perception , Boston: Houghton-Mifflin. Gibson, James J. 1980 "A Prefatory Essay on the Perception of Surfaces versus the Perception of Markings on a Surface", in M. Hagen (ed.), The Perception of Pictures , Volume I: Alberti's Window , New York: Academic Press, xi–xvii. Gibson, James J. 1982 Reasons for Realism. Selected Essays of James J. Gibson , Edward Reed and Rebecca Jones (eds.), Hillsdale, NJ and London: Lawrence Erlbaum. Gombrich, E. H. (1975) "Mirror and Map: Theories of Pictorial Representation", Philosophical Transactions of the Royal Society of London , 270, 119–49, reprinted in E. H. Gombrich, The Image and the Eye , Ithaca: Cornell University Press, 1982. Husserl, Edmund 1970 Logical Investigations , London: Routledge and Kegan Paul, 1970. Kennedy, John Miller 1974 A Psychology of Picture Perception: Images and Information , San Francisco: Jossey Bass. Lefebvre, Henri 1974 La Production de L'Espace , Paris: Editions Anthropos. Millikan, Ruth Garrett 1998 "A common structure for concepts of individuals, stuffs, and real kinds: More Mama, more milk, and more mouse", Behavioral and Brain Sciences , 9: 1, 55–100. EXCERPT #ZK6NVW p. 16 Mulligan, Kevin 1995 "Perception" in B. Smith and D. W. Smith, eds., The Cambridge Companion to Husserl , Cambridge and New York: Cambridge University Press, 168-238. Panofsky, Erwin 1927 "Die Perspektive als 'symbolische Form'", Vorträge der Bibliothek Warburg , 258-330. English translation: Perspective as Symbolic Form , New York: Zone Books, 1991. Pirenne, M. H. 1952 "The Scientific Basis for Leonardo Da Vinci's Theory of Perspective", British Journal for the Philosophy of Science , 3: 10, 169-185. Pylyshyn, Zenon W. 1989 "The Role of Location Indexes in Spatial Perception: A Sketch of the FINST Spatial-Index Model", Cognition , 32, 65-97. Smith, Barry, 2001 "Fiat Objects", Topoi , 20: 2. Smith, Barry and Brogaard, Berit 2001 "Quantum Mereotopology", Annals of Mathematics and Artificial Intelligence (in press). Smith, Barry and Brogaard, Berit 2001a "A Unified Theory of Truth and Reference", Logique et Analyse , 43 (in press). Spencer, John R. 1956 "Introduction" to the English translation of Alberti 1435-36, 1-31. Veltman, Kim H. 1977 Review of Edgerton 1975, The Art Bulletin , 59: 2, 281-282. Veltman, Kim H. 1986 Linear Perspective and the Visual Dimensions of Science and Art (Leonardo da Vinci Studies I), Munich: Deutscher Kunstverlag, 1986. Veltman, Kim H. 1999 Continuity and Discovery in Optics and Astronomy (Leonardo da Vinci Studies II), http://www.sumscorp.com . See also http://www.mmi.unimaas.nl . Westfall, Carroll William 1974 In This Most Perfect Paradise: Alberti, Nicholas V, and the Invention of Conscious Urban Planning in Rome, 1447-55 , University Park: Pennsylvania State University Press. Wittgenstein, Ludwig 1961 Tractatus Logico-Philosophicus , with English translation by D. F. Pears and B. F. McGuinness, London: Routledge and Kegan Paul. DOCUMENT #DT9Y7X True Grid SECTION #UNR4PK 6 How to Tell the Truth with Maps EXCERPT #8NZETC p. 8 A good map casts a transparent net over the surface of the earth in just the way Alberti's reticolato casts a transparent net over some portion of objective reality. As the painter's grid casts into relief a certain visual scene, so the grid of the map casts into relief a certain spatial region. There is a deep-seated analogy here; but it is an analogy that has nothing to do with perspective – for it obtains even in relation to maps and plans of strictly two-dimensional planar arrays. It has to do, rather, with the highly general concept of a transparent grid and with the associated highly general notion of projection , both of them notions which (as Figure 3 makes clear) can be applied even to types of organization which are entirely non-spatial. (Bittner and Smith, in this volume.) EXCERPT #PZG2L9 p. 9 To see what all of these cases have in common we need to boil Alberti's reticolato down to its essential elements, which we can list as follows: EXCERPT #ALTNLS p. 9 1. the eye (or point of projection), 2. projective rays, 3. the artist's grid, 4. the constituent cells of the artist's grid, 5. the totality of objective visible surfaces, 6. the target grid: the artist's grid as projected onto the objective visible surfaces, 7. the constituent cells of the target grid EXCERPT #VH67DQ p. 9 Extending from the point of projection (1.), projective rays (2.) bring about a one-one correspondence between the two arrays of cells (4. and 7.), within the artist's grid (3.) and the target grid (6.), respectively. We shall call a structure of this sort a true grid . A true grid is transparent to the corresponding objects in reality. EXCERPT #M3LQQH p. 9 Almost all our customary maps are true grids in the sense defined. In this case the term for the eye or station-point (1.) is replaced by that of the user of the map. The projective rays (2.) are replaced by relations of rigid designation (to be discussed below). The counterpart of the artist's grid (3.) is constituted not just by the rectilinear grid of the map but more generally by whatever is the pattern of contour and border lines and cartographic icons (4.) to be found on the map. The counterpart of (5.) is the corresponding portion of the earth's surface, and the counterparts of (6.) and (7.) are the results of projecting the grid of the map onto this more or less planar region. EXCERPT #5TT3SV p. 9 The image is a black and white diagram illustrating the concept of cartographic projection. On the left side, there is a map of the British Isles, including Great Britain and Ireland, overlaid with a grid of irregular cells representing the map's grid. On the right side, there is a grayscale image of a portion of the Earth's surface, showing land and water. A grid of cells is also visible on the Earth's surface, representing the target grid. Several straight lines, representing projective rays, originate from a point on the map's grid and extend towards the Earth's surface, showing how the map's grid is projected onto the real-world surface. A diagram illustrating cartographic projection. It shows a map of the British Isles on the left, with a grid of cells. On the right, a portion of the Earth's surface is shown, also with a grid. Projective rays originate from a point on the map and extend towards the Earth's surface, demonstrating the mapping process. EXCERPT #X3N6YB p. 9 Fig. 4 Cartographic Projection EXCERPT #PR2R7Q p. 9 As in the reticolato , so also here we need to distinguish in the ontological structure of the map between two distinct grids. On the one hand is the grid of the map itself, which is in the simplest case a system of regular or irregular cells, each cell enjoying a certain intrinsic position within the grid and thus also standing in certain determinate relations to its neighboring cells. On the other hand is an isomorphic grid on the side of the target portion of the surface of the earth (in Figure 4 a grid of English counties). Regular cells in such grids will standardly be identified by their coordinates within the grid itself; irregular cells will standardly be assigned proper or common noun labels such as 'Berkshire' or 'lateral geniculate nucleus'. EXCERPT #FBAZDY p. 10 In the case of the reticolato , the projective rays can be made to point in whichever direction the user might desire. In the case of a map, on the other hand, the projective rays tie the cells of the map rigidly to corresponding portions of reality. This means that when a user buys a map, he buys not any simple piece of paper but rather a complex cognitive device from out of which, as soon as he begins to use the map, there will project invisible arrows – rays of marvelous subtlety – which tie its constituent cells rigidly to corresponding features on the ground. 7 DOCUMENT #DT9Y7X True Grid SECTION #DFMTJ5 7 Semantic Projection EXCERPT #5QETQ9 p. 10 Such lines of projection are at the basis, too, of the so-called picture theory of meaning defended by Wittgenstein in the Tractatus : EXCERPT #Q2ZKRW p. 10 The pictorial relationship consists of the assignments of the picture's elements to the things. These correlations are, as it were, the feelers of the picture's elements, with which the picture touches reality . (2.1514 f., italics added) EXCERPT #JVXZAW p. 10 A true proposition, for Wittgenstein, is a picture or map of a state of affairs in reality. It is a propositional sign in its projective relation to the world . (3.12) Each (atomic) proposition in the Tractarian framework consists of simple signs (names), which stand in a projection relation to corresponding simple objects. If the proposition is true, then these simple signs stand to each other in the propositional picture as the corresponding objects stand to each other in the world. Here the counterpart of the artist's grid, in Alberti's terminology, is the propositional sign, a complex of names arranged in a certain order, the names serving as the equivalent of the constituent cells. The counterpart of the target grid is a state of affairs in the world. EXCERPT #9TGCJ7 p. 10 7 We can now see our way to resolving a thorny problem highlighted by Ernst Gombrich, which turns on the apparent irreversibility of the projective function involved in maps and pictures. The theory of perspective representation, as Gombrich notes, EXCERPT #ED6SWG p. 10 was treated as if it were a mapping procedure. It was claimed that it enabled the artist to represent what has been called 'measurable space'. Yet ... it is clear from the theory of central projection that you cannot reverse the process: while we can work out what the projection of a three-dimensional object will be like on a given plane, the projection itself does not give us adequate information about the object concerned, since not one but an infinite number of related configurations would result in the same image, just as not one but an infinite number of related objects would cast the same shadow if placed in the beam emanating from a one-point source. (1974, pp. 190 f.) EXCERPT #RBPADL p. 10 Gombrich extends this indeterminacy of projection also to maps: the answer to the question 'how can we ever know whether a picture or a map represents a particular building?' is, he says, simple: we cannot . (Op. cit., p. 175) But surely something has gone wrong with Gombrich's thinking here. Millions of people are, after all, using maps perfectly successfully every day to find particular token buildings. To do justice to this fact we need to recognize that the lines of projection emanating from maps, as also from representational paintings, and photographs and other similar projective devices are not purely geometrical. Rather, they involve also a combination of semantic, perceptual and other projective elements and some of their constituent cells are analogous to proper names, which tie their users rigidly to specific object tokens. EXCERPT #NS3K67 p. 11 It is a complex question how far Wittgenstein's picture theory of meaning can be extended to language in general. Where it does unproblematically apply is in relation to simple lists, which constitute true grids in the sense here intended – provided only that the items listed do indeed exist in reality. Even a single name, for example 'Mama', constitutes a true grid under the obvious ('Mama' – Mama) projective relation. EXCERPT #ALZU85 p. 11 Moreover, as Figure 3 once more reveals, a system of concepts, too, can form a true grid in the sense defined. The idea of a projective relation from concepts to corresponding categories on the side of target objects in reality is at work in the following passage from Millikan: EXCERPT #DMED94 p. 11 The membership of the category 'cat,' like that of 'Mama,' is a natural unit in nature, to which the concept cat does something like pointing , and continues to point despite large changes in the properties the thinker represents the unit as having. ... The difficulty is to cash in the metaphor of 'pointing' in this context. (Millikan 1998) EXCERPT #S4GD5V p. 11 The generalized reticolato and the associated notion of projection can, I suggest, help to cash in Millikan's pointing metaphor in precisely the way required. EXCERPT #2SJXAV p. 11 The generalization from Alberti's reticolato to maps is in one sense simple. Both involve grids which are recognizable as such; both involve relations between spatial neighborhoods which can easily be defined in topological terms; both types of grid can also be subject to the same types of geometrical transformations. It is at first sight difficult to see how we can generalize beyond these sorts of cases to talk of semantic or conceptual grids. In light of recent advances in mereotopology and in the study of so-called 'conceptual neighborhoods' or 'continuity networks', and also in light of the work on granular partitions outlined in the paper by Bittner and Smith (in this volume), we can more readily understand what such generalized grids involve. They all share in common – in the ideal case – the presence of a domain (the user's grid) and a co-domain (the target grid), with systems of mereotopological relations defined on each, and with a notion of correspondence or mapping connecting the two of a sort which – in the case of a true grid – preserves mereotopology. EXCERPT #YBAGQV p. 11 But not all grids are true. For while the examples dealt with so far have involved an isomorphism between cells in the grid and corresponding objects, grids may fall short of such perfection by involving some mismatch between user's grid and target grid. Such a mismatch can come about either because the projective relation is not well-defined (cells in the grid are putatively projected onto objects where there are no such objects) or because the cells of the grid do not stand to each other in relations isomorphic to the relations between the corresponding target objects. EXCERPT #7NJEH5 p. 11 Even grids which satisfy both of these requirements may still fall short of the sort of perfection that is manifested in the examples of optical, cartographic and conceptual projection referred to above. The grids of the latter satisfy a requirement to the effect that the cells within the target grid fit exactly to the corresponding cells of the user's grid. This condition can in various ways be weakened. Bittner and Stell (1998) offer an approach to spatial grids otherwise similar to the one advanced here but within which the restriction on cell-object fit is relaxed through the notion of 'rough' location. Smith and Brogaard (2001, 2001a) show how the theory of true grids can be used to develop a new version of the supervaluationist theory of vagueness. Their work turns on the idea that grids are always such as to involve a certain coarse-grainedness, which implies that their cells trace over parts or features of reality which fall beneath a certain size. This in turn means that the latter can vary while the user's grid – which represents our cognitive access to the relevant objects – remains the same. The phenomenon of vagueness, from this perspective, is just the other side of the coin from the phenomenon of granularity. It arises because of the possibility of a variation that falls, in a given context, beneath the threshold of salience. DOCUMENT #DT9Y7X True Grid SECTION #KHBJYM 10 Fit Happens EXCERPT #HXBEYN p. 14 There is nothing subjective in Alberti’s reticolato . As Pirenne makes clear, the geometrical relationship between an object and its projection on the picture plane obtains quite independently of whether there is an eye at the vanishing point. As the technology of laser-guided missiles reveals, the laws of perspective hold independently of the existence of subjects, observers, artists or cultures: they are laws governing the way light, space and the surfaces of objects are related together. The laws of perspective are laws of geometrical optics; they have nothing to do with neurology or psychology. Correspondingly, the picture drawn in perspective aims not at representing anything like the retinal image or any pattern of nervous stimulation on the side of an observer. Rather it aims to send to the eye the same distribution of light as that which the object itself would send. EXCERPT #868TWE p. 14 This corresponds to the theory of picture perception sketched by the great theorist of realism J. J. Gibson and encapsulated by his student Kennedy (1974) in the form of a definition of a picture as: ‘a surface treated so that it yields light to a particular station point, usually on a normal to the picture surface, which could have come from a scene in the real world.’ (Compare Gibson 1978.) EXCERPT #K446N4 p. 14 Gibson naturally recognizes that there are other sorts of pictures (including maps), some of which involve conventional elements (symbols, icons), which have nothing to do with the conveyance of light to the eye in a way which simulates the light that is projected from surfaces in three-dimensional space. Even these pictures can, however, be interpreted in realist fashion on the basis of the general theory of projection sketched above. There are of course also many cases of pictorial images in which perspectival or other features of the represented scene are distorted in one or other way. As the Gibsonian realist can insist, however, the fact that pictures are sometimes made, for whatever reason, in such a way as to embody such distortions does not imply that all pictures are lacking in the sort of transparency for which the followers of Alberti strove. EXCERPT #MHVS9U p. 14 All maps must be of a certain scale or combination of scales, just as every grid must have a certain resolution or granularity of cells. And since reality itself (as Gibson 1979 emphasizes) contains entities accessible at many different scales, it follows that no single grid can be complete. Rather, as scientific practice shows, we need grids of many different resolutions if we are to do justice to reality in its many aspects. This implies, as the enemies of realism are fond of pointing out, that there is no ‘God’s eye perspective’ or ‘view from nowhere’. This does not, however, mean that we are justified in drawing the conclusion that every single one of the myriad perspectives which we have at our disposal embodies a false view of reality. The inference from partiality to falsehood might indeed be valid, but only in a world without windows – a world in which no single one of our grids enjoys the condition of transparency. EXCERPT #Q8JTQD p. 15 The fact that there are maps which deviate, for whatever reason, from the strictly veridical representation of reality does not take away from the fact that – leaving aside any small errors which may have been made in the application of the relevant projection system – almost all maps are true of the corresponding portion of reality. This applies to Mercator's map, and it even applies to Saul Steinberg's View of the World from Ninth Avenue . Maps must of course embody some projection system in representing three dimensions on a planar surface. Yet those who see in this an argument to the effect that all maps must necessarily involve some form of systematic distortion are simply revealing their own misunderstanding of the nature of projection. They are like those who, on noticing that the Circle Line is represented on maps of the London Underground as a yellow band, complain of 'distortion' because yellow, rather than some other color, has been used. ### 47. Tool result: read DOCUMENT #KY3Y9U Naïve Physics: An Essay in Ontology SECTION #N99NM2 Introduction EXCERPT #YJGJH2 p. 1 In the works of Aristotle or of the medievals, as also in the writings of later common-sense philosophers such as Thomas Reid or G. E. Moore, we find a family of different attempts to come to grips with the structures of common sense and of the common-sense world that is given to us in normal, pre-theoretical experience. We shall argue in what follows that the theory of such structures provides an important and hitherto unappreciated link between early Gestalt psychology on the one hand and contemporary developments in philosophy and in artificial intelligence research on the other. EXCERPT #6AMU4E p. 1 The notion of providing an adequate theory of the common-sense world has been taken seriously of late above all by those, such as Patrick Hayes or Kenneth Forbus, who see in such a theory of what they call 'naive' or 'qualitative physics' the foundations of future practical successes in robotics. (2) This naive physics is, however, like cognitive science in general, in a state of flux, and a serious philosophical investigation of its presuppositions and achievements has hardly been attempted. Yet it is already at this stage possible to point to a certain apparent defect or one-sidedness of current research in this field that is due to the predominant assumption that it is set theory and related instruments of ontology that are to provide the basis for naive-physical theorizing. The defect arises, we shall suggest, in virtue of the fact that naive physicists working in the A.I. sphere are for obvious reasons concerned with certain specific sorts of formal implementations. Their motivations are in the first place pragmatic, and so their aim is not so much a theory of the common-sense world that could be defended as being true , but rather a theory that has certain sorts of practical advantages from the point of view of implementation. Both of these factors, we shall argue, lead the naive physicists to neglect important detailed contributions to the theory of common sense that have been made by both psychologists and philosophers, contributions which it will be the business of the present paper to describe. EXCERPT #JJN4U7 p. 1 What will be surprising to those who are acquainted mainly with the standard artificial intelligence literature on the topic of naive or commonsensical physics is the extent to which it is among the Gestalt psychologists, above all, that some of the most important and original work in this respect is to be found. Indeed one could argue that the Gestalt-theoretical approach to external reality is in its entirety a variety of naive physics, something which is brought out clearly for example in the pronouncements of Wolfgang Köhler to the effect that there seems to be 'a single starting point for psychology, exactly as for all the other sciences: the world as we find it, naively and uncritically'. Our naive experience, as Köhler points out, 'consists first of all of objects, their properties and changes, which appear to exist and to happen quite independently of us' (1947, p. 1, 2). We can compare, on this very issue, Gibson: EXCERPT #E3QAV5 p. 1 Some thinkers, impressed by the success of atomic physics, have concluded that the terrestrial world of surfaces, objects, places, and events is a fiction. They say that only the particles and their fields are "real" . . . But these inferences from microphysics to the perception of reality are thoroughly misleading. The world can be analyzed at many levels, from atomic through terrestrial to cosmic. There is physical structure on the scale of millimicrons at one extreme and on the scale of light years at another. But surely the appropriate scale for animals is the intermediate one of millimeters to kilometers, and it is appropriate because the world and the animal are then comparable. (1966, p. 21f) EXCERPT #WYNBUR p. 2 It is this intermediate world, the world of common sense, which will be our concern in what follows. DOCUMENT #KY3Y9U Naïve Physics: An Essay in Ontology SECTION #BQHK5F III. BRANCHES OF NAIVE PHYSICS EXCERPT #TPQF6K p. 7 The task of naive physics, which is that of establishing an adequate theory of the structures and relations captured in such descriptions, is for a variety of reasons not an easy one. Hayes, above all, has stressed the extent to which the concepts of naive physics are subject to a massive holistic interconnectedness in the sense that each is interwoven with all the others in ways which make it difficult, if not impossible, to distinguish distinct and separable branches of the discipline at hand. (Hayes 1979, 175ff) One should speak instead, he argues, of only loosely discriminable conceptual clusters, bearing in mind always that the concepts in each cluster are capable of being understood only by appeal to concepts in other, neighbouring clusters. One partial and provisional list of such sub-branches of the discipline might read as follows: EXCERPT #52SFAR p. 7 1. Objects, Natural Units and Natural Kinds 2. Events, Processes and Causality 3. Stuffs, States of Matter, Qualities 4. Surfaces, Limits, Boundaries, Media 5. Motivation, Requiredness, Value EXCERPT #M8CV6L p. 7 The first four of these are standard (compare e.g. the list supplied by Hayes 1979, pp. 187-97). The fifth, however, which derives from phenomenology and from the Gestalt-theoretical perspective on naive-physical reality, is non-standard, in the sense that phenomena of value are not normally classed as belonging to 'physics' in either the naive or sophisticated senses. EXCERPT #8MAHXT p. 7 In what follows we shall sketch a range of illustrative examples of early contributions to the field of naive physics. These early contributions are unaffected by the predominance of the desire to achieve ('quick and dirty') formal representations: they are contributions to the sophisticated and non-reductionistic theory of common sense, rather than contributions to the computerized representation thereof. On the other hand however these early contributions remain in many cases at the level of isolated insights and much of the work of combining them into a full and adequate theory has still to be done. In our remarks in what follows we shall proceed always in keeping with the spirit of a realistic ontology. Thus we shall take the world of common sense as serving at one and the same time as (1) an object of a sophisticated theory and also (2) as that to which we have ready access in straightforward and non-theoretical everyday experience. For as already Avenarius (in his fashion) saw, naive physics is part of an answer to the question: what do we (straightforwardly) perceive? What, then, are the branches of the theory of the world of straightforward perception? SECTION #CYFWW4 1. Objects, Natural Units and Natural Kinds EXCERPT #5QP2WR p. 8 The common-sense world is from the formal-ontological perspective first of all a world of things, of stable material bodies that are given to us as things in the sense that they are given as inert, as complete and as three-dimensional. Each thing is present in the flesh, as something which has surfaces and an inside, that is filled with matter. Things are perceived also as manipulable units, and as potential subjects of fragmentation (splitting, cutting) and of unification (gluing, bonding). The articulation of the world of things now follows along natural lines: objects inanimate as well as animate are grouped together according to their typical patterns of behaviour and qualitative determination into natural kinds . The common-sense world is further such that in all its spheres and dimensions we can distinguish what is 'normal' and what is to a greater or lesser degree 'abnormal'. Thus the natural kinds in commonsensical reality have both standard and non-standard instances. Both Gestaltists and phenomenologists have insisted from the start further on the optimality of perceived objects; even where the objects themselves are marked by various deviations from the norm, there is a tendency to discount such deviations in our straightforward experience of things and in our assignment of things to kinds or categories (a notion linked also to Mach's principle of the 'economy of thought', as also to the familiar phenomenon whereby even the scientific image of reality must in every case be rooted in the categories of common sense (16) ). EXCERPT #WY54ZA p. 8 There is normal and abnormal also among experiences and among the conditions of experience. Consider, for example, the ways in which colour-appearances differ under different lighting conditions. 'Normal', here, is EXCERPT #XBS567 p. 8 seeing in sunlight, on a clear day, without the influence of other bodies which might affect the colour appearance. The 'optimum' which is thereby attained then counts as the colour itself , in opposition, for example, to the red light of the sunset which 'outshines' all proper colours. (Husserl 1952, p. 59) EXCERPT #YW5UPW p. 8 In normal experience, then, we take ourselves as having access to the things themselves and to their real states. Other appearances are taken by common sense as secondary to or as deformations of that optimal appearance which alone counts as an appearance of reality. 'The features which pertain to the thing "itself" are the "optimal" ones. This applies to all features, to the geometrical as well as to the sensuous qualities.' (Husserl 1952, p. 76f.) EXCERPT #WEECXT p. 8 All families (kinds, species) of objects in the common-sense world are subject to the opposition between normal (standard, typical) and abnormal (non-standard, non-typical) instances. (17) EXCERPT #JMNF6N p. 8 And the normal instances of such species are marked by familiarity, they are understood by common sense, both in regard to what they are and also in regard to what they will do (in regard to their regular patterns of behaviour in normal and regular circumstances). Thus I grasp a door, or a leaf, in one stroke , and I know already the sorts of future ways in which this thing will behave. SECTION #LZAZAB 2. Events, Processes and Causality EXCERPT #SKZ67P p. 8 The common-sense world of material entities is bicategorical: in spite of certain revisionary attempts on the part of Whitehead (1929), Kotarbinski (1955), Quine (1960, § 36), and others traces of which appear in Hayes' treatment of histories (1979, p. 189ff, 1985a) we still find it necessary to insist that common sense takes material objects and processes/events as belonging to two utterly different though interdependent categories. (18) EXCERPT #888GVJ p. 9 The work of the Polish phenomenologist Roman Ingarden (1935, 1964/65/74) includes what is probably the most detailed bicategorical ontology of things and processes/events to date, regarding processes as extended in time and events as boundaries (beginnings, endings or crossings) of processes. He thus stands opposed not only to monocategorical ontologies in the spirit of Kotarbinski or Quine but also to the Whiteheadian conception of processes as series of events. The Ingardenian classification can be supplemented by those which one finds in Thom (on confluences and convergences of processes, etc. (19) ). Thus instantaneous events can be sub-divided into culminations (a sudden turn) and achievements (a victory). A widely exploited analogy (e.g. Bach 1986, Galton 1984) is that between instantaneous events and unitary things on the one hand and between processes and masses or stuffs on the other. Thus processes (of growth or disintegration) are like stuffs in that they can be divided into parts which are themselves processes. EXCERPT #MCG3EL p. 9 Mention must be made in this connection also of the Gestaltists' work on process/event perception in the tradition of Gibson and G. Johansson. Thus Cutting (1981) sets out a number of conditions on event perception, which can also be taken as salient features of events themselves. An event or process if it is to be salient (to be discriminated as this or that event within a whole dynamic situation) must have an underlying invariant structure of properties that does not change (this might be the shape of the object involved, for example). These invariants concern both the whole event and also parts thereof, and they are hierarchically organized in the sense that some are essential, others inessential or such as to depend upon the former, essential properties. Each whole dynamic situation has one or more centre, for example the fulcrum of an acting lever, which are picked up and tracked in perception. (20) EXCERPT #WHQ7MZ p. 9 The common-sense world is causally organized -- as was recognized for example by Husserl, whose account of the common-sense world put forward in his 1952 is built around the two central notions of cause and change . To know a thing, Husserl argues, is to know its causal dependencies: it is to know how it will change under given influences, how it will behave when heated or bent. But it is to know also in what respects it will remain the same through given series of changes, and it is part of our common-sense understanding of reality that its denizens are such as to manifest a limited repertoire of systematic regularities in this respect, in the sense that under similar circumstances similar series of changes occur. EXCERPT #35MRQA p. 9 There are different sorts of change in the realm of common sense. Thus for example there is change that is internal , as e.g. when a person gets angrier of his own accord. Cases of this sort can be contrasted with changes caused by external circumstances, for example when a thing is dented or bruised. Changes can be divided further into changes in mere appearance (as when objects appear lighter through a change in external lighting conditions) and real changes (as when the apple ripens or a piece of metal expands). EXCERPT #2J4NF8 p. 9 Our bodies, too, of course are involved in causal dependencies, and yield the most important family of examples of real change, both internal and external. The body is a thing in space, with its form (extension) and its stock of qualities. The system of causalities into which my body is interwoven in normal experience is moreover such that my body retains an identity of type and of function through all its changes. Thus my limbs return again and again to the same basic positions. They can again and again accomplish the same sorts of things (lifting, turning, running) in the same sorts of regular ways. (Cf. Husserl 1952, pp. 61, 73) EXCERPT #3J5K22 p. 9 Among the most important changes in the body, now, are those changes we call sensory perceptions. The network of sensory changes in the body is interwoven with other networks of changes, above all with changes of position and orientation, and more generally with the body's movements. Husserl in fact anticipates here the later position of J. J. Gibson (as also of EXCERPT #K65G9X p. 10 Merleau-Ponty and others) concerning the necessary interwovenness of perception as a naturally occurring phenomenon with bodily movements on the part of the perceiving subject. (21) SECTION #272Z8G 3. Stuffs, States of Matter, Qualities EXCERPT #UVCRW9 p. 10 The world of experience is characterized by the fact that it has a qualitative aspect: its basic unities (things) are in such and such qualitative states and are filled through and through by sensory qualities. Not everything that we perceive is a thing. We perceive also the gaps between things, (22) holes, (23) the media (for example water, smoke) in which things move, and we can perceive holograms as well as rainbows and similar phenomena. EXCERPT #898Z97 p. 10 Things, now, are in common-sense experience spontaneously correlated with discriminable areas of organization within the continuum of what is given in sensory experience. Within this continuum centres are picked out, centres of accumulation of sensory qualities (where accumulation, here, is to be understood in the usual topological sense (24)). As Husserl pointed out, when we perceive a thing, then we perceive also sensuous qualities. But the latter are not there as it were alongside the physical thing; what is there before us is a unity, something which has physical and sensible properties as one. Moreover, the different strata of sensible properties are themselves bound intimately together: the things we experience are not built out of separate or separable seen, heard and touched constituents. Rather, there is but one thing, along with its properties, 'some of which are predominantly or exclusively (as, e.g., colours and their distinctions) grasped by vision, others by touch.' (Husserl 1952, p. 70) EXCERPT #ZC8YH6 p. 10 The multidimensional sensory continuum with its various centres of accumulation is marked further by the feature of extension. Everything that belongs to a material thing is related as a matter of essence to its extension. Extension is, as it were, the axial determination of the thing. Whatever other determinations the thing has, both as a whole and in its parts, it has these determinations across the whole relevant extent they fill its corporeal space. (Cf. Husserl 1952, p. 30). Thus the coloration of an opaque thing covers the entire outer surface of the corporeal thing in its specific fashion. Warmth fills the warm body in another, quite different fashion, and matters are different again as concerns hardness, texture, weight, and so on. (25) EXCERPT #YMENJS p. 10 The complexity of the relationship between colour and extension was hinted at already by Hering (1905), who talks of colour as a sort of primitive stuff. Bits of this stuff, he holds, are colour expanses, three-dimensional entities which are made up of colour as such (an idea taken up again by Quine in his Word and Object (§ 19) via the thesis that colour terms are mass terms). In his 1911 David Katz puts forward a taxonomy of the modes of appearance of colour in space which are in fact modes of diffusion or filling of space by different sorts of sensible qualities. Thus for instance surface colour densely occupies a plane; it has texture and is disposed on planes of various intrinsic orientation; volume colour is lacking in texture; film colour is disposed on a plane that is always orthogonal to the line of sight of the viewer; and so on. The French perceptual psychologist Jean Nogu, generalizing Katz's results, went so far as to classify the ways in which different sensuous qualities (colours, sounds, odours) fill space. If one interprets this classification from the point of view of space itself, one can claim that space is sensorily organized following different topologies. The typical mode of diffusion of odours given off by a source, for example, organizes the relevant space into non-oriented olfactory tracks; the recognition of a sound source, on the other hand, organizes the space of the auditor in oriented auditory paths; colours, in contrast, enclose or envelope space. (26) EXCERPT #KW94QR p. 10 A part of this programme is developed also by Husserl's doctoral student Wilhelm Schapp, who published in 1910 his Contributions to the Phenomenology of Perception , an attempt to defend a much extended variety of direct realism in the theory of perception. Visual perception, Schapp argues, gives us immediate access not only to things and their colour and form, but also to elasticity, solidity and other dispositional properties: EXCERPT #VAZ4H7 p. 11 We see whether a thing is smooth, as we see whether the brass of the lamp is rough like our suit or whether it is liquid like the water or the coffee or whether it is solid like the cup; whether it is homogeneous like the brass, or grainy like the table; whether it is sticky like the honey or runny like the ink. (1910, p. 19) EXCERPT #2SAUHP p. 11 Schapp especially contrasts cases in which some parts only of an object are seen as moving with cases in which the whole of an object moves: EXCERPT #LH7C5Y p. 11 The case where the whole thing moves offers us little insight into the 'inner structure' of the thing. We then see for example only the lightness or the heaviness of the thing. (1910, p. 21) EXCERPT #GJ4VRL p. 11 When, on the contrary, some parts of the object move whereas others do not, and this in a way which follows some lawlike pattern, then we can see whether a body is elastic or whether it is composed of viscous or solid matter (p. 22f). The configuration that is manifested by a given qualitative filling of space both in dynamic cases, as in the perception of elasticity, and also in static ones, as in the perception of surface qualities such as lustre gives us access to certain structural properties of the perceived thing. And this kind of knowledge which is employed by the craftsman, for example Schapp contrasts explicitly with that of the natural scientist (pp. 19, 21-26). EXCERPT #MP8P2Q p. 11 What is most interesting about the structural properties picked out by Schapp is that they are properties relating to the stuff of things: to their solidity, fluidity, and the like. Hedwig Conrad-Martius, another early phenomenologist, offers complementary investigations of phenomena linked to stuffs in her Realontologie of 1924. What differentiates stuffs, according to Conrad-Martius, is their qualitative structure in space: EXCERPT #GVGZKP p. 11 Material being is substantial fullness in space. And it is precisely the manner in which this fullness is put together in space which leads to the range of different modalities of material constitution (§ 122). EXCERPT #2ENPMS p. 11 In chapter 3 ("Concrete Forms of Stuff") Conrad-Martius then analyzes the ways in which sound and noise bear witness to the internal organization of stuffs. She also analyzes the qualitative features of temperature and light and offers a discussion of the different states of matter (§§ 135-70), of naive atomistic explanations (§ 162), of such dispositional properties of stuffs as elasticity, fragility and so on (§§ 171-80), and of aggregates (§ 176). EXCERPT #2HSWWV p. 11 In this connection it is worth pointing out also that as early as 1902 Pierre Duhem traced the history of the scientific notion of 'mixture' and provided an outline of its common-sense background in certain elementary human operations. The concept of mixture, as he notes, serves to link conceptually the two notions of aggregate or assembly on the one hand and stuff in the strict sense on the other. SECTION #5ZPALD 4. Surfaces, Limits, Boundaries, Media EXCERPT #9FSQAD p. 11 A systematic ontology of surfaces has been put forward in Stroll's classic (1988), where he also investigates the role that is played by surfaces from the point of view of epistemology. Stroll contrasts two conceptions of surfaces: as two-sided interfaces (the surface of an apple would in this sense involve both thing and medium); and as outermost layers (where only the apple itself is involved). EXCERPT #CLW7LV p. 12 Descriptive details of the theory of surfaces are to be found primarily in Gibson (1986), in the section entitled "Surface and the ecological laws of surfaces". As Gibson writes: EXCERPT #PGJPDY p. 12 According to classical physics, the universe consists of bodies in space. We are tempted to assume, therefore, that we live in a physical world consisting of bodies in space and that what we perceive consists of objects in space. But this is very dubious. The terrestrial environment is better described in terms of a medium , substances , and the surfaces that separate them. (1986, p. 16) EXCERPT #XA8AMM p. 12 The medium, then, is separated from the substances of the environment by surfaces, each surface being such as to have a characteristic texture depending on the composition of the stuff of the relevant underlying substance. (27) Gibson seeks accordingly 'a theory of surface layout, a sort of applied geometry that is appropriate for the study of perception and behavior' and which would investigate concepts such as: ground, open environment, enclosure, detached object, attached object, hollow object, place, sheet, fissure, stick, fibre, dihedral, etc. (1986, p. 33) EXCERPT #SMY28Y p. 12 Husserl, on the other hand, describes media as the normal environment for solid objects; they are amorphous , in the sense that they receive their form from the presence of material bodies in them. (28) Media are furthermore the vehicles of causality, and as a by-product of this they carry information about causal sources of all kinds. They are usually transparent in the sense that they do not themselves become objects of cognition in normal cases, though they can, in special circumstances, be properly representable in experience and they can be turned into such non-standard things as clouds of smoke, and so on. EXCERPT #QCMR2R p. 12 Fritz Heider's "Thing and Medium" (1926), an elaboration of part of his doctoral dissertation written in Graz under Meinong's direction, seeks an answer to a question central to causal theories of perception: why, when we look at an object, do we perceive the object and not the illuminating source, when the latter is after all causally responsible for the perceptual experience? (29) Heider then analyzes the ambient conditions under which remote objects can be perceived. Not satisfied with the simple statement of a causal relation between the distal and proximal stimuli, he introduces concepts such as relative dominance , order and disorder to account for the unaffectedness of the medium in the course of the transfer of information. A solid thing, he holds, is normally unsuited for the transmission of information which requires a certain causal independence of the vehiculating parts involved. Heider's work then finds echoes in Gibson's notion of perception as a picking up of information in the ambient light (cf. esp. 1986, ch. 2). (30) SECTION #AX6HDN 5. Motivation, Requiredness, Value EXCERPT #XBEZHD p. 12 The world of common sense in contrast to the naive- physical worlds described by Hayes, et al. is both salient and valuable: it is shot through with complex gradients of preferability . The relevance of this fact to a treatment of naive physics, now, turns on the fact that one central aim of naive-physical investigations is to find a means of simulating human action by means of intelligent artifacts. For it seems clear that our human capacity successfully to find our way around the physical world depends crucially upon the spontaneous ways in which we take such value-differentials into account. EXCERPT #T3GM9C p. 12 Our perceptual experiences are caused by objects and they are grasped as such from the perspective of common sense. Experiences are thereby bound together dynamically with the objects of this world through relations of causality. Experiences and the objects of the common-sense world are also bound together dynamically in a second sense, however, in that the objects of this world, on being experienced, exert positive and negative forces upon me belonging not to the sphere of causality but to that of human salience and value. The common-sense world is in this sense a meaningful dynamic whole that is shaped in manifold ways by forces of attraction and repulsion. EXCERPT #N5RQRQ p. 13 One is reminded in this connection of the Gestalt-theoretical notion of 'requiredness' introduced by Köhler. Requiredness is a form of reference, it is a relation from one thing to another. Requiredness differs from other forms of reference, however, by its demanding character. 'It involves acceptance or rejection of the present status of the context in question, often more particularly, acceptance or rejection of some part by the remainder of the context.' (1938, p. 336) EXCERPT #LTGUPQ p. 13 When I apprehend things and persons and surrounding circumstances I am determined by what Husserl calls 'motivations'. One object steers my regard onto itself through its special form. Another draws attention to itself through its beautiful colour or texture. The noise out there makes me close the window. The glass of beer over here makes me reach out my arm to grasp it: EXCERPT #7MPZTC p. 13 In short, in my theoretical, emotional, and practical behaviour in my theoretical experience and thinking, in my position-taking as to pleasure, enjoyment, hoping, wishing, desiring, wanting I feel myself conditioned by the matter in question (Husserl 1952, p. 140, cf. also p. 219). EXCERPT #2C73TH p. 13 It is an invariant feature of our straightforward experience that the objects motivate us in this sense. There are 'effects' on the subject emanating from the objects, effects of greater or lesser intensity. And then, as Köhler points out: 'The lower this intensity, the more will a condition of merely factual [i.e. physical] relation, juxtaposition, or sequence be realized.' (1938, p. 337) EXCERPT #NXN7X4 p. 13 We can consider as a thought experiment the idea that we might present to ourselves the objects of the common-sense world merely perceptually. As subjects of this world, however, we are not merely perceiving but also acting beings, and thus constantly subject to corresponding motivations. Thus in normal conditions we effect spontaneous evaluations of the objects by which we are confronted in a way which amounts to a sort of value-perception : 'the value-character itself is given in original intuition.' (Husserl 1952, p. 186) We directly experience the world as containing values, and thereby also we acquire mediate and immediate goals: objects 'afford' action, in Gibson's phrase. These affordances give rise in turn to new motivational connections in light of the interrelations between the various different goals and sub-goals in whose realization we are at any given moment engaged. These values and goals can then be seen as a new dimension of being within the common-sense world itself, a dimension which, we should argue, is crucial to our capacity to find our way around this world in a physical sense. DOCUMENT #KY3Y9U Naïve Physics: An Essay in Ontology SECTION #LS7S69 Conclusion EXCERPT #CA57TA p. 13 As most workers in the field of artificial intelligence have recognized, naive physics is far from being a single, unified discipline. Clusters and sub-clusters of concepts are investigated in a piecemeal way, without much concern for their relation to the whole, in spite of the fact that, as we stressed earlier, this conceptual network is marked by strong holistic features which are reinforced by the pervasiveness of spatial concepts and by the focus on those interactions which are relevant to the concerns of our everyday human behaviour. EXCERPT #V5HBPD p. 14 We have also seen that contemporary representations of common-sense experience in the sphere of naive physics are over-narrow to a degree which has had dramatic consequences for their reliability as representations. We suggested earlier that this narrowness depends on too quick a jump to implementationally attractive features of certain special means of representing naive-physical knowledge, means derived, in effect from the fundamentally atomistic (non-holistic) world of set theory. Against this tendency we wish to stress once more the need for a wider, and deeper, and more painstaking phenomenological investigation of the naive-physical realm and of the associated value-laden dimensions of the world of common-sense experience. The work of the Gestalists and of Gibson, taken together with work in naive physics and in formal ontology in the tradition of the early phenomenologists, has the chance of providing a unifying theoretical framework for the development of a realistic account of the structures here involved, in ways which can, we suggest, be of value also in the construction of more adequate theories of the sort that are still needed by naive physicists in the field of artificial intelligence. ### 48. Tool result: read DOCUMENT #FJ5KCA More Things in Heaven and Earth SECTION #DVWAMH MORE THINGS IN HEAVEN AND EARTH EXCERPT #8Q7M7C p. 0 Barry SMITH SUNY Buffalo EXCERPT #M7YYL4 p. 0 What follows is an exercise in hunter-gatherer ontology. More precisely, the region of space and of spatial objects will be adopted as a happy hunting ground for the purposes of Meinongian metaphysics. Meinong, notoriously, struggled against the prejudice in favour of the actual and fought on behalf of the ontological rights of incomplete, impossible, and indeterminate objects. A parallel struggle, as we shall see, can be waged in the domain of spatial objects. Meinong's ideas can in this way be seen to have relevance for studies of the philosophical foundations of the theories of land-surveying and of international law. SECTION #WSZ4H4 1. Heaven EXCERPT #GZT52W p. 0 Heaven, for our (initially purely illustrative) purposes, is simply empty space; it is the three-dimensional counterpart of the territory that is represented by the Bellman's blank Ocean Chart in Lewis Carroll's Hunting of the Snark : EXCERPT #LTSM68 p. 0 A simple black rectangular outline, representing a blank chart or map. A blank rectangular box representing the Bellman's blank Ocean Chart. EXCERPT #UZNRQ7 p. 0 Figure One: Bellman's Blank Ocean Chart EXCERPT #RNKHZK p. 1 EXCERPT #FK589U p. 1 Candidate denizens of empty space are the parts of this space. These include: three-dimensional spatial volumes, two-dimensional surfaces, one-dimensional lines, zero-dimensional points. (We shall ignore such further options as would arise in case a temporal dimension in the realm of heavenly objects were taken into account, or in case heaven were allowed more generally to contain topoids of larger numbers of dimensions. We shall ignore also the issue of deviant geometries, space-filling curves, Klein-bottle- and Sierpinski-Menger-sponge-shaped regions, and the like. 1 ) We shall concentrate our attentions further on finite portions of space, though we acknowledge that, if the empty universe is itself infinite, then infinite spatial volumes, too, for example the western hemisphere of the universe, would have strong claims to be countenanced as existing within it. EXCERPT #6KW8MQ p. 1 Suppose, now, that empty space as here defined exists. Do all abstractly (geometrically) conceivable finite portions of this space exist also? Imagine, for example, that portion of space which consists of two disjoint and non-connected spheres. Does this double sphere exist in the same sense (have the same ontological rights) as its separate spherical parts? Or imagine a perforated spatial region that has the form of a sphere of two-unit radius, in the interior of which is a one-unit radius spherical hole. Does this perforated sphere exist in the same sense as does the corresponding solid sphere of two-unit radius? EXCERPT #3CX898 p. 1 Or imagine some single spherical volume of unit radius. Imagine further that this spatial volume is topologically closed (or in other words: includes as proper part its outer boundary or skin). Does this skin itself exist with the same rights as does the closed spherical volume with which we began? And what of the corresponding open spherical volume (the residue which remains when the skin is conceived, abstractly, as having been removed from the sphere as a whole)? Does this open sphere exist as an object additional to its closed counterpart? And what of the infinitely many partially open and partially closed unit spheres, the results of subtracting different EXCERPT #WF7MG9 p. 1 1. Our aim is to depart not too far from space as given intuitively, while at the same time leaving open the possibility of applying a version of these reflections to space as described by the mathematician. EXCERPT #3Y8CEE p. 2 EXCERPT #TY69NM p. 2 fragments of this skin from the original spatial volume – all of which, we would do well to bear in mind, occupy the very same spatial region as does the closed sphere with which we began? Consider moreover the fact that, if the unit sphere exists as a closed spatial region, then its complement – that object which results when we abstractly conceive the sphere in question as having been removed from the universe – is itself open. Do heavenly complements exist with the same civil rights as do the objects which they are the complements of? EXCERPT #NW75W2 p. 2 In heaven, as we see, there are many questions. EXCERPT #HWMNW6 p. 2 Some, more brutally minded ontologists (the practitioners of ontological force majeure ), might want to resolve these questions by conceiving heaven set-theoretically, so that the only heavenly entities which existed with full civil rights would turn out to be extensionless spatial points. In addition to these one would recognize, as entities existing in some second sense, all sets of points, all sets of sets of points, etc. This set-theoretic account and the system of coordinate geometry which goes along with it have familiar advantages. But it also brings problems connected not least with the failures of the set-theoretical project associated with Cantor's continuum hypothesis. A further family of problems arises when we consider how the set-theoretic treatment of space is to be understood. If, on the one hand, it is conceived as yielding a mere model of space, then it leaves open the very ontological questions which are here at issue. If, on the other hand, it is treated not as a model but as an exercise in serious ontology – if, in other words, it is accepted that spatial regions are sets, then it would follow that such regions are abstract objects. But how, then, could they be such that concrete things are able to occupy them? EXCERPT #S7LVNT p. 2 The set-theoretic account dictates finally a controversial metaphysical thesis to the effect that space is built up out of points. In the absence of secure intuitions as to the truth or falsehood of this thesis we should surely seek a more neutral theory – such as mereology – which is consistent with both the postulate of atomism and its negation. 2 EXCERPT #JSX9BX p. 2 2. See Simons 1987. EXCERPT #WC748R p. 3 EXCERPT #ZMLCDJ p. 3 Mereology proceeds as it were from the top down, taking as its starting point in our present case extended spatial continua. An extreme version of the mereological top-down approach is generated by what might be called Aristotelian mereological potentialism, a view to the effect that the part of a whole can never be an actual thing if the whole is. 3 My arm, as part of me, is real or actual on this view, but it is not a real or actual thing ; rather, it is merely a potential thing: it would become an actual thing only through physical separation. Similarly a collective of bodily wholes, say of separate coral reefs, would become an actual thing only if the bodies in question were fused together to form a unitary object. These constituent bodies would then themselves thereby cease to be actual things. (My arm as part of me is, we might say, sub-unitary; the collection of non-connected coral reefs is super-unitary.) Since heaven is, by assumption, everywhere homogeneous, there is on the potentialist view only one candidate heavenly object (only one place), namely the universe as a whole. Each putative constituent place exists only potentially (i.e. it would exist if, counterfactually, heaven were reduced in size in corresponding fashion). EXCERPT #2J32AA p. 3 Mereological potentialism thus avoids the embarrassments of an over-generous ontology; it recognizes only one (actual) object. It appeals to those of our intuitions which suggest that our answers to the considered questions can be a matter of convention only, so that such questions might surely be ignored for any purposes of an ultimate ontological assay. These advantages of the potentialist view are spurious, however; for the very questions raised above reappear in modified form in the potentialist framework: do all those candidate denizens (open and closed regions, solid and perforated regions, and all the possible sums, differences and complements thereof) exist potentially in the same sense and in such a way as to enjoy equal ontological rights? In what follows, therefore, we shall defend the (Brentanian) doctrine of mereological actualism, a doctrine to the effect that parts exist with the same ontological standing as do their respective wholes. If you have a single spherical region, then you thereby also already have infinitely many pairs of hemispherical regions, infinitely many quadruples of quarter-spherical EXCERPT #L76VSY p. 3 3. On this terminology see Smith 1987. EXCERPT #JFCEW6 p. 4 EXCERPT #Y9ZGB2 p. 4 regions, and so on ad indefinitum . EXCERPT #X6MK6N p. 4 The doctrine of mereological actualism asserts that the parts of things are as actual as the things themselves. With Brentano (1981), we shall extend mereological actualism to boundaries also, both external (the outer surface of the closed spherical region as a whole), and internal (interior surfaces, interior lines, interior points, and so on). We shall however deny the presupposition that is at the heart of set theory to the effect that boundaries – for example isolated points – can exist independently of the entities of higher dimension which they are the boundaries of. Boundaries are actual things, but they are dependent entities; they can exist only in tandem with the larger things or regions which are their hosts. Already Abelard had remarked that ‘A line ... is unable to exist apart from some bodily subject,’ and as Chisholm points out, ‘Brentano makes the same point with respect to every type of boundary.’ (1992/93) In Smith (1993) I refer to the principle that boundaries cannot exist except in consort with the higher-dimensional entities which they are the boundaries of as ‘Brentano’s thesis’. SECTION #9XN2MV 2. Earth EXCERPT #RPMU2F p. 4 I have not the foggiest notion as to how to go about answering the various questions raised and left open in the foregoing. One thing, however, seems clear: the empty space that is described above is from the ontological (as from every other) point of view thoroughly homogeneous. What holds in one corner of heaven holds identically in every other corner, and for this reason also empty space has no dynamics and no history. EXCERPT #NKDQ4B p. 4 Where space is not empty, however, matters are entirely different. Here a range of different sorts of spatial objects can be distinguished, in addition to that stock of homogeneous spatial objects (the stock of places ) which exists purely in virtue of the underlying geometry of space. Following the terminology advanced in Smith 1994 we can distinguish above all between: EXCERPT #SEDFE8 p. 4 1. Bona fide spatial objects (for example planets, moons, islands, lakes) which exist in virtue of intrinsic physical discontinuities EXCERPT #BA23T8 p. 5 EXCERPT #7BWQZZ p. 5 in the material constitution of the earth. EXCERPT #BVQWKG p. 5 2. Fiat spatial objects (for example states, counties, land-parcels) of a sort which reflect no intrinsic physical discontinuities but are rather the product of boundaries drawn on the basis of human fiat or convention or are otherwise the artefacts of human geographical practices. EXCERPT #AYLC7W p. 5 Earth, unlike heaven, contains conventional parts. Fiat spatial objects, in contrast to the purely geometrical denizens of heaven and in contrast to the bona fide ('natural') spatial objects here on earth, seem to be human creations : they are entities which come to be superadded to the world in consequence of human cognitive acts and practices. EXCERPT #MFXMHD p. 5 The opposition between what is found or discovered and what is made or created is of course nothing new in the history of metaphysics. For present purposes we might distinguish, in the range of possible ontologies, between: EXCERPT #ZFU9E2 p. 5 Extreme idealism : the doctrine that all objects are created, or in other words that all objects exist exclusively as the products or figments of human cognition. EXCERPT #TY2R9K p. 5 Moderate (or 'creationist' or 'Ingardenian' 4 ) realism : the doctrine that some objects are created, some discovered. EXCERPT #H95HNC p. 5 Extreme (or 'platonist' or 'Meinongian') realism : the doctrine that all objects are discovered, or more particular that all objects are found and not made. EXCERPT #ES3XAH p. 5 I shall dismiss immediately the extreme idealist alternative (or is there really some extreme idealist who believes sincerely that the ground on which he stands, or the meteor speeding towards the building in which he sits, is a mere product of human cognition?). The important debate, I would argue, is that between extreme and moderate realism. Consider, in this light, the case of Wyoming which, like many political and administrative spatial objects in the United States, has a shape roughly as follows: EXCERPT #CK96SA p. 5 4. See Smith 1980. EXCERPT #5AUY27 p. 6 EXCERPT #E5K2L9 p. 6 A simple black rectangular outline, representing a geometric shape. It is a rectangle with a thin black border and no internal details or text. A simple black rectangular outline, representing a geometric shape. EXCERPT #5RG3RV p. 6 Figure Two: Wyoming EXCERPT #EEW8GP p. 6 From the Meinongian, extreme realist perspective, which we might also call 'geometrical Platonism' or 'geometrical objectivism', Wyoming existed long before man first set foot on the American continent, but so also did infinitely many alternative Wyomings (Wyoming displaced 1 inch to the east, Wyoming displaced 1 furlong to the west, Wyoming minus Crook County, and so on). Wyoming is thus to be conceived along the lines of the heavenly objects discussed above. Wyoming as it is at present ( anno 1995) geometrically constituted will on this view continue to exist even if Wyoming and one or more of its neighbours should agree to some exchange of territory (though our present geometrical Wyoming would then no longer be called 'Wyoming' and would likely not have any name at all). Surprisingly, this Meinongian view can claim the advantages of ontological economy – at least for those who have already embraced a suitably rich ontology of spatial objects distributed purely geometrically across the surface of the earth. For it conceives political and administrative spatial objects of the sort which might otherwise be seen as being created by acts of human fiat as mere logical constructions out of geometrical objects, and it is exclusively the latter which are granted full ontological rights. The Meinongian account can even do justice to changes in geopolitical and administrative borders: entities like Bosnia, or Poland, or the Netherlands would turn out from this perspective to be entia successiva , whose successive real parts are corresponding purely geometrical bits of space. (I am here clearly leaving out of account issues pertaining to the fact that the earth itself is such as to occupy distinct portions of space at different times.) EXCERPT #RG4PQQ p. 7 EXCERPT #XLMN67 p. 7 The competing, Ingardenian view, on the other hand, can claim the advantages of naturalness. This view asserts that, in the year 1890, a new spatial object called 'Wyoming' came into being as a result of human fiat and that this object has since enjoyed a certain history of its own; thus in the intervening period Wyoming might have changed its size or shape or location in relation to other spatial objects on the surface of the earth. Political and administrative entities are comparable in this respect to organisms – they may grow and develop, yet in such a way as to preserve their identity. (Unlike organisms they may even, as occurred in the case of Poland and Austria, enjoy a period of non-existence after which their identity is once again recovered: perhaps we might refer in such cases, in Meinongian spirit, to the 'implexive existence of the pure spatial object'.) Land-parcels, political and administrative entities may also fuse and split, in such a way that new entities are produced out of parts existing earlier. 5 Certainly Wyoming in this historical sense is at any given moment coincident with some region of space of the purely geometrical sort; but as we shall argue below it is never identical with any such region of space. SECTION #MHPRY4 3. Performative Maps EXCERPT #2ADGE7 p. 7 The Ingardenian ontology of historically existing political-administrative spatial objects is an extension of the theory of multi-dimensional continua elaborated by Brentano in the papers collected together as Philosophical Investigations on Space, Time and the Continuum (1988). Brentano there sketches a conception of the realm of spatial objects as a lasagna-like, many-layered edifice, with realms of heterogeneous ('secondary') spatial objects built up on the basis of a homogeneous 'primary' spatial continuum on the lowest level. But the theory can also be seen as an extension of the theory of performatives worked out by Adolf Reinach in his A Priori Foundations of the Civil Law in 1913 (a theory subsequently rediscovered, though with none of Reinach's ontological sophistication, EXCERPT #QM2MSZ p. 7 5. They are comparable, in this respect to holes and other superficial entities of the sort described in Casati and Varzi 1994. EXCERPT #KVMGQ4 p. 8 EXCERPT #GUK8BU p. 8 by Anglosaxophone speech act theorists in the 1950s). EXCERPT #ZWTU8X p. 8 We distinguish first of all between two classes of speech acts, giving rise to two sorts of products or consequences, which we shall call abstract and concrete , respectively. Commandings, thankings, forgivings, warnings and threatenings are performative uses of language which yield concrete consequences – above all actions, attitudes and feelings on the part of real people, entities which are fully a part of the real, historical, world of causal change. Promisings, legislatings, contractings, plightings, baptisings, ennoblings on the other hand, are performative uses of language which give rise to abstract consequences, to entities sui generis which are not (or not directly) subject to causal influences. Examples of such abstract consequences are: claims, obligations, laws, rights (including property-rights), troths, knighthoods, names, etc. 6 EXCERPT #FA6NFD p. 8 Having drawn this distinction in the realm of linguistic acts, we can now point to the existence of a parallel distinction in the realm of what we might call performative uses of maps . That is, we can distinguish between EXCERPT #3U6PU8 p. 8 1. concrete consequences of uses of maps, for example actions (above all actions of way-finding, acts of war, etc.) and feelings (of being threatened, overawed, offended, etc.: see Monmoyer 1991, ch. 7); 2. abstract consequences of uses of maps, for example the creation of state-, county- and property-boundaries, as also of such entities as the International Date Line, the Mason-Dixon line, and so on. EXCERPT #GLHSW5 p. 8 Abstract consequences are distinguished by the fact that they are entities of such a sort that they can exist only as the fruits or products of corresponding performative acts. They are distinguished further by the fact that, like claims, obligations and other legal entities, they fall midway between Platonic objects, which lie outside the realm of time and change, and real objects of the causal flux. (See Twar- EXCERPT #832TCC p. 8 6. Certainly such abstract consequences may give rise in their turn to concrete consequences. The terms 'abstract' and 'concrete' may from this point of view be misleading. EXCERPT #9692US p. 9 EXCERPT #QX6CH9 p. 9 dowski 1979.) In this respect they are comparable to the natural kinds of biology as also to linguistic kinds (such as adverb or phoneme ) and to the other entities treated of by Husserl (1973) under the heading 'bound ideality'. EXCERPT #JMSFQS p. 9 The feature of dependence upon specific acts of human fiat seems to be characteristic of political and administrative spatial objects in general, though some manifest this feature to a higher degree than others. Thus there are maximally conventional objects of this sort whose boundaries are exact geometrical figures, normally straight lines (though part of the Delaware-Pennsylvania border is an arc of a circle). Straight borders are associated especially with colonialism: they are borders drawn by governments in (Washington, Ottawa, London, or Mexico) before they know how things look on the ground. Such borders can be quite stable and peaceful (this applies also to the colonially drawn borders in the sub-Sahara region), in contrast to the carefully drawn boundaries of Europe based on the idea of a "self-determination of nations", or to the boundaries insisted upon by Irish nationalists, for whom 'Ireland cannot shift her frontiers. The Almighty traced them beyond the cunning of man to modify.' (Bowman 1982, p. 11) – God made Ireland, we might say, but all the rest is the work of man. EXCERPT #D52FDA p. 9 Even those island nations which seem to be blessed with maximally natural borders are abstract consequences in our sense, however, which is to say that they are products of human convention or fiat. This is because their apparent naturalness disappears when we take into account the status of all nations as historical products. Certainly any given political or administrative entity may at any given time be loosely identified with some given portion of land (either a two-dimensional surface or a three-dimensional slab of a certain thickness). That this identification is at best loose, however, i.e. that we do not have before us here a case of identity , is shown by the fact that the surface or slab in question typically existed long before the corresponding political or administrative entity came into being. The political or administrative entity is marked further by the fact that it may change in shape or location, may in other words become similarly loosely identified with a different slab or surface in the course of time, yet in such a way as to remain itself one and identical. EXCERPT #VVCWQD p. 10 EXCERPT #F5DV73 p. 10 At least in many of the cases standardly put forward as natural political unities the appearance of naturalness is diminished still further in view of the fact that we are dealing not with some one single land-mass, but rather with more complex products of human demarcation. Ireland, even the unitary Ireland conceived in the minds of Irish Republicans, is still in Meinong's terminology an 'object of higher order': it is a super-unitary entity built up on the basis of constituent parts such as Inishkea, Inishmore, Gorumna Island, and so on. Other even more conspicuous examples of higher order geographical objects are: the Holy Roman Empire, the European Union, the United States of America and so on. Each of the latter is a super-unitary spatial whole made up of smaller and relatively more unitary parts. On the other side we can distinguish sub-unitary spatial objects: spatial parts which can be distinguished within larger (and more or less 'natural') unitary wholes: the non-coastal states and nations of South America and of continental Europe are sub-unitary in this sense (and Catalonia and Cornwall might be regarded as sub-sub-unitary spatial objects along the same lines). 7 Denmark, the Helvetian Confederation, the Commonwealth of Independent States are examples of spatial objects which manifest both super- and sub-unitary features, which is to say they are at one and the same time the products of unification of scattered parts some or all of which are at the same time the products of carving out of smaller parts within a larger spatial whole. EXCERPT #LUWSPL p. 10 That sub-unitary spatial objects such as Catalonia or the Czech Republic are fiat or created objects is shown further by the fact that, even where the exterior boundaries of such objects coincide in large degree with rivers or other natural topographical features, these boundaries are still not identical with the given features; rather, the boundaries in question will standardly be identified with some non-natural surrogate. The boundary will run, for example, along the middle of a river. All legal and political boundaries must, it seems, be infinitely thin; they must take up no space, since otherwise EXCERPT #VHEXMH p. 10 7. A unitary spatial whole is analogous, if one will, to a single organism; the super-unitary whole to a family of organisms; the sub-unitary whole to undetached limbs or organs within a single organism. For more on these distinctions and their applications to geography see Smith 1995. EXCERPT #WK5VGJ p. 11 EXCERPT #V2A7W6 p. 11 disputes would constantly arise in relation to the no-mans-land which the boundaries themselves would then occupy. This 'middle' will in the first place be geometrically defined; should the river change its course, however, then it may have to be determined by negotiation or by some other non-trivial means where its 'middle' now lies. 8 EXCERPT #FH6MLJ p. 11 A final reason for conceiving political and administrative boundaries as created entities (rather than as entities picked out or discovered within the pre-existing totality of all relevant geometrically determined possibilities) turns on the fact that there are political and administrative boundaries which coincide (occupy an identical spatial location) throughout their total length. The name 'Vienna' refers on the one hand to a certain Austrian city, on the other hand it refers to one of the constituent states ( Bundesländer ) of the Austrian Republic. As it happens the boundaries of the city and of the state of Vienna coincide exactly, and both serve as boundaries in the same direction. But they are for all that not identical, as is seen in the fact that the two might in principle diverge (as is currently true, for example, in relation to the otherwise analogous case of the city and state of Salzburg). SECTION #L3379F 4. Conclusion: Impossible and Incomplete Spatial Objects EXCERPT #AEZPKK p. 11 We shall conclude, briefly, with a discussion of a spatial analogue of what Meinong referred to as 'impossible' and 'incomplete objects'. Nothing can be red and green all over. And so, also, we might conclude, with our eyes on a map of troop-movements on the Indo-Chinese border, nothing can be both Indian and Chinese all over. The fundamental principles of international law seem after all to dictate, for each given state, exclusive jurisdiction over its national territory and the permanent population living there together with a duty of non-intervention in the area of exclusive jurisdiction of all other states. A moment's reflection reveals, however, that parts of the earth's surface can indeed be both Indian and Chinese (or British and Argentine) all over: something like this applies even under EXCERPT #Q7B5BZ p. 11 8. See Prescott 1978. EXCERPT #W5ZDK4 p. 12 EXCERPT #PURP2F p. 12 present political conditions to international waters and to Antarctica, and outcomes of this sort were earlier the standard product of one favoured method for resolving border-disputes à la Austria-Hungary, namely through interdynastic marriage and fusion of territories. EXCERPT #56KMEH p. 12 As to incomplete spatial objects, which is to say spatial objects lacking crisp exterior boundaries, here a range of examples present themselves, beginning with spatial objects depicted on weather maps ('an area of high pressure over the Atlantic') and ending with territorial regions ('the Khanate of the Golden Horde ( circa 1350)', 'the Seljuk Kingdom of Iconium ( circa 1140)') on the fringes of areas of settled political power. Objects of these sorts might be said to constitute a third category of spatial object, in addition to the fiat and bona fide spatial objects distinguished above. They are found most conspicuously in the extra-human world (of deserts, clouds, gulf-streams, of flocks of birds and shoals of fish), but they are to be found depicted also in language- and dialect-atlases, as also in maps of religious observance and political behavior. Objects of this sort are clearly not the fiat products of some deliberate drawing of conventional borders, but nor are they the products of any tracking of underlying autonomous contours in nature. As they are depicted on maps we might think of them rather as the products of sampling (of temperature, air-pressure, rainfall, etc.), to the results of which certain standard algorithms are applied to generate bounded regions, regions which will however often overlap (and which may be represented, pictorially, by means of cross-hatching). Historical reflection will tell us, now, that objects of this third sort must in fact precede the tidily demarcated fiat spatial objects (nations, states, empires) with which we have grown familiar in the course of time. As the historian Owen Lattimore expresses it (1962, p. 471): EXCERPT #YKWD3S p. 12 · Frontiers are of social, not geographic origin. Only after the concept of a frontier exists can it be attached by the community that has conceived it to a geographical configuration. The consciousness of belonging to a group, a group that includes certain people and excludes others, must precede the conscious claim for that group of the right to live or move about within a particular territory. EXCERPT #K4AF5D p. 12 The metaphysical treatment of boundaries and frontiers is still, unfortunately, in its early stages. One of its tasks will be to do justice to these 'frontiers of social origin' and to the processes by which, not singly but in more or less harmonious consort, they become attached to specific regions of space. EXCERPT #L8WTSU p. 13 DOCUMENT #FJ5KCA More Things in Heaven and Earth SECTION #MHPRY4 3. Performative Maps EXCERPT #2ADGE7 p. 7 The Ingardenian ontology of historically existing political-administrative spatial objects is an extension of the theory of multi-dimensional continua elaborated by Brentano in the papers collected together as Philosophical Investigations on Space, Time and the Continuum (1988). Brentano there sketches a conception of the realm of spatial objects as a lasagna-like, many-layered edifice, with realms of heterogeneous ('secondary') spatial objects built up on the basis of a homogeneous 'primary' spatial continuum on the lowest level. But the theory can also be seen as an extension of the theory of performatives worked out by Adolf Reinach in his A Priori Foundations of the Civil Law in 1913 (a theory subsequently rediscovered, though with none of Reinach's ontological sophistication, EXCERPT #QM2MSZ p. 7 5. They are comparable, in this respect to holes and other superficial entities of the sort described in Casati and Varzi 1994. EXCERPT #KVMGQ4 p. 8 EXCERPT #GUK8BU p. 8 by Anglosaxophone speech act theorists in the 1950s). EXCERPT #ZWTU8X p. 8 We distinguish first of all between two classes of speech acts, giving rise to two sorts of products or consequences, which we shall call abstract and concrete , respectively. Commandings, thankings, forgivings, warnings and threatenings are performative uses of language which yield concrete consequences – above all actions, attitudes and feelings on the part of real people, entities which are fully a part of the real, historical, world of causal change. Promisings, legislatings, contractings, plightings, baptisings, ennoblings on the other hand, are performative uses of language which give rise to abstract consequences, to entities sui generis which are not (or not directly) subject to causal influences. Examples of such abstract consequences are: claims, obligations, laws, rights (including property-rights), troths, knighthoods, names, etc. 6 EXCERPT #FA6NFD p. 8 Having drawn this distinction in the realm of linguistic acts, we can now point to the existence of a parallel distinction in the realm of what we might call performative uses of maps . That is, we can distinguish between EXCERPT #3U6PU8 p. 8 1. concrete consequences of uses of maps, for example actions (above all actions of way-finding, acts of war, etc.) and feelings (of being threatened, overawed, offended, etc.: see Monmoyer 1991, ch. 7); 2. abstract consequences of uses of maps, for example the creation of state-, county- and property-boundaries, as also of such entities as the International Date Line, the Mason-Dixon line, and so on. EXCERPT #GLHSW5 p. 8 Abstract consequences are distinguished by the fact that they are entities of such a sort that they can exist only as the fruits or products of corresponding performative acts. They are distinguished further by the fact that, like claims, obligations and other legal entities, they fall midway between Platonic objects, which lie outside the realm of time and change, and real objects of the causal flux. (See Twar- EXCERPT #832TCC p. 8 6. Certainly such abstract consequences may give rise in their turn to concrete consequences. The terms 'abstract' and 'concrete' may from this point of view be misleading. EXCERPT #9692US p. 9 EXCERPT #QX6CH9 p. 9 dowski 1979.) In this respect they are comparable to the natural kinds of biology as also to linguistic kinds (such as adverb or phoneme ) and to the other entities treated of by Husserl (1973) under the heading 'bound ideality'. EXCERPT #JMSFQS p. 9 The feature of dependence upon specific acts of human fiat seems to be characteristic of political and administrative spatial objects in general, though some manifest this feature to a higher degree than others. Thus there are maximally conventional objects of this sort whose boundaries are exact geometrical figures, normally straight lines (though part of the Delaware-Pennsylvania border is an arc of a circle). Straight borders are associated especially with colonialism: they are borders drawn by governments in (Washington, Ottawa, London, or Mexico) before they know how things look on the ground. Such borders can be quite stable and peaceful (this applies also to the colonially drawn borders in the sub-Sahara region), in contrast to the carefully drawn boundaries of Europe based on the idea of a "self-determination of nations", or to the boundaries insisted upon by Irish nationalists, for whom 'Ireland cannot shift her frontiers. The Almighty traced them beyond the cunning of man to modify.' (Bowman 1982, p. 11) – God made Ireland, we might say, but all the rest is the work of man. EXCERPT #D52FDA p. 9 Even those island nations which seem to be blessed with maximally natural borders are abstract consequences in our sense, however, which is to say that they are products of human convention or fiat. This is because their apparent naturalness disappears when we take into account the status of all nations as historical products. Certainly any given political or administrative entity may at any given time be loosely identified with some given portion of land (either a two-dimensional surface or a three-dimensional slab of a certain thickness). That this identification is at best loose, however, i.e. that we do not have before us here a case of identity , is shown by the fact that the surface or slab in question typically existed long before the corresponding political or administrative entity came into being. The political or administrative entity is marked further by the fact that it may change in shape or location, may in other words become similarly loosely identified with a different slab or surface in the course of time, yet in such a way as to remain itself one and identical. EXCERPT #VVCWQD p. 10 EXCERPT #F5DV73 p. 10 At least in many of the cases standardly put forward as natural political unities the appearance of naturalness is diminished still further in view of the fact that we are dealing not with some one single land-mass, but rather with more complex products of human demarcation. Ireland, even the unitary Ireland conceived in the minds of Irish Republicans, is still in Meinong's terminology an 'object of higher order': it is a super-unitary entity built up on the basis of constituent parts such as Inishkea, Inishmore, Gorumna Island, and so on. Other even more conspicuous examples of higher order geographical objects are: the Holy Roman Empire, the European Union, the United States of America and so on. Each of the latter is a super-unitary spatial whole made up of smaller and relatively more unitary parts. On the other side we can distinguish sub-unitary spatial objects: spatial parts which can be distinguished within larger (and more or less 'natural') unitary wholes: the non-coastal states and nations of South America and of continental Europe are sub-unitary in this sense (and Catalonia and Cornwall might be regarded as sub-sub-unitary spatial objects along the same lines). 7 Denmark, the Helvetian Confederation, the Commonwealth of Independent States are examples of spatial objects which manifest both super- and sub-unitary features, which is to say they are at one and the same time the products of unification of scattered parts some or all of which are at the same time the products of carving out of smaller parts within a larger spatial whole. EXCERPT #LUWSPL p. 10 That sub-unitary spatial objects such as Catalonia or the Czech Republic are fiat or created objects is shown further by the fact that, even where the exterior boundaries of such objects coincide in large degree with rivers or other natural topographical features, these boundaries are still not identical with the given features; rather, the boundaries in question will standardly be identified with some non-natural surrogate. The boundary will run, for example, along the middle of a river. All legal and political boundaries must, it seems, be infinitely thin; they must take up no space, since otherwise EXCERPT #VHEXMH p. 10 7. A unitary spatial whole is analogous, if one will, to a single organism; the super-unitary whole to a family of organisms; the sub-unitary whole to undetached limbs or organs within a single organism. For more on these distinctions and their applications to geography see Smith 1995. EXCERPT #WK5VGJ p. 11 EXCERPT #V2A7W6 p. 11 disputes would constantly arise in relation to the no-mans-land which the boundaries themselves would then occupy. This 'middle' will in the first place be geometrically defined; should the river change its course, however, then it may have to be determined by negotiation or by some other non-trivial means where its 'middle' now lies. 8 EXCERPT #FH6MLJ p. 11 A final reason for conceiving political and administrative boundaries as created entities (rather than as entities picked out or discovered within the pre-existing totality of all relevant geometrically determined possibilities) turns on the fact that there are political and administrative boundaries which coincide (occupy an identical spatial location) throughout their total length. The name 'Vienna' refers on the one hand to a certain Austrian city, on the other hand it refers to one of the constituent states ( Bundesländer ) of the Austrian Republic. As it happens the boundaries of the city and of the state of Vienna coincide exactly, and both serve as boundaries in the same direction. But they are for all that not identical, as is seen in the fact that the two might in principle diverge (as is currently true, for example, in relation to the otherwise analogous case of the city and state of Salzburg). DOCUMENT #FJ5KCA More Things in Heaven and Earth SECTION #L3379F 4. Conclusion: Impossible and Incomplete Spatial Objects EXCERPT #AEZPKK p. 11 We shall conclude, briefly, with a discussion of a spatial analogue of what Meinong referred to as 'impossible' and 'incomplete objects'. Nothing can be red and green all over. And so, also, we might conclude, with our eyes on a map of troop-movements on the Indo-Chinese border, nothing can be both Indian and Chinese all over. The fundamental principles of international law seem after all to dictate, for each given state, exclusive jurisdiction over its national territory and the permanent population living there together with a duty of non-intervention in the area of exclusive jurisdiction of all other states. A moment's reflection reveals, however, that parts of the earth's surface can indeed be both Indian and Chinese (or British and Argentine) all over: something like this applies even under EXCERPT #Q7B5BZ p. 11 8. See Prescott 1978. EXCERPT #W5ZDK4 p. 12 EXCERPT #PURP2F p. 12 present political conditions to international waters and to Antarctica, and outcomes of this sort were earlier the standard product of one favoured method for resolving border-disputes à la Austria-Hungary, namely through interdynastic marriage and fusion of territories. EXCERPT #56KMEH p. 12 As to incomplete spatial objects, which is to say spatial objects lacking crisp exterior boundaries, here a range of examples present themselves, beginning with spatial objects depicted on weather maps ('an area of high pressure over the Atlantic') and ending with territorial regions ('the Khanate of the Golden Horde ( circa 1350)', 'the Seljuk Kingdom of Iconium ( circa 1140)') on the fringes of areas of settled political power. Objects of these sorts might be said to constitute a third category of spatial object, in addition to the fiat and bona fide spatial objects distinguished above. They are found most conspicuously in the extra-human world (of deserts, clouds, gulf-streams, of flocks of birds and shoals of fish), but they are to be found depicted also in language- and dialect-atlases, as also in maps of religious observance and political behavior. Objects of this sort are clearly not the fiat products of some deliberate drawing of conventional borders, but nor are they the products of any tracking of underlying autonomous contours in nature. As they are depicted on maps we might think of them rather as the products of sampling (of temperature, air-pressure, rainfall, etc.), to the results of which certain standard algorithms are applied to generate bounded regions, regions which will however often overlap (and which may be represented, pictorially, by means of cross-hatching). Historical reflection will tell us, now, that objects of this third sort must in fact precede the tidily demarcated fiat spatial objects (nations, states, empires) with which we have grown familiar in the course of time. As the historian Owen Lattimore expresses it (1962, p. 471): EXCERPT #YKWD3S p. 12 · Frontiers are of social, not geographic origin. Only after the concept of a frontier exists can it be attached by the community that has conceived it to a geographical configuration. The consciousness of belonging to a group, a group that includes certain people and excludes others, must precede the conscious claim for that group of the right to live or move about within a particular territory. EXCERPT #K4AF5D p. 12 The metaphysical treatment of boundaries and frontiers is still, unfortunately, in its early stages. One of its tasks will be to do justice to these 'frontiers of social origin' and to the processes by which, not singly but in more or less harmonious consort, they become attached to specific regions of space. EXCERPT #L8WTSU p. 13 ### 49. Tool result: read DOCUMENT #TQPVBD New Foundations for Qualitative Physics SECTION #BWMMSE Preamble EXCERPT #RBT2RS p. 0 Physical reality is all the reality we have, and so physical theory in the standard sense is all the ontology we need. This, at least, was an assumption taken almost universally for granted by the advocates of exact philosophy for much of the present century. Every event, it was held, is a physical event, and all structure in reality is physical structure. The grip of this assumption has perhaps been gradually weakened in recent years as far as the sciences of mind are concerned. When it comes to the sciences of external reality, however, it continues to hold sway, so that contemporary philosophers – even while devoting vast amounts of attention to the language we use in describing the world of everyday experience – still refuse to see this world as being itself a proper object of theoretical concern. EXCERPT #M3LVGB p. 0 Here, however, we shall argue that the usual conception of physical reality as constituting a unique bedrock of objectivity reflects a rather archaic view as to the nature of physics itself and is in fact incompatible with the development of the discipline since Newton. More specifically, we shall seek to show that the world of qualitative structures, for example of colour and sound, or the commonsense world of coloured and sounding things, can be treated scientifically (ontologically) on its own terms, and that such a treatment can help us better to understand the structures both of physical reality and of cognition. EXCERPT #Z862PZ p. 0 A number of recent moves have been made by workers in the field of artificial intelligence in the direction of a theoretical account of the qualitative level of objective reality. We can point, for example, to the idea of a 'naive physics' as this has been propagated by Patrick Hayes, 1 and to the qualitative physics of Kleer and Brown. 2 Parallel ideas are present also in the project of a 'semiophysics' – a physics of the salient structures in reality – that has been advanced by the French mathematician René Thom. Thom's ideas are propounded, interestingly enough, in the form of a commentary on Aristotle's Physics . 3 For it was not always the case that philosophers were disposed to cast aspersions on the project of a science of the qualitative world. To Aristotle and his disciples physics itself was indeed a EXCERPT #7BFSSH p. 0 1 . Patrick J. Hayes, "The Second Naive Physics Manifesto", in J. R. Hobbs und R. C. Moore (eds.), Formal Theories of the Commonsense World , Norwood, NJ: Ablex, 1985, 1-36. EXCERPT #UXB5FW p. 0 2 . J. D. Kleer and J. S. Brown, "A Qualitative Physics Based on Confluences", Artificial Intelligence , 24 (1984), 7-84 and in Hobbs and Moore (eds.), op. cit. , 109-183. EXCERPT #8UEJLN p. 0 3 . R. Thom, Esquisse d'une Sémiophysique. Physique aristotélicienne et Théorie des Catastrophes , Paris: Interditions, 1988. EXCERPT #TAJTGJ p. 0 EXCERPT #FAYGSQ p. 1 EXCERPT #MA7E26 p. 1 EXCERPT #WLM83L p. 1 qualitative discipline, and modern-day practitioners in the field of naive physics have recognized that there are valuable insights to be gained from the work of medieval thinkers such as Buridan and Oresme, still working within a broadly Aristotelian framework. 4 Thomas Reid and other Scottish common sense philosophers can likewise be seen as having explored the world of qualitative reality in ways relevant to more recent experiments. In the writings of thinkers such as Reid, however, as also in the work of the medievals, the issue is for obvious reasons not addressed as to the proper relation between the (qualitative) description of commonsensical reality and physical science in the modern (quantitative) sense. EXCERPT #RPG9BD p. 1 The question thus arises as to who, in the philosophical tradition, was the first exponent of what might be called a sophisticated naive physics , which is to say a theory of the commonsensical domain whose relations to physics proper are made the subject of explicit theoretical concern. Claims might be made in this respect for Whitehead, whose "On Mathematical Concepts of the Material World" 5 stands at the beginning of a long and valuable tradition of formal ontology embracing also, inter alia , the work of J. H. Woodger. 6 It seems, however, to have been Husserl's Crisis of European Sciences of 1936 which first addressed in explicit fashion the relation between the ontology of the commonsense world – called by Husserl the 'theory of the structures of the life-world' – and post-Galilean physics. 7 And Husserl's ideas as presented both in this work and also in his earlier writings on formal ontology 8 will surely be recognized by future researchers in the area of naive physics as one crucial philosophical pillar of their discipline. EXCERPT #4FEW5C p. 1 It might, for a number of reasons, seem somewhat incongruous to run together such diverse intellectual currents under the single umbrella of what we are still somewhat loosely calling 'naive' or 'qualitative' physics. There is, first of all, an important divide between those, like Thom, who are concerned to develop the physics of salience as a mathematical discipline, and those, like Husserl, who see the structures of the life-world as demanding a theoretical treatment of a quite different sort. There is a deep divide also between those thinkers – such as Aristotle – who see the discipline of naive physics as a science with its own distinctive subject-matter, EXCERPT #NLBGFX p. 1 4 . See e.g. John H. Holland, Keith J. Holyoak, Richard E. Nisbett and Paul R. Thagard, Induction. Processes of Inference, Learning, and Discovery , Cambridge, Mass. and London: MIT Press, 1986, p. 208. EXCERPT #DTJHM3 p. 1 5 . Philosophical Transactions of the Royal Society of London , series A, vol. 205, 465-525, repr. in F. S. C. Northrop and M. W. Gross (eds.), Alfred North Whitehead. An Anthology , Cambridge: Cambridge University Press, 1953, 7-82. EXCERPT #AYGDLG p. 1 6 . See e.g. his 1937 The Axiomatic Method in Biology , Cambridge: Cambridge University Press, 1937 and The Technique of Theory Construction (International Encyclopedia of Unified Science , vol. II, no. 5, 1939), Chicago: University of Chicago Press. EXCERPT #6KQ6N9 p. 1 7 . See E. Husserl, The Crisis of European Sciences and Transcendental Phenomenology. An Introduction to Phenomenological Philosophy , trans. by D. Carr, Evanston: Northwestern University Press, 1970. EXCERPT #DNMM32 p. 1 8 . These works and their influence are treated at length in B. Smith (ed.), Parts and Moments. Studies in Logic and Formal Ontology , Munich: Philosophia, 1982. EXCERPT #N8EF4W p. 1 EXCERPT #YQTTGD p. 2 EXCERPT #ABUMM7 p. 2 EXCERPT #9QAYDV p. 2 and those – like most contemporary workers in the field – who see naive physics in quite other (cognitive or psychological) terms. Of course, given the assumption mentioned at the head of this paper, it is not difficult to see why the first alternative should nowadays prove so unpopular. If reality an sich is conceived as being captured exclusively and exhaustively in the (suitably perfected) equations of a purportedly monolithic discipline of standard physics, then there would seem to be no room for any additional science of the structures of commonsensical reality – unless, that is, such a science should be a sort of psychology in disguise, a science of ‘knowledge-simulations’ or of ‘mental models’, readily associable with investigations e.g. in the sphere of ‘children’s physics’. 9 There is a danger, however, that the idea of a science of commonsensical reality will in this way be confused with the quite different and patently absurd idea of a ‘commonsensical science’ – a confusion of the sort which seems to lie at the heart of discussions of ‘folk psychology’ by Churchland and others. 10 Folk psychology is of course not a science, but a matter of sheer popular prejudice. From this, however, we clearly cannot conclude that there cannot be a sophisticated science of mind, and nor can we conclude from the muddled state of many folk beliefs about the commonsensical world that a sophisticated science of the structures of (the objective component of) this world is ruled out a priori . EXCERPT #NAXWMQ p. 2 Workers in the field of artificial intelligence may be able to afford to ignore such issues and concentrate on the practical job of simulating relevant human beliefs and processes of reasoning in formal theories, irrespective of the issue as to whether the propositions which thereby result are true or false of any independent reality. Thus they may take the view that all that matters, from their practical point of view, is merely the extent to which one obtains desired end-results in the sphere of automated reasoning. Neither the psychological nor the pragmatic conception of naive physics can be ultimately satisfying on their own terms however. For both leave open the question why it is that just these mental models or systems of beliefs should have arisen as they did and why they should have survived so long. Moreover, they leave open the question as to why it is that they should have the power to sustain so remarkable a facility of both thought and action. In order to answer these questions one must, it seems, adopt a wider theoretical focus, taking account of the structures of the world in which such thought and action is realized. One needs, that is, to place one’s theories or simulations of the psychology of human thought processes within the wider framework of an ontology. One very tempting hypothesis then consists in the idea that the remarkable facility which humans manifest in thinking and acting on the level of everyday experience is made explicable, at least in part, precisely by the existence of corresponding stable structures on the side of reality. EXCERPT #HDN6MT p. 2 It is this hypothesis – a hypothesis which comes down in the end to the view that there is a level of reality which enjoys a certain sort of intrinsic intelligibility – EXCERPT #5VQV3S p. 2 9 . See Holland, et al. , op. cit. , pp. 206ff. EXCERPT #WN2D4E p. 2 10 . See e.g. P. Churchland, Scientific Realism and the Plasticity of Mind , Cambridge: Cambridge University Press, 1979. EXCERPT #TXCAXG p. 2 EXCERPT #6EMD7Z p. 3 EXCERPT #K2Z2XT p. 3 EXCERPT #ZQGXKB p. 3 which we shall pursue in what follows. Thus we shall be concerned to establish (the foundations of) a theory of the qualitative or commonsensical world conceived as a relatively autonomous level of reality confronting us in everyday experience. 11 Such a theory must rest on one or other form of Aristotelian ontology, in the sense of an ontology recognizing enduring animate and inanimate substances manifesting an opposition between form and matter, possessing sensible and non-sensible qualities and undergoing changes (events and processes) of various sorts. On the appropriateness of such an ontology there is wide agreement among all practitioners, whether they work in the field of philosophy or in artificial intelligence, and whether or not they are willing to give credence to the corresponding propositions as propositions which are true of some independent reality. Thus it is remarkable to observe the extent to which Hayes' list of conceptual 'clusters' or sub-theories of the discipline of naive physics as he conceives it 12 corresponds to the original master-list of categories supplied by Aristotle. EXCERPT #26QHUW p. 3 The Aristotelian ontology must in addition recognize species and genera (or 'natural kinds') which these entities, both substances and their accidents, instantiate, and it must recognize further that the instances in each kind are divided into circles of more and less standard or prototypical instances. The prototypical instances in each species can then be expected to be more readily discriminable (salient, prägnant ) than their non-prototypical counterparts, and also more readily able to give rise to correspondingly skilled responses on the parts of perceiving and acting subjects. All of these features were investigated extensively by successive generations of philosophers inspired by Aristotle – up to and including Husserl. The Aristotelian qualitative ontology was however called into question by Galileo and his successors. Above all, substances and sensible ('secondary') qualities came to be eliminated from the view of the world accepted by the physicists, along with the whole apparatus of form and matter, natural kinds, prototypical instances, and so on. EXCERPT #NMJUF4 p. 3 Clearly, though, the qualitative or commonsensical ontology can be Aristotelian only in a broad sense. Thus the space of this ontology must be three-dimensional and global in type, as contrasted with the purely local space of Aristotle. Substances occupy volumes of this space and move continuously through it. They have closed spatial boundaries which delimit and separate them from other substances and they are capable of communicating impetus. And, most importantly for our present purposes, the sensible qualities inhering in such substances will manifest qualitative discontinuities which may or may not coincide with the boundaries which mark their exterior surfaces in space. Consider, for example, the case of a black dog with brown spots. Here, two sorts of qualitative discontinuities EXCERPT #USAKYR p. 3 11 . We are not, for the moment, interested in the precise relation between 'qualitative' and 'commonsensical'. Suffice it to say that the qualitative domain as specified below extends more widely than does the domain of commonsensical experience, for example in including non-spectral colours. On the other hand the world of commonsensical experience embraces dimensions of ontological form, above all the dimension of substance , which are skew to the strictly qualitative sphere. EXCERPT #ZHWDFH p. 3 12 . See Hayes, op. cit. , pp. 18ff. EXCERPT #M8YC8N p. 3 EXCERPT #67CNSE p. 4 EXCERPT #9TDBGQ p. 4 EXCERPT #42LL8M p. 4 can be distinguished. On the one hand are the discontinuities corresponding to the exterior apparent contours of the dog; and on the other hand are the internal discontinuities on the surface of the dog (for example the contours of the spots on his back). DOCUMENT #TQPVBD New Foundations for Qualitative Physics SECTION #YNPWYT A Theory of the Commonsense World EXCERPT #M457E2 p. 10 We claim that one can succeed in this way in solving the problem (in principle, at least) of relating physics and the qualitative world from the point of view of the mathematics of morphologies. Of course our treatment of qualitative discontinuities, too, does not yield a description of the qualitative world as some monolithic bedrock of reality. In this sense it is at the same distance from an ontology in the strict sense as are classical mechanics, quantum mechanics, etc. For our theory deals after all not with objects (qualities, etc.) in the world, but rather with products of mathematical reconstruction. The difference, however, is that this reconstruction turns out to allow on its own terms a mimicking of just those central features of the Aristotelian commonsensical ontology that were so fatefully abandoned by Galileo and his successors. That is, it offers not only a theory of qualities, but also, in the long run, a theory of substance, of change or process, of typicality, species and categorization, and so on – or in other words an entire theory of the commonsensical world. EXCERPT #TGVXS9 p. 10 But does this theory constitute a science in the strict and proper sense? Certainly it is not predictive in the usual (causal) sense; but then the aim of the qualitative ontology is not the aim of standard physics. The approach does lead to prediction, but only in the sense that it leads to the possibility of our explicating mathematical constraints for different sorts of empirical morphologies. This is a 'prediction' of exactly the same sort as the predictions to the effect that if you have, for example, a crystal, or the envelope of a virus, or a snowflake, or honeycomb, or an ornamentation of the Alhambra of Granada, then the symmetry of the structure is necessarily one of the 'Platonic' symmetries which is allowed by geometry. There exist theorems which make the same type of structural predictions for the possible morphologies K . These predictions can be interpreted as abstract mathematical constraints upon the universe of morphological phenomena. EXCERPT #A7BLJ7 p. 10 EXCERPT #8MTWQU p. 11 EXCERPT #RRGRW3 p. 11 DOCUMENT #TQPVBD New Foundations for Qualitative Physics SECTION #DBCBET Qualitative Ontology and the Science of Cognition EXCERPT #SPRF8Z p. 11 We have seen how a theory of the commonsense world can be rooted in the physics of the material substrates. But in order to have a plausible theory of the qualitative world we clearly need in addition a psychological-cognitive theory of perception and an account of the link between this theory and the substrate theory, or between perceiver and object of perception. How is the perceiving subject involved in the perceptual explication and cognitive interpretation of the qualitative structure of the commonsensical world? As far as qualities such as colour is concerned, we already dispose of considerable work on these problems and we know something about the steps which lead from physics to the mind. EXCERPT #TEUQWC p. 11 We have first of all, at the microlevel, the absorption-emission spectra of the atoms making up the substrate. At the macrolevel we have the reflectance of the object, which gives rise in its turn to transmission of light of certain wavelengths. At the level of the retina, the light excites the cones and the information (pattern of wavelengths) it bears is processed by these transducers, which is to say it is transformed into neuronal information (frequencies of neuron-firings codifying the wavelengths). This gets processed further on its way to the visual cortex, where there occurs the registering of a sensible quality of colour. Through all these steps something is preserved, and from our present point of view it is clear that at least part of what is preserved can be very well explained via the concept of qualitative discontinuity. For this is a concept which applies equally to qualities as realized physically and as apprehended in patterns of sensation in the mind. Wave optics explains (in a highly non-trivial manner) how the very special type of information which concerns qualitative discontinuities can come to be encoded in light (that is to say how singularities can be propagated by light). And the theory of visual perception (for example as propounded by David Marr) clarifies the perceptive endowment which allows us as cognitive systems to detect and to process this information. Briefly we can say that qualitative discontinuities on the side of the object are qualitatively salient for the subject because the concept of qualitative discontinuity is at one and the same time an objective and a subjective concept. EXCERPT #NTKF5J p. 11 The interest of the point of view of Marr and his successors is that it reconciles two apparently antagonistic approaches: the information-processing point of view, and the ecological point of view of J. J. Gibson, et al. In the classical cognitivist paradigm (as exemplified by Fodor, Pylyshyn, et al. ), information processing is essentially reduced to the operations of calculation on symbolic mental representations. These operations are essentially syntactic: the cognitivists focus exclusively on algorithms and neuronal implementations thereof; they leave no room for any link between the cognitive system and those objective, stable features of the qualitative world which we have placed at the centre of our concerns. If, however, one wants to introduce such objective structures of the environment into one's account of perception and cognition, as the ecologists do, then one is committed to making such structures constrain the information-processing devices. EXCERPT #PDHZ2P p. 11 EXCERPT #NM8T43 p. 12 EXCERPT #L26VAL p. 12 EXCERPT #35XM2V p. 12 Marr explains that what Gibson considered as the 'extraction' (pick-up) of invariants from the environment can be understood as a form of information processing capable of being explicated in computational terms. But a computational theory in Marr's sense must not only focus on algorithms and neuronal implementations; it must in addition comprehend each algorithm in relation to the type of object-derived information which it processes. But then, because the algorithms employed are determined by objective properties of the environment, it follows that we have a means of reconciling the syntactic status of the algorithms with an ecological semantics. Methodological solipsism is hereby overcome, and the world is once more made safe for qualitative experience. EXCERPT #RPV3L2 p. 12 ### 50. Assistant tool calls #### Tool call: `read` ```json { "blocks": [ "UR75WH", "JM424D", "F7CTDR", "4TYZWW" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "DY7KEM", "F997NM", "48D2BK" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "6KMTK8", "74LR6W", "VPLR2G" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "XBYXRY", "WKUZRV", "89CNFV", "8Y4CDY", "G6S8LB" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "JMETCH", "BZ6CLE", "EMC9A9", "ADGTJ6", "NGNLAC", "6LAAZD" ], "expand": true } ``` ### 51. Tool result: read DOCUMENT #XZX6PE Vague Reference and Approximating Judgments SECTION #UR75WH Vague Reference and Approximating Judgments EXCERPT #K89VNA p. 0 Thomas Bittner University of Leipzig EXCERPT #AY7JT5 p. 0 Barry Smith University of Leipzig and State University of New York at Buffalo EXCERPT #Y649ZY p. 0 ‘Mount Everest’ is a vague name. That is (on the account here defended) there are many portions of reality all of which have equal claims to serve as its referent. We propose a new account of such vagueness in terms of a theory of what we shall call granular partitions . We distinguish different kinds of crisp and non-crisp granular partitions and we describe the relations between them, concentrating especially on spatial examples. In addition, we describe the practice whereby subjects use systems of reference grids as a means for tempering the vagueness of their judgments, for example when they say that Libya straddles the Equator or that the meeting will take place between 2 and 3pm. We then demonstrate how the theory of reference partitions can yield a natural account of this practice, which is referred to in the literature as ‘approximation’. EXCERPT #CLTN9A p. 0 Keywords: ontology, granular partitions, vagueness, semantic partitions, partition theory, approximation EXCERPT #Y55QNW p. 0 Consider the proper name ‘Mount Everest’. This refers to a mereological whole, a certain giant formation of rock. A mereological whole is the sum of its parts, and Mount Everest certainly contains its summit as part. But it is not so clear which parts along the foothills of Mount Everest are parts of the mountain and which belong to its surroundings. Thus it is not clear which mereological sum of parts of reality actually constitutes Mount Everest. One option is to hold that there are multiple candidates, no one of which can claim exclusive rights to serve as the referent of this name. All of these candidates are involved, in some sense, when we use the name 'Mount Everest.' We are however not conscious of this multiplicity of candidate referents, effectively because we simply do not care about the question where, precisely, the boundaries around Mount Everest are to be drawn. EXCERPT #Q5W3VU p. 0 EXCERPT #UEMJTQ p. 1 EXCERPT #NMHUD7 p. 1 Each of the many candidates has the summit, with its height of 29,028 feet, as part. Each is also a perfectly determinate portion of reality. The candidates differ only regarding which parts along the foothills are included and which are not. EXCERPT #9S9CFB p. 1 Varzi (2001) refers to the above as a de dicto view of vagueness. It treats vagueness not as a property of objects but rather as a semantic property of names and predicates, a property captured formally in terms of a supervaluationistic semantics (Fraassen 1966), (Fine 1975). We shall concentrate our attentions in what follows on the case of singular reference , i.e., reference via names and definite descriptions to concrete portions of reality such as mountains and deserts. We shall also concentrate primarily on spatial examples. As will become clear, however, it is one advantage of the framework here defended that it can be generalized automatically beyond the spatial case. EXCERPT #47WBPF p. 1 In order to understand vague reference we use the theory of granular partitions we advanced in our earlier papers: (Bittner and Smith 2001a), (Bittner and Smith 2003), (Smith and Brogaard 2002). The fundamental idea is that every use of language to make a judgment about reality brings about a certain granular partition , a grid-like system of cells conceived as projecting onto reality in something like the way in which a bank of flashlights projects onto reality when it carves out cones of light in the darkness. Each judgment, J , can then be conceived as a pair consisting of a sentence, S , and an associated granular partition P J . EXCERPT #B69NA4 p. 1 We consider reference as a two-step-process . Language tokens are associated with cells in a grid-like structure, and these cells are projected onto reality in the way suggested by our flashlight metaphor. Granular partitions can then be conceived as the cognitive artifacts whereby language gains its foothold in reality. (They thus play a role somewhat similar to that of set-theoretical models in more standard treatments.) In our earlier papers, we showed how this two-step-process allows us to explain the features of selectivity and granularity of reference in judgments. In this paper, we show how the same machinery can help us to understand the phenomena of vagueness and approximation. SECTION #L35HYH Crisp Granular Partitions EXCERPT #5Q9KNR p. 1 The theory of granular partitions has two parts: (A) a theory of the relations between cells and the structures they form, and (B) a theory of the relations between cells and objects in reality. Consider Figure 1. The left part shows a very simple cell structure, with cells labeled Everest , Lhotse and The Himalayas . The right part shows portions of reality onto which those cells project. EXCERPT #G2YZHN p. 2 EXCERPT #P2SCV2 p. 2 The figure consists of two parts. The left part is a diagram of a partition. It features a large rectangle labeled 'The Himalayas' at the bottom. Above this rectangle are two smaller, side-by-side rectangles labeled 'Lhotse' and 'Everest'. The right part is a satellite photograph of a mountainous region, likely the Himalayas. Two specific peaks are outlined with thin black lines. The peak on the left is labeled 'Mount Lhotse' and the peak on the right is labeled 'Mount Everest'. Figure 1: Left: A partition diagram with three cells labeled 'Lhotse', 'Everest', and 'The Himalayas'. Right: A satellite image of the Himalayas with two regions outlined and labeled 'Mount Lhotse' (left) and 'Mount Everest' (right). EXCERPT #CC6VVM p. 2 Figure 1: Left: a partition, with cells Lhotse , Everest and The Himalayas . Right: A part of the Himalayas seen from space, with admissible candidate referents for 'Mount Lhotse' (left) and 'Mount Everest' (right). SECTION #U33NC7 Language EXCERPT #T2ANET p. 2 In what follows, we use lower case roman letters o, o_1, o_2, \dots to symbolize objects in reality; z, z_1, z_2, \dots to symbolize cells of granular partitions; upper case roman letters from the beginning of the alphabet A, B, C, \dots to symbolize sets of cells; upper case roman letters from the middle of the alphabet L, P, \dots to symbolize sets of ordered tuples; and upper case Greek letters \Delta, \Delta_1, \dots to symbolize sets of objects in reality. SECTION #A7KYQ7 Theory A EXCERPT #N2UWQL p. 2 A granular partition Pt = ((A, \sqsubseteq), (\Delta, \leq), P, L) is a quadruple such that (A, \sqsubseteq) is a system of cells or a cell-structure , (\Delta, \leq) is a target domain , L \in \text{Pow}(\Delta \times A) is a location relation, and P \in \text{Pow}(A \times \Delta) is a projection relation. The target domain (\Delta, \leq) is hereby understood as a mereological structure with \Delta a set of objects and \leq a part-of relation defined on \Delta which satisfies the axioms of general extensional mereology (GEM). A cell structure, (A, \sqsubseteq) , is a finite set of cells, z_0, z_1, \dots, z_n with a binary subcell relation \sqsubseteq . We say that z_1 is a subcell of z_2 in A if and only if the first is contained in the latter. We then impose four axioms (or 'master conditions') on cell structures as follows: EXCERPT #QWLR4J p. 2 MA1: The subcell relation \sqsubseteq is reflexive, transitive, and antisymmetric. EXCERPT #T8RAF5 p. 2 MA2: The cell structure of a partition is always such that chains of nested cells are of finite length. EXCERPT #K2MPH3 p. 2 MA3: If two cells have subcells in common, then one is a subcell of the other. EXCERPT #3DA4WX p. 2 MA4: Each partition contains a unique maximal cell. EXCERPT #F82JL5 p. 2 These conditions, which are explored further in our earlier papers, together ensure that each cell structure can be represented as a tree (a directed graph with a root and no cycles). EXCERPT #DAPNJU p. 3 SECTION #58M7S8 Theory B EXCERPT #W4WVSM p. 3 Theory (B) arises in reflection of the fact that partitions are more than just systems of cells. They are constructed in such a way as to project upon reality in the way names and other referring expressions in natural and scientific languages project onto entities in reality. Projection and location then are relations between cells in a cell structure on the one hand and objects in a target domain on the other. We write ' P(z, o) ' as an abbreviation for: cell z is projected onto object o , and ' L(o, z) ' as an abbreviation for: object o is located in cell z . The partitions of interest in this paper are transparent , which means that MB1 and MB2 hold: EXCERPT #ZXKRU8 p. 3 \text{MB1} \quad L(o, z) \rightarrow P(z, o). EXCERPT #C29P3W p. 3 \text{MB2} \quad P(z, o) \rightarrow L(o, z). EXCERPT #4HUGHN p. 3 (Here and in what follows initial universal quantifiers are taken as understood. We preserve L and P as distinct relations in order to hold open the possibility of dealing with certain sorts of breakdown in the relation between granular partitions and their targets.) EXCERPT #VC79WF p. 3 We demand further that projection and location be functional relations, i.e., that every cell projects onto just one object and every object is located in just one cell: EXCERPT #BDBLC7 p. 3 \text{MB3} \quad P(z, o_1) \text{ and } P(z, o_2) \rightarrow o_1 = o_2 EXCERPT #BXNJ38 p. 3 \text{MB4} \quad L(o, z_1) \text{ and } L(o, z_2) \rightarrow z_1 = z_2 EXCERPT #J6UPE2 p. 3 The partitions of interest in this paper are in addition complete , in the sense that every cell projects onto at least one object, i.e., they satisfy an axiom to the effect that they contain no empty cells (no cells projecting outwards into the void): EXCERPT #YMUB2Z p. 3 \text{MB5} \quad z \in A \rightarrow \exists o: L(o, z) EXCERPT #YAEBU2 p. 3 We require also that projection, considered as a function p: A \rightarrow \Delta between two partially ordered domains (A \text{ and } \Delta) , be an order homomorphism: EXCERPT #XB7V58 p. 3 \text{MB6:} \quad z_1 \subseteq z_2 \rightarrow p(z_1) \leq p(z_2) EXCERPT #FNTVGQ p. 3 The root or maximal cell in the cell structure is then the maximal object (the universal or total fusion) in \Delta . EXCERPT #8Q6WGG p. 3 The resulting class of partitions is quite narrow. For a more general treatment, embracing also less well-behaved granular partitions, see (Bittner and Smith 2003). Note also that our axioms MB1-6 have been formulated for easy understandability and the system they form is not minimal. (Thus MB2 already follows from MB1, MB3 and MB5.) In order to simplify the notation in what follows, we write Pt = (A, P, L) as an abbreviation for Pt = ((A, \subseteq), (\Delta, \leq), P, L) . At the same time we assume a fixed target domain \Delta , which the reader can think of as the whole of reality. EXCERPT #REVKHC p. 4 SECTION #JM424D Vague Granular Partitions SECTION #Y9BDQJ The Theory EXCERPT #WKFKYU p. 4 What, now, of vagueness? A vague granular partition Pt^V = ((A, \subseteq), (A, \leq), P^V, L^V) is a quadruple in which the cell structure and target domain are defined as above, and P^V and L^V are classes of projection and location relations (Bittner and Smith 2001b). Again, we will write Pt^V = (A, P^V, L^V) in order to keep the notation simple. EXCERPT #KY49KP p. 4 Consider Figure 2, which depicts a vague partition Pt^V = (A, P^V, L^V) of the Himalayas. This has a cell structure A , as shown in the left part of Figure 2, which is in fact identical to the corresponding part of Figure 1. In the right part of the figure, in contrast, there is a multiplicity of possible candidate projections for the cells in A , indicated by boundary regions depicted via cloudy ovoids. The boundaries of the actual candidates onto which the cells ‘Lhotse’ and ‘Everest’ are projected under the various P_i in P^V are continuous ovoids included somewhere within the cloudy regions depicted in the Figure. EXCERPT #SY3X7V p. 4 The projection and location relations in these classes form pairs (P_i, L_j) , which are such that each P_i has a corresponding unique L_j and vice versa, satisfying the following conditions (where the notation ‘ \exists! ’ abbreviates: ‘there exists one and only one!’): EXCERPT #A5Y6GR p. 4 \begin{aligned} MB1^V & \quad \forall j: L_j(o, z) \rightarrow \exists! i: P_i(z, o) \\ MB2^V & \quad \forall i: P_i(z, o) \rightarrow \exists! j: L_j(o, z) \end{aligned} EXCERPT #36VZ4E p. 4 We also demand that all P_i and all L_j are functional in the sense discussed in the crisp case: EXCERPT #ZMQYAR p. 4 \begin{aligned} MB3^V & \quad P_i(z, o_1) \text{ and } P_i(z, o_2) \rightarrow o_1 = o_2 \\ MB4^V & \quad L_j(o, z_1) \text{ and } L_j(o, z_2) \rightarrow z_1 = z_2 \end{aligned} EXCERPT #6BT7CZ p. 4 Lhotse Everest The Himalayas EXCERPT #QADS89 p. 4 A grayscale image of a mountain range, likely the Himalayas, with two specific peaks highlighted by dark, circular, cloud-like ovoids. These ovoids represent the boundary regions for the cells 'Lhotse' and 'Everest' in the vague partition. EXCERPT #HW75H4 p. 4 Figure 2: A vague partition of the Himalayas EXCERPT #F6LAMJ p. 5 EXCERPT #4XCF8X p. 5 We demand further that cells project onto some object (are non-empty) under every projection: EXCERPT #KGRZ4U p. 5 \text{MB5}^V: Z(z, A) \rightarrow \forall j \exists o: L_j(o, z) EXCERPT #Q3G6RU p. 5 Again, every particular projection considered as a function from A to \Delta is an order homomorphism: EXCERPT #VUTPPH p. 5 \text{MB6}^V: z_1 \subseteq z_2 \rightarrow p_i(z_1) \leq p_i(z_2) EXCERPT #MBWS9R p. 5 We now add an axiom that governs the interrelations between projection relations with distinct indexes in the vague partition \text{Pt}^V . Recall that all projection relations operate on the same cell structure. We need to ensure that if the same object is targeted by two cells z_1 and z_2 under different projections P_i and P_j then the targeting cells must be identical: EXCERPT #N7ZFNN p. 5 \text{MB7}^V \quad P_i(z_1, o) \text{ and } P_j(z_2, o) \rightarrow z_1 = z_2 SECTION #UHNNJV Equivalence of Candidate Referents EXCERPT #BA26P4 p. 5 Given a vague partition as defined above, we can define an equivalence relation between entities in the target domain \Delta as EXCERPT #MVNFM6 p. 5 D \approx \quad o_1 \approx o_2 \equiv \exists z, i, j : P_i(z, o_1) \text{ and } P_j(z, o_2). EXCERPT #98ENN5 p. 5 Clearly, \approx is symmetric and reflexive. To see that \approx is also transitive, assume o_1 \approx o_2 and o_2 \approx o_3 . This means that there exist z_1, z_2, i, j, k, m such that P_i(z_1, o_1) , P_j(z_1, o_2) , P_k(z_2, o_2) and P_m(z_2, o_3) . From P_j(z_1, o_2) and P_k(z_2, o_2) it follows by \text{MB7}^V that z_1 = z_2 , and hence for some cells z and some projections P_i and P_m it holds that P_i(z, o_1) and P_m(z, o_3) , i.e., o_1 \approx o_3 . In the remainder, we write [o]_z to denote the set \{o \mid \exists i : P_i(z, o)\} . EXCERPT #ZH6UHR p. 5 Let \text{Pt}^V = (A, P^V, L^V) be a vague granular partition. We call all partitions \text{Pt} = (A, P_b, L_j) with P_i \in P^V and L_j \in L^V which satisfy the axioms MB1–MB6 crispings of the vague partition \text{Pt}^V . Consider a partition with cells labeled with vague proper names. Intuitively, each crisping (A, P_b, L_j) then recognizes exactly one candidate precisified referent for each such cell. The precise candidates carved out by the separate (A, P_b, L_j) are all slightly different. But each is perfectly crisp and thus it has all of the properties of crisp partitions discussed in the previous sections. All those different candidate referents are equivalent in the sense of our relation \approx . This captures the de dicto view of vagueness. SECTION #FA6PAW Semantic partition EXCERPT #SSW2K9 p. 5 Given a vague partition \text{Pt}^V = (A^V, \subseteq), (A, \leq), P^V, L^V) , we can for each cell z \in A^V classify corresponding portions of reality with respect to the vague projection of the cell z into three zones: the determinate zone, the indeterminate zone, and the exterior zone. EXCERPT #F7AAZB p. 6 EXCERPT #RA6P43 p. 6 We say that x is part of the determinate core of the vague projection P^V of the cell z if and only if, under all projections p_i(z) in P^V , x is a part of the targeted candidate referent: EXCERPT #PX3Z8X p. 6 \text{determinate}_V(x, z) \equiv \forall i: x \leq p_i(z) EXCERPT #LYXKFQ p. 6 Thus the summit of Mount Everest is part of the determinate core of the vague projection of the name 'Everest' and its associated cell. EXCERPT #NVZT8B p. 6 We say that x is a part of the indeterminate zone of the vague projection P^V of the cell z if and only if there are some projections p_i in P^V under which x is part of the targeted candidate referent and other projections p_j(z) in P^V under which this is not the case: EXCERPT #FEGF5R p. 6 \text{indeterminate}_V(x, z) \equiv \exists i: x \leq p_i(z) \text{ and } \exists j: \neg(x \leq p_j(z)) EXCERPT #6NNBUR p. 6 The dotted region in Figure 2 illustrates the indeterminate zone of the projection of the cell associated with the vague name 'Mount Everest'. EXCERPT #RTMR9N p. 6 We say that x is a part of the zone exterior to the vague projection P^V of the cell z if and only if x is not reached by any projection p_i(z) in P^V . EXCERPT #3UV5GK p. 6 \text{exterior}_V(x, z) \equiv \forall i: \neg(x \leq p_i(z)) EXCERPT #D78WCR p. 6 There are parts of reality – such as Berlin – that are not reached by any projections of the cell 'Everest' in the partitions used by humans projecting in transparent fashion. EXCERPT #73N5CY p. 6 Reference via a vague name 'N' creates a partition of reality into determinate core, indeterminate zone, and exterior zone. Let \sigma x \psi(x) denote the mereological sum of all x satisfying \psi(x) and let z be the cell in the partition P^V which projects onto the candidate referents for 'N'. We then define determinate core, indeterminate zone, and exterior zone as the mereological sums of all determinate, indeterminate, and exterior parts of reality with respect to the vague projection of the cell z : EXCERPT #DD5ND9 p. 6 \text{det}_V(z) = \sigma x \text{determinate}_V(x, z) EXCERPT #QJ5SPC p. 6 \text{indet}_V(z) = \sigma x \text{indeterminate}_V(x, z) EXCERPT #TGLT6H p. 6 \text{ext}_V(z) = \sigma x \text{exterior}_V(x, z) EXCERPT #9FNY6M p. 6 We define the semantic partition of reality with respect to the vague name N as a triple of determinate zone, indeterminate zone, and exterior zone. In general \text{det}_V(z) is a partial function since there does not necessarily exist a portion of reality which is a part of all projections of the cell z . EXCERPT #WBEUY7 p. 7 SECTION #F7CTDR Approximating Judgments SECTION #RAK36R Approximation in Egg-yolk Partitions EXCERPT #6AJ3ZT p. 7 Consider, again, Figure 1. The cells labeled 'Everest' and 'Lhotse' carve mountain-candidates out of a certain formation of rock. They do not do this physically, but rather by establishing fiat boundaries in reality, represented by the black lines in the right part of the figure (Smith 1995), (Smith 2001), (Bittner and Smith 2001a). But how are we to understand the phenomenon whereby judging subjects are able to impose spatial boundaries vaguely ? EXCERPT #C3C2BS p. 7 Suppose you are an expert mountain guide hiking through the Himalayas with your friends and you assert: EXCERPT #LBXRDB p. 7 [A]: We will cross the boundary of Mount Everest within the next hour. EXCERPT #J2HTXV p. 7 We shall assume that through your use of the phrase 'within the next hour' you successfully delimit a range of admissible candidates for the boundary of Mount Everest along the trajectory of your hike. Consider the left part of Figure 3. Here boundaries delimiting admissible candidates are imposed by specifying a time interval that translates to travel distance along a path; time serves here as frame of reference . The boundaries are defined by your current location (marked: 'now') and your location after the specified time has passed (marked: 'in one hour'). The boundary of each admissible candidate referent crosses the path at some point between these two boundaries, called the exterior and the interior boundaries, respectively. EXCERPT #AQL8V2 p. 7 The general case is illustrated in the right part of Figure 3, which is intended to depict how judging subjects project egg-yolk-like granular partitions onto reality involving three cells: an exterior, a core, and an intermediate region within which the boundary candidates lie. (See (Cohn and Gotts 1996) and (Roy and Stell 2001).) This granular partition serves as the frame of reference in terms of which the judging subject is able at the same time to both specify the range of admissible entities to which he (vaguely) refers and also to constrain this range. SECTION #PHAMTM Egg-yolk Partitions vs. Semantic partition EXCERPT #Z6Y5HP p. 7 Consider the egg-yolk partition in the right part of Figure 3. It is important to see that this is not a semantic partition in the sense discussed above. This is because it was created by a judging subject by imposing boundaries onto reality in order to constrain admissible candidate referents and at the same time to serve as a frame of reference. EXCERPT #R6ZVZV p. 7 A semantic partition on the other hand is induced by classifying portions of reality into determinate zone, indeterminate zone, and exterior zone with respect to the vague projection of a certain cell in a vague partition. Ideally, when corresponding to the same vague name, both partitions coincide but, as we shall see below, this is not the case in general. We will discuss the relationships between the two kinds of partitions in a later section. Until then we ignore the notion of semantic partition and focus on the kinds of partitions shown in the right part of Figure 3 and their use as frames of reference in approximations. EXCERPT #RD44CN p. 8 EXCERPT #XTM7P7 p. 8 Figure 3: Egg-yolk like reference partitions. The left diagram shows a cross-section of an egg with concentric regions: a central 'core', an 'interior boundary', an 'exterior boundary', and a 'candidate boundary' between them. The right diagram shows a curved path with a 'core' at the start, an 'interior boundary' labeled 'in one hour', and an 'exterior boundary' further along. An arrow indicates the 'direction of travel' along the path, and a label points to the area between the boundaries: 'where-the-boundary-candidates are'. EXCERPT #LEGRTR p. 8 Figure 3: Egg-yolk like reference partitions SECTION #GJDGHT Approximation in Complex Partitions EXCERPT #WK2NUD p. 8 In the case just discussed, new fiat boundaries are created by judging subjects in ad hoc fashion in order to delimit vagueness. But there are also cases where already existing systems of boundaries are re-used for this same purpose. There is one crisp granular partition of this sort with which we are all familiar. It has exactly 50 cells, which project onto the 50 United States of America. A fragment of this partition is presented in the left and right parts of Figure 4. In the foreground of the figure we see in addition an area of bad weather (also called ‘Hurricane Walter’), represented by a dark dotted region that is subject to vagueness de dicto in the sense discussed above. Wherever the boundaries of this object might be located, they certainly lie skew to the boundaries of the relevant states. But the figure also indicates (with the help of suitable labeling) that: EXCERPT #D8UZDS p. 8 [B] Hurricane Walter extends over parts of Wyoming, parts of Montana, parts of Utah, and parts of Idaho. EXCERPT #SDAG7A p. 8 In the sorts of contexts which we humans normally inhabit, it is impossible to refer to any crisp boundary when making judgments about the location of a region of bad weather of the sort described. However, it is possible to describe its (current) location relative to the grid of a map in the manner illustrated in judgment [B]. EXCERPT #QDM45X p. 8 We, the judging subjects, then deliberately employ a corresponding partition as our frame of reference and we describe the relationships that hold between all admissible referents of the vague term ‘Hurricane Walter’ and the cells of this partition. In terms of spatial relations, this means in the given case that all admissible candidates partially overlap the states of Wyoming, Montana, Utah, and Idaho and that they do not overlap any other state. Consequently, if a judging subject can specify for every partition cell a unique relation – for example part of – that holds for all admissible candidate referents of a vague term, then this is a determinate way to effect vague reference . The technical name for this phenomenon is approximation . For details, see (Bittner and Stell 2002). EXCERPT #EGLWSG p. 9 EXCERPT #RBG5PM p. 9 A meteorologist may achieve a finer approximation by employing a finer-grained partition as frame of reference in order to make a more specific judgment about the current location of the bad weather region. Thus she might use cells labeled Eastern Idaho, Southern Montana, Western Wyoming, and Northern Utah, and so on, yielding a fiat boundary of the sort depicted in the right part of Figure 4. EXCERPT #A9ZR5P p. 9 Notice that all these boundaries predate the judgments which use them as frames of reference in relation to this particular bad weather system. They are there to be used over and over again in formulating constraints on the possible locations of admissible candidate referents corresponding to vague referring terms. They represent a convenient and determinate way to make vague reference, which has even greater utility when the frame of reference is a commonly accepted one, as in the present case. SECTION #UCKBZM Approximation and Judgments EXCERPT #6N7U6P p. 9 Approximating judgments are a special class of judgments that contain both vague names and a (relatively) crisp reference to boundaries that delimit this vagueness. [A], too, is an approximating judgment which contains the vague name 'Everest' and also a reference to boundaries delimiting the vagueness of this term via the phrase '[crossable] within the next hour'. In this paper, we consider approximating judgments which contain a single vague name and a crisp reference frame. More complex cases are possible – including the case where the reference frame itself involves a certain degree of vagueness – but formal consideration of the latter is omitted here since its treatment follows the same basic pattern. EXCERPT #8A543U p. 9 The image contains two side-by-side maps of the United States, focusing on the western region. Both maps show a grid of state boundaries. A dark, irregular, cloud-like shape representing Hurricane Walter is shown moving from the southwest towards the northeast. - The left map has a coarse grid where each cell covers a large area (e.g., an entire state or a large portion of one). The hurricane's path is somewhat fuzzy and overlaps several of these large cells. - The right map has a finer grid where each cell covers a much smaller area (e.g., a smaller portion of a state). The same hurricane path is shown, but it is more precisely defined by the boundaries of these smaller cells, illustrating a 'finer-grained partition' as mentioned in the text. Figure 4: Two maps of the United States showing the path of Hurricane Walter. The left map shows a coarse partition of the states, while the right map shows a finer partition. Both maps show a dark, irregular shape representing the hurricane's path, which is more precisely defined in the right map. EXCERPT #CV32ME p. 9 Figure 4: States of the United States with Hurricane Walter EXCERPT #TV44Q9 p. 10 EXCERPT #8HLD7C p. 10 An approximating judgment J^A , if uttered successfully, imposes two partitions onto reality: a vague partition Pt^V and a reference partition Pt^R , along the lines above, whereby the latter serves to delimit the vagueness of the former. An approximating judgment J^A is thus a triple (S, Pt^V, Pt^R) , consisting of a sentence, S , together with two granular partitions, Pt^V and Pt^R . EXCERPT #3NWSUN p. 10 In the approximating judgment J^A = ([B], Pt^V, Pt^R) , expressed by the sentence: 'Hurricane Walter extends over parts of Wyoming, Montana, Utah, and Idaho', the corresponding vague partition Pt^V contains a cell labeled 'Hurricane Walter', which projects onto a multiplicity of admissible candidates. At the same time this judgment reuses the partition depicted in the left part of Figure 4 as its reference partition Pt^R . The latter constrains the admissible projections of the cell labeled 'Hurricane Walter' in Pt^V in such a way that each candidate referent that is targeted by a projection P_i of Pt^V must extend over parts of reality targeted by the cells of Pt^R labeled 'Wyoming', 'Utah', 'Montana', and 'Idaho' respectively. SECTION #EUKWZB Partition Theory and Approximation EXCERPT #3HQVJF p. 10 The idea underlying the partition-theoretic view of approximation is that a (crisp) granular partition can be used as a frame of reference (a generalized coordinate frame (Bittner 1997)), which allows us (a) to describe the approximate location of objects and thus (b) to project onto portions of reality in an approximate way. We call a granular partition which is used as a frame of reference in this manner a reference partition . EXCERPT #TAV47T p. 10 Consider a vague name such as 'Hurricane Walter' (hereafter: 'HW') and the corresponding multiplicity of admissible candidate referents for this name formed by crisp portions of reality in the domain of the northwestern United States at some given point in time. Consider some crisp partition structuring this same domain but without recognizing any of the candidates referred to by the name 'HW' directly. This might be the partition created by the boundaries of the separate States of the sort used in Figure 4, or it might be a partition formed by a raster of cells aligned to lines of latitude and longitude. EXCERPT #V4PL3D p. 10 To understand the formal details of how the latter can serve as reference partition in relation to the former we introduce the three concepts of full overlap ( fo ), partial overlap ( po ), and non-overlap ( no ), concepts which we shall now use to generalize the notions of projection and location, as follows. Consider a reference partition whose cells are projected onto regions of space on the surface of the Earth. Let o be a portion of reality that straddles the boundaries of the cells of this reference partition. The constants fo , po , no will now be used to measure the degree of mereological coverage of the object o by the corresponding regions of space. EXCERPT #LB8ZVQ p. 11 EXCERPT #HDZZQ9 p. 11 We call the relation L^R(o, z, \omega) the rough location of the portion of reality o with respect to the cell z and the relation P^R(z, o, \omega) the rough projection of the cell z onto o . (We use the phrases 'rough location' and 'rough projection' in order to emphasize our indebtedness to the account of approximation in terms of rough sets advanced in (Pawlak 1982).) In both relations, \omega stands for the degree of mereological overlap of the portion of reality targeted by the cell z with the actual portion of reality o , i.e., it takes one or other of the values fo , po , or no . Consider the left part of Figure 4. There the relation po holds between all admissible candidate referents HW_i and Montana, i.e., \forall i: L^R(HW_i, Montana, po) . The relation no holds between all the HW_i and Oregon, i.e., \forall i: L^R(HW_i, Oregon, no) . EXCERPT #YGR664 p. 11 We can characterize the relationships between exact and rough location and exact and rough projection in reference partitions as follows: EXCERPT #EYBZNH p. 11 \begin{aligned} L^R(o, z, fo) &\equiv \exists x (L(x, z) \text{ and } x \leq o) \\ P^R(z, o, fo) &\equiv \exists x (P(z, x) \text{ and } x \leq o) \\ L^R(o, z, po) &\equiv \exists x (L(x, z) \text{ and } \exists y (y \leq x \text{ and } y \leq o) \text{ and } \\ &\quad \exists y (y \leq x \text{ and } \neg(y \leq o))) \\ P^R(z, o, po) &\equiv \exists x (P(z, x) \text{ and } \exists y (y \leq x \text{ and } y \leq o) \text{ and } \\ &\quad \exists y (y \leq x \text{ and } \neg(y \leq o))) \\ L^R(o, z, no) &\equiv \exists x (L(x, z) \text{ and } \neg \exists y: y \leq x \text{ and } y \leq o) \\ P^R(z, o, no) &\equiv \exists x (P(z, x) \text{ and } \neg \exists y: y \leq x \text{ and } y \leq o) \end{aligned} EXCERPT #RN7G42 p. 11 The notion of rough location gives rise to an equivalence relation in the domain of objects (portions of reality), with respect to a given reference partition Pt^R with rough location relation L^R , as follows: EXCERPT #3BRV55 p. 11 D\sim \quad o_1 \sim o_2 \equiv \forall z, \omega: L^R(o_1, z, \omega) \leftrightarrow L^R(o_2, z, \omega). EXCERPT #LGJNKV p. 11 Thus two objects are equivalent with respect to the granular partition Pt^R if and only if they have an identical rough location with respect to all cells of this partition. The relation \sim can thus be interpreted as meaning: indiscernibility with respect to the frame of reference provided by Pt^R . In an approximating judgment J^A = (S, Pt^V, Pt^R) , the reference partition Pt^R will be chosen in such a way that the candidate referents targeted by a single cell in Pt^V are equivalent with respect to \sim . Often there may be a number of possible choices for reference partitions, all of which have the feature that all candidate referents of the vague name in question are equivalent with respect to the indiscernibility relation \sim induced by the reference partition. For example in Figure 4 we could also have used a regular (raster-shaped) reference partition of some appropriate resolution. However, it is more appropriate in a weather forecast to use the reference partition defined by the boundaries of the separate States because of its familiarity. We will discuss different choices of reference partitions in later sections. EXCERPT #EVNBR6 p. 12 EXCERPT #Q56CFN p. 12 We define a reference partition as a quintuple, Pt^R = ((A, \subseteq), (\Delta, \leq), P^R, L^R, \Omega) where (A, \subseteq) and (\Delta, \leq) are a cell structure and target domain as specified above, P^R and L^R are rough projection and location relations, and \Omega is the set of values \{fo, po, no\} indicating degrees of overlap (coarser and finer distinctions are possible, as discussed in (Bitner and Stell 2003)). We then can prove that the following counterparts of MB1-3 hold for reference partitions: EXCERPT #2NN6W4 p. 12 \begin{aligned} \text{TR1} \quad & L^R(o, z, \omega) \rightarrow P^R(z, o, \omega) \\ \text{TR2} \quad & P^R(z, o, \omega) \rightarrow L^R(o, z, \omega) \\ \text{TR3} \quad & (\forall z, \omega: P^R(z, o_1, \omega) \leftrightarrow P^R(z, o_2, \omega)) \rightarrow o_1 \sim o_2 \end{aligned} EXCERPT #KMS6VE p. 12 TR1 follows from MB1. To see this assume L^R(o, z, \omega) and let \omega = fo . We have \exists x (L(x, z) \text{ and } x \leq o) . By MB1 we have \exists x (P(z, x) \text{ and } x \leq o) , hence P^R(z, o, fo) and similarly for \omega = po and \omega = no . TR2 follows from MB2 in a similar manner. To see TR3 assume \forall z, \omega (P^R(z, o_1, \omega) \leftrightarrow P^R(z, o_2, \omega)) . By TR1 and TR2 L^R and P^R are logically equivalent and can be substituted for each other. Therefore we have L^R(o_1, z, \omega) \leftrightarrow L^R(o_2, z, \omega) , i.e., o_1 \sim o_2 . EXCERPT #GEYDLA p. 12 Corresponding to MB4 we now demand that if all objects have the same relation \omega \in \{fo, po, no\} with respect to the cells z_1 and z_2 then these two cells are identical: EXCERPT #9PLATP p. 12 \text{R1} \quad (\forall o, \omega: L^R(o, z_1, \omega) \leftrightarrow L^R(o, z_2, \omega)) \rightarrow z_1 = z_2. SECTION #H67PPK Constraining Approximation SECTION #B4N4D4 Well-Formed Approximations EXCERPT #7W9WD7 p. 12 If an approximating judgment like ([A], Pt^V, Pt^R) is to succeed, that is if a true judgment of this form is to have been made, then the reference partition needs to project onto reality in such a way that all admissible candidate referents are equivalent with respect to the indiscernibility relation imposed by Pt^R . Thus, in the hiker case, each value of p^V ("Everest") must be such that its boundary can be crossed in one hour from the time when the judgment is made (Figure 3). EXCERPT #4T58AH p. 12 Let (S, Pt^V, Pt^R) be an approximating judgment and let Pt^V be a vague partition with a cell for each vague name in the sentence S . We then demand that in such an approximating judgment the reference partition Pt^R and the vague partition Pt^V be related to each other in such a way that candidate referents which are targeted by the same cell (i.e., are equivalent in the sense of \approx ) have the same rough approximation in the underlying reference partition (i.e., are also equivalent in the sense of \sim ): EXCERPT #WR7NBE p. 12 \text{EP} \quad o_1 \approx o_2 \rightarrow o_1 \sim o_2. EXCERPT #GC9QKL p. 13 EXCERPT #QS4FA3 p. 13 We call EP the equivalence principle . EP rules out many reference partitions Pt^R which cannot be used for constraining the vagueness of the vague partition Pt^V . Thus it rules out, for example, reference partitions with resolutions too fine for the degree of vagueness of the corresponding vague partition. Consider Figure 4. A raster-cell-partition with cell size of 1m^2 would violate the equivalence principle, since not all candidate referents of the vague name 'Hurricane Walter' would be indiscernible with respect to this reference partition. If the reference partition is too fine then equivalence in the sense of \approx does not imply equivalence in the sense of \sim . EXCERPT #4HLZRV p. 13 Notice that the converse of the equivalence principle does not hold. This is because there might be portions of reality ( x and y ) that are equivalent with respect to the reference partition ( x \sim y ), but which are such that neither is a candidate referent targeted by the cell in question. Consider Figure 5. The approximation of the mereological sum of Yellowstone National Park and Zion National Park ( YNP + ZNP ) with respect to the Federal State reference partition (left) is identical to the approximation of the candidate referents of the name 'Hurricane Walter' (right) but surely ( YNP + ZNP ) is not a candidate referent for the name 'Hurricane Walter'. EXCERPT #CFE25N p. 13 Figure 5 consists of two side-by-side diagrams. The left diagram shows a map of the western United States with a grid overlay. A small rectangular area is highlighted and labeled 'YNP' (Yellowstone National Park). Below it, another small rectangular area is labeled 'ZNP' (Zion National Park). The right diagram shows a similar grid overlay, but instead of a map, it shows a dark, irregular, cloud-like shape that fills a portion of the grid cells, representing the candidate referents of the name 'Hurricane Walter'. EXCERPT #3PUAJ5 p. 13 Figure 5 Left: Yellowstone National Park (YNP) and Zion National Park (ZNP). Right: Hurricane Walter. SECTION #V7FNPA Precise Approximation EXCERPT #VEQL6M p. 13 In this section, we discuss the relationship between semantic partition and the kinds of reference partitions previously discussed. EXCERPT #JEK4VV p. 13 Consider the approximating judgment J = ([A], Pt^V, Pt^R) . The reference partition Pt^R shown in the left part of Figure 3 imposes two flat boundaries onto reality: an interior boundary of the approximation and an exterior boundary of the approximation. As discussed above, this often results in a partition structure similar to the one depicted in the right part of the figure. The projection of this partition onto the path the judging subject takes on her journey towards the summit of Mount Everest results in the reference partition Pt^R . EXCERPT #TKDBSP p. 13 Consider now the semantic partition imposed by the cell labeled 'Everest' in the vague partition Pt^V . We can see that the relationship between Pt^V and Pt^R satisfies the equivalence principle EP, which demands that all candidate referents of the vague name 'Everest' are equivalent under \sim . Consider now the location of two pairs of boundaries: (a) the interior and exterior boundaries imposed by the judging subject as a frame of reference for her approximation; and (b) the boundaries imposed by the semantic partition into determinate zone, indeterminate zone, and exterior zone via the cells of P^A . We say that the approximating judgment is precise if and only if (1) the interior boundary of the approximation coincides with the boundary separating the determinate zone from the surrounding parts of the semantic partition; and (2) the exterior boundary of the approximation coincides with the boundary separating the exterior zone from the indeterminate zone of the semantic partition. This means that the semantic partition and the reference partition coincide. EXCERPT #VWCUR5 p. 14 EXCERPT #D39GKV p. 14 In order to take more complex reference partitions into account, we now define upper and lower approximations of an object o with respect to such partitions. (Again, we use the notions of lower and upper approximation in order to emphasize the correspondence to rough set theory of Pawlak (1982).) The lower approximation of an object o with respect to a reference partition P^A is the mereological sum of all those portions of reality which are targeted by cells of P^A and which are contained in o : EXCERPT #SEKWJW p. 14 \text{Lower}(o) = \text{co}(\exists x(o' = p(x) \ \& \ P^A(x, o, f(o))), EXCERPT #RUZZF4 p. 14 where \text{co}(x) is defined as above. EXCERPT #SP7YKZ p. 14 The upper approximation is the mereological sum of all those portions of reality which are targeted by cells of the reference partition and which overlap o : EXCERPT #28M527 p. 14 \text{Upper}(o) = \text{co}(\exists x(o' = p(x) \ \& \ (P^A(x, o, f(o)) \ \vee \ P^A(x, o, po)))) EXCERPT #TUW7K8 p. 14 Consider now the reference partitions shown in the left part of Figure 6, which is a refined version of the egg-yolk reference partition in the right part of Figure 5. The core cell of the latter is subdivided into eastern and western subcells (ec and wc for eastern part of the core region and western part of the core region, respectively). Moreover the region where the boundaries are is subdivided into a northern and southern region (nb and sb, respectively). The lower approximation of the candidate referent signified by its outer boundary is the mereological sum of those portions of reality which are targeted by the cells 'ec' and 'wc'. Its upper approximation has as parts in addition portions of reality targeted by the cells 'nb' and 'sb'. EXCERPT #RLT7FT p. 14 The diagram consists of two parts. The left part shows a 'refined egg-yolk partition' with concentric regions. The outermost region is labeled 'exterior'. Inside it is a ring labeled 'nb' (north boundary) and 'sb' (south boundary). The center is divided into two cells: 'ec' (eastern core) and 'wc' (western core). A line labeled 'Candidate boundary' points to the boundary between the 'nb/sb' ring and the 'ec/wc' core. The right part shows a 'raster partition' as a 4x4 grid of cells labeled A through P. A blue-shaded irregular shape is overlaid on the grid, representing a complex object. The shape covers cells B, C, D, E, F, G, H, I, J, K, L, M, N, O, and P, with some cells partially covered. Figure 6: Approximation in complex partitions. Left: a refined egg-yolk partition. Right: a raster partition. EXCERPT #B5QMJX p. 14 Figure 6: Approximation in complex partitions. Left: a refined egg-yolk partition. Right: a raster partition. EXCERPT #E4QM45 p. 15 EXCERPT #57LXKF p. 15 For another example of lower and upper approximations, consider the right part of Figure 6. The upper approximation of the depicted region with respect to the raster-shaped partition is the mereological sum of the targets of the cells [B,..., L, O, P]. The lower approximation is identical to the target of the cell K. EXCERPT #FBCQFT p. 15 Notice that upper approximations are always defined. Lower approximations, however, are only defined if the reference partition is sufficiently fine grained, so that P^R(z, o, fo) holds for some cell z and some portion of reality o . This is because in mereology there is no counterpart to the empty set. EXCERPT #VGHKWF p. 15 With respect to these more complex reference partitions we now say that an approximating judgment is precise if and only if (1) the boundary of the lower approximation of any candidate referent of 'N' coincides with the boundary separating the determinate zone from the surrounding parts of the semantic partition imposed by the vague projection of the cell associated with 'N'; and (2) the boundary of the upper approximation of any of the candidate referents coincides with the boundary separating the exterior zone from the indeterminate zone of the vague projection of the cell associated with 'N'. EXCERPT #87PDSK p. 15 In formal terms, we describe this as follows. Let J = (N, Pt^V, Pt^R) be an approximating judgment with Pt^V = ((A^V, \subseteq), (\Delta, \leq), P^V, L^V) and Pt^R = ((A^R, \subseteq), (\Delta, \leq), P^R, L^R, \Omega) . We then call J^A precise if and only if EXCERPT #5MJLZV p. 15 \forall o \in [o]_z: \text{Lower}(o) = \text{det}^V(z) \text{ and } \text{Upper}(o) = (\text{det}^V(z) + \text{indet}^V(z)). EXCERPT #HZCEA6 p. 15 Here z \in A^V is the cell in the vague partition corresponding to the vague name 'N', [o]_z is the set of all objects targeted by the vague projection of z , and + is the mereological sum. SECTION #HDWEPC Constraining Approximation EXCERPT #42BMVA p. 15 We now discuss constraining approximations, defined as those approximations that do not have the property of being precise but still satisfy the equivalence principle. EXCERPT #ELWWPB p. 15 Let z be the cell in the vague partition Pt^V which corresponds to the vague name 'N', and let [o]_z be the set of all candidate referents of 'N', i.e., portions of reality targeted by z under P^V . The approximation of candidate referents o \in [o]_z with respect to the approximating partition Pt^R is called constraining if and only if the following holds: EXCERPT #2WKNLC p. 15 \forall o \in [o]_z: EXCERPT #H5CEL5 p. 16 EXCERPT #LNDVAA p. 16 either: EXCERPT #SX8YEU p. 16 Lower(o) is defined and \text{Lower}(o) \leq \text{det}_V(z) \leq (\text{det}_V(z) + \text{indet}_V(z)) \leq \text{Upper}(o) or: EXCERPT #7XPPNJ p. 16 \text{det}_V(z) \leq (\text{det}_V(z) + \text{indet}_V(z)) \leq \text{Upper}(o) EXCERPT #QPUTNJ p. 16 Consider Figure 4 and assume that the determinate zone of the vague reference of the name ‘Hurricane Walter’, \text{det}_V(\text{HW}) , is situated along the border between Idaho and Wyoming. It follows that the lower approximation is undefined for any of the candidate referents because the federal state partition is too coarse. The upper approximation however is defined, since any candidate referent is part of the mereological sum of Idaho, Montana, Wyoming, and Utah. Hence the resulting approximation is constraining. EXCERPT #AK2DNU p. 16 Consider the class of constraining approximations. As already Aristotle repeatedly emphasized (at 1094b11 sq. , 1098a26, 1103b34 sq. , 1165a13), judging subjects will characteristically use those approximating judgments which are constraining but which are as precise as necessary in whatever is the context in hand. In (Bittner and Smith 2001b), we argue that this will imply that the limits imposed on vagueness by an approximation will normally be such that the resulting judgment is not subject to truth-value indeterminacy. The judgments we actually make in normal contexts (as contrasted with those types of artificial judgments invented by philosophers) are determinately either true or false even in spite of the vague terms which they contain. SECTION #PT3REE Properties of Reference Partitions EXCERPT #9VSR8R p. 16 Reference partitions are of central importance for approximating judgments. Examples of reference partitions include: any political subdivision, raster-shaped partitions adjusted to latitude and longitude, the block structure in American cities, the subdivision of Vienna into Bezirke and of France into Départements , etc. Other important groups of reference partitions are partitions imposed by quantity-scales of all kinds (Johansson 1989, chapter 4), including temporal partitions like calendars (Bittner 2002). EXCERPT #QCPKTL p. 16 Consider again the judgment [B]: ‘Hurricane Walter extends over parts of Wyoming, Montana, Utah, and Idaho’ and the corresponding structure J^R = ([B], \text{Pt}^V, \text{Pt}^R) . The skeleton of the reference partition \text{Pt}^R is the partition \text{Pt}^S , which recognizes the United States (Figure 4) and thereby establishes the frame of reference for the approximation. Consider Figure 3. Here the skeleton \text{Pt}^S of the reference partitions is an egg-yolk structure containing the cells labeled ‘core’, ‘exterior’, and ‘where the candidate boundaries are’. EXCERPT #4ZCB5M p. 16 Every reference partition \text{Pt}^R = ((A, \sqsubseteq), (\Delta, \leq), \text{P}^R, \text{L}^R, \Omega) has a crisp partition \text{Pt}^S = ((A, \sqsubseteq), (\Delta, \leq), \text{P}^S, \text{L}^S) called the skeleton of \text{Pt}^R . Both, \text{Pt}^S and \text{Pt}^R share the cell structure (A, \sqsubseteq) and the target domain (\Delta, \leq) . In order to ensure that the intuitions sketched in the previous paragraph (and implicitly assumed in our definitions of rough location L^R and rough projection P^R ) are satisfied we demand that the skeleton has following properties: EXCERPT #ENMN8X p. 17 EXCERPT #3MTCEX p. 17 i. If P^S(z, o) holds in Pt^S then so does P^R(z, o, fo) in Pt^R . For all other cells z_1 in the shared cell structure A we have P^R(z_1, o, no) . That is: P^S(z, o) \rightarrow (P^R(z, o, fo) \text{ and } (\forall z_1 \in A: z_1 \neq z \rightarrow P^R(z_1, o, no))). ii. If L^S(o, z) holds in Pt^S then so does L^R(o, z, fo) in Pt^R . For all other cells z_1 in A we have L^R(o, z_1, no) . That is: L^S(o, z) \rightarrow (L^R(o, z, fo) \text{ and } (\forall z_1 \in A: z_1 \neq z \rightarrow L^R(o, z_1, no))). iii. Skeletons satisfy MB1–6. EXCERPT #AK9MDN p. 17 Often skeletons are also full, exhaustive, and complete in the sense of (Bittner and Smith 2003), which means in effect that they create subdivisions of the targeted domain into jointly exhaustive and pairwise disjoint portions. EXCERPT #DWRT8F p. 17 The skeletons of reference partitions which serve as frames of reference are often spatial or temporal in nature. They are relatively stable, i.e., they do not change over time. This implies in turn: (a) that the pertinent cell structure is fixed and (b) that the objects onto which the skeleton projects do not change (they are, for example, spatial regions tied to the surface of the Earth). Consider again the examples in Figure 4. The granular partition projecting onto the United States has existed for more than one hundred years without significant changes. (Hurricane Walter, in contrast, changes continuously throughout the course of its (brief) existence.) In fact, Figure 4 itself needs to be considered as a snapshot of reality at some determinate point in time (Smith and Brogaard, 2002, Bittner and Smith 2003a, Grenon 2003). It provides us with useful information when we are told that Hurricane Walter was located in parts of Montana, Idaho, Wyoming, and Utah at such and such a time. Every American child learns the corresponding reference partition in school, and uses it for all sorts of purposes thereafter (Stevens and Coupe 1978). Reference partitions are characteristically built out of boundaries with which human beings can become easily familiar, objects which facilitate easy learning. SECTION #4TYZWW Conclusions EXCERPT #PPF9QH p. 17 We have proposed an application of the theory of granular partitions to the phenomenon of vagueness, a phenomenon which is itself seen in de dicto terms, i.e. as a semantic property of names and predicates. We defended a supervaluationistic theory of the underlying semantics and expressed it in terms of the theory of granular partitions. We showed that the use of frames of reference in making approximating judgments can be formulated very naturally in partition-theoretic terms, and that the framework of granular partitions then helps us to understand the relationships between vagueness and approximation. While the bulk of our examples were derived from the spatial domain, the generality of the theory of granular partitions allows an easy generalization to other sorts of cases. EXCERPT #LCA3T8 p. 18 SECTION #W4CE7E Acknowledgments EXCERPT #ZDQWWA p. 18 The authors thank Wolfgang Heydrich and the reviewers for helpful comments. Part of this work was carried out under the auspices of the Wolfgang Paul Program of the Alexander von Humboldt Foundation. It was also supported by DARPA under the Command Post of the Future program, and by the National Science Foundation under its Research on Learning and Education program and also under NSF Research Grant BCS-9975557: "Geographic Categories: An Ontological Investigation". SECTION #TG3Z68 References EXCERPT #XU664D p. 18 Bittner, T. (1997). A qualitative coordinate language of location of figures within the ground. In S. Hirtle & A. U. Frank (Eds.), Spatial information theory: A theoretical basis for GIS , COSIT 99, 1329 Lecture Notes in Computer Science, Berlin: Springer, 223–240. Bittner, T. (2002). "Approximate Temporal Reasoning." Annals of Mathematics and Artificial Intelligence , 35, 1–2. Bittner, T. and B. Smith (2001a). A taxonomy of granular partitions. In D. Montello (Ed.), Spatial information theory: Foundations of geographic information science , COSIT 01, Lecture Notes in Computer Science, 2205, Berlin: Springer, 28–43. Bittner, T. and B. Smith (2001b). Vagueness and granular partitions. In C. Welty & B. Smith (Eds.), Formal ontology and information systems , New York: ACM Press, 309–321. Bittner, T. and B. Smith (2003). A theory of granular partitions. In M. Duckham, M. F. Goodchild & M. F. Worboys (Eds.), Foundations of geographic information science , London: Taylor & Francis, 117–151. Bittner, T. and B. Smith (2003a). Granular spatio-temporal ontologies. AAAI Spring Symposium on Foundations and Applications of Spatio-Temporal Reasoning (FASTR). Bittner, T. and J. Stell (2002). Approximate qualitative spatial reasoning. Spatial Cognition and Computation , 2, 435–466. Clementini, E. and P. D. Felice (1996). An algebraic model for spatial objects with undetermined boundaries. In P. Burrough and A. U. Frank (Eds.), Geographic objects with indeterminate boundaries . London: Taylor and Francis. Cohn, A. G. and N. M. Gotts (1996). The egg-yolk representation of regions with indeterminate boundaries. In P. Burrough & A. U. Frank (Eds.), Geographic objects with indeterminate boundaries . London: Taylor and Francis. Fine, K. (1975). Vagueness, truth and logic. Synthese , 30, 265–300. EXCERPT #MX7TDH p. 19 EXCERPT #44VEDB p. 19 Grenon, P. (2003) The spatio-temporal ontology of reality and its formalization. AAAI Spring Symposium on Foundations and Applications of Spatio-Temporal Reasoning (FASTR). Johansson, I. (1989). Ontological investigations: An inquiry into the categories of nature, man, and society . New York: Routledge. Pawlak, Z. (1982). Rough sets. International Journal of Computation Information . 11, 341–356. Roy, A. J. and J. G. Stell (2001). Spatial relations between indeterminate regions. Journal of Approximate Reasoning , 27(3), 205–234. Smith, B. (1995). On drawing lines on a map. In A. U. Frank & W. Kuhn (Eds.), Spatial information theory: A theoretical basis for GIS , COSIT 95, Lecture Notes in Computer Science 988 , Berlin: Springer, 475–484. Smith, B. (2001). Fiat objects. Topoi , 20(2), 131–148. Smith, B. and B. Brogaard (2002). Quantum mereotopology. Annals of Mathematics and Artificial Intelligence , 35, 1–2, 153–175. Smith, B. and B. Brogaard (to appear). A unified theory of truth and reference. Logique et Analyse . Stevens, A. and P. Coupe (1978). Distortions in judged spatial relations. Cognitive Psychology , 10, 422–437. Varzi, A. (2001). Vagueness in Geography. Philosophy and Geography , 4, 49–65. DOCUMENT #XZX6PE Vague Reference and Approximating Judgments SECTION #JM424D Vague Granular Partitions SECTION #Y9BDQJ The Theory EXCERPT #WKFKYU p. 4 What, now, of vagueness? A vague granular partition Pt^V = ((A, \subseteq), (A, \leq), P^V, L^V) is a quadruple in which the cell structure and target domain are defined as above, and P^V and L^V are classes of projection and location relations (Bittner and Smith 2001b). Again, we will write Pt^V = (A, P^V, L^V) in order to keep the notation simple. EXCERPT #KY49KP p. 4 Consider Figure 2, which depicts a vague partition Pt^V = (A, P^V, L^V) of the Himalayas. This has a cell structure A , as shown in the left part of Figure 2, which is in fact identical to the corresponding part of Figure 1. In the right part of the figure, in contrast, there is a multiplicity of possible candidate projections for the cells in A , indicated by boundary regions depicted via cloudy ovoids. The boundaries of the actual candidates onto which the cells ‘Lhotse’ and ‘Everest’ are projected under the various P_i in P^V are continuous ovoids included somewhere within the cloudy regions depicted in the Figure. EXCERPT #SY3X7V p. 4 The projection and location relations in these classes form pairs (P_i, L_j) , which are such that each P_i has a corresponding unique L_j and vice versa, satisfying the following conditions (where the notation ‘ \exists! ’ abbreviates: ‘there exists one and only one!’): EXCERPT #A5Y6GR p. 4 \begin{aligned} MB1^V & \quad \forall j: L_j(o, z) \rightarrow \exists! i: P_i(z, o) \\ MB2^V & \quad \forall i: P_i(z, o) \rightarrow \exists! j: L_j(o, z) \end{aligned} EXCERPT #36VZ4E p. 4 We also demand that all P_i and all L_j are functional in the sense discussed in the crisp case: EXCERPT #ZMQYAR p. 4 \begin{aligned} MB3^V & \quad P_i(z, o_1) \text{ and } P_i(z, o_2) \rightarrow o_1 = o_2 \\ MB4^V & \quad L_j(o, z_1) \text{ and } L_j(o, z_2) \rightarrow z_1 = z_2 \end{aligned} EXCERPT #6BT7CZ p. 4 Lhotse Everest The Himalayas EXCERPT #QADS89 p. 4 A grayscale image of a mountain range, likely the Himalayas, with two specific peaks highlighted by dark, circular, cloud-like ovoids. These ovoids represent the boundary regions for the cells 'Lhotse' and 'Everest' in the vague partition. EXCERPT #HW75H4 p. 4 Figure 2: A vague partition of the Himalayas EXCERPT #F6LAMJ p. 5 EXCERPT #4XCF8X p. 5 We demand further that cells project onto some object (are non-empty) under every projection: EXCERPT #KGRZ4U p. 5 \text{MB5}^V: Z(z, A) \rightarrow \forall j \exists o: L_j(o, z) EXCERPT #Q3G6RU p. 5 Again, every particular projection considered as a function from A to \Delta is an order homomorphism: EXCERPT #VUTPPH p. 5 \text{MB6}^V: z_1 \subseteq z_2 \rightarrow p_i(z_1) \leq p_i(z_2) EXCERPT #MBWS9R p. 5 We now add an axiom that governs the interrelations between projection relations with distinct indexes in the vague partition \text{Pt}^V . Recall that all projection relations operate on the same cell structure. We need to ensure that if the same object is targeted by two cells z_1 and z_2 under different projections P_i and P_j then the targeting cells must be identical: EXCERPT #N7ZFNN p. 5 \text{MB7}^V \quad P_i(z_1, o) \text{ and } P_j(z_2, o) \rightarrow z_1 = z_2 SECTION #UHNNJV Equivalence of Candidate Referents EXCERPT #BA26P4 p. 5 Given a vague partition as defined above, we can define an equivalence relation between entities in the target domain \Delta as EXCERPT #MVNFM6 p. 5 D \approx \quad o_1 \approx o_2 \equiv \exists z, i, j : P_i(z, o_1) \text{ and } P_j(z, o_2). EXCERPT #98ENN5 p. 5 Clearly, \approx is symmetric and reflexive. To see that \approx is also transitive, assume o_1 \approx o_2 and o_2 \approx o_3 . This means that there exist z_1, z_2, i, j, k, m such that P_i(z_1, o_1) , P_j(z_1, o_2) , P_k(z_2, o_2) and P_m(z_2, o_3) . From P_j(z_1, o_2) and P_k(z_2, o_2) it follows by \text{MB7}^V that z_1 = z_2 , and hence for some cells z and some projections P_i and P_m it holds that P_i(z, o_1) and P_m(z, o_3) , i.e., o_1 \approx o_3 . In the remainder, we write [o]_z to denote the set \{o \mid \exists i : P_i(z, o)\} . EXCERPT #ZH6UHR p. 5 Let \text{Pt}^V = (A, P^V, L^V) be a vague granular partition. We call all partitions \text{Pt} = (A, P_b, L_j) with P_i \in P^V and L_j \in L^V which satisfy the axioms MB1–MB6 crispings of the vague partition \text{Pt}^V . Consider a partition with cells labeled with vague proper names. Intuitively, each crisping (A, P_b, L_j) then recognizes exactly one candidate precisified referent for each such cell. The precise candidates carved out by the separate (A, P_b, L_j) are all slightly different. But each is perfectly crisp and thus it has all of the properties of crisp partitions discussed in the previous sections. All those different candidate referents are equivalent in the sense of our relation \approx . This captures the de dicto view of vagueness. SECTION #FA6PAW Semantic partition EXCERPT #SSW2K9 p. 5 Given a vague partition \text{Pt}^V = (A^V, \subseteq), (A, \leq), P^V, L^V) , we can for each cell z \in A^V classify corresponding portions of reality with respect to the vague projection of the cell z into three zones: the determinate zone, the indeterminate zone, and the exterior zone. EXCERPT #F7AAZB p. 6 EXCERPT #RA6P43 p. 6 We say that x is part of the determinate core of the vague projection P^V of the cell z if and only if, under all projections p_i(z) in P^V , x is a part of the targeted candidate referent: EXCERPT #PX3Z8X p. 6 \text{determinate}_V(x, z) \equiv \forall i: x \leq p_i(z) EXCERPT #LYXKFQ p. 6 Thus the summit of Mount Everest is part of the determinate core of the vague projection of the name 'Everest' and its associated cell. EXCERPT #NVZT8B p. 6 We say that x is a part of the indeterminate zone of the vague projection P^V of the cell z if and only if there are some projections p_i in P^V under which x is part of the targeted candidate referent and other projections p_j(z) in P^V under which this is not the case: EXCERPT #FEGF5R p. 6 \text{indeterminate}_V(x, z) \equiv \exists i: x \leq p_i(z) \text{ and } \exists j: \neg(x \leq p_j(z)) EXCERPT #6NNBUR p. 6 The dotted region in Figure 2 illustrates the indeterminate zone of the projection of the cell associated with the vague name 'Mount Everest'. EXCERPT #RTMR9N p. 6 We say that x is a part of the zone exterior to the vague projection P^V of the cell z if and only if x is not reached by any projection p_i(z) in P^V . EXCERPT #3UV5GK p. 6 \text{exterior}_V(x, z) \equiv \forall i: \neg(x \leq p_i(z)) EXCERPT #D78WCR p. 6 There are parts of reality – such as Berlin – that are not reached by any projections of the cell 'Everest' in the partitions used by humans projecting in transparent fashion. EXCERPT #73N5CY p. 6 Reference via a vague name 'N' creates a partition of reality into determinate core, indeterminate zone, and exterior zone. Let \sigma x \psi(x) denote the mereological sum of all x satisfying \psi(x) and let z be the cell in the partition P^V which projects onto the candidate referents for 'N'. We then define determinate core, indeterminate zone, and exterior zone as the mereological sums of all determinate, indeterminate, and exterior parts of reality with respect to the vague projection of the cell z : EXCERPT #DD5ND9 p. 6 \text{det}_V(z) = \sigma x \text{determinate}_V(x, z) EXCERPT #QJ5SPC p. 6 \text{indet}_V(z) = \sigma x \text{indeterminate}_V(x, z) EXCERPT #TGLT6H p. 6 \text{ext}_V(z) = \sigma x \text{exterior}_V(x, z) EXCERPT #9FNY6M p. 6 We define the semantic partition of reality with respect to the vague name N as a triple of determinate zone, indeterminate zone, and exterior zone. In general \text{det}_V(z) is a partial function since there does not necessarily exist a portion of reality which is a part of all projections of the cell z . EXCERPT #WBEUY7 p. 7 DOCUMENT #XZX6PE Vague Reference and Approximating Judgments SECTION #F7CTDR Approximating Judgments SECTION #RAK36R Approximation in Egg-yolk Partitions EXCERPT #6AJ3ZT p. 7 Consider, again, Figure 1. The cells labeled 'Everest' and 'Lhotse' carve mountain-candidates out of a certain formation of rock. They do not do this physically, but rather by establishing fiat boundaries in reality, represented by the black lines in the right part of the figure (Smith 1995), (Smith 2001), (Bittner and Smith 2001a). But how are we to understand the phenomenon whereby judging subjects are able to impose spatial boundaries vaguely ? EXCERPT #C3C2BS p. 7 Suppose you are an expert mountain guide hiking through the Himalayas with your friends and you assert: EXCERPT #LBXRDB p. 7 [A]: We will cross the boundary of Mount Everest within the next hour. EXCERPT #J2HTXV p. 7 We shall assume that through your use of the phrase 'within the next hour' you successfully delimit a range of admissible candidates for the boundary of Mount Everest along the trajectory of your hike. Consider the left part of Figure 3. Here boundaries delimiting admissible candidates are imposed by specifying a time interval that translates to travel distance along a path; time serves here as frame of reference . The boundaries are defined by your current location (marked: 'now') and your location after the specified time has passed (marked: 'in one hour'). The boundary of each admissible candidate referent crosses the path at some point between these two boundaries, called the exterior and the interior boundaries, respectively. EXCERPT #AQL8V2 p. 7 The general case is illustrated in the right part of Figure 3, which is intended to depict how judging subjects project egg-yolk-like granular partitions onto reality involving three cells: an exterior, a core, and an intermediate region within which the boundary candidates lie. (See (Cohn and Gotts 1996) and (Roy and Stell 2001).) This granular partition serves as the frame of reference in terms of which the judging subject is able at the same time to both specify the range of admissible entities to which he (vaguely) refers and also to constrain this range. SECTION #PHAMTM Egg-yolk Partitions vs. Semantic partition EXCERPT #Z6Y5HP p. 7 Consider the egg-yolk partition in the right part of Figure 3. It is important to see that this is not a semantic partition in the sense discussed above. This is because it was created by a judging subject by imposing boundaries onto reality in order to constrain admissible candidate referents and at the same time to serve as a frame of reference. EXCERPT #R6ZVZV p. 7 A semantic partition on the other hand is induced by classifying portions of reality into determinate zone, indeterminate zone, and exterior zone with respect to the vague projection of a certain cell in a vague partition. Ideally, when corresponding to the same vague name, both partitions coincide but, as we shall see below, this is not the case in general. We will discuss the relationships between the two kinds of partitions in a later section. Until then we ignore the notion of semantic partition and focus on the kinds of partitions shown in the right part of Figure 3 and their use as frames of reference in approximations. EXCERPT #RD44CN p. 8 EXCERPT #XTM7P7 p. 8 Figure 3: Egg-yolk like reference partitions. The left diagram shows a cross-section of an egg with concentric regions: a central 'core', an 'interior boundary', an 'exterior boundary', and a 'candidate boundary' between them. The right diagram shows a curved path with a 'core' at the start, an 'interior boundary' labeled 'in one hour', and an 'exterior boundary' further along. An arrow indicates the 'direction of travel' along the path, and a label points to the area between the boundaries: 'where-the-boundary-candidates are'. EXCERPT #LEGRTR p. 8 Figure 3: Egg-yolk like reference partitions SECTION #GJDGHT Approximation in Complex Partitions EXCERPT #WK2NUD p. 8 In the case just discussed, new fiat boundaries are created by judging subjects in ad hoc fashion in order to delimit vagueness. But there are also cases where already existing systems of boundaries are re-used for this same purpose. There is one crisp granular partition of this sort with which we are all familiar. It has exactly 50 cells, which project onto the 50 United States of America. A fragment of this partition is presented in the left and right parts of Figure 4. In the foreground of the figure we see in addition an area of bad weather (also called ‘Hurricane Walter’), represented by a dark dotted region that is subject to vagueness de dicto in the sense discussed above. Wherever the boundaries of this object might be located, they certainly lie skew to the boundaries of the relevant states. But the figure also indicates (with the help of suitable labeling) that: EXCERPT #D8UZDS p. 8 [B] Hurricane Walter extends over parts of Wyoming, parts of Montana, parts of Utah, and parts of Idaho. EXCERPT #SDAG7A p. 8 In the sorts of contexts which we humans normally inhabit, it is impossible to refer to any crisp boundary when making judgments about the location of a region of bad weather of the sort described. However, it is possible to describe its (current) location relative to the grid of a map in the manner illustrated in judgment [B]. EXCERPT #QDM45X p. 8 We, the judging subjects, then deliberately employ a corresponding partition as our frame of reference and we describe the relationships that hold between all admissible referents of the vague term ‘Hurricane Walter’ and the cells of this partition. In terms of spatial relations, this means in the given case that all admissible candidates partially overlap the states of Wyoming, Montana, Utah, and Idaho and that they do not overlap any other state. Consequently, if a judging subject can specify for every partition cell a unique relation – for example part of – that holds for all admissible candidate referents of a vague term, then this is a determinate way to effect vague reference . The technical name for this phenomenon is approximation . For details, see (Bittner and Stell 2002). EXCERPT #EGLWSG p. 9 EXCERPT #RBG5PM p. 9 A meteorologist may achieve a finer approximation by employing a finer-grained partition as frame of reference in order to make a more specific judgment about the current location of the bad weather region. Thus she might use cells labeled Eastern Idaho, Southern Montana, Western Wyoming, and Northern Utah, and so on, yielding a fiat boundary of the sort depicted in the right part of Figure 4. EXCERPT #A9ZR5P p. 9 Notice that all these boundaries predate the judgments which use them as frames of reference in relation to this particular bad weather system. They are there to be used over and over again in formulating constraints on the possible locations of admissible candidate referents corresponding to vague referring terms. They represent a convenient and determinate way to make vague reference, which has even greater utility when the frame of reference is a commonly accepted one, as in the present case. SECTION #UCKBZM Approximation and Judgments EXCERPT #6N7U6P p. 9 Approximating judgments are a special class of judgments that contain both vague names and a (relatively) crisp reference to boundaries that delimit this vagueness. [A], too, is an approximating judgment which contains the vague name 'Everest' and also a reference to boundaries delimiting the vagueness of this term via the phrase '[crossable] within the next hour'. In this paper, we consider approximating judgments which contain a single vague name and a crisp reference frame. More complex cases are possible – including the case where the reference frame itself involves a certain degree of vagueness – but formal consideration of the latter is omitted here since its treatment follows the same basic pattern. EXCERPT #8A543U p. 9 The image contains two side-by-side maps of the United States, focusing on the western region. Both maps show a grid of state boundaries. A dark, irregular, cloud-like shape representing Hurricane Walter is shown moving from the southwest towards the northeast. - The left map has a coarse grid where each cell covers a large area (e.g., an entire state or a large portion of one). The hurricane's path is somewhat fuzzy and overlaps several of these large cells. - The right map has a finer grid where each cell covers a much smaller area (e.g., a smaller portion of a state). The same hurricane path is shown, but it is more precisely defined by the boundaries of these smaller cells, illustrating a 'finer-grained partition' as mentioned in the text. Figure 4: Two maps of the United States showing the path of Hurricane Walter. The left map shows a coarse partition of the states, while the right map shows a finer partition. Both maps show a dark, irregular shape representing the hurricane's path, which is more precisely defined in the right map. EXCERPT #CV32ME p. 9 Figure 4: States of the United States with Hurricane Walter EXCERPT #TV44Q9 p. 10 EXCERPT #8HLD7C p. 10 An approximating judgment J^A , if uttered successfully, imposes two partitions onto reality: a vague partition Pt^V and a reference partition Pt^R , along the lines above, whereby the latter serves to delimit the vagueness of the former. An approximating judgment J^A is thus a triple (S, Pt^V, Pt^R) , consisting of a sentence, S , together with two granular partitions, Pt^V and Pt^R . EXCERPT #3NWSUN p. 10 In the approximating judgment J^A = ([B], Pt^V, Pt^R) , expressed by the sentence: 'Hurricane Walter extends over parts of Wyoming, Montana, Utah, and Idaho', the corresponding vague partition Pt^V contains a cell labeled 'Hurricane Walter', which projects onto a multiplicity of admissible candidates. At the same time this judgment reuses the partition depicted in the left part of Figure 4 as its reference partition Pt^R . The latter constrains the admissible projections of the cell labeled 'Hurricane Walter' in Pt^V in such a way that each candidate referent that is targeted by a projection P_i of Pt^V must extend over parts of reality targeted by the cells of Pt^R labeled 'Wyoming', 'Utah', 'Montana', and 'Idaho' respectively. SECTION #EUKWZB Partition Theory and Approximation EXCERPT #3HQVJF p. 10 The idea underlying the partition-theoretic view of approximation is that a (crisp) granular partition can be used as a frame of reference (a generalized coordinate frame (Bittner 1997)), which allows us (a) to describe the approximate location of objects and thus (b) to project onto portions of reality in an approximate way. We call a granular partition which is used as a frame of reference in this manner a reference partition . EXCERPT #TAV47T p. 10 Consider a vague name such as 'Hurricane Walter' (hereafter: 'HW') and the corresponding multiplicity of admissible candidate referents for this name formed by crisp portions of reality in the domain of the northwestern United States at some given point in time. Consider some crisp partition structuring this same domain but without recognizing any of the candidates referred to by the name 'HW' directly. This might be the partition created by the boundaries of the separate States of the sort used in Figure 4, or it might be a partition formed by a raster of cells aligned to lines of latitude and longitude. EXCERPT #V4PL3D p. 10 To understand the formal details of how the latter can serve as reference partition in relation to the former we introduce the three concepts of full overlap ( fo ), partial overlap ( po ), and non-overlap ( no ), concepts which we shall now use to generalize the notions of projection and location, as follows. Consider a reference partition whose cells are projected onto regions of space on the surface of the Earth. Let o be a portion of reality that straddles the boundaries of the cells of this reference partition. The constants fo , po , no will now be used to measure the degree of mereological coverage of the object o by the corresponding regions of space. EXCERPT #LB8ZVQ p. 11 EXCERPT #HDZZQ9 p. 11 We call the relation L^R(o, z, \omega) the rough location of the portion of reality o with respect to the cell z and the relation P^R(z, o, \omega) the rough projection of the cell z onto o . (We use the phrases 'rough location' and 'rough projection' in order to emphasize our indebtedness to the account of approximation in terms of rough sets advanced in (Pawlak 1982).) In both relations, \omega stands for the degree of mereological overlap of the portion of reality targeted by the cell z with the actual portion of reality o , i.e., it takes one or other of the values fo , po , or no . Consider the left part of Figure 4. There the relation po holds between all admissible candidate referents HW_i and Montana, i.e., \forall i: L^R(HW_i, Montana, po) . The relation no holds between all the HW_i and Oregon, i.e., \forall i: L^R(HW_i, Oregon, no) . EXCERPT #YGR664 p. 11 We can characterize the relationships between exact and rough location and exact and rough projection in reference partitions as follows: EXCERPT #EYBZNH p. 11 \begin{aligned} L^R(o, z, fo) &\equiv \exists x (L(x, z) \text{ and } x \leq o) \\ P^R(z, o, fo) &\equiv \exists x (P(z, x) \text{ and } x \leq o) \\ L^R(o, z, po) &\equiv \exists x (L(x, z) \text{ and } \exists y (y \leq x \text{ and } y \leq o) \text{ and } \\ &\quad \exists y (y \leq x \text{ and } \neg(y \leq o))) \\ P^R(z, o, po) &\equiv \exists x (P(z, x) \text{ and } \exists y (y \leq x \text{ and } y \leq o) \text{ and } \\ &\quad \exists y (y \leq x \text{ and } \neg(y \leq o))) \\ L^R(o, z, no) &\equiv \exists x (L(x, z) \text{ and } \neg \exists y: y \leq x \text{ and } y \leq o) \\ P^R(z, o, no) &\equiv \exists x (P(z, x) \text{ and } \neg \exists y: y \leq x \text{ and } y \leq o) \end{aligned} EXCERPT #RN7G42 p. 11 The notion of rough location gives rise to an equivalence relation in the domain of objects (portions of reality), with respect to a given reference partition Pt^R with rough location relation L^R , as follows: EXCERPT #3BRV55 p. 11 D\sim \quad o_1 \sim o_2 \equiv \forall z, \omega: L^R(o_1, z, \omega) \leftrightarrow L^R(o_2, z, \omega). EXCERPT #LGJNKV p. 11 Thus two objects are equivalent with respect to the granular partition Pt^R if and only if they have an identical rough location with respect to all cells of this partition. The relation \sim can thus be interpreted as meaning: indiscernibility with respect to the frame of reference provided by Pt^R . In an approximating judgment J^A = (S, Pt^V, Pt^R) , the reference partition Pt^R will be chosen in such a way that the candidate referents targeted by a single cell in Pt^V are equivalent with respect to \sim . Often there may be a number of possible choices for reference partitions, all of which have the feature that all candidate referents of the vague name in question are equivalent with respect to the indiscernibility relation \sim induced by the reference partition. For example in Figure 4 we could also have used a regular (raster-shaped) reference partition of some appropriate resolution. However, it is more appropriate in a weather forecast to use the reference partition defined by the boundaries of the separate States because of its familiarity. We will discuss different choices of reference partitions in later sections. EXCERPT #EVNBR6 p. 12 EXCERPT #Q56CFN p. 12 We define a reference partition as a quintuple, Pt^R = ((A, \subseteq), (\Delta, \leq), P^R, L^R, \Omega) where (A, \subseteq) and (\Delta, \leq) are a cell structure and target domain as specified above, P^R and L^R are rough projection and location relations, and \Omega is the set of values \{fo, po, no\} indicating degrees of overlap (coarser and finer distinctions are possible, as discussed in (Bitner and Stell 2003)). We then can prove that the following counterparts of MB1-3 hold for reference partitions: EXCERPT #2NN6W4 p. 12 \begin{aligned} \text{TR1} \quad & L^R(o, z, \omega) \rightarrow P^R(z, o, \omega) \\ \text{TR2} \quad & P^R(z, o, \omega) \rightarrow L^R(o, z, \omega) \\ \text{TR3} \quad & (\forall z, \omega: P^R(z, o_1, \omega) \leftrightarrow P^R(z, o_2, \omega)) \rightarrow o_1 \sim o_2 \end{aligned} EXCERPT #KMS6VE p. 12 TR1 follows from MB1. To see this assume L^R(o, z, \omega) and let \omega = fo . We have \exists x (L(x, z) \text{ and } x \leq o) . By MB1 we have \exists x (P(z, x) \text{ and } x \leq o) , hence P^R(z, o, fo) and similarly for \omega = po and \omega = no . TR2 follows from MB2 in a similar manner. To see TR3 assume \forall z, \omega (P^R(z, o_1, \omega) \leftrightarrow P^R(z, o_2, \omega)) . By TR1 and TR2 L^R and P^R are logically equivalent and can be substituted for each other. Therefore we have L^R(o_1, z, \omega) \leftrightarrow L^R(o_2, z, \omega) , i.e., o_1 \sim o_2 . EXCERPT #GEYDLA p. 12 Corresponding to MB4 we now demand that if all objects have the same relation \omega \in \{fo, po, no\} with respect to the cells z_1 and z_2 then these two cells are identical: EXCERPT #9PLATP p. 12 \text{R1} \quad (\forall o, \omega: L^R(o, z_1, \omega) \leftrightarrow L^R(o, z_2, \omega)) \rightarrow z_1 = z_2. SECTION #H67PPK Constraining Approximation SECTION #B4N4D4 Well-Formed Approximations EXCERPT #7W9WD7 p. 12 If an approximating judgment like ([A], Pt^V, Pt^R) is to succeed, that is if a true judgment of this form is to have been made, then the reference partition needs to project onto reality in such a way that all admissible candidate referents are equivalent with respect to the indiscernibility relation imposed by Pt^R . Thus, in the hiker case, each value of p^V ("Everest") must be such that its boundary can be crossed in one hour from the time when the judgment is made (Figure 3). EXCERPT #4T58AH p. 12 Let (S, Pt^V, Pt^R) be an approximating judgment and let Pt^V be a vague partition with a cell for each vague name in the sentence S . We then demand that in such an approximating judgment the reference partition Pt^R and the vague partition Pt^V be related to each other in such a way that candidate referents which are targeted by the same cell (i.e., are equivalent in the sense of \approx ) have the same rough approximation in the underlying reference partition (i.e., are also equivalent in the sense of \sim ): EXCERPT #WR7NBE p. 12 \text{EP} \quad o_1 \approx o_2 \rightarrow o_1 \sim o_2. EXCERPT #GC9QKL p. 13 EXCERPT #QS4FA3 p. 13 We call EP the equivalence principle . EP rules out many reference partitions Pt^R which cannot be used for constraining the vagueness of the vague partition Pt^V . Thus it rules out, for example, reference partitions with resolutions too fine for the degree of vagueness of the corresponding vague partition. Consider Figure 4. A raster-cell-partition with cell size of 1m^2 would violate the equivalence principle, since not all candidate referents of the vague name 'Hurricane Walter' would be indiscernible with respect to this reference partition. If the reference partition is too fine then equivalence in the sense of \approx does not imply equivalence in the sense of \sim . EXCERPT #4HLZRV p. 13 Notice that the converse of the equivalence principle does not hold. This is because there might be portions of reality ( x and y ) that are equivalent with respect to the reference partition ( x \sim y ), but which are such that neither is a candidate referent targeted by the cell in question. Consider Figure 5. The approximation of the mereological sum of Yellowstone National Park and Zion National Park ( YNP + ZNP ) with respect to the Federal State reference partition (left) is identical to the approximation of the candidate referents of the name 'Hurricane Walter' (right) but surely ( YNP + ZNP ) is not a candidate referent for the name 'Hurricane Walter'. EXCERPT #CFE25N p. 13 Figure 5 consists of two side-by-side diagrams. The left diagram shows a map of the western United States with a grid overlay. A small rectangular area is highlighted and labeled 'YNP' (Yellowstone National Park). Below it, another small rectangular area is labeled 'ZNP' (Zion National Park). The right diagram shows a similar grid overlay, but instead of a map, it shows a dark, irregular, cloud-like shape that fills a portion of the grid cells, representing the candidate referents of the name 'Hurricane Walter'. EXCERPT #3PUAJ5 p. 13 Figure 5 Left: Yellowstone National Park (YNP) and Zion National Park (ZNP). Right: Hurricane Walter. SECTION #V7FNPA Precise Approximation EXCERPT #VEQL6M p. 13 In this section, we discuss the relationship between semantic partition and the kinds of reference partitions previously discussed. EXCERPT #JEK4VV p. 13 Consider the approximating judgment J = ([A], Pt^V, Pt^R) . The reference partition Pt^R shown in the left part of Figure 3 imposes two flat boundaries onto reality: an interior boundary of the approximation and an exterior boundary of the approximation. As discussed above, this often results in a partition structure similar to the one depicted in the right part of the figure. The projection of this partition onto the path the judging subject takes on her journey towards the summit of Mount Everest results in the reference partition Pt^R . EXCERPT #TKDBSP p. 13 Consider now the semantic partition imposed by the cell labeled 'Everest' in the vague partition Pt^V . We can see that the relationship between Pt^V and Pt^R satisfies the equivalence principle EP, which demands that all candidate referents of the vague name 'Everest' are equivalent under \sim . Consider now the location of two pairs of boundaries: (a) the interior and exterior boundaries imposed by the judging subject as a frame of reference for her approximation; and (b) the boundaries imposed by the semantic partition into determinate zone, indeterminate zone, and exterior zone via the cells of P^A . We say that the approximating judgment is precise if and only if (1) the interior boundary of the approximation coincides with the boundary separating the determinate zone from the surrounding parts of the semantic partition; and (2) the exterior boundary of the approximation coincides with the boundary separating the exterior zone from the indeterminate zone of the semantic partition. This means that the semantic partition and the reference partition coincide. EXCERPT #VWCUR5 p. 14 EXCERPT #D39GKV p. 14 In order to take more complex reference partitions into account, we now define upper and lower approximations of an object o with respect to such partitions. (Again, we use the notions of lower and upper approximation in order to emphasize the correspondence to rough set theory of Pawlak (1982).) The lower approximation of an object o with respect to a reference partition P^A is the mereological sum of all those portions of reality which are targeted by cells of P^A and which are contained in o : EXCERPT #SEKWJW p. 14 \text{Lower}(o) = \text{co}(\exists x(o' = p(x) \ \& \ P^A(x, o, f(o))), EXCERPT #RUZZF4 p. 14 where \text{co}(x) is defined as above. EXCERPT #SP7YKZ p. 14 The upper approximation is the mereological sum of all those portions of reality which are targeted by cells of the reference partition and which overlap o : EXCERPT #28M527 p. 14 \text{Upper}(o) = \text{co}(\exists x(o' = p(x) \ \& \ (P^A(x, o, f(o)) \ \vee \ P^A(x, o, po)))) EXCERPT #TUW7K8 p. 14 Consider now the reference partitions shown in the left part of Figure 6, which is a refined version of the egg-yolk reference partition in the right part of Figure 5. The core cell of the latter is subdivided into eastern and western subcells (ec and wc for eastern part of the core region and western part of the core region, respectively). Moreover the region where the boundaries are is subdivided into a northern and southern region (nb and sb, respectively). The lower approximation of the candidate referent signified by its outer boundary is the mereological sum of those portions of reality which are targeted by the cells 'ec' and 'wc'. Its upper approximation has as parts in addition portions of reality targeted by the cells 'nb' and 'sb'. EXCERPT #RLT7FT p. 14 The diagram consists of two parts. The left part shows a 'refined egg-yolk partition' with concentric regions. The outermost region is labeled 'exterior'. Inside it is a ring labeled 'nb' (north boundary) and 'sb' (south boundary). The center is divided into two cells: 'ec' (eastern core) and 'wc' (western core). A line labeled 'Candidate boundary' points to the boundary between the 'nb/sb' ring and the 'ec/wc' core. The right part shows a 'raster partition' as a 4x4 grid of cells labeled A through P. A blue-shaded irregular shape is overlaid on the grid, representing a complex object. The shape covers cells B, C, D, E, F, G, H, I, J, K, L, M, N, O, and P, with some cells partially covered. Figure 6: Approximation in complex partitions. Left: a refined egg-yolk partition. Right: a raster partition. EXCERPT #B5QMJX p. 14 Figure 6: Approximation in complex partitions. Left: a refined egg-yolk partition. Right: a raster partition. EXCERPT #E4QM45 p. 15 EXCERPT #57LXKF p. 15 For another example of lower and upper approximations, consider the right part of Figure 6. The upper approximation of the depicted region with respect to the raster-shaped partition is the mereological sum of the targets of the cells [B,..., L, O, P]. The lower approximation is identical to the target of the cell K. EXCERPT #FBCQFT p. 15 Notice that upper approximations are always defined. Lower approximations, however, are only defined if the reference partition is sufficiently fine grained, so that P^R(z, o, fo) holds for some cell z and some portion of reality o . This is because in mereology there is no counterpart to the empty set. EXCERPT #VGHKWF p. 15 With respect to these more complex reference partitions we now say that an approximating judgment is precise if and only if (1) the boundary of the lower approximation of any candidate referent of 'N' coincides with the boundary separating the determinate zone from the surrounding parts of the semantic partition imposed by the vague projection of the cell associated with 'N'; and (2) the boundary of the upper approximation of any of the candidate referents coincides with the boundary separating the exterior zone from the indeterminate zone of the vague projection of the cell associated with 'N'. EXCERPT #87PDSK p. 15 In formal terms, we describe this as follows. Let J = (N, Pt^V, Pt^R) be an approximating judgment with Pt^V = ((A^V, \subseteq), (\Delta, \leq), P^V, L^V) and Pt^R = ((A^R, \subseteq), (\Delta, \leq), P^R, L^R, \Omega) . We then call J^A precise if and only if EXCERPT #5MJLZV p. 15 \forall o \in [o]_z: \text{Lower}(o) = \text{det}^V(z) \text{ and } \text{Upper}(o) = (\text{det}^V(z) + \text{indet}^V(z)). EXCERPT #HZCEA6 p. 15 Here z \in A^V is the cell in the vague partition corresponding to the vague name 'N', [o]_z is the set of all objects targeted by the vague projection of z , and + is the mereological sum. SECTION #HDWEPC Constraining Approximation EXCERPT #42BMVA p. 15 We now discuss constraining approximations, defined as those approximations that do not have the property of being precise but still satisfy the equivalence principle. EXCERPT #ELWWPB p. 15 Let z be the cell in the vague partition Pt^V which corresponds to the vague name 'N', and let [o]_z be the set of all candidate referents of 'N', i.e., portions of reality targeted by z under P^V . The approximation of candidate referents o \in [o]_z with respect to the approximating partition Pt^R is called constraining if and only if the following holds: EXCERPT #2WKNLC p. 15 \forall o \in [o]_z: EXCERPT #H5CEL5 p. 16 EXCERPT #LNDVAA p. 16 either: EXCERPT #SX8YEU p. 16 Lower(o) is defined and \text{Lower}(o) \leq \text{det}_V(z) \leq (\text{det}_V(z) + \text{indet}_V(z)) \leq \text{Upper}(o) or: EXCERPT #7XPPNJ p. 16 \text{det}_V(z) \leq (\text{det}_V(z) + \text{indet}_V(z)) \leq \text{Upper}(o) EXCERPT #QPUTNJ p. 16 Consider Figure 4 and assume that the determinate zone of the vague reference of the name ‘Hurricane Walter’, \text{det}_V(\text{HW}) , is situated along the border between Idaho and Wyoming. It follows that the lower approximation is undefined for any of the candidate referents because the federal state partition is too coarse. The upper approximation however is defined, since any candidate referent is part of the mereological sum of Idaho, Montana, Wyoming, and Utah. Hence the resulting approximation is constraining. EXCERPT #AK2DNU p. 16 Consider the class of constraining approximations. As already Aristotle repeatedly emphasized (at 1094b11 sq. , 1098a26, 1103b34 sq. , 1165a13), judging subjects will characteristically use those approximating judgments which are constraining but which are as precise as necessary in whatever is the context in hand. In (Bittner and Smith 2001b), we argue that this will imply that the limits imposed on vagueness by an approximation will normally be such that the resulting judgment is not subject to truth-value indeterminacy. The judgments we actually make in normal contexts (as contrasted with those types of artificial judgments invented by philosophers) are determinately either true or false even in spite of the vague terms which they contain. DOCUMENT #XZX6PE Vague Reference and Approximating Judgments SECTION #4TYZWW Conclusions EXCERPT #PPF9QH p. 17 We have proposed an application of the theory of granular partitions to the phenomenon of vagueness, a phenomenon which is itself seen in de dicto terms, i.e. as a semantic property of names and predicates. We defended a supervaluationistic theory of the underlying semantics and expressed it in terms of the theory of granular partitions. We showed that the use of frames of reference in making approximating judgments can be formulated very naturally in partition-theoretic terms, and that the framework of granular partitions then helps us to understand the relationships between vagueness and approximation. While the bulk of our examples were derived from the spatial domain, the generality of the theory of granular partitions allows an easy generalization to other sorts of cases. EXCERPT #LCA3T8 p. 18 ### 52. Tool result: read DOCUMENT #88BVY3 Categories in Top-Level Ontologies: Revisiting the Aristotelian Background SECTION #DY7KEM Categories in Top-Level Ontologies: Revisiting the Aristotelian Background EXCERPT #ZJY3G9 p. 0 Ludger Jansen 1 , Barry Smith 2 EXCERPT #PWGMZ6 p. 0 Abstract. In the field of applied ontology, it is now commonplace to refer to a top-level ontology, and the Basic Formal Ontology (BFO) has even been recognized as an ISO standard. Like other contemporary top-level ontologies, BFO makes use of distinctions that have been developed in philosophy, many of which are already to be found in Aristotele. In this essay, we revisit Aristotele's metaphysics and discuss the similarities and differences with BFO. EXCERPT #3VQT6C p. 0 Traditionally, the task of ontology has been to represent reality, for example in terms of a division between different modes of being. More recently, ontologies are being used to support not only philosophers but also scientists and others in their representation of reality. Indeed, with the advent of computers there has arisen a new discipline of applied ontology, whose task is to support the integration and discoverability of data deriving from different sources, nowadays including financial, industrial, governmental and other organizations, by providing logically supported classification systems (Munn 2008). 3 EXCERPT #R6E2X4 p. 0 An important instrument for all these purposes is the technique of classification. But, in any classification we have to select what will be the classes or kinds we place at the very top. What should the top level of an ontology, or indeed of any classification, look like? What are the most general classes of all classifications? It is upon questions such as this that we shall focus in what follows. EXCERPT #J2KENE p. 0 1 Cusanus Professor for Philosophy at the PTH College Brixen in Bressanone, Italy. Supernumery Professor at the University of Rostock. Email: ludger.jansen@pthsta.it. Affiliation: PTH Brixen College; University of Rostock. ORCID: https://orcid.org/0000-0002-0097-6359 . EXCERPT #PWSLWU p. 0 2 Distinguished Professor of Philosophy, Professor of Biomedical Informatics. Professor of Computer Science and Engineering. Director of the National Center for Ontological Research. Email: phismith@buffalo.edu. Affiliation: University at Buffalo, NY, USA. ORCID: https://orcid.org/0000-0003-1384-116X . EXCERPT #X82DHV p. 0 3 This essay is an updated and extended version of Jansen, Ludger. "Categories: The top-level ontology". Applied Ontology. An Introduction, eds. K Munn, B. Smith, Frankfurt: Ontos 2008. The text has been heavily revised, and several old sections have been deleted. Introduction and Sect. 7 are totally new. To a fair extend, references originate from this first version, though they have been supplemented by pointers to the more recent literature. EXCERPT #4MS9GN p. 0 EXCERPT #SSNL3H p. 1 EXCERPT #K42RSM p. 1 EXCERPT #X8HUM6 p. 1 EXCERPT #YS4B7A p. 1 Authors in the fields of informatics and knowledge representation have suggested various answers to these questions. An early candidate for the role was SUMO, short for ‘Suggested Upper Merged Ontology’, 4 which was developed from an open-source project bringing together freely available, non-commercial ontologies into a common system. Nowadays Basic Formal Ontology (BFO) is a popular candidate top-level ontology, with over 600 domain ontologies defined in its terms (Otte 2022). 5 BFO has been recognized by the International Standards Organization (ISO) and the International Electrotechnical Commission (IEC) as an ISO standard (ISO/IEC: 21838-2), confirming that it satisfies the requirements for being a top-level ontology set forth in standard ISO/IEC:21838-1 (2021). In addition, use of BFO as top-level ontology has been mandated by the Joint Enterprise Standards Committee of the US Department of Defense and Intelligence Community. BFO also provides the architecture for the agency wide data repository of the US Department of Homeland Security. EXCERPT #83YWML p. 1 The first thinkers to address the idea of a standard ontology were, in fact, philosophers, most notably Aristotle in his short treatise on the Categories . This classical text can be read as addressing exactly those questions that matter to applied ontologists today when they think about top-level ontologies (Jansen 2007). From the point of view of traditional philosophy, the question of a top-level ontology is tantamount to the question of the most basic categories of being, and our strategy for addressing this question is to examine Aristotle’s theory of categories and describing some of the ways in which this theory has influenced current work in the discipline of applied ontology. EXCERPT #Q7G3D9 p. 1 There were three principal influences which helped to shape the earliest versions of BFO: Aristotle’s Categories ; the literature of geographic information science (cf. Smith & Mark 2001); and the realist phenomenological approach to ontology pioneered by Edmund Husserl, Adolf Reinach, Roman Ingarden and other members of the so-called Munich-Göttingen school (cf. Smith 1996). We will here address only the first of these influences by presenting the ontology laid out by Aristotle in his Categories and showing how it maps to the philosophical foundations underlying BFO. We first clarify what a category is (section 1). From there, we continue to introduce three ontological dichotomies that are at the root of both Aristotle’s ontology in the Categories and of BFO: the distinction between universals and particulars (section 2), the distinction between dependent and independent entities (section 3), and the distinction between continuants and occurrents (section 4). We then present two diagrams that synthesize various ontological views that can be generated along the lines of these three distinctions: the EXCERPT #PYN5H7 p. 1 4 See Jansen 2008 for a discussion of other early suggestions, such as OpenCyc and the Sowa Diamond. EXCERPT #74VTPZ p. 1 5 For a regularly updated list, cf. http://basic-formal-ontology.org/users.html . EXCERPT #TD6RVA p. 1 EXCERPT #WQ4KLB p. 2 EXCERPT #GFH6NE p. 2 EXCERPT #R9A8PF p. 2 EXCERPT #DR73QZ p. 2 ontological square, inspired by Aristotle's Categories , and the ontological sextet, which is, as we claim, a more adequate representation of the ontological structure of reality, and represents the basic structure of BFO (section 5). We conclude by pointing out some points where we need to go beyond the simple picture of Aristotle's Categories in order to represent adequately our world as described, for example, by biology and medicine (section 6). SECTION #Y994UA 1. What are Categories? SECTION #HC4LC5 1.1 The word “category” EXCERPT #6DC88W p. 2 As far as we know, Aristotle was the first to use the Greek word kategoria as a technical term in the context of philosophy. Originally, the noun kategoria and its corresponding verb, katēgorein , belonged to legal discourse. There, kategoria means the accusation in front of the judge, and katēgorein means to accuse someone . Probably because an accusation asserts something of someone, the verb can also mean to make known or to assert , and it was used in this way by Plato. 6 EXCERPT #Q6EFRY p. 2 Aristotle uses the active verb phrase katēgorein ti tinos in the sense of: to assert something about something, but even more often he uses the passive katēgoreisthai ti tinos or katēgoreisthai ti kata tinos in the sense of: is said of something. The noun kategoria is used by him in a variety of ways, including using the plural of the noun in the sortal sense to mean ‘kinds of predicates’ or ‘kinds of predication’. Such kinds of predicates are for example quantitative or qualitative predicates such as ‘3 foot long’ or ‘wet’. It is in this sense that the Greek word kategoria can be translated into English as category (Jansen, 2006). EXCERPT #UBFQDR p. 2 We have evidence that Aristotle's conception of the categories developed in three phases. First, as in Topics I 9, the distinction of different categories was meant exclusively as a classification of predicates. In this first phase, the categories served as aids for finding arguments and for avoiding or discovering false inferences, and it was in this way that talk of categories found a place in the theory of argumentation, which is what the Topics are about. EXCERPT #ZRPF6K p. 2 The second phase is represented in Aristotle's Categories . Here the division of categories encompasses, not only predicate terms such as ‘is tall’ or ‘is hungry’, but also subject terms, including proper names such as ‘Socrates’ or ‘Plato’, which function in sentences only as the subject of predication but never as predicates in their own right EXCERPT #6NGW77 p. 2 6 See for example Theaetetus 208b; Phaedrus 73b. Theaetetus 167a links the two meanings. EXCERPT #ATZMRT p. 2 EXCERPT #3DKJYE p. 3 EXCERPT #FEHGNJ p. 3 EXCERPT #A7SZG8 p. 3 EXCERPT #QPD8KG p. 3 ( Categories 5, 3a 36-37). This represents a step in the direction of ontology and away from concerns with the theory of argumentation. EXCERPT #8UXKWJ p. 3 The third phase finds its expression in the Metaphysics . There we find Aristotle's famous observation that 'to be' and 'a being' are used in as many different ways as there are categories ( Metaphysics V 7, 1017a 22-23). Here, the division into separate categories becomes a full-fledged part of one of the most important of Aristotle's ontological works. SECTION #5JW3Z5 1.2 The Interpretation of Aristotle's Categories EXCERPT #5PWHAP p. 3 Aristotle's theory of categories was the subject of much dispute in antiquity, and it has been interpreted in a variety of ways in the history of philosophy ever since. Partly, this has to do with the fact that category theory had many different facets even in the works of Aristotle himself. This came about because Aristotle repeatedly subjected his ideas to further development and highlighted different aspects when presenting his theory. But we can distinguish four prototypical conceptions of what categories are (which often appear in combination), according to whether they classify: EXCERPT #9SQTMY p. 3 (1) subject and predicate terms and their associated meanings , (2) mental or extra-mental concepts , (3) meanings of the copula 'is', or (4) beings or entities . 7 EXCERPT #MPUNV9 p. 3 Here, we draw on the last of these, which was certainly the main conception of the late Aristotle, namely that categories are the highest species of beings. EXCERPT #T8S5AC p. 3 One reason why the question of the highest species of beings is important turns on the way in which definitions are conceived by Aristotle and his successors. Aristotle himself was interested in real definitions , that is, in definitions of things. A definition is then a phrase indicating a thing's essence ( ti esti , what it is). Only substances have essences, and so only substances are definable – an outcome that is confusingly bolstered by the fact that the same word ousia is used to mean both 'substance' and 'essence' (Jansen 2017). The essence of a thing is what is expressed in its definition. EXCERPT #KJY27Q p. 3 One standard technique for creating definitions has its roots in Aristotle's thinking on this topic, and thus in some circles of contemporary applied ontology its results are referred to as 'Aristotelian definitions' (Rosse, Mejino & Jose 2003). Such Aristotelian definitions are constructed by joining a genus term with a specific difference, following the template: EXCERPT #WD8PWD p. 3 7 For these four options cf. Bonitz 1853, Ebert 1985, Kahn 1978, Oehler 1986. EXCERPT #444ZK8 p. 3 EXCERPT #T6Y7GC p. 4 EXCERPT #P5NJAH p. 4 EXCERPT #ZVWSYB p. 4 EXCERPT #9LNDRD p. 4 An S is a G which Ds EXCERPT #3R2NDN p. 4 Here, ‘ S ’ stands for the species to be defined, ‘ G ’ for the genus term, and ‘ D ’ for the differentia , or in other words for the specific difference which picks out all and only those instances of G that are also instances of S , as for example, in the classical definition of human beings as rational animals, where “animal” is the genus term and rationality the specific difference. EXCERPT #XBVZR4 p. 4 A problem arises for such an approach to definitions, however, since it works only where the term to be defined has some more general term which can be used as starting point in creating its definition. Hence the need for a top-level ontology comprising ‘top-level general terms’, 8 which are primitive in the sense that they cannot be defined, though they can in various ways be elucidated , for example by specification of necessary conditions for instantiation, and by provision of examples. Aristotle’s ideas on categories represent the first attempt to create a top-level ontology so conceived. SECTION #CM3E3X 1.3 Aristotle’s Ten Categories EXCERPT #QDNE4G p. 4 There are many lists of categories in the extant works of Aristotle, of different length and content. 9 In Topics I 9, Aristotle says explicitly that there are ten categories, which he then proceeds to delineate. A list of ten categories can also be found in the Categories (see Table 1). The ontology thereby envisaged is nicely summarized in the following passage: EXCERPT #CY4UMJ p. 4 Expressions which are in no way composite signify substance, quantity, quality, relation, place, time, position, state, action, or affection. To sketch my meaning roughly, examples of substance are ‘man’ or ‘the horse’, of quantity, such terms as ‘two cubits long’ or ‘three cubits long’, of quality, such attributes as ‘white’, ‘grammatical’. ‘Double’, ‘half’, ‘greater’, fall under the category of relation; ‘in the market place’, ‘in the Lyceum’, under that of place; ‘yesterday’, ‘last year’, under that of time. ‘Lying’, ‘sitting’, are terms indicating position, ‘shod’, ‘armed’, state; ‘to lance’, ‘to cauterize’, action; ‘to be lanced’, ‘to be cauterized’, affection. (Aristotle, Categories 4, 1b25–2a4, transl. Edghill) EXCERPT #DS3UMG p. 4 8 As Bonaventure describes the problem in his Itinerarium mentis in Deum c. 3, 3: “The function of the intellectual faculty consists in understanding the meaning of terms [...]. Now, the intellect grasps the meanings of terms when it comprehends in a definition what a thing is. But definitions are constructed by using more universal terms; and these are defined by more universal terms until we come to the highest and most universal. Consequently, unless these latter are known, the less universal cannot be grasped in a definition.” EXCERPT #PFHPZK p. 4 9 A synopsis of these lists can be found in the appendix of Oehler 1986. EXCERPT #CSLK77 p. 4 EXCERPT #SB56WF p. 5 EXCERPT #39URVQ p. 5 EXCERPT #LV8X64 p. 5 EXCERPT #VYMPU5 p. 5 Table 1 lists the Greek terms Aristotle uses for his ten categories, together with their verbal translations, Latin equivalents, and some modern terms now in use. Aristotle presents this system of categories by adverting to how each category is represented in ordinary language. More particularly, Aristotle would sometimes name categories not directly, but rather by using questions whose answers would make reference to entities in the respective categories. Many of the names we currently use for these categories then have their origins in the corresponding Latin interrogative expressions (see Figure 3). EXCERPT #562MGJ p. 5 Table 1: Different Terms for Aristotle's Categories EXCERPT #BC8YN6 p. 5 Aristotle's Term English Translation Latin Term Modern Terms ti esti ousia What is it? Essence quod est, quiditas, essentia Essence, Substance poson How much? quantum, quantitas Quantum, Quantity poion How is it? quale, qualitas Quality pros ti Related to what? relativum Relative, Relation pou Where? ubi Place pote When? quando Time keisthein Lying, Being situated situs Position, Posture echein Having habitus poiein Doing agere Action paschein Suffering pati Passion EXCERPT #9W29QS p. 5 Kant accused Aristotle of choosing his categories in a 'rhapsodic manner', without any guiding principles. It is for this reason, or so Kant argues, that Aristotle could never be certain that his list of categories was complete ( Critique of Pure Reason , A 81 = B 106- EXCERPT #9TB2DR p. 5 EXCERPT #U7YJ7V p. 6 EXCERPT #C7AWW4 p. 6 EXCERPT #A857VN p. 6 EXCERPT #FHNAWL p. 6 107). Later Aristotelians, such as Thomas Aquinas 10 or Franz Brentano (1862) 11 , undertook the task of reconstructing a system that yields the Aristotelian categories in the precise order in which they are named and discussed in the Categories . 12 EXCERPT #Z2S7P7 p. 6 To do justice to Aristotle, however, we should note that the works by him that survived are mainly lecture notes and not polished works ready for publication. This helps to explain the disparities between Aristotle's various lists, disparities which draw attention also to the fact that the elements in his lists are not all of the same standing. EXCERPT #MMNMKM p. 6 There are three important dichotomies in terms of which we can understand how Aristotle's categories are organized: EXCERPT #NL95G9 p. 6 • they make room for both universals (kinds, types) and the particulars which are the instances of these universals (section 2 below); • they encompass dependent as well as independent entities (section 3); • they divide reality into continuants and occurents (section 4). EXCERPT #6S36FX p. 6 Taken together, these dichotomies help to systematise Aristotle's list of categories, as can be seen in Figure 1. EXCERPT #QAZCZE p. 6 10 See Aquinas, In Physicorum Aristotelis expositio III, lectio 5, Nr. 322 [15] and In Metaphysicorum Aristotelis expositio V, lectio 9, Nr. 891-892. EXCERPT #PVXEK3 p. 6 11 On Brentano's idea see Smith 1987. EXCERPT #5QTYMC p. 6 See also Simons 1992 and Jansen 2007 for new proposals for the hierarchical organization along the lines suggested in this communication. EXCERPT #CQ2DBG p. 6 12 See Jansen 2007 for a new suggestion of a hierarchy of Aristotle's categories along the lines suggested here. EXCERPT #U4EK9D p. 6 EXCERPT #4R7HC7 p. 7 EXCERPT #YFDCHE p. 7 EXCERPT #FUN5Y2 p. 7 EXCERPT #HSZQU3 p. 7 graph TD Particular[Particular] --> Independent[Independent] Particular --> Dependent[Dependent] Independent --> Substance[Substance] Dependent --> Occurrent[Occurrent] Dependent --> Continuant[Continuant] Occurrent --> WithoutChange[Without change] Occurrent --> InvolvingChange[Involving change] Occurrent --> WithinWhichChange[Within which change] Continuant --> InASingleBearer[In a single bearer] Continuant --> InAPlurality[In a plurality of bearers] Continuant --> WhereinBearers[Wherein bearers can be] WithoutChange --> Position[Position] InvolvingChange --> Action[Action] InvolvingChange --> Passion[Passion] WithinWhichChange --> Time[Time] InASingleBearer --> Quality[Quality] InASingleBearer --> Quantity[Quantity] InAPlurality --> Relation[Relation] Relation --> Having[Having] WhereinBearers --> Place[Place] A hierarchical tree diagram representing Aristotle's categories. The root is 'Particular', which branches into 'Independent' and 'Dependent'. 'Independent' leads to 'Substance'. 'Dependent' branches into 'Occurrent' and 'Continuant'. 'Occurrent' branches into 'Without change', 'Involving change', and 'Within which change'. 'Without change' leads to 'Position'. 'Involving change' branches into 'Action' and 'Passion'. 'Within which change' leads to 'Time'. 'Continuant' branches into 'In a single bearer', 'In a plurality of bearers', and 'Wherein bearers can be'. 'In a single bearer' branches into 'Quality' and 'Quantity'. 'In a plurality of bearers' leads to 'Relation', which then leads to 'Having'. 'Wherein bearers can be' leads to 'Place'. EXCERPT #J2G55G p. 7 Figure 1: A schematic representation of Aristotle's categories. EXCERPT #PMBZCT p. 7 Source: Smith 2022, modified after Jansen 2007. SECTION #9UCDSE 2. Universals and Particulars EXCERPT #CDAA5E p. 7 Concerning the first of these dichotomies, we note that universal and particular are not themselves categories in Aristotle's sense, and nor do they divide the categories into distinct groups. Rather, both universals and particulars can be classified according to the categories; hence the distinction between universals and particulars is orthogonal to Aristotle's categorial distinctions. We can call it 'transcategorical' (Lowe, 2006, 21). EXCERPT #8BRF4X p. 7 This dichotomy is given systematic treatment in the second chapter of the Categories , where Aristotle distinguishes between what can and what cannot be predicated of another entity. Predication requires an aspect of generality. Particulars, such as Socrates or my height, cannot be predicated of other entities. Sentences that contain as predicates expressions such as 'is Cicero' or 'is my height' are not predications in the technical sense at work here. Rather, they are identity claims comparable with 'Tully is Cicero' or 'My height is 5 feet'. A general expression such as 'human' can, in contrast, appear both as the subject and as the predicate of predicating assertions, as in 'A human is a vertebrate', and 'Cicero is a human'. EXCERPT #84SX4H p. 7 EXCERPT #FXKPUA p. 7 EXCERPT #JCTBJ2 p. 8 EXCERPT #X8M5EZ p. 8 EXCERPT #V3HHFL p. 8 SECTION #FHJA5L 3. Dependent and Independent Entities SECTION #9FLUAK 3.1 The Priority of Particular Substances EXCERPT #YSZF8P p. 8 Aristotle is quite clear that his ten categories are not to be viewed as equals. Pride of place is given to the so-called substances, most notably material objects like organisms. They are called 'substances' because their existence is in a sense basic – substances allow entities of all other categories to exist: qualities are always qualities of substances, relations are ultimately relations between substances, and so on. Qualities and relations are, thus, ontologically dependent on substances. From Aristotle's perspective, this dependence on substances is even that which guarantees the unity of ontology ( Metaphysics IV 2). EXCERPT #9G8EKR p. 8 Customarily, the dependent categories are called accidents and are placed in opposition to substances. Examples of accidents are being hot , being hungry , being seated , or being asleep . A traditional criterion for the opposition of substances and accidents can be found in the second chapter of the Categories : qualities and quantities are in a substance , while substances are not in but are, rather, identical with a substance. But it is not entirely clear how this 'being in something else' is to be understood (Smith & Mulligan 1982). A heart is in a body and a tapeworm is in its host, but these are not cases of the 'being in something else' relation that Aristotle had in mind. Thus he explicitly excludes 'being-in' in the sense in which a part is in a whole (as the heart is in the body). And a parasite such as a tapeworm is not even a part of its host, any more than a foetus is a part of its mother, or a tub of yogurt is a part of a refrigerator. EXCERPT #S3UG6W p. 8 The criterion of ontological dependence helps to solve this problem. The tapeworm could leave its host and move into another host. Both tapeworm and host are independent entities. A headache or an instance of colour, in contrast, cannot leave its bearer in this way and continue to exist. It is not possible for the Cheshire Cat to disappear and leave its grin behind. 13 The headache and the colour are dependent for their existence upon a specific bearer, a substance which has this headache, or this colour, among its properties. These properties cannot migrate from one substance to another. EXCERPT #YHEPTD p. 8 In addition, Aristotle also sees a similar dependence between universals and particulars. For him, universals are ontologically dependent on particulars that instantiate EXCERPT #3W5QMB p. 8 13 We discuss this topic further under the heading of 'tropism in subsection 5.1 below. EXCERPT #3VTH9K p. 8 EXCERPT #GGKHXC p. 9 EXCERPT #6N3PJ8 p. 9 EXCERPT #V2T7F5 p. 9 EXCERPT #8E8W8H p. 9 them. In the Categories , Aristotle distinguishes between primary substance ( protē ousia ), that is, a particular substance, such as an organism, and secondary substance ( deutera ousia ), that is, a kind of substance, like the kind Man or the kind Bed . Of these two, Aristotle accords special ontological status to the particular substances. Every universal is then predicated either of some particular substance, or of some particular accident that is in some particular substance in the sense explained. 14 SECTION #94L6NZ 3.2 The Relation of Dependence EXCERPT #DL4QYK p. 9 Substances do not require entities of the other categories further down the list in order to exist. But entities of these other categories do require some entity in the category of substance to serve as ground or fundament for their existence. It is in this sense that substances are said to be ontologically independent entities, where accidents are ontologically dependent . More precisely: substances are ontologically independent of accidents, while accidents are ontologically dependent upon substances. The notion of ontological dependence can be formally captured through a counterfactual criterion. The rough idea is that an entity x is ontologically dependent upon an entity y if x could not exist if y did not exist. EXCERPT #Y6NAHH p. 9 To elaborate further on this relation, we examine how it applies within the categorial framework laid down by Aristotle. First, we note that substances are independent entities. That is to say, they do not rely upon anything else in order to exist. This relation is an ontological one – it is a constraint which operates only in the plane of existence (Ingarden 1965, II/1; Koslicki 2012). Thus our usage of ‘dependence’ and ‘independence’ here is distinct from what we find in other domains, for example when we talk of dependent children or of someone’s being drug dependent. EXCERPT #J3VZ7H p. 9 Such dependence relations hold, now, for all instances of accident categories. More precisely, it holds that, if s is a substance and a is one of s ’s accidents, then a cannot exist unless s exists. This means that a is ontologically dependent on s , or, in a more traditional formulation used already by Aristotle, that a is ontological ‘prior’ to s . In the case of substances and their accidents it is also often said that accidents ‘inhere’ in their substance. This specific form of dependence is a matter of the ways a and s exist – they are different sorts of beings in reality, or, as ontologists express the matter, they belong to different ontological categories. 15 EXCERPT #AE49X7 p. 9 14 Categories 5. 2a 34-35; 2b 3-5; 2b 15-17. In later texts, Aristotle affirms the centrality of particular substances and their special importance with respect to the other categories, which he then also calls ‘affections of the substances’. Metaphysics IV 2, 1003b6: ousiai – pathē ousias ; see also Metaphysics XIV 2, 1089 b 23: ousiai – pathē – pros ti . EXCERPT #YE4XL8 p. 9 15 Aquinas refers in this connection to different degrees of being, with God at the highest degree. EXCERPT #Y9JR9H p. 9 EXCERPT #ESWENC p. 10 EXCERPT #X96ZBN p. 10 EXCERPT #382DK5 p. 10 EXCERPT #5A5P4D p. 10 Quantities and qualities specifically depend on the one substance they inhere in; they cannot switch their bearers. This many-one pattern (many accidents in one substance) can be modified in various ways, as we will see in the next subsection. EXCERPT #U8LL5E p. 10 Second, there are relational entities, which are ontologically dependent on two or more bearers, possibly at the same time (see Smith et al. , 2006). Relational accidents – such as being owner of or being in the marketplace – also show that not all things that are ontologically dependent on a certain entity do in fact inhere in that entity. Relational processes, such as kisses or hits involving two persons, are ontologically dependent upon each of their relata taken singly, but they inhere not in their relata taken singly, but in the totality which these relata form. It is also possible for two or more entities to be mutually ontologically dependent. There can only be a timbre of a musical tone, for example, if there is also a pitch and a loudness (Smith 1997). This also applies for reciprocal relational accidents. There can only be a husband if there is a wife, there can only be an employer if there are employees, and in each case vice versa . SECTION #T27VSK 3.3 Generic Dependence EXCERPT #G65AZM p. 10 Accidents are in every case dependent on the specific substance or substances in which they inhere. This variety of dependence is often called ‘specific dependence’. Not all dependence relations are of this kind. Remember that, according to Aristotle, universals depend on their instances. They do not, however, depend on any one specific instance. It suffices for the universal human to exist that any instance of human exists, and in fact there is no instance of this species that exists now and already existed in Aristotle’s time. EXCERPT #WFPQWG p. 10 Similarly, being a doctor is not dependent upon the existence of any particular individual patient; any patient at all would be sufficient. By the same token, the existence of a patient does not end when there is no longer any doctor treating him. Only if there are no more doctors at all would there be no more patients. And while being an employer requires having at least one employee, the employees can vary over time. This variety of dependence is normally called ‘generic dependence’. Universals are thus generically dependent on their instances. Doctors are generically dependent on their patients, employers on their employees, and so on. This relation can be viewed also at the level of universals, so that, for two universals F and G , being F is generically dependent upon being G if and only if nothing can be F unless something is G . EXCERPT #9T7UEK p. 10 Taken together with Aristotle’s idea that the existence of a universal depends on the existence of at least one instance thereof, we can then carry this idea over to the instances of universals, and define: EXCERPT #92X9SC p. 10 EXCERPT #X5BFAA p. 11 EXCERPT #AK79MH p. 11 EXCERPT #P7YG6U p. 11 EXCERPT #JXWUUH p. 11 An instance of the universal F is generally dependent on the universal G =def. an instance of F cannot exist unless there exists some instance of G . EXCERPT #PE4PQA p. 11 This kind of dependence is important for a wide range of phenomena. Organisms are generically (but not specifically) dependent on their constituent cells and molecules. Literary texts cannot exist without their exemplars. A poem can be written or printed on paper, stored on a computer, learned by heart, or stored as the recording of a recitation. The poem exists as soon and as long as one of these ‘concretizations’ of it exists, but it does not need to be the same concretization for the entire duration of its existence (Arp, Smith & Spear 2015, 105–107, following Ingarden 1974). SECTION #T97WK5 4. Continuants and Occurrents SECTION #SC7ZP5 4.1 Time and Existence EXCERPT #D5JD5J p. 11 There is another way in which Aristotle’s list of categories can be divided into two sub-groups. Note, first, that, where a substance such as a bacterium, a quantity such as a length of 20 meters, or a quality such as an instance of redness, exists in toto at every point in time at which it exists at all, actions and passions are as it were spread out over the course of some time interval. Whenever we encounter a bacterium, we encounter the whole bacterium – and this is so at each point in time over the course of the bacterium’s life. The process (action) by which a bacterium reproduces, by contrast, or by which it moves through a medium, takes place in time and is manifested over a time span. The process of bacterial reproduction has a beginning and an end; it is composed of various phases that follow one another in time. Instances of process universals like reproduction and movement have temporal parts. By contrast, the bacterium itself has spatial parts – for example, a nucleus, a membrane, a cytoplasm – each of which exists as one and the same entity through time, even while potentially gaining and losing qualities, and even while gaining and losing parts. EXCERPT #X6Z9MC p. 11 Hence, we see that there are two kinds of entities. First, there are for example organisms, which continue to exist through time, and for this reason they are called continuants . Next to an organism, however, there is its life or history . It and its successive phases occur in time , and for this reason they are called occurrents . The organism itself is present as a whole at every time at which it exists. For the organism’s life, in contrast, there is no time it is wholly present. Rather, it unfolds itself in successive phases. EXCERPT #54HV8M p. 11 The words ‘continuant’ and ‘occurrent’ can be traced back to the Cambridge logician William Johnson. Johnson defines ‘continuant’ as ‘that which continues to exist while its states or relations may be changing’ (1921, 199). More recently, David Lewis (1986, 202) drew a similar distinction between endurers and perdurers : EXCERPT #BGL97J p. 11 EXCERPT #ZH7SW4 p. 12 EXCERPT #3BX74Z p. 12 EXCERPT #L8SWZA p. 12 EXCERPT #X2U7GH p. 12 Something perdures iff it persists by having different temporal parts, or stages, at different times, though no one part of it is wholly present at more than one time; whereas it endures iff it persists by being wholly present at more than one time. EXCERPT #2YBAJ6 p. 12 Distinguishing between these two modes of existence is often seen as marking a distinction between two competing theories of the ontology of reality, referred to as endurantism and perdurantism, respectively (Donnelly 2011). Lewis, for example, is a perdurantist. He held that all entities are four-dimensional occurrents, 16 a view which is accordingly often referred to as four-dimensionalism, reflecting the fact that occurrents are seen as occupying four-dimensional chunks of spacetime. There is no David Lewis, on the perdurantist view, but rather a process of davidlewis filling out a certain spacetime region. EXCERPT #CRQXDG p. 12 Other philosophers, in contrast, follow Aristotle – and common sense – in embracing a view according to which there are two very different modes of existence exemplified by the bacterium on the one hand and its movement on the other (McCall & Lowe 2009). We, too, hold a view of this sort – namely that we need to acknowledge both continuants and occurrents in order to achieve an accurate representation of reality. EXCERPT #TA62SS p. 12 However, the opposition between entities which continue to exist over time and entities whose existence is spread across successive regions of time does not present an exhaustive classification. This is because it captures only those entities whose existence is, in fact, extended in one or other way over multiple points in time. But there are in addition also what we can think of as temporal boundary entities, including instants of time, on the one hand, and also instantaneously existing qualities and quantities (see Johansson, 2005). If, for example, the temperature of a body is increasing continuously from 2° to 3°C across a certain stretch of time, then it has at just one time point somewhere in the middle the instantaneous quality of exactly 2.5°C. If a tumour grows continuously during its growth process and maintains constant density, then there are no two points at which the tumour has the same weight. If a surface changes its colour continuously from, say, blue to red, then there are no two points in time at which this surface has the same colour. EXCERPT #SJUD4G p. 12 Temporal instants and the temporal boundaries of processes are limit cases of occurrent and are therefore included in the occurrent class. And we must similarly define ‘continuant’ in such a way as to comprehend also instantaneous existents in the realm of specifically dependent continuants, for example the instantaneous quality of 2.5°C in the body temperature example above. EXCERPT #6BXF5W p. 12 16 For an overview of this discussion, see for example Lowe, 2002, 49–58. EXCERPT #43AZ4H p. 12 EXCERPT #4YZ58X p. 13 EXCERPT #B3EFUM p. 13 EXCERPT #KMK7GF p. 13 SECTION #HNHSLF 4.2 SNAP and SPAN EXCERPT #QQXLXP p. 13 If we picture the world at any single point in time, we will discover in our picture planets, people, animals, artifacts, colours, sizes, and relations. But changes, processes, events and happenings that are taking place at that point in time will not be visible in the picture. In order to represent the latter, we need a sequence of pictures; we need something like a film. In order to obtain an all-inclusive picture of our ever-changing world, we thus need two kinds of representation. On the one hand, in order to capture the continuants, we need snapshots of the world at particular points in time. We might call such snapshots SNAP ontologies (following Grenon and Smith, 2004). Included among SNAP entities are substances, quantities, qualities, relations and positions. It will include also the boundaries of substances, collections of substances, spatial regions such as points, lines, surfaces, and spatial volumes, as well as places such as niches and holes, and also the environments in which substances are to be found (Smith 2001). Over and above the traditional category of continuants, SNAP ontologies comprise also the merely instantaneously existing instances of qualities and quantities which would otherwise be ontologically homeless. EXCERPT #78PQP9 p. 13 On the other hand, we need a representation of change, something like a film which represents entire time spans. Grenon and Smith (2004) called these representations SPAN ontologies . Included among SPAN entities are processes and temporal regions, as well as time instants and instantaneous process boundaries which serve as their boundaries. Spatiotemporal regions and the boundaries of such regions are also included, since they too exist along the temporal dimension. Boundaries of processes include for example the beginning and ending of a race, the beginning and ending of a millennium, or the beginning and ending of your life as a 2-year old. SECTION #L5HS8A 5. Putting It All Together SECTION #C8ATRQ 5.1 The Ontological Square EXCERPT #K6EMKY p. 13 Joining together the universal–particular dichotomy and the dichotomy between inhering and non-inhering entities yields a fourfold distinction of entities represented by the so-called ontological square (Figure 4), which captures the core structure of Aristotle’s early ontology. 17 We can think of both Aristotle’s list of categories and the ontological square as transparent partitions of reality (Bittner & Smith 2001). That is, if we look through (as it were) the respective cells in these partitions, then it is as if we can see the corresponding entities in each cell. EXCERPT #3ERY6E p. 13 17 See Smith, 2003a. On the history of such diagrams see Angelelli, 1967, 12; see also Wachter 2000, 149. An alternative interpretation, drawing on Categories 2 is given in Jansen 2014/15, where the ontological square is seen as combining the universal–particular dichotomy with the concrete–abstract dichotomy. EXCERPT #HVUFXA p. 13 EXCERPT #7HJUBL p. 14 EXCERPT #S6TGH4 p. 14 EXCERPT #VYDYS2 p. 14 EXCERPT #EGMVDA p. 14 substantial (not in a subject) accidental, non-substantial (in a subject) universal (predicated of a subject) III. substance universals human being horse IV. accident universals being white knowing particular (not predicated of a subject) I. individual substances this human being this horse II. individual accidents this individual whiteness this individual knowing EXCERPT #P8FD9M p. 14 Figure 4: Aristotle's Ontological Square EXCERPT #5Y394C p. 14 One of the most important contemporary exponents of the idea of a four-category ontology is E. J. Lowe (2006), whose account in his The Four-Category Ontology provides an overhaul of the ontological square designed to match the needs of contemporary philosophers but still very much in Aristotle's spirit. On the other hand, many contemporary philosophers reject some of the fields recognized by proponents of the ontological square. These rejections take a number of different forms, including: EXCERPT #A896Z3 p. 14 • Nominalist philosophers accept only particulars, i.e., only entities from the two lower fields, I and II. Some nominalist philosophers even try to make do with only one of these two categories. For example, the so-called tropists accept only the existence of particular accidents in field II, which they call 'tropes'. This would be a view close to one in which the world consists exclusively of accident instances, a view under which individual substances such as you and me are viewed as more or less loosely connected bundles of such tropes. This view has been defended, for example, by Donald C. Williams (1953, 2018) and Keith Campbell (1990). • Plato, in contrast, ascribed real being only to universals, i.e., to the entities in the two upper fields, III. And IV. A modern defender of such a position was Bertrand Russell, EXCERPT #3C6N6M p. 14 EXCERPT #V5AYDM p. 15 EXCERPT #V99UJ2 p. 15 EXCERPT #BRLE3B p. 15 EXCERPT #DVT8RW p. 15 who wanted to eliminate the level of individuals, 18 most likely under the influence of Leibniz's theory of individual concepts. 19 EXCERPT #A4KK9J p. 15 • Very many twentieth-century philosophers embraced a view amounting to the acceptance of only cells I. and IV. This is because they see the language of First-Order Logic as their tool for understanding reality, as though they regard its syntax as providing a mirror of reality. The particulars in cell I. correspond, on this account, to the individual constants (' a ', ' b ', ' c ' ...), and the property universals in cell IV. to the predicate variables (' F ', ' G ', ' R ' ...). The idea that the formula ' F(a) ' is the key to ontology has been dubbed fantology by Smith (2005a). Representatives of fantology include the Wittgenstein of the Tractatus (Wittgenstein 1960) and an elaborated version of this view, traces of which can be found in the works of almost all of the principal figures of 20th-century analytic philosophy, in the work of David Armstrong, who accepts only particular substances and what he calls property universals (Armstrong, 1978 and 1997). • As concerns the ways in which analytic philosophers treat entities in what we are calling cell IV., different accounts exist as to whether terms such as 'horse' or 'human being' refer (i) to genuine entities (e.g., Bigelow and Leckey 2022), or (2) to one or other logical or set-theoretic constructions or to concepts in people's minds (e.g., Goodman and Leonhard 1940), or (3) such terms are merely façons de parler and so lacking in referents of any sort (e.g., Quine 1964). EXCERPT #JHQPST p. 15 Ontologists who want to eliminate one or more of the fields of the ontological square represent one or other kind of reductionist position. They are required to produce an alternative explanation for why we suppose in our everyday understanding that these things – people, their lives, the species mus musculus , the colour red, the number 2, the redness of this apple – exist. They do this mainly through explaining our reference to entities in these fields as merely a roundabout way of talking about entities in other, philosophically more highly favoured, fields. SECTION #F997NM 5.2 Adding Processes: The Ontological Sextet EXCERPT #TN4GYR p. 15 But can Aristotle truly fit all his categories into the ontological square? In particular, does he deal adequately with the role of processes in his category system? He does list 'doing' EXCERPT #EJDJFG p. 15 18 See for example Russell, 1940, ch. 6; and 1948, Part II, ch. 3 und Part IV ch. 8; 1959, ch. 9. For a similar position see Hochberg 1965, 1966, and 1969. EXCERPT #7CT7EG p. 15 19 Russell (1948) attributes this conception explicitly to Leibniz. See also Armstrong, 1978, I 89: "while the influence of Leibniz on Russell is clear, it is less clear that Leibniz held this theory of the nature of particulars." EXCERPT #ZQA884 p. 15 EXCERPT #EPM68W p. 16 EXCERPT #E5ZQF3 p. 16 EXCERPT #PZ7E3Z p. 16 EXCERPT #DQGJR7 p. 16 (action) and ‘suffering’ (passion) among his ten categories, and seems to understand them as the active and passive sides of a change ( kinesis , De anima III 2, 426a2, cf. Physics III 3, 202a13 –21). Moreover, he refers elsewhere to processes that are not associated with change, as for example in Metaphysics IX 6, where he calls these non-change processes energeiai . It is difficult to say where processes such as seeing or living are to be put in the list of ten categories. It seems, indeed, that Aristotle resists the idea that processes should be treated as fundamental entities. EXCERPT #EBUSET p. 16 On the side of mainstream analytic philosophy, Donald Davidson appears against this background as something of a hero, thanks to his argument in favour of the need for what he called an ‘ontology of events’ (Davidson 1970, 1980). He starts with a problem we face in understanding the semantics of a sentence such as ‘John buttered the toast slowly’. Here the ‘slowly’ requires an entity with which the corresponding adverbial characteristic could be associated. This is why Davidson analyses the sentence as predicating something of a certain event, namely, a buttering event that is slow. Davidson hereby breaks out from the radically simplifying fantologically reduced ontological square by admitting a cell devoted to events as particulars. EXCERPT #2JSE2L p. 16 A picture of the world which did not provide a special place for occurrents would indeed be incomplete. We need to account for the dynamic, processual character of the world, and for this reason we need to add occurrents, including not only processes but also temporal intervals in our ontology. There are also of course important relations that obtain between occurrents and continuants, for example individual substances participate in individual processes. There are also important kinds of processes, for example nuclear decay or apoptosis, which have all of the features of universals in the continuant sphere. All of which suggests that we expand the ontological square to an ontological sextet , as illustrated in Figure 5 (Smith, 2005a). EXCERPT #L9N8YC p. 16 EXCERPT #69DW37 p. 17 EXCERPT #T89QF5 p. 17 EXCERPT #4Q5RZ6 p. 17 EXCERPT #K3K3RN p. 17 graph TD SU[Substance Universal] PU[Property Universal] PUS[Process Universal] SP[Substance Particular] IP[Individual Property] IPr[Individual Process] PU -- "predicated of" --> SU SU -- "instantiates" --> SP SP -- "exemplifies" --> PU PU -- "instantiates" --> IP IP -- "inheres in" --> SP PUS -- "instantiates" --> IPr IPr -- "participates in" --> SP Figure 5: The Ontological Sextet and Associated Formal-ontological Relations. A diagram showing six entities in a 2x3 grid. Top row: Substance Universal, Property Universal, Process Universal. Bottom row: Substance Particular, Individual Property, Individual Process. Relations: 'predicated of' from Property Universal to Substance Universal; 'instantiates' from Substance Universal to Substance Particular; 'exemplifies' from Substance Particular to Property Universal; 'instantiates' from Property Universal to Individual Property; 'inheres in' from Individual Property to Substance Particular; 'instantiates' from Process Universal to Individual Process; 'participates in' from Individual Process to Substance Particular. EXCERPT #3URGS7 p. 17 Figure 5: The Ontological Sextet and Associated Formal-ontological Relations SECTION #CTHD63 5.3 Ontological relations EXCERPT #9SKJDB p. 17 The discussion of ontological dependence, the ontological square and the ontological sextet already revealed that there is a number of relations that are of utmost relevance for ontology. First and foremost, these are the basic relations that obtain among entities in the six fields of Figure 5, namely: EXCERPT #2JRGVP p. 17 • individual accidents inhere in individual substances. • accident universals are predicated of substance universals. • individual substances instantiate substance universals. • individual accidents instantiate accident universals. • individual substances exemplify accident universals. EXCERPT #7SP7SE p. 17 The relations of inheritance, exemplification, instantiation, and participation govern the relations among the entities in the four fields of the ontological square. If we look at the ontological sextet, we can add relations to deal with processes ('events' in Davidson's terminology) drawing on the participates_in relation, as follows: EXCERPT #PUJ8AL p. 17 • individual substances participate in processes. EXCERPT #BMT6NV p. 17 EXCERPT #HUUEQ5 p. 18 EXCERPT #6FJVE3 p. 18 EXCERPT #E6DL2H p. 18 EXCERPT #CFU5J6 p. 18 • individual processes instantiate process universals. EXCERPT #ZMV78W p. 18 One important feature of ontological relations is their generality. Regardless of which area of reality we want to represent, we must take relations like inherence and instantiation into account. In all domains that display changes over time, the relation of participation will be needed. EXCERPT #9CDU9Y p. 18 A second feature of ontological relations is their ‘formal’ nature. This means that these relations hold in a way that does not involve any additional ‘matter’ in the world. Rather, they hold simply because of the very ontological nature of their relata. For more familiar ‘material’ relations something different holds. If Mary is in love with Peter, this is not simply because there are Mary and Peter, but because there is this third thing, namely, Mary’s love for Peter. Similarly, if Peter and Mary are married, this is because there was this processual entity, their marriage, that made them a married couple. It is because of their generality that ontological relations are so important for applied ontology. Because they apply in virtually all domains of reality, much work has been done in recent decades to formalize and standardize the representation of such relations, which are being re-used in myriad ontology applications. SECTION #K2VRML 6. Complex Entities EXCERPT #XJTQWL p. 18 In addition to the categories we have discussed thus far, contemporary ontologists have considered other candidate categories, such as states of affairs, sets and classes, and mereological sums. As they are regularly referred to in writings on applied ontology, we introduce them briefly here. SECTION #7AVA8H 6.1 States of Affairs EXCERPT #2NTMAA p. 18 The idea of states of affairs (in Latin status rerum , German Sachverhalte ) has a long history, starting out from its usage in clinical trials (Smith 1992), but it reached the apogee of its influence in the era of Lotze (Milkov 2023), and then of Lotze’s student Carl Stumpf (Chrudzimski 2015), who, in turn, inspired Husserl and through him Adolf Reinach. Husserl’s and Reinach’s work on states of affairs then runs in (partial) parallel with Wittgenstein’s thinking on this topic as laid down in his Tractatus (Smith 1978). Since then, interest in the idea has waned, having been pushed aside, by set-theoretic approaches to semantics on the one hand, and by truthmaker approaches on the other (see for example Faroldi & Van De Putte 2023). EXCERPT #WMHZTR p. 18 States of affairs are complex entities that can be represented in normal language by means of ‘that’ clauses such as that the ball is green or that the cat is on the mat . The state of affairs that John is sick is a complex entity composed of a substance (this person), and a certain quality or disposition (the sickness). The state of affairs that a certain molecule is attached to a receptor is composed of a substance (the molecule), a part of a substance (the receptor), and the two-place relation of being attached. EXCERPT #H84UBT p. 18 EXCERPT #ZXFCHD p. 19 EXCERPT #NQSYEB p. 19 EXCERPT #R36NPZ p. 19 EXCERPT #6GKE8E p. 19 One problem with attempts to specify the ontology of states of affairs turns on an apparent redundancy which arises from the fact that, if we are representing (for example) the aforementioned ball, then we are already representing something that is green. If we now represent the state of affairs that the ball is green , and if we assume that this state of affairs includes all that exists on the side of reality that is salient to (part of the truthmaker for) our representation, then the ball's greenness quality – in contrast to its other qualities of being round, weighing 4 ounces, being made of plastic – would seem to figure twice in the state of affairs that we are representing. It figures once as it were coiled up inside the object alongside all its other qualities, and then again as somehow uncoiled, or made in some sense explicit, by our representation (Ingarden 1965, II/1 §§ 39–42; Smith 1978). This seems, however, to add an unfortunate epistemological or cognitive or language-dependent dimension 20 to the idea of a state of affairs, which makes such entities not strictly ontological. Part of the background behind the idea of 'truthmakers' (Mulligan, Simons & Smith 1984) was the aim of providing the means to solve this problem. EXCERPT #6ZNJRT p. 19 On the other hand there are states of affairs where an epistemological dimension is ontologically in order – where the epistemological dimension is part of the objective reality which makes the corresponding assertion true. Consider the state of affairs that the doctor believes that her patient has the flu . This is composed of the doctor and the intentional relation of believing, and (if the doctor has a true belief) of the further state of affairs that the patient has the flu . EXCERPT #TJTDPW p. 19 From the perspective of applied ontology we can now identify at least part of the reason for the contemporary disregard of states of affairs as lying in the difficulty we face if we attempt to construct logically coherent taxonomies of states of affairs in the face of the sorts of combinatorial explosions we encounter in taking account of intentional relations such as this. SECTION #78RVGJ 6.2 Sets EXCERPT #PFB668 p. 19 Sets are well known from mathematics, where the term (in German ' Menge ') was introduced by Cantor in 1883 to refer to 'a collection M of definite, well-differentiated EXCERPT #JS9CYT p. 19 20 Compare Strawson (1950): "If you prise the statements off the world you prise the facts off it too; but the world would be none the poorer." EXCERPT #57E39J p. 19 EXCERPT #H8TSJH p. 20 EXCERPT #YPKCHM p. 20 EXCERPT #CY9NZV p. 20 EXCERPT #GTUUKT p. 20 objects m [...] into a whole'. 21 Mathematical views of sets proceed axiomatically, and a number of different axiomatic set theories have been devised. We are interested here, however, in how set theory in general might help us in understanding the sorts of classification or categorization of real-world entities in which ontologists are interested. Does it make sense, for example, to try to understand Aristotle's list of categories, or the ontological square, in set-theoretic terms? Does it make sense to understand real-world phenomena such as a horse race or a ride on the L Train in terms of set theory? Can even a stamp collection be correctly described as a set of stamps? EXCERPT #UWZP25 p. 20 To answer this question, we note that sets can be represented either by simply listing their elements or by pointing to a common feature of these elements. As examples of the first, we can write '{2, 3, 5, 7}' or '{Aristotle, 2, my stethoscope}', reflecting the fact that sets can be built out of unrelated elements. As an example of the second, we can write 'set of prime numbers less than 10'. Here we represent a set by specifying certain necessary and sufficient conditions for membership. Further examples would be 'the set of all patients in Leipzig at noon on November 1, 2008', or 'the set of all such patients with a fever'. EXCERPT #7N3TX8 p. 20 Sets are identical if and only if they contain the same elements. From this it follows that sets are in a certain sense timeless; hence: sets can include elements which exist at different times and at no times. They are also outside space (if the elements of a set move about in space the set is not affected in any way). Sets are in all of these respects distinguished from other collections – for example, my stamp collection or your collection of cancer tissue samples – in that its members are fixed, though this feature that is not clearly captured in Cantor's definition). This means that sets are something abstract; they live outside the world of what happens and is the case. EXCERPT #LS9K5S p. 20 In his "Against Set Theory", Peter Simons quotes a number of authorities in set theory who have, as he says, pulled the wool over the eyes of philosophers and others by presenting sets as something wholly natural and uncontroversial: EXCERPT #M6MYCA p. 20 The technique, usually applied on or about page 1 of a textbook of set theory, is to claim that we are already familiar with sets under some other names or guises, and then trade on this supposed familiarity to sell us a bill of fare which is ontologically far from neutral and far from benign. EXCERPT #2ADQMM p. 20 Simons quotes several authorities from mathematics and logics: EXCERPT #58A5BZ p. 20 21 Translation taken from Oliver & Smiley 2018. The German original (Cantor 1895, 481 and 1932, 282) reads: "Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterscheidbaren Objecten m [...] zu einem Ganzen." EXCERPT #N85QXV p. 20 EXCERPT #UCM7YG p. 21 EXCERPT #Z9BZYY p. 21 EXCERPT #9M2QA4 p. 21 EXCERPT #MK6QXZ p. 21 Consider a collection of concrete objects, for instance of the apples, oranges etc. in a fruit shop. We may call it a set of fruit, the individual apples etc. being the members (or elements) of the set. Conceiving the collection as a new single concept is an elementary intellectual act. (Fraenkel 1953, 4) EXCERPT #GLWD4D p. 21 Or: EXCERPT #CP8GAW p. 21 In our examples, sets consisted of concrete and familiar objects, but once we have sets, we can form sets of sets, for example the set of all football teams. (Van Dalen, Doets and de Swart 1978, 1) EXCERPT #22A7LJ p. 21 So, as Simons goes on, “one can buy a set from a fruiterer, or we can buy a set called ‘Manchester United’ or ‘Juventus’” (Simons 2005). Because they exist outside space and time, however, sets are peculiar entities, in ways which would forestall their use for the sorts of ontological purposes pursued by Aristotle or Lowe or by contemporary applied ontologists. SECTION #5QFVKE 6.3 Mereology: Wholes and Their Parts EXCERPT #L2NEMN p. 21 The mentioned problems have induced some logicians and philosophers to develop an alternative to the set-theoretic approach under the heading of ‘mereology’ 22 , and like set theory, mereology has been subjected to a number of different axiomatizations (Simons 1992; Varzi & Cotnoir 2021). EXCERPT #6RWWF7 p. 21 In the world of sets, every element is just what it is, and has the granularity that it has – whether this be of molecules, of cells, of whole organisms or of entire populations. Mereological sums, in contrast, are concrete entities that can be partitioned on various levels of granularity. Each human being is an organism, and a mereological sum of cells, and a mereological sum of molecules – and all of these are at any given time identical. Moreover, in mereology – as contrasted with set theory – there is no requirement that an ultimate bottom layer of mereological simples needs to be specified or accepted (Smith & Brogaard 2002). EXCERPT #SPJP8W p. 21 My stomach, my sandwich, and the Midwestern US states can each comprise such a mereological sum. Just as with sets, there is, on some formalizations, virtually no limitation to the building of mereological sums. And just as with sets, many mereological sums (such as the sum of Napoleon and the pebbles in this bowl) have an artificial character, though some have sought axiomatic ways of restricting mereology in such a EXCERPT #HR4H8Y p. 21 22 Simons, 1987; Ridder, 2002; see also Husserl’s third Logical Investigation , “On the theory of wholes and parts” (Husserl 1970). EXCERPT #QPSG6R p. 21 EXCERPT #RG9MBL p. 22 EXCERPT #W4KRRF p. 22 EXCERPT #25X3PS p. 22 EXCERPT #4MNMTE p. 22 way as to allow credence only to what we can think of as ‘natural wholes’ (see Simons 2006). EXCERPT #4424YK p. 22 While sets are abstract entities even when composed of concrete elements, mereological sums composed of concrete elements are themselves concrete. Mereological sums exist in space and time, but only – on most formalizations – for so long as all of their parts exist. Like membership in set theory, parthood is temporally rigid in classical mereology: A mereological sum does not survive the loss or destruction of even one of its parts. Gaining or losing a part will result in another mereological sum. EXCERPT #HY5HYB p. 22 In many ontologies, part-whole relations are used as formal-ontological relations. The theory of granular partitions (Bittner & Smith 2001, 2003) introduces an approach which attempts to blaze a third trail between set theory and mereology by linking the concreteness of mereological sums with the hierarchical nature of the element-of relation. SECTION #CCUHN7 6.4 Classes EXCERPT #JKMKQ2 p. 22 Although the words ‘set’ and ‘class’ are often used as synonyms, we will here use them to signify different things. In many standard mathematical treatments, sets can be composed arbitrarily by placing singular terms between curly brackets as in: {Aristotle, the year 1969, New York}. But there are also sets that are defined by means of a uniform property, or a conjunction of such properties, as in: like the class of all things that are red, or the class of all humans, or the class of all electric charges. We will reserve the term ‘class’ for collectives of the latter sort. EXCERPT #9GTMDP p. 22 This is the approach followed by Smith and Ceusters (2006, 60) for whom ‘class’ signifies ‘a collection of all and only the particulars to which a given general term applies’. When the general term connected to a class represents a universal, we can speak of a natural class , which is the totality of instances of a universal. Where sets may be constructed by enumeration of their members, natural classes require that there be universals of which they are the extension. Two natural classes are identical if their defining general term represents the same universal. Because not all general expressions correspond to universals, not all classes are natural classes. The non-natural classes are called ‘defined classes’, as for example: the class of diabetics in Paris on a certain day, or the class of Italian restaurants in San Diego. EXCERPT #9GN24P p. 22 Where sets can have members of arbitrarily different sorts, ‘class’ on this reading refers to collections of members which are in some sense constrained, as for example in: the class of mammals, the class of red things, the class of electrons. The account of classes suggested here thus attempts to mitigate the arbitrary features of set construction. 23 EXCERPT #4CSHA9 p. 22 EXCERPT #XV9D9N p. 23 EXCERPT #HW3QYB p. 23 EXCERPT #VG39BQ p. 23 EXCERPT #SMKM8W p. 23 Unlike what is the case in set theory, class theory does not require us to know what things there are in the world in order to say, for example, that the class of red things and the class of round things are different from one another. EXCERPT #SFNX9H p. 23 In addition, natural classes, but not sets, can survive the destruction or coming into existence of new instances; for sets are individuated by their elements, where natural classes are individuated by a (possible) universal which stays the same even as it has different instances at different times. 24 EXCERPT #JYSAX8 p. 23 The result of dividing entities into classes is called a classification . Instead of speaking of a class we sometimes speak of a taxon , which is derived from the Greek tattein meaning: to place in order, which form in turn the nodes of a taxonomy . A taxonomy must be distinguished from a partonomy . While a classification or a taxonomy divides a universal into species or kinds, a partonomy divides a whole into its parts. SECTION #48D2BK 7. Top-Level Ontologies Today EXCERPT #LUNL5W p. 23 What should an ontology look like at the highest level? In this essay, we used Aristotle's Categories as a guideline for our understanding the development of crucial ontological distinctions that underlie many modern top-level ontologies, including BFO. We already pointed out that Aristotle focusses mainly on continuants and does not really develop the analysis of occurrents. BFO addresses this problem by enriching the ontological square to form the ontological sextet. EXCERPT #YRWGR9 p. 23 There is another respect in which Aristotle's analysis should be supplemented. As presented in the foregoing, Aristotle's theory of categories conforms to a high degree with our common-sense understanding of reality. It shares with common sense above all a view of the world as of a single granularity – the granularity of organisms and their qualities – where our contemporary scientific understanding requires a multi-granular approach, incorporating cells and cell components, molecules, atoms, electrons, and so forth, as well as planets, stars, galaxies, black holes, and other entities dealt with by cosmology. EXCERPT #A5U9YD p. 23 23 There are earlier attempts to link intensional elements with set theory; for example, in Feibleman, 1974. The remarks presented here draw on Johansson 2006. See also Smith 2005b and Smith et al. 2005. EXCERPT #MG2EAP p. 23 24 We here leave open the question as to how one might deal with the natural class corresponding to the universal dodo . EXCERPT #2Z79HP p. 23 EXCERPT #GFCNL5 p. 24 EXCERPT #84RU88 p. 24 EXCERPT #9JEQUW p. 24 EXCERPT #P3ERPC p. 24 Reality, on this approach, appears as a complex hierarchy of levels that are nested within each other. Molecules are embedded in the interior of cells, cells in leaves, leaves in trees, trees in forests and so on (Smith 2001). As our everyday perceptions and actions are tuned to the entities appearing on the level of the common-sense world, so various sciences are tuned to entities on other levels within this complex hierarchy. For example, there is not only macroscopic anatomy, with offshoots such as clinical, surgical and radiological anatomy, but also microscopic anatomy, with sub-disciplines such as histology, cytology and secondary branches such as anatomical embryology, anatomical genomics, neuroanatomy, and so on. Astronomy, similarly, incorporates multiple disciplines such as planetary science, stellar astronomy, galactic astronomy, radio astronomy, and infrared astronomy, which focus on different aspects of the cosmos at different levels of granularity. EXCERPT #9A66DM p. 24 Basic Formal Ontology shares many of the features of the Aristotelian ontology set forth above, but as is made clear throughout the BFO handbook (Arp, Smith and Spear 2015), BFO embraces the principle of perspectivalism, whereby the BFO ontology can be implemented in association with domain ontologies at many different levels of granularity. For example, the BFO class object might comprehend cells and cell components in one implementation, and planets and their satellites in another. In this way BFO can serve as an ontology that simultaneously supports both common-sense and scientific realism, doctrines which are otherwise seen as being incompatible. EXCERPT #7UQEDS p. 24 BFO and its many users thereby provide support for the case that the progress of science is not a step away from Aristotle towards something better, any more than quantum physics is a step away from classical Newtonian physics. Rather, just as quantum physics incorporates the physics of Newton as a limit case, so contemporary science as a whole incorporates many features of Aristotle's ontological approach. SECTION #NHHNAK References EXCERPT #8PKFLB p. 24 Angelelli, Ignazio. Studies on Gottlob Frege and Traditional Philosophy. Dordrecht: Reidel, 1967. Aquinas. In octo libros Physicorum Aristotelis exposition, ed. Maggiolo, Turin-Rom: 1965. Aristotle. Categoriae , transl. by E.M. Edghill, in: The Works of Aristotle, ed. W. D. Ross, vol. 1, Oxford: Oxford University Press, 1928, repr. 1994. Aristotle. The Complete Works of Aristotle (Revised Oxford translation). ed. Jonathan Barnes. Princeton, 1984. EXCERPT #LGVQTE p. 24 EXCERPT #L5LUEH p. 25 EXCERPT #DAXQGP p. 25 EXCERPT #2GVW8G p. 25 EXCERPT #HYHK7B p. 25 Austin, John L. “Truth”, Proceedings of the Aristotelian Society, Supplementary Volumes 24 (1950), 111–128. Chrudzinski, Arkadiusz. “Carl Stumpf über Sachverhalte”. In: Denis Fisette & Riccardo Martinelli (eds.), Philosophy from an Empirical Standpoint: Essays on Carl Stumpf . Rodopi 2015. Armstrong, David M. Nominalism and Realism: Universals and Scientific Realism, Volume I . Cambridge 1978. Armstrong, David M. A World of States of Affairs . Cambridge: Cambridge University Press, 1997. Arp, Robert; Smith, Barry; Spear, Andrew D. Building Ontologies with Basic Formal Ontology , Cambridge, MA: MIT Press, 2015. BFO, Basic Formal Ontology, http://basic-formal-ontology.org/ . Bigelow, John; Leckey, Martin, “Two Kinds of Platonism and Categorical Semantics”. In: Helen Beebe, A. R. J. Fisher (eds), Perspectives on the Philosophy of David K. Lewis , Oxford: Oxford University Press, 2022, 154–173. Bittner, Thomas, and Barry Smith. "A taxonomy of granular partitions." International Conference on Spatial Information Theory , Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. 28–43. Bittner, Thomas and Barry Smith, A Theory of Granular Partitions , Foundations of Geographic Information Science , M. Duckham, M. F. Goodchild and M. F. Worboys, eds., London: Taylor & Francis Books, 2003, 117–151 Brentano, Franz. Von der mannigfachen Bedeutung des Seienden nach Aristoteles . Freiburg/Brsg: 1862. Bucher, Thomas. Einführung in die angewandte Logik , 2nd edition, Berlin 1998 Campbell, Keith. Abstract Particulars . Oxford: Basil Blackwell, 1990. Cantor, Georg. “Beiträge zur Begründung der transfiniten Mengenlehre.” Mathematische Annalen 46 (1895) 481–512. Cantor, Georg, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts , ed. Richard Dedekind, Berlin: Springer 1932. Davidson, Donald. “Events and Particulars”, Noûs , 4 (1), 1970, 25–32 Davidson, Donald. “The Logical Form of Action Sentences.” Essays on Actions and Events . ed. Donald Davidson. Oxford: Clarendon Press, 1980. Donald Davidson, “Events and Particulars”, Noûs 4:1 (1970): 25–32. EXCERPT #K642SS p. 25 EXCERPT #SUSZWC p. 26 EXCERPT #4J2T6L p. 26 EXCERPT #RNAZEL p. 26 EXCERPT #SMDFV4 p. 26 Donnelly, M. Endurantist and perdurantist accounts of persistence. Philosophical Studies 154, 27–51 (2011). https://doi.org/10.1007/s11098-010-9526-z . Faroldi, Federico LG, and Frederik Van De Putte. Kit Fine on Truthmakers, Relevance, and Non-classical Logic . Springer, 2023. Feibleman, James K. “Professor Quine and Real Classes.” Notre Dame Journal of Formal Logic 15 (1974): 207–224. Fraenkel, Abraham A. 1953 Abstract Set Theory , Amsterdam: North-Holland. Frege, Gottlob. “Begriff und Gegenstand.” Breslau: Koebner 1884, in: Vierteljahrschrift für wissenschaftliche Philosophie 16 (1892): 192–205. Frege, Gottlob. The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number . Trans. J. L. Austin. Oxford: Basil Blackwell, 1980. Goodman, Nelson; Leonard, Henry S., “The Calculus of Individuals and Its Uses”, Journal of Symbolic Logic 5 (1940): 45–55. 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Hochberg, Herbert. “Universals, Particulars and Predication.” Review of Metaphysics 19 (1965): 87–102. Husserl, Edmund, Logical investigations . Vol. 2, transl. by J. N. Findlay from the second German edition, London : Routledge & Kegan Paul, 1970. Ingarden, Roman. Der Streit um die Existenz der Welt . Vol. II/1, Niemeyer, Tübingen 1965. English translation: Controversy over the Existence of the World . Vol. II, translated by Arthur Szylewicz, Bern: Peter Lang, 2016. EXCERPT #ZE2YU5 p. 26 EXCERPT #N94FNY p. 27 EXCERPT #VCKCBN p. 27 EXCERPT #UW9529 p. 27 EXCERPT #FKP8AY p. 27 Ingarden, Roman. The Literary Work of Art , Evanston IL: Northwestern University Press 1974. Jansen, Ludger. “Aristoteles’ Kategorie des Relativen zwischen Dialektik und Ontologie.” Philosophiegeschichte und logische Analyse 9 (2006): 79–104. Jansen L, “Aristotle’s Categories”, in: Topoi 26 (2007) 151–158. Jansen, Ludger. “Categories: The top-level ontology”. Applied Ontology. An Introduction , eds. K Munn, B. Smith, Frankfurt: Ontos 2008. Jansen L, “The Fullness of Being. Why Property Nominalism and Physicalism Are Falling Short of the Demands of Philosophy of Religion.” International Journal for Philosophy and Public Affairs 2-3 (2014/15) 17–30. https://www.interjournalphilpubaffairs.com/Contents_Contribu.aspx?cId=1007 . Jansen L, Art. “Substance”, in: Hans Burkhardt, Johanna Seibt, Guido Imaguire (eds.), Handbook of Mereology , München: Philosophia 2017. Johansson, Ingvar. “Four Kinds of ‘Is_A; Relations: Genus-subsumption, Determinable-subsumption, Specification, and Specialization.” Contributions to the Third International Workshop on Philosophy and Informatics, Saarbrücken 2006 (IFOMIS Reports 14) . eds. Ingvar Johansson, Bertin Klein, Thomas Roth-Berghofer. WSPI (2006: 47–62. Johansson, Ingvar. “Qualities, Quantities, and the Endurant-Perdurant Distinction in Top-Level Ontologies.” WSPI ’05. Proceedings of the Second International Workshop on Philosophy and Informatics. 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EXCERPT #BMNPAS p. 27 EXCERPT #6PNYUP p. 28 EXCERPT #C3795F p. 28 EXCERPT #DSSFQL p. 28 EXCERPT #RRJXSN p. 28 Macdonald, Cynthia. "Tropes and Other Things." Contemporary Readings in the Foundations of Metaphysics. eds. Stephen Laurence and Cynthia Macdonald. Oxford: Basil Blackwell, 1998. 329–350. McCall, Storrs and E.J. Lowe. "The Definition of Endurance, Analysis 69 (2009): 277–280. Milkov, Nikolay, Hermann Lotze's Influence on Twentieth Century Philosophy, Berlin: de Gruyter 2023 Mulligan, Kevin, Peter Simons & Barry Smith, "Truth-Makers", Philosophy and Phenomenological Research 44 (3):287–321 (1984) Munn, Katherine, "Introduction: What is Ontology for", in K. Munn and B. Smith (eds.), Applied Ontology: An Introduction, Berlin: de Gruyter, 2008, 7–19. Oliver, Alex and Timothy Smiley, "Cantorian set theory", Bulletin of Symbolic Logic 24 (4):393–451 (2018). Ontology portal, as of August 8, 2006, http://www.ontologyportal.org/ (as of August 8, 2006). Plato. "Phaedrus." 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EXCERPT #53KFLD p. 30 DOCUMENT #88BVY3 Categories in Top-Level Ontologies: Revisiting the Aristotelian Background SECTION #F997NM 5.2 Adding Processes: The Ontological Sextet EXCERPT #TN4GYR p. 15 But can Aristotle truly fit all his categories into the ontological square? In particular, does he deal adequately with the role of processes in his category system? He does list 'doing' EXCERPT #EJDJFG p. 15 18 See for example Russell, 1940, ch. 6; and 1948, Part II, ch. 3 und Part IV ch. 8; 1959, ch. 9. For a similar position see Hochberg 1965, 1966, and 1969. EXCERPT #7CT7EG p. 15 19 Russell (1948) attributes this conception explicitly to Leibniz. See also Armstrong, 1978, I 89: "while the influence of Leibniz on Russell is clear, it is less clear that Leibniz held this theory of the nature of particulars." EXCERPT #ZQA884 p. 15 EXCERPT #EPM68W p. 16 EXCERPT #E5ZQF3 p. 16 EXCERPT #PZ7E3Z p. 16 EXCERPT #DQGJR7 p. 16 (action) and ‘suffering’ (passion) among his ten categories, and seems to understand them as the active and passive sides of a change ( kinesis , De anima III 2, 426a2, cf. Physics III 3, 202a13 –21). Moreover, he refers elsewhere to processes that are not associated with change, as for example in Metaphysics IX 6, where he calls these non-change processes energeiai . It is difficult to say where processes such as seeing or living are to be put in the list of ten categories. It seems, indeed, that Aristotle resists the idea that processes should be treated as fundamental entities. EXCERPT #EBUSET p. 16 On the side of mainstream analytic philosophy, Donald Davidson appears against this background as something of a hero, thanks to his argument in favour of the need for what he called an ‘ontology of events’ (Davidson 1970, 1980). He starts with a problem we face in understanding the semantics of a sentence such as ‘John buttered the toast slowly’. Here the ‘slowly’ requires an entity with which the corresponding adverbial characteristic could be associated. This is why Davidson analyses the sentence as predicating something of a certain event, namely, a buttering event that is slow. Davidson hereby breaks out from the radically simplifying fantologically reduced ontological square by admitting a cell devoted to events as particulars. EXCERPT #2JSE2L p. 16 A picture of the world which did not provide a special place for occurrents would indeed be incomplete. We need to account for the dynamic, processual character of the world, and for this reason we need to add occurrents, including not only processes but also temporal intervals in our ontology. There are also of course important relations that obtain between occurrents and continuants, for example individual substances participate in individual processes. There are also important kinds of processes, for example nuclear decay or apoptosis, which have all of the features of universals in the continuant sphere. All of which suggests that we expand the ontological square to an ontological sextet , as illustrated in Figure 5 (Smith, 2005a). EXCERPT #L9N8YC p. 16 EXCERPT #69DW37 p. 17 EXCERPT #T89QF5 p. 17 EXCERPT #4Q5RZ6 p. 17 EXCERPT #K3K3RN p. 17 graph TD SU[Substance Universal] PU[Property Universal] PUS[Process Universal] SP[Substance Particular] IP[Individual Property] IPr[Individual Process] PU -- "predicated of" --> SU SU -- "instantiates" --> SP SP -- "exemplifies" --> PU PU -- "instantiates" --> IP IP -- "inheres in" --> SP PUS -- "instantiates" --> IPr IPr -- "participates in" --> SP Figure 5: The Ontological Sextet and Associated Formal-ontological Relations. A diagram showing six entities in a 2x3 grid. Top row: Substance Universal, Property Universal, Process Universal. Bottom row: Substance Particular, Individual Property, Individual Process. Relations: 'predicated of' from Property Universal to Substance Universal; 'instantiates' from Substance Universal to Substance Particular; 'exemplifies' from Substance Particular to Property Universal; 'instantiates' from Property Universal to Individual Property; 'inheres in' from Individual Property to Substance Particular; 'instantiates' from Process Universal to Individual Process; 'participates in' from Individual Process to Substance Particular. EXCERPT #3URGS7 p. 17 Figure 5: The Ontological Sextet and Associated Formal-ontological Relations DOCUMENT #88BVY3 Categories in Top-Level Ontologies: Revisiting the Aristotelian Background SECTION #48D2BK 7. Top-Level Ontologies Today EXCERPT #LUNL5W p. 23 What should an ontology look like at the highest level? In this essay, we used Aristotle's Categories as a guideline for our understanding the development of crucial ontological distinctions that underlie many modern top-level ontologies, including BFO. We already pointed out that Aristotle focusses mainly on continuants and does not really develop the analysis of occurrents. BFO addresses this problem by enriching the ontological square to form the ontological sextet. EXCERPT #YRWGR9 p. 23 There is another respect in which Aristotle's analysis should be supplemented. As presented in the foregoing, Aristotle's theory of categories conforms to a high degree with our common-sense understanding of reality. It shares with common sense above all a view of the world as of a single granularity – the granularity of organisms and their qualities – where our contemporary scientific understanding requires a multi-granular approach, incorporating cells and cell components, molecules, atoms, electrons, and so forth, as well as planets, stars, galaxies, black holes, and other entities dealt with by cosmology. EXCERPT #A5U9YD p. 23 23 There are earlier attempts to link intensional elements with set theory; for example, in Feibleman, 1974. The remarks presented here draw on Johansson 2006. See also Smith 2005b and Smith et al. 2005. EXCERPT #MG2EAP p. 23 24 We here leave open the question as to how one might deal with the natural class corresponding to the universal dodo . EXCERPT #2Z79HP p. 23 EXCERPT #GFCNL5 p. 24 EXCERPT #84RU88 p. 24 EXCERPT #9JEQUW p. 24 EXCERPT #P3ERPC p. 24 Reality, on this approach, appears as a complex hierarchy of levels that are nested within each other. Molecules are embedded in the interior of cells, cells in leaves, leaves in trees, trees in forests and so on (Smith 2001). As our everyday perceptions and actions are tuned to the entities appearing on the level of the common-sense world, so various sciences are tuned to entities on other levels within this complex hierarchy. For example, there is not only macroscopic anatomy, with offshoots such as clinical, surgical and radiological anatomy, but also microscopic anatomy, with sub-disciplines such as histology, cytology and secondary branches such as anatomical embryology, anatomical genomics, neuroanatomy, and so on. Astronomy, similarly, incorporates multiple disciplines such as planetary science, stellar astronomy, galactic astronomy, radio astronomy, and infrared astronomy, which focus on different aspects of the cosmos at different levels of granularity. EXCERPT #9A66DM p. 24 Basic Formal Ontology shares many of the features of the Aristotelian ontology set forth above, but as is made clear throughout the BFO handbook (Arp, Smith and Spear 2015), BFO embraces the principle of perspectivalism, whereby the BFO ontology can be implemented in association with domain ontologies at many different levels of granularity. For example, the BFO class object might comprehend cells and cell components in one implementation, and planets and their satellites in another. In this way BFO can serve as an ontology that simultaneously supports both common-sense and scientific realism, doctrines which are otherwise seen as being incompatible. EXCERPT #7UQEDS p. 24 BFO and its many users thereby provide support for the case that the progress of science is not a step away from Aristotle towards something better, any more than quantum physics is a step away from classical Newtonian physics. Rather, just as quantum physics incorporates the physics of Newton as a limit case, so contemporary science as a whole incorporates many features of Aristotle's ontological approach. ### 53. Tool result: read DOCUMENT #GN66WW Ontologies of Common Sense, Physics and Mathematics SECTION #6KMTK8 Ontologies of common sense, physics and mathematics EXCERPT #VXDXJ6 p. 0 Jobst Landgrebe      Barry Smith EXCERPT #25HQHG p. 0 May 3, 2023 EXCERPT #GWLWW5 p. 0 University at Buffalo EXCERPT #38UJ9N p. 0 jobstlan@buffalo.edu, phismith@buffalo.edu EXCERPT #W2DQDT p. 0 The view of nature we adopt in the natural attitude is determined by common sense, without which we could not survive. Classical physics is modelled on this common-sense view of nature, and uses mathematics to formalise our natural understanding of the causes and effects we observe in time and space when we select subsystems of nature for modelling. But in modern physics, we do not go beyond the realm of common sense by augmenting our knowledge of what is going on in nature. Rather, we have measurements that we do not understand, so we know nothing about the ontology of what we measure. We help ourselves by using entities from mathematics, which we fully understand ontologically. But we have no ontology of the reality of modern physics; we have only what we can assert mathematically. In this paper, we describe the ontology of classical and modern physics against this background and show how it relates to the ontology of common sense and of mathematics. EXCERPT #EC8VKE p. 0 Before Kepler and Galileo, physics was a science based on arithmetic and trigonometry and with a correspondingly limited reach. Today, however, physics provides the theoretical foundation of the technosphere in which we live and it provides our models for the understanding of the universe and matter. Experience teaches us that we can use the laws of physics to create reliable technological artefacts such as cars, planes and MRI devices. Such devices have enhanced human lives to a degree which would have been unimaginable in former times. How did this evolution take place? And why is the physics we have today so useful, given that, as we shall see, the models it uses are all in a strict sense wrong? 1 The attempt to answer this and related questions has led to a EXCERPT #QN7XLA p. 0 1 We here follow the arguments of Cartwright [5] to the effect that the models of physics are either merely approximations of reality made under certain simplifying conditions or such as to apply only to certain restricted systems which are either deliberately chosen from the vast realm of inanimate nature or built artificially. A model is a mind-dependent representation of an aspect of reality using abstract symbols (and potentially also text and figures) that is created to describe, explain, or predict EXCERPT #E76UUM p. 0 EXCERPT #49HDCA p. 1 huge body of work in the philosophy of science and quantum physics. 2 Missing from all of this work, however, is any systematic attempt to develop an ontology of physics. EXCERPT #PQ6EWJ p. 1 An ontology in the sense used in this paper is a formal representation of ‘the kinds and structures of objects, properties, events, processes, and relations’ in some domain of reality. The applied ontologist seeks to provide a definitive and exhaustive classification of entities in specific domains and of the relations between them [26]. Such classifications have become increasingly important in many areas. They are now used routinely in the life sciences [7, 24], where they promote reuse and exchange of data deriving from different, heterogeneous sources by providing common systems for data annotation. EXCERPT #ENF2A4 p. 1 Why, then, has no convincing ontology of physics been created thus far? First , physicists and applied mathematicians working in physics do not feel the need for such an ontology. This is because they already have mathematical models of reality which have been validated over and over again for the established body of knowledge in physics through experiments. These models are well understood and accepted as a matter of course by all major groups of physicists and the associated experimental data are annotated using commonly accepted mathematical formalisms. This means that there is no major impediment to the communication between different communities of both theories and data – thus no impediment of the sort that we find in the life sciences and in other areas where work in ontology plays an important role. Nowadays, indeed, the experiments in physics are themselves often planned and conducted by huge international consortia such as CERN, ISS or ITER, with the aim of ensuring consistent collection and unproblematic exchange of data across national and linguistic boundaries. EXCERPT #9V88ND p. 1 Second , physics is distinguished from biology or chemistry by the fact that, where the latter are descriptive sciences dealing with billions of different types of entities, physics is a systematic science describing general principles 3 and laws of nature. 4 EXCERPT #Y4EVEA p. 1 And lastly , relationships found in one domain of physics are frequently reused also in other domains, for example where analogues of the phenomenon of mechanical resonance (forced oscillator with dampening) reappear in the phenomenon of electrical resonance. Chemistry can be seen as a branch of physics, and physical properties 5 of the elements have been used as basis for classification systems of chemical entities such as PubChem or ChEBi [14]. Furthermore, classification in physics is often performed using mathematical structures. For example, in quantum field theory, the mathematics of group theory has EXCERPT #VRCYM2 p. 1 the aspect of reality in question. More details are given in section 2.3. EXCERPT #CD7QYE p. 1 2 For examples see: [4, 9, 10, 17]. EXCERPT #XEBT9U p. 1 3 A principle of physics is a highly abstract, universal type of relation between material objects that is represented in corresponding models. Examples are the principle of least action or the symmetries represented in models of nature by symmetry operations such as translations in time or space. EXCERPT #V8VAAK p. 1 4 A law of nature is a mathematical model of an interaction of material system elements that has universal validity. Examples are the models defined by the field equations of the General Theory of Relativity or by the Schrödinger equation for quantum systems with a small number of particles. EXCERPT #6BJY6F p. 1 5 Here in the sense of: matter-related properties, by which we mean, when not stated otherwise, physics-(science)-related. [16, sect. 2.2.2] EXCERPT #U2Y7UH p. 1 EXCERPT #TCS5N9 p. 2 been used to create the classification scheme that now forms the basis of the standard model of particle physics [28]. For these reasons, the classification problems of physics are, already to a large degree solved. EXCERPT #95QARH p. 2 The need for an ontology of physics arises, rather, at the point where physics is applied , for example, in materials science, which deals with millions of different types of entities which need to be taxonomically organised in order to enable data exchange and system interoperability in areas such as solid state engineering. EXCERPT #N9ATQT p. 2 However, to create physics domain ontologies which address this need in a way that can promote interoperability 6 , a common upper-level ontology of physics is required which defines the basic kinds of entities and relations that applied scientists and engineers have to deal with 7 . We proceed as follows. First, we briefly state our view of the nature of theories in physics and describe our proposed upper ontologies of physics (PhysO) and mathematics (MathO) and document also how they relate to the BFO ontology of common sense. 8 We then sketch applications of these upper ontologies to two highly contrasting examples, one from the domain of classical mechanics (the harmonic oscillator), the other from quantum physics (photon entanglement). Our rationale for this choice of examples is that it will allow us to demonstrate that phenomena from two of the most important domains of physics, domains which require fundamentally different mathematical formalisms, can be dealt with coherently within a single ontological framework. Success in this regard will, we believe, lend support to our hypothesis that all other domains of physics can be described using the proposed framework. Based on these examples, we then summarise our view of the relationships between the ontologies of, respectively, common sense (BFO), physics (PhysO) and mathematics (MathO). SECTION #MFV3NU 1 Ontology of physics SECTION #JJQFJY 1.1 The nature of theories and models in physics EXCERPT #5S2Z5B p. 2 We view physics as the science of inanimate natural and technical systems, where a system, in the sense relevant to our purposes here, is a totality of dynamically interrelated material elements. EXCERPT #UQNTR4 p. 2 Systems are parts of nature delimited by fiat. To delimit a system is to select a level of granularity of its elements, from microphysical particles to entire galaxies, and of the interactions between these elements, from gravitational attraction to galactic collision. It EXCERPT #M9VD62 p. 2 6 This is the ability to exchange data between machines and enable automated computation (search, aggregation modelling) on the data. EXCERPT #JC5B2V p. 2 7 For example, see https://industrialontologies.org/iof-charter/ EXCERPT #EXXX82 p. 2 8 This is a top-level ontology used especially in the life sciences which has been documented as an international standard in ISO/IEC 21838-2. See [2]. The authors of BFO did not see the need to include mathematical entities in the coverage domain of the ontology. In particular the authors of BFO did not include any room in the ontology for the treatment of sets. EXCERPT #4UA62Q p. 2 EXCERPT #VHCW2P p. 3 is typically also to select a boundary around those elements which will form the system in question. Physics takes such delimited systems as its object because the complexity of nature in its undelimited totality would go far beyond what can be achieved by any theoretical approach. SECTION #ZUFVAD 1.1.1 Classical and modern physics EXCERPT #3QPA9P p. 3 Classical physics is the physics that was pursued from the inception of the discipline in the early 17th century (Kepler, Galileo) to 1905 – the decisive year in which Planck described the quantization of energy levels of the electron and Einstein the special theory of relativity. These two discoveries mark the beginning of modern physics, which is distinguished from classical physics through its use of abstract mathematical structures which cannot be imagined using the natural attitude: Hilbert spaces for quantum mechanics and Riemannian manifolds for the general theory of relativity. SECTION #ZHEERU 1.1.2 Laws as models EXCERPT #FMNURY p. 3 Since the late 18th century, it is mathematical equations which generate the descriptive, explanatory or predictive power of the laws of physics. Equations can express two sorts of laws: fundamental and phenomenological [5, essay 6]. A fundamental law is a conjunction of one or more equations created to contribute to a model that (i) describes the behaviour of systems of a certain type and (ii) is ‘universal’ in the sense that it describes the behavior of a system that is found throughout the universe and thus also models something that is independent of the human mind. Examples are the equations of Einstein’s field theory or Maxwell’s equations of electromagnetism. Although such equations are sometimes also called ‘causal laws’, they do not in fact cause anything. Rather, they describe causal relationships. Both principles and laws are, in our view, mere models. The first have explanatory power which involves an appeal to causality; the second have predictive power. EXCERPT #6DKR5R p. 3 A phenomenological law is a model of inanimate nature 9 describing the behaviour of some specific type of natural system, and are thus without universal validity. An example is the conjunction of equations expressing the quantum theory of the laser. EXCERPT #8BSH8L p. 3 Models in physics typically contain not only equations and figures but also text defining the assumptions of the model, for example in descriptions of a natural system using ‘as if’ clauses (as in: ‘helium gas behaves as if it is a collection of molecules which interact only on collision’ [5, p. 128f.]). Text of the sort that occurs in models may be used also to justify the fundamental laws of a theory, or to justify the use of mathematical approximations introduced to make the equations of the model solvable (for example, by neglecting terms of equations under conditions specified in the text). EXCERPT #338ES8 p. 3 9 Nature is the totality of material entities of the world not created by humans; the totality of what is created by humans forms culture (τέχνη ὀνείας in the sense of Aristotle). EXCERPT #PQKLVD p. 3 SECTION #CAHVQY 1.1.3 The scope of physics EXCERPT #55E6N8 p. 4 Each of these cases illustrates how the most prestigious models in physics idealize reality . Indeed, the fundamental laws of physics are not, as was supposed by 17th century authors such as Boyle or Hooke, laws written by God into the book of nature. Rather, as Cartwright puts it, they model simplifying abstractions of natural processes; they ‘do not govern objects in reality; they govern only objects in models.’ [5, p. 129f.] In this connection Cartwright speaks of ‘physics as theatre’ (p. 139). The laws are indeed themselves mathematical models of nature. But they do not accurately describe the complexity of nature itself; nor can we learn from them what nature is really like. Hence verbal descriptions in physics often draw attention to the fact that a representation of the full complexity of the systems under study is impossible, followed by documentation of the simplifying assumptions. EXCERPT #PR6G6Y p. 4 But we do of course learn something from models: What they provide is a consistent and, with regard to the mathematical part, logically structured approximation of certain aspects of nature, above all of those which allow us to make predictions about systems and events that are relevant to the realization of human goals because they allow the creation of technical artifacts. In other words, they allow us to explain 10 aspects of nature in ways that are instrumental uses of physics. 11 SECTION #FR2Q9H 1.1.4 The need for approximations EXCERPT #ETUUP5 p. 4 We must use approximations, in all of this, because there is a discrepancy between the causality of nature and our ability to do justice to this causality when we formalize the laws of nature using mathematics. This arises because the causalities are too complex to model exactly using mathematics. This is in conformity with our assumption, widely accepted among physicists, that all events in nature are caused by combinations of the four fundamental interactions of electromagnetism, gravitation, and the strong and weak forces. 12 Sound, for example, can be reduced to the electromagnetic interactions of the particles involved in a sound wave travelling through a medium. For some natural systems, the behavior can be formalized by means of universal laws using these interactions to describe the relationships between their elements. The assumption of fundamental laws of physics is powerful for prediction and explanation especially for those natural systems whose behavior can be observed unproblematically, as was shown most consequentially by Newton in his treatment of the solar system as a gravitational system. EXCERPT #DBVT6F p. 4 10 An explanation is the description of the causality of a system. For example, Newton’s laws explain the movements of the planets in the solar system by describing them causally. EXCERPT #SCT85U p. 4 11 Our view is thus in contrast to Cartwright’s later view that fundamental laws do not apply to nature at all [6, ch. 1]. EXCERPT #JEMFR4 p. 4 12 Where the first two are the only forces we can observe at the mesoscopic level of granularity of common sense, the views of physics on the weak and strong forces are extremely well grounded in experiments and theory, though our assumptions concerning the fundamental role of the four forces may change in the future if different paradigms for experimentally analysing matter would become available. EXCERPT #5JA5YJ p. 4 EXCERPT #R9U7FQ p. 5 This assumption can yield valuable results also for those natural systems in relation to which we can perform laboratory experiments. However, for the vast majority of natural systems, including nearly all systems in organic nature, we cannot model mathematically the ways this causation occurs [16, chapter 8]. Thus while the common-sense view of nature, central elements of which are reflected in Basic Formal Ontology, is very powerful – it provides us with all of the knowledge that is relevant to our daily lives – it does not extend into the realm of physical science. SECTION #9BTD4U 2 Physics upper ontology EXCERPT #KCASJC p. 5 The PhysO ontology we propose can therefore not be an ontology of nature simpliciter . Rather, it is an ontology of nature as it is modelled mathematically at different levels of abstraction – such as classical geometry and calculus for classical physics, and abstract mathematical entities out of scope for common sense for modern physics. The ontology of physics thus depends always on the ontology of mathematics. EXCERPT #VU6BED p. 5 The ontology of physics comprises three main axes: system entities, magnitudes, and models. An overview is shown in figure 1, where we follow the standard strategy of representing an ontology as a directed acyclic graph with nodes representing entities and edges representing binary relations between these entities (in this figure: Aristotelian genus-species (class-subclass) relations 13 ). EXCERPT #RE8X8L p. 5 Importantly, in classical physics, the system entity branch of the physics ontology falls within the domain of the common-sense ontology BFO. In modern physics, in contrast, no branch contains BFO entities. 14 EXCERPT #Z2WY6H p. 5 We now proceed to describe the three branches of the physics ontology. SECTION #NYRFHH 2.1 System elements EXCERPT #CMTY6U p. 5 A system is a part of reality. It is a totality of entities (called ‘system elements’ in figure 1), each made of matter, which interact with each other; thus of elements which participate in processes of interaction. To delimit a system is to select a level of granularity of elements and a system boundary. For example, the solar system can be seen as a gravitational system in which the elements interact via the force of gravity. Its elements are planets, satellites and other objects bound by the sun’s gravitational force. EXCERPT #TADE8C p. 5 Different systems are delimited according to what are taken as elements: just the planets; or the planets together with their respective satellites; or also pieces of space debris bound by the sun’s gravitational force. As these examples again make clear, systems are EXCERPT #AF878V p. 5 13 Often designated as ‘is_a’ relations in the applied ontology literature. EXCERPT #XWY9DD p. 5 14 See section 5 and figure 10 for more details. EXCERPT #CX47YD p. 5 EXCERPT #JBECM6 p. 6 graph TD PE[physics entity] --> SE[system entity] PE --> M[magnitude] PE --> MO[model] SE --> SElem[system element] SE --> SR[system relation] SE --> S[system] SElem --> W[weight] SR --> EM[electro-magnetic interaction] S --> WOS[weight-on-spring] M --> CQ[continuant quality] M --> PC[process characteristic] CQ --> M1[mass] PC --> A[acceleration] MO --> TM[textual model] MO --> GM[graphic model] MO --> MM[mathematical model] GM --> GEM[geometric model] GEM --> OM[orbit model] MM --> FOD[forced oscillator with dampening] A hierarchical tree diagram showing the top-level entities of PhysO. The root is 'physics entity', which branches into 'system entity', 'magnitude', and 'model'. 'system entity' branches into 'system element' (leading to 'weight'), 'system relation' (leading to 'electro-magnetic interaction'), and 'system' (leading to 'weight-on-spring'). 'magnitude' branches into 'continuant quality' (leading to 'mass') and 'process characteristic' (leading to 'acceleration'). 'model' branches into 'textual model', 'graphic model' (leading to 'geometric model' and then 'orbit model'), and 'mathematical model' (leading to 'forced oscillator with dampening'). EXCERPT #VRW2LD p. 6 Figure 1: Top-level entities of PhysO, lowest nodes here show only examples, which are documented in section 4.1. Note that the system entities and the magnitudes shown here can also be classified using BFO, and that the mathematical models consist of representations of mathematical entities classified using the mathematics ontology described in section 3. EXCERPT #M5DF43 p. 6 always delimited by fiat. The sun itself can also be seen as a system of electromagnetic interactions, whose elements are the particles emitted by the sun. EXCERPT #MSM5ZN p. 6 System elements can range from elementary particles to entire galaxies. The interactions between the elements are governed in every case by one or more of the four fundamental forces listed above (see ‘system relation’ in figure 1). 15 We can observe and measure each of the four forces and physicists have not identified any other force that is not composed of them. When several effects of one force or several forces are overlayed in complex systems, it is in general impossible to contrive mathematical models that are of sufficient quality for exact description or high-quality prediction [16, 27]. Examples we have from inanimate nature, such as the solar system as a gravitational system, are very close to what we have called logic systems [16, chapter 7] and can be modelled very well using mathematics. SECTION #XLSYCZ 2.2 Magnitudes EXCERPT #8NKC2N p. 6 In classical physics, a magnitude is a phenomenon in reality – for example mass, distance, velocity, acceleration or energy – which has the feature that its instances can be measured . EXCERPT #5FCDYY p. 6 15 We have inductive, positive knowledge about the four forces, but we know little about their fundamental nature. We observe them, we understand and model them as causes of events, but in the end, as Feynman puts it, we do not really know what a force is [11, ch. 12]. EXCERPT #BM72HV p. 6 EXCERPT #LWV4KJ p. 7 The phenomenon in question exists both on the level of instances and on the level of universals. 16 EXCERPT #5GBMKR p. 7 Classical physics magnitudes, the associated processes of measurement, and the underlying qualities are all such as to fall within the realm of what can be imagined by common sense (that is, in the realm of Basic Formal Ontology). The magnitude universals of non-relativistic mass, time, distance, and so forth, have existed since the beginning of the universe, and thus long before any measurements occurred. The mathematical treatment of magnitudes, on the other hand, is mind-dependent. For example, the terms ' F ', ' m ' and ' a ' in the equation ' F = ma ' which are all common-sense magnitudes in classical physics, when used in a mathematical context are mere variables, 17 and they are treated as such for example when you use ' F/m ' to substitute for ' a ' in another equation. EXCERPT #Y9JR3W p. 7 In modern physics, in contrast, for example in quantum physics and the general theory of relativity (GTR), magnitudes no longer fall within the domain of what can be imagined using common sense. In GTR, for example, time is conceived as one inseparable dimension of a four-dimensional spacetime manifold, a structure adapted not from our experience of reality – either in common sense or in experiment – but rather from mathematics. Thus time in GTR is not a universal in the sense outlined in footnote 16, and it is not something that we measure like we measure distance in classical physics. We can certainly identify particular times in GTR, such as the moment a supernova begins to form. But we cannot separate it from the spacetime manifold and consider it as something existing in indefinitely many interchangeable copies. Moments in time in GTR go hand in hand not with a framework of universals of the sort with which we are familiar in our common-sense experience, but rather with a structure that is mathematical in nature – a structure that we ourselves make. EXCERPT #YAR97Z p. 7 In GTR the time of classical physics, the time compatible with common sense, is gone . Similarly, the quantum-theoretical magnitude spin 18 is a mathematical entity used to model measurement results of magnitudes we cannot understand using our common-sense-based view of the world or our natural (non-mathematical) imagination. EXCERPT #XCKN2E p. 7 As for time in GTR, so also for spin, we have measurements (spin direction) for particular particles, but no corresponding universal. We thus have the following situation: EXCERPT #VR4MS5 p. 7 • classical physics: (universal – instance), • modern physics: (mathematical entity – measurement), EXCERPT #9GYQA2 p. 7 16 A universal is an abstract entity that is instantiated in reality by an indefinite number of instances. EXCERPT #452E7F p. 7 We gain knowledge of universals by abstracting from their instances, for example, going from this concrete atom of hydrogen in this water molecule in this glass to the universal ' hydrogen atom '. EXCERPT #YME3YJ p. 7 17 A variable (as contrasted with an associated symbol) is a mathematical entity which serves as typed placeholder in a mathematical term ranging over a set of mathematical entities. 'Typed', here, means that the typed placeholder can only take values from its associated set. EXCERPT #RXKQYN p. 7 18 See section 4.1.2 below. EXCERPT #98RNS6 p. 7 EXCERPT #2NHV6Z p. 8 where the instance of classical physics is replaced in modern physics by observation or measurement. The second type of tuple is a matter of artificial declaration (attribution). 19 This is because, when we measure, for example, spin, there is nothing which we can identify as instance of a universal. In other words, the tuples of type (mathematical entity – measurement) are created by declaration because we are unable to understand the phenomena related to their associated magnitudes using common sense, we cannot think of them as universals by abstracting from their instances as we do, for example, when we start out from individual beeches and oaks to obtain the universal ‘tree’. EXCERPT #WCYEB3 p. 8 Therefore, modern physics has declared that a mathematical entity (for example a structure like the manifold in terms of which spacetime is defined in GTR) 20 takes the place of a universal in classical physics, while the instance of the magnitude, that which is represented by a measurement result (including the measurement error), is real. Thus, magnitudes in modern physics can be understood only via mathematics, which means: through the use of models. EXCERPT #PTQCVU p. 8 Magnitudes are in every case, in classical as in modern physics, associated with material entities. This means that magnitudes are either (i.) continuant magnitudes 21 (such as mass or density), which are qualities of material bodies; or they are (ii.) process magnitudes , 22 such as velocity or force, where the processes measured involve material bodies as participants and interactions mediated by particles (as for example photons mediate the electromagnetic force). 23 EXCERPT #6UWAVK p. 8 Each process magnitude at the instance level is what we call a process profile [25]. This means that it is a chain of process characteristics, for example a chain of values of the expression \ddot{x}|_{t=j} , for the acceleration of a mass determined at different time points j . 24 EXCERPT #XFKEE3 p. 8 In physics, magnitudes are also called ‘physical quantities’. Here, however, we distinguish for clarity’s sake between ‘magnitudes’ on the one hand, and ‘quantities’ on the other. We define a magnitude as an entity which can be quantified using measurements. Magnitudes are then of two sorts: (1) qualities of continuants (for example mass , length ), or characteristics of processes (for example velocity , acceleration ). 25 The magnitude is EXCERPT #LMJWVS p. 8 19 A tuple \langle x, y \rangle is an ordered pair, in set-theory it is written as follows: \{\{\emptyset, \{x\}\}, \{y\}\} . EXCERPT #NHHDH3 p. 8 20 A manifold is a topological space. It is called an n -manifold if it is a Hausdorff-space, fulfils the axiom of countability, and is locally Euclidean. EXCERPT #P3Q3GF p. 8 21 A continuant is an entity that persists, endures, or continues to exist through time while maintaining its identity [1]. EXCERPT #BPRV9K p. 8 22 Processes are occurents , in BFO terms, which means that they are entities that unfold through time, in other words entities that have temporal parts (parts along the temporal dimension). EXCERPT #CSR9JQ p. 8 23 Only in the case of gravitation we are unable to identify a material particle mediating the interaction (the graviton is a hypothetical particle). EXCERPT #AZEC4T p. 8 24 An example is an ECG monitor used on a patient in an ICU. The electric currents of the cardiac conduction system visualized as lines on the monitor are such process profiles. EXCERPT #P4W9GA p. 8 25 A quality , in BFO, is defined as ‘a specifically dependent continuant that, in contrast for example to dispositions , does not require any further process in order to be realized’ [1]. Only continuants may have qualities , in BFO, and the term ‘ characteristic ’ is used as the counterpart of ‘ quality ’ for EXCERPT #95VL9Z p. 8 EXCERPT #E4PY99 p. 9 something repeatable that we find in reality in many particulars – in classical physics, again, it is what was traditionally called a universal. 26 EXCERPT #3R8J26 p. 9 A physical quantity , in contrast, is the count of how many of this or that particular magnitude are found in a specific case of measurement. It is that about which we speak when we collect measurement data and express it in terms of, for example, numbers (quantities) of meters or joules. The term ‘quantity’ thereby captures also how much of or how many of a particular measurement magnitude we have in a system we observe or on which we perform measurements. In physics, the quantity is expressed using a mathematically defined magnitude that can be referenced using a variable in an equation. EXCERPT #J2996J p. 9 Physicists model the real magnitudes encountered in nature by means of mathematical structures which are models of natural processes. To see what this means, consider an equation such as Newton’s second law: EXCERPT #S22WRK p. 9 F = m\ddot{x} = \frac{md^2x}{dt^2} \quad (1) EXCERPT #M4LPKQ p. 9 We have terms on the left, the middle and the right hand side of this double equation. A mathematical term is a meaningful (syntactically and semantically valid) combination of numbers, variables and symbols that is used to designate structural entities in mathematics. An analogy in natural language is a noun or noun phrase in a sentence. There can be no false terms, but there are invalid terms, such as ‘ \frac{1}{0} ’ or ‘1.’. EXCERPT #RKZP4B p. 9 The referring expressions of the equation, ‘ F ’, ‘ m ’, and ‘ \ddot{x} ’, play a dual role. First , they may refer to real magnitudes, in this case to the force, mass and acceleration, of some specific system element (a particle or body with a mass), thus describing an aspect of the reality of this element. When measurements of a system described by equation (1) are made and the results plugged into the variables of the equation, a law of nature is applied to particular measurements and the calculation prescribed by the equation is performed to yield a calculation result, for example in the form of a vector or – in the above case – a scalar with a unit of measurement. The set of permissible values depends on the context in which the variable is used. For example in the term \frac{1}{x} , x \in \mathbb{R} \setminus 0 . EXCERPT #DQYCY4 p. 9 Secondly , each variable of equation (1) also represents a certain non-instantiable, abstract mathematical entity on the side of the human-created model itself. This second nature of the expression becomes evident when physicists manipulate equations using mathematical operations without using the equation with measurement results to calculate a value. The equation itself, and the referring expressions within it, are mind-dependent structures that are used by physicists to refer to measurable features of reality that exist independently of the human mind and of any measurement process. EXCERPT #CTN499 p. 9 occurents. EXCERPT #XWAR32 p. 9 26 Magnitudes are assigned in each case to some particular. We can also say that a magnitude is a dimension of a phase space, namely of the phase space which models the behaviour of the underlying particular in the modelled system. EXCERPT #WTCYQU p. 9 SECTION #MSMT2E 2.2.1 Quantification via constants EXCERPT #AG9JKR p. 10 Constants, for example the gravitational constant, are magnitudes with a quantity that is fixed in a certain context. Universal constants are thought to have fixed quantities in the entire universe. EXCERPT #HLYHA4 p. 10 An example of a universal constant is the reduced Planck constant, measured in joule seconds, which is a continuant quality constant which captures the relationship between the energy of a photon and its frequency, \hbar \approx 6.582 \times 10^{-16} \text{ eV} \cdot \text{s} . An example of a universal process characteristic constant is the speed of light c = 299792458 \frac{\text{m}}{\text{s}} . SECTION #KTYY4Y 2.2.2 Quantities and units of measurement EXCERPT #UV27XM p. 10 Magnitude quantities – the results of acts of measurement – are expressed by means of a count together with a measurement unit. Occasionally we have pure counts (the mole 27 ). A measurement unit is a fiat entity, which is a sub-entity of the corresponding quantity intuitively resulting from carving out equal divisions along a scale. It is itself a kind of quality or characteristic. Measurement units are used in quantifying instances of the corresponding magnitudes [15]. EXCERPT #84X9SK p. 10 The act of measuring consists in identifying the quantity of a physical magnitude (in the simplest cases by answering the question: How many ) of a given quality such as length or mass with either its discrete number of instances using natural numbers ( \mathbb{N} ) or using rational numbers ( \mathbb{Q} ) to approximate real numbers \mathbb{R} . EXCERPT #GTW8P6 p. 10 Leaving the mole aside, measurements resulting from use of measurement units are expressions which consist of two parts referring respectively to: EXCERPT #BN23EE p. 10 (i) a rational number approximating a real number, and (ii) the unit of measurement itself, EXCERPT #58EJLJ p. 10 joined together in expressions such as ‘4.449 kg’. EXCERPT #VBYE2V p. 10 That rational numbers (symbol: \mathbb{Q} ) are used as approximations reflects practical constraints, including limits on precision of our measurement instruments, on the size of a display or on the storage capacity of a computer. 28 Each number, wherever it appears in a model in physics, whether \mathbb{N} , \mathbb{Q} , \mathbb{R} or any other type of number (including complex EXCERPT #EB8RXK p. 10 27 While the kilogram and metre are examples of measurement units used to quantify continuous quantities , the mole is a measurement unit that is used for counting discrete quantities . The mole is in fact not a unit of measurement, but rather a dimensionless counting measure – a standardization of a count of particles – introduced into the SI system for reasons of counting convenience. It appears in expressions such as ‘2 mol’ or ‘2.00175 mol’, where the latter is still, appearances notwithstanding, a discrete count (natural number) \in \mathbb{N} , because 2.00175 \text{ mol} = 2.00175 \times 6.02214076 \times 10^{23} particles. EXCERPT #YRN2Y6 p. 10 28 This is the reason why there are the floating point data types, which are formulaic representations of real numbers. EXCERPT #8AVJMB p. 10 EXCERPT #VFM2JK p. 11 numbers in \mathbb{C} ), is a mathematical entity. This is so whether the number in question is the result of a counting or measurement process or is inferred from an equation. 29 SECTION #NEM8TT 2.3 Models EXCERPT #YRFUYC p. 11 A model in physics is a mind-dependent representation of an aspect of reality using abstract symbols that is created to describe, explain, or predict the aspect of reality in question. Abstract symbols are used primarily in the context of equations, but they may be accompanied by textual descriptions which may in turn be complemented by graphical, often geometrical, representations. EXCERPT #S98ZZT p. 11 The subject of the model is either a system, a system element, or an interaction. Examples are: for system elements: an electron; for systems: the solar system, or the two-element hydrogen atom system consisting of a proton and an electron; and for interactions: gravity, in the case of the solar system; or the electromagnetic force through which, in the case of the hydrogen atom, proton and electron interact. When two systems interact, they are modelled as system elements on a more coarse-grained level. This is how the granularity levels of systems treated by physics range from elementary particles to galaxies. The equations in the model represent the model's subjects by means of mathematical entities. The equations describe the relationships between these entities using mathematical relations. They thereby model physical reality in an idealised form. Therefore, models are not exact representations of associated system entities in reality. They merely approximate such entities. Just as real-world shapes are approximated by mathematical shapes, so real-world processes are approximated by the models of physics. EXCERPT #595UMS p. 11 In classical physics, we have systems in reality which are modelled using universals, magnitudes, which have individual instances but are represented in the models as mathematical entities. In modern physics, in contrast, we have system entities, magnitudes and models which contain no universals, but only mathematical entities (see section 5). We now turn to the mathematics upper ontology needed to describe the formal models in physics. SECTION #JGRA8D 3 A mathematics upper ontology SECTION #E3Q2KL 3.1 The nature of mathematical entities EXCERPT #KP3G8G p. 11 Mathematical entities are ideal, which means that they are not part of the causal world of time and change. They are intrinsically intelligible entities (which means that propositions about them can be known a priori ). Further, they are mind-dependent; but at EXCERPT #6AGE4J p. 11 29 Many numbers in \mathbb{N} , \mathbb{Q} , \mathbb{R} have real-world counterparts (for example the number of coins in my pocket is 3, the length of the hypotenuse drawing of a right-angled triangle with two sides of length \approx 1 is \approx \sqrt{2} \in \mathbb{R} ). This is however not the case for numbers in \mathbb{C} . EXCERPT #YRWUMS p. 11 EXCERPT #VV9NBC p. 12 the same time they are independent of our sensory experience of the world outside our mind [22, pp. 69f.]. EXCERPT #TNGD89 p. 12 We saw that they are not universals in anything like the Aristotelian sense in which we use this term, and they are also not instances, and nor do they have instances, unless, as we shall see in section 4.2, instances are assigned. In physics, chemistry and also in computer science and other branches of applied mathematics, mathematical entities are used to model reality; but this does not mean that there is something in reality to which these models directly correspond. EXCERPT #XKC6UE p. 12 The domain of mathematical entities ranges from very simple examples such as numbers and basic shapes, which we appeal to when performing acts of counting or of describing real entities such as tabletops or pieces of string, to highly abstract entities such as Hilbert spaces or Lebesgue integrals, which can be understood only with the aid of equations. SECTION #YFJ8WQ 3.2 Mathematics upper ontology EXCERPT #S3P823 p. 12 There are three types of mathematical entities: monads, constructors and structures. Monads are atomic primitives from which structures are made through the application of constructors. 30 Examples of monads are zero, one, or the point. EXCERPT #5FWWKZ p. 12 Constructors are used to obtain structures from monads and structures. For example, the number 2 is constructed with the equality constructor and the addition operator (a constructor) like this: 1 + 1 = 2 . 31 Simple operators 32 are constructors used to perform basic mathematical operations such as addition and multiplication. Their relations are determined by the axioms of mathematics (for example, the Peano axioms) chosen. Structures in mathematics are essential structures, which means that they are entities whose component parts stand in necessary relations to each other. We are influenced here by the German philosopher Adolf Reinach, who spoke of ‘essential connections’ ( Wesenszusammenhänge ) thereby defending the idea that there is a wide class of material necessities which can be known a priori in domains such as color and shape, rational psychology, and above all social ontology. Reinach’s uniqueness consists in the fact that he was one of the first to recognize essential structures not only in the timeless realm of mathematics à la Plato, but also in the historically changing realm of debt and ownership. 33 A helpful example of an essential structure in the Reinachian sense, slightly EXCERPT #DM7BJP p. 12 30 Our ontology of mathematics differs from mathematical structuralism, for example as described by Reck and Schiemer [21], because we postulate, like Frege, that there are primitives (monads) which are not merely determined by the context of other mathematical entities, but have an existence in their own right. EXCERPT #BH2LLJ p. 12 31 Note that the monad 1 is here used twice. EXCERPT #38VCFK p. 12 32 An example of an operator in the sense of functional analysis is discussed in section 4.2.1.4. EXCERPT #JWZGGH p. 12 33 There is an essential connection, for example, between a promise, a claim and an obligation. The latter exist as nodes in a system of necessary connections alongside other nodes such as intention (to act as promised), realization (of the promise), waiving (of the claim) and so forth [23]. EXCERPT #2MUNKF p. 12 EXCERPT #WB2A4Z p. 13 more involved than 1 + 1 = 2 is the constant structure \pi . 34 EXCERPT #WU6TG2 p. 13 In mathematics, all of these entities are intrinsically intelligible (= a priori ) entities which have no instances in reality. 35 Using constructors and monads, all mathematical entities can be obtained. EXCERPT #7J7JVR p. 13 Figure 2: Top-level entities of mathematics with examples of entities at lower levels. The diagram shows three hierarchical structures. 1. 'mathematical entity' branches into 'constructor', 'monad', and 'essential structure'. 'monad' has dashed edges to '0', '1', 'a', and 'x'. 'essential structure' has a solid edge to 'point'. 2. 'essential structure' branches into 'R ⊆ (A × B)' and 'field'. 'R ⊆ (A × B)' has solid edges to 'f : ℝ^n ↦ ℝ' and 'O : ℝ^n ↦ ℝ^k'. 'field' has a solid edge to 'ℝ'. 3. 'constructor' branches into '(in)equation c.', 'set c.', 'quantifier c.', and 'simple operator c.'. '(in)equation c.' has solid edges to '=', '≠', '<', and '>'. 'set c.' has a solid edge to '∈' and a dotted edge to '{x ∈ ... | ...}'. 'quantifier c.' has solid edges to '∀' and '∃'. 'simple operator c.' has solid edges to '+' and '·'. EXCERPT #J5V6HF p. 13 Figure 2: Top-level entities of mathematics with examples of entities at lower levels. A \times B = \{(a, b) \mid a \in A, b \in B\} , R : relation with relata (domain and range), \times : cross product, \mathbb{R} : real numbers. \forall and \exists are fully expressed as constructors like this: \exists x_1, \dots, x_n \mid \dots , where the second ‘ \dots ’ indicates a syntactically correct term of predicate logic obtained using zero or more of the connectives \neg, \vee, \wedge or \rightarrow and using the variables x_1, \dots, x_n, n \geq 1 , possibly with n-ary predicates R_1, \dots, R_m as well as constants a_1, \dots, a_\ell . f – functional, O – operator (functional analysis). Note that the solid edges of the graph mean \subseteq , whereas the dotted edges mean \in , so that the monads, which are primitive elements not regarded as sets here, are shown as elements of the set of monads (and not as subsets). In the constructor taxonomy at the bottom ‘c.’ means ‘constructor’. EXCERPT #G8SDLX p. 13 Figure 2 shows some important constructors with their associated monads and essential EXCERPT #ASFVPT p. 13 34 \pi is a structure that is obtained from the monad ‘1’ by first defining the structure plane ( E = \{x, y, z \in \mathbb{R} \mid ax + by + cy = d\} ) using the constructors =, \in , monadic variables and the set constructor \neg, \vee, \wedge as well as the addition and multiplication operators and distance (a structure defined as the segment of the line \alpha x + \beta y = \gamma ) and then by constructing the circle as the set \{X \in E \mid \overline{MX} = r\} in the plane E with central point M \in E and a distance r \in E called radius with circumference C . \pi is then defined as \pi = C/2r . EXCERPT #8HLR79 p. 13 35 Other Reinachian a priori structures, such as the promise, have instances. EXCERPT #LK4ZFM p. 13 EXCERPT #6SH3C8 p. 14 structures. The latter are always sets in the mathematical sense. 36 Therefore, unlike the ontology of physics, in which the edges represent the ‘is a’ relation, in the mathematics ontology, it signifies ‘subset of’ ( \subseteq ): The entire realm of mathematical entities is built up from monads (0 and 1, the geometrical point, etc.) using set theory, and all the relations in the ontology are set-theoretical (using the constructors). As in mainstream mathematics, we assume a set theory (in almost all cases using Zermelo-Fraenkel axiomatization with the axiom of choice (ZFC) will suffice for our purposes [8]). For example, the type R \subseteq A \times B ranging over the sets A and B as its relata is a superset of one of its subset types, the set of functionals f (functional is the designation of function in functional analysis). The type R is also the superset of the type operator (in the sense of functional analysis), an example of which is \nabla , the gradient operator, or its special case, the univariate differential \frac{d}{dx} . EXCERPT #UCHX59 p. 14 Note that we do not discuss the expressions we use to formulate mathematical models. We are interested in the models themselves. When we use expressions such as ‘equation’, we are referring not to a string of symbols but rather to mathematical entities which the symbols represent. Using examples in the next section, we will see how mathematical entities can be classified using the upper ontology shown in Figure 2. SECTION #KHZPSR 4 Examples from classical and quantum physics EXCERPT #T3GM7K p. 14 We proceed with an ontological analysis of one case from classical physics, the harmonic oscillator, and another from quantum physics, the entangled photons. SECTION #AWHTTL 4.1 The harmonic oscillator EXCERPT #6Y6AR3 p. 14 The simple harmonic oscillator is a purely idealized model which is used as a template for further models which successively approximate real oscillators more closely, for example the harmonic oscillator with dampening used to model electric circuits. In the simple harmonic oscillator, a mass that is displaced from its equilibrium position undergoes a restoring force proportional to the displacement and undergoes changes of distance along only one dimension. EXCERPT #VA8VQP p. 14 An example for such an idealized, imaginary system – imaginary not least in that friction is entirely ignored – is a weight suspended from a spring that is subjected to displacement by means of an imagined force. Here the spring provides a restoring force that is proportional to the displacement of the weight, as defined in the differential equation EXCERPT #CLL8MJ p. 14 \frac{md^2x}{dt^2} = -kx, \quad (2) EXCERPT #DNWSFN p. 14 36 Or in some cases they are classes in the Neumann-Bernays-Gödel (NBG) axiomatization of set theory [19, ch. 4]. EXCERPT #UXBRH5 p. 14 EXCERPT #3WUG9K p. 15 Here m is the mass of the oscillating weight, t is the time interval defining the acceleration of the mass which constitutes the exerted force, x is the displacement distance defined as a function of t (so ‘ x ’ would be written out in full as ‘ x(t) ’), and k is the constant of the retraction force, also called ‘stiffness’. EXCERPT #H4K36A p. 15 The equation is part of a model that represents an imaginary harmonic oscillator, whose movement is assumed to be without friction. This means that its total energy is constant, and when averaged over time (over one period, 2\pi ) the kinetic energy equals the potential energy. EXCERPT #HEDB4R p. 15 On the left-hand side of the equation, we have a representation of the force F as defined by Newton’s second law (see equation (1) above). On the right-hand side, we have a representation of this same force exerted by the spring. 37 EXCERPT #YV7DQL p. 15 The solution to equation (2) is EXCERPT #VHAEG5 p. 15 x = a \cos(\omega t + \Delta), \quad (3) EXCERPT #3LF7J8 p. 15 with constant amplitude a (the maximal extent of change over a period), angular frequency \omega = \sqrt{\frac{k}{m}} , and constant phase \Delta , the starting point of the periodic function. EXCERPT #LK4TFX p. 15 The values of a and \Delta depend on with how much force the motion 38 is initiated. 39 The value of \omega depends on the properties of the idealised oscillator itself (namely on its mass and on the retraction force). EXCERPT #NHMT4U p. 15 With these variables and this solution, we can calculate the idealised position x of the oscillating mass at any time t in an imaginary phase space. A phase space is the algebraic field which is used by the model of the system to obtain the required model entities which are elements of or defined over the field. SECTION #J9BHWH 4.1.1 Mathematical ontology of the oscillator model EXCERPT #58E8TL p. 15 Like all models (in the sense of this term that concerns us here), the oscillator model consists of equations, text and (optionally) figures. This model is built to represent a physical system, albeit one that is imaginary. But it can also be viewed as representing a mathematical structure. We discuss ontologically only the linear differential equation (2) here. As we saw, it has the form \frac{md^2x}{dt^2} = -kx . This equation asserts the equality (which is created using the constructor = , see figure 2) of two essential structures: (1) the EXCERPT #HQEZ57 p. 15 37 We note in passing that the minus sign on the right-hand side denotes the opposing directions of the two forces this equation describes. We note in passing, too, that we are viewing the ideal harmonic oscillator as an inertial frame of reference, that is, as a system imagined to be moving at a constant velocity. We make this assumption in order to avoid the need to enter into any relativistic discussion of spacetime. EXCERPT #3B6RZ8 p. 15 38 Recall that we are here dealing with idealized magnitudes and system entities only. EXCERPT #UNPNZ3 p. 15 39 Once initiated it remains constant – recall that this is an idealized model. EXCERPT #HKSBEN p. 15 EXCERPT #558ZZU p. 16 product of the mass variable m with the double derivative relative to the time variable t of the distance function x(t) , and (2) the negative of the product of a constant k and the distance function x . The negation and product are operators. x(t) is a function f : \mathbb{R} \mapsto \mathbb{R} (see Figure 2). Its derivative is an essential structure, a binary relation from the tangent space 40 of a functional to \mathbb{R} , and this is also true for the double derivative \frac{d^2}{dx^2} . Both yield a scalar. EXCERPT #UUFWXJ p. 16 Because the idealized harmonic oscillator only moves up and down in one direction, vectors are not needed to model its motion. The phase space which is required for the model is just the simple space of real numbers \mathbb{R} , an algebraic field. EXCERPT #64HVDR p. 16 Because of the set-theoretic relationships which define the taxonomy of mathematical entities, the ontological structure is shown in Fig. 3 (all edges mean \subseteq ). EXCERPT #44VGPZ p. 16 graph TD A["R ⊆ (A × B)"] --> B["f : ℝ^n ↦ ℝ"] A --> C["O : ℝ^n ↦ ℝ^k"] B --> D["x(t)"] C --> E["d/dx : T_x M ↦ ℝ"] C --> F["d^2/dx^2 : T_x M ↦ ℝ"] Figure 3: Mathematical relations: Functionals and operators. A hierarchical diagram showing the relationships between mathematical entities. At the top is a box labeled 'R ⊆ (A × B)'. Two lines descend from it to two boxes: 'f : ℝ^n ↦ ℝ' on the left and 'O : ℝ^n ↦ ℝ^k' on the right. From 'f : ℝ^n ↦ ℝ', a line descends to a box labeled 'x(t)'. From 'O : ℝ^n ↦ ℝ^k', two lines descend to two boxes: 'd/dx : T_x M ↦ ℝ' on the left and 'd^2/dx^2 : T_x M ↦ ℝ' on the right. EXCERPT #DTJJG4 p. 16 Figure 3: Mathematical relations: Functionals and operators. EXCERPT #GSWKDS p. 16 We will see in the quantum physics example below a radically more complicated and also more abstract (MathO) modelling space. SECTION #RTGZW9 4.1.2 Magnitudes in the harmonic oscillator model EXCERPT #QKTQW3 p. 16 The harmonic oscillator model (2) contains four MathO entities, which in turn represent four PhysO magnitudes. Note that because all mathematical entities other than monads are sets, magnitudes mapped to mathematical entities in physics equations are sets. EXCERPT #E2ALGQ p. 16 The magnitudes are part of the system that is represented by the physics and mathematics entities: mass ( m ), distance ( x ), time ( t ), and acceleration ( \ddot{x} ) and the spring constant (stiffness) k . Mass, distance and the spring constant are continuant magnitudes. Mass is a quality of a body that ‘does not require any further process in order to be realized’ [1]. The distance x is a length, a primitive quality in our common sense understanding of the world that is also a mathematical entity, see the definition of \pi in footnote 34. The EXCERPT #9GY2DT p. 16 40 The tangent space T_x M of a manifold M is the set of all tangential vectors v = \frac{d\gamma}{dt}(0) \in T_x M with the differentiable curve \gamma(0) = x and curve parameter t for all points x \in M . This, too, is a structure in MathO, but we do not analyse it further here. EXCERPT #AFS8V6 p. 16 EXCERPT #EY4FKK p. 17 retraction (spring) constant k is the quality of a spring which accounts for its retraction force. EXCERPT #UJCPHS p. 17 These magnitudes have the ontological relationships shown in Fig. 4. EXCERPT #76Z2AJ p. 17 graph TD magnitude[magnitude] --> continuant_quality[continuant quality] magnitude --> process_characteristic[process characteristic] continuant_quality --> continuant_constant[continuant constant] continuant_quality --> scalar_distance[scalar distance] continuant_constant --> retraction_constant[retraction constant] process_characteristic --> mass[mass] process_characteristic --> force[force] process_characteristic --> acceleration[acceleration] Figure 4: Ontological relationships of harmonic oscillator magnitudes. A hierarchical diagram showing 'magnitude' at the top, branching into 'continuant quality' and 'process characteristic'. 'continuant quality' further branches into 'continuant constant' (which points to 'retraction constant') and 'scalar distance'. 'process characteristic' branches into 'mass', 'force', and 'acceleration'. EXCERPT #3QMCRG p. 17 Figure 4: Harmonic oscillator magnitudes. EXCERPT #C5G8DK p. 17 The acceleration referred to on the left hand side of equation (2), \ddot{x} = \frac{d^2x}{dt^2} , is a measurable process profile 41 of a process of motion. It depends on the motion and can exist only if the motion exists. Without motion, there is no acceleration. Acceleration is a process magnitude. EXCERPT #ALBL7G p. 17 The force F shown in equation (1) is also a process magnitude. It is the effect on a body (mass) which it accelerates, which means that it changes the amount or direction of velocity of the moving mass or deforms its body. ‘Process’ is defined in such a way that all processes have temporal parts. This applies to acceleration also, which is a derivative of velocity with respect to time, 42 and so has temporal parts also from its mathematical definition. When acceleration is measured at an instant, then a measurement result, a continuant magnitude, is obtained. All such results are merely approximations to their real-world counterparts. EXCERPT #F35K2T p. 17 What, now, does an ontological analysis of the equation for the harmonic oscillator (3) and its solution yield? From the perspective of physics, the model describes the relationships between physical magnitudes which are used to model natural systems. When experiments are performed, these magnitudes are measured, and units of measure have been invented for this purpose. But the way the magnitudes relate to each other is described using mathematical equations. EXCERPT #D42YRA p. 17 41 For an account of the term ‘process profile’ see BFO 2.0 [25]. To say that velocity and acceleration are process profiles is to say that they are proper parts of some (intuitively) larger process of motion. They are that part upon which our focus is directed when we carry out the corresponding measurement. EXCERPT #CT9ASP p. 17 42 Or spacetime, in the language of the general theory of relativity. EXCERPT #9Z3LN6 p. 17 EXCERPT #ARRYBH p. 18 From the perspective of mathematics, the magnitudes used in the models are mathematical structures. From the perspective of physics, they describe properties of system elements and their interactions. 43 SECTION #FJ2UE9 4.2 Entangled photons model EXCERPT #QA8M5A p. 18 We now consider a very simple, highly artificial (but experimentally realised) quantum system consisting of two particles (entangled low-energy photons). We use this system as our example, because it enables us to discuss many important ontological questions relating to the boundary between classical and modern physics. We believe that an ontology that can cope with this system can be used for any system in the quantum domain. It can be used also for the general theory of relativity, because there, too, the main problem is an adequate representation – and provides an understanding – of its mathematical components. EXCERPT #6Y5ZJA p. 18 A photon is simply a system element in the physics ontology, a particle. The existence of an entangled system is commonly seen as proving that quantum physics leads to the metaphysical conclusion that there are non-local dynamic effects in nature, or in other words that there is “action-at-a-distance.” [18, p. 486] To see whether this conclusion follows, we now analyse this type of system ontologically. EXCERPT #7QLH9L p. 18 Using a technique called ‘spontaneous parametric down-conversion’ one single high energy photon of spin one can be converted into a pair of entangled low-energy photons a and b each of spin half , as outlined in [3]. The photons obtained in such experiments are highly artificial, in the sense that a process of the given sort can be realized only in very special artificially contrived circumstances. EXCERPT #P8376H p. 18 Parametric 44 down-conversion “is a nonlinear instant optical process that converts one photon of higher energy (namely, a pump photon), into a pair of photons (namely, a signal photon, and an idler photon) of lower energy, in accordance with the law of conservation of energy and the law of conservation of momentum.” 45 EXCERPT #DBPZR9 p. 18 The spin is an immutable inner quantum property of a particle that has no universal. Rather, in place of the universal, we have a mathematical entity (see page 9). Nevertheless, we can perform a measurement of spin experimentally, for example by exploiting the real magnetic moment caused by it, an occult magnitude. When we say that we measure spin, we declare that the measurement is related to certain mathematical entities we have decided to use in building up quantum mechanics. Spin has the characteristics of the classic angular momentum, namely it satisfies a conservation law and it can undergo geometric transformation. Yet, it cannot be explained as the rotation of a mass. That is just a pretty picture. EXCERPT #EAMDUB p. 18 43 An overview is given in figure 10. EXCERPT #PNJ7ZU p. 18 44 The process is called parametric because its underlying quantum effect can be modelled using a parametric (exponential family derived) non-linearity. EXCERPT #2H6EMJ p. 18 45 Source: https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion . EXCERPT #FWHJ9H p. 18 EXCERPT #HK95XG p. 19 When we say that the two particles are entangled, what we mean is that the spin of the particles is complementary, in the sense that if a is spin up in z -direction, then b is spin down in that direction and vice versa. Because the particles are in a state of superposition, which one is spin up and which one down is not known before a measurement on one of the respective particles is performed. This state can be modelled using the singlet 46 wave function: EXCERPT #Z6558T p. 19 |\phi_0\rangle = \frac{1}{\sqrt{2}} (|z_a^+ z_b^-\rangle - |z_a^- z_b^+\rangle), \quad (4) EXCERPT #H5T29S p. 19 where |z_a^+ z_b^-\rangle is a quantum state in which particle a has the physical property S_{az} = +1/2 , particle b has the physical property S_{bz} = -1/2 , and |z_a^- z_b^+\rangle has the analogous meaning with the opposite sign. The qualities S_{az} and S_{bz} are the spin quantities of the particles in the z -direction. In MathO, they are classified as projectors (see 4.2.1.4). The equation says that these properties are superposed in opposing particle states (the particle state at a point in time, i.e. the set of measurable values of the particle's non-invariant properties at a point in time). To understand what this means, we need to analyse the ontologies of mathematics (MathO) and of magnitudes (PhysO) as they apply in this entangled particle model, to which we now turn. SECTION #2LXWR4 4.2.1 Mathematical ontology of the entangled photon model EXCERPT #58MAY8 p. 19 In the entangled photon model formed by equation (4), the following entities appear which are not present in the harmonic oscillator model (eqn. (2)): EXCERPT #SCZNA9 p. 19 (1) fraction and square root. These are operators like multiplication or addition. (2) parentheses which are paired precedence operators used to change or highlight precedence in equations. Here they allow the distributive usage of the factor 1/\sqrt{2} to simplify the appearance of the expression. EXCERPT #WHME9H p. 19 Given the set-theoretical nature of the mathematics ontology, we have the ontological structure shown in Fig. 5. EXCERPT #NAQ4WZ p. 19 But there is also the Dirac notation |\dots\rangle , which is used to denote a state (vector) in a finite Hilbert space \mathcal{H} as follows: 47 EXCERPT #EQ89SF p. 19 |\psi\rangle = \sum_j |j\rangle \langle j|\psi\rangle, \quad (5) EXCERPT #2MXEX5 p. 19 46 Singlet because the entangled photons resulted from one single photon and conserve its energy although they are separated in space. EXCERPT #Q6YPLB p. 19 47 Chapters 1 to 6 of [12] give an excellent introduction to the mathematics of quantum mechanics on which our account here is based. EXCERPT #EJKQK7 p. 19 EXCERPT #H6LH7J p. 20 graph TD constructor[constructor] --> simple_operator[simple operator] simple_operator --> xy["x/y"] simple_operator --> sqrt_x["√x"] simple_operator --> precedence_operator[precedence operator] precedence_operator --> dots["(...)"] A hierarchical diagram showing the structure of simple operators. At the top is a box labeled 'constructor'. A line connects it to a box labeled 'simple operator'. From 'simple operator', three lines branch out to three boxes: 'x/y', 'sqrt(x)', and 'precedence operator'. From 'precedence operator', a line connects to a box labeled '(...)'. EXCERPT #NBXAF7 p. 20 Figure 5: Simple operators EXCERPT #C5FL98 p. 20 so that the state vector |\psi\rangle is the sum of all the amplitudes (a quantum mechanics specific pre-probability defined in next paragraph (eqn. 6)) of state |\psi\rangle to be in each of the base states |j\rangle of \mathcal{H} multiplied by |j\rangle . 48 EXCERPT #E4USW8 p. 20 \mathcal{H} is equipped with an inner product [12, ch. 3-4] between two wave function vectors |\phi\rangle and |\psi\rangle given by EXCERPT #W3A28D p. 20 \mathcal{I}(|\phi\rangle, |\psi\rangle) = \langle\phi|\psi\rangle = \sum_m \phi^*(m)\psi(m), \quad (6) EXCERPT #QZRULS p. 20 where the m are the base vectors of a finite Hilbert space (see paragraph 4.2.1.1 and [12, ch. 3]). This inner product is used in physics to express the probability amplitude of state |\psi\rangle to move to state |\phi\rangle . It expresses a relationship between the two states as a probabilistic measure in the form of a complex number. 49 The square of its absolute |\langle\phi|\psi\rangle|^2 is the corresponding quantum probability (Born rule). For example, the inner product \mathcal{I}(|\phi\rangle, |\psi\rangle) (eqn. 6) is used to express the amplitude of a particle in state |\psi\rangle with a momentum p to be found at a position x as EXCERPT #LXJ3DA p. 20 \langle x|\psi\rangle = \psi(x) \propto e^{+ipx/\hbar}, \quad (7) EXCERPT #U6CGN2 p. 20 which is a complex number. EXCERPT #FXPKQ5 p. 20 Here we have the constructor \propto which expresses proportionality, the dependence of one EXCERPT #C56TWH p. 20 48 For an introduction to quantum mechanics, see volume 3 of [11]; Dirac notation is explained in detail in [12], chapters 3-4, 6-7. EXCERPT #TKQ5QZ p. 20 49 The probabilistic view of quantum mechanics has been formalised thoroughly, for example in Griffiths [12, ch. 5]. We do not show the ontological representation of this formalism here, but it can be obtained in a straight-forward manner. EXCERPT #UGAQ8S p. 20 EXCERPT #3LZ34Q p. 21 magnitude (output) upon one or more others (input) 50 , the exponential function, a relation, and the imaginary number i = \sqrt{-1} . EXCERPT #5W4FJD p. 21 The denominator of the exponential term is the reduced Planck constant \hbar which we encountered in section 2.2.1. From the perspective of mathematics, the Planck constant is just a scalar. EXCERPT #CUBAMR p. 21 How are these entities treated ontologically? EXCERPT #9DWSA5 p. 21 4.2.1.1 Hilbert space A Hilbert space \mathcal{H} used in quantum physics is a vector space over the field of complex numbers \mathbb{C} endowed with an inner product (shown in eqn. (6) above, and in 4.2.1.3) that turns it into a metric space. As a metric space, it is complete with respect to the norm induced by the inner product, i.e. every Cauchy sequence 51 of points in \mathcal{H} has a limit that is also in \mathcal{H} . In many highly idealised models of physics, it is conceived as finite, but there is an infinite Hilbert space for quantum mechanics as well, which was conceived by Dirac. This conception is problematic, however, because to define it one must use the Dirac function, whose value is zero everywhere except at zero, yet whose integral over the entire real line is equal to 1. EXCERPT #JUD5HK p. 21 The problem is that a function cannot have an integral of 1 on an infinitely small domain interval. 52 So this function is mathematically irregular (it cannot be represented as a locally integrable function \delta(f) = \int_{\Omega} \delta(x)f(x)dx = f(0), \Omega \subset \mathbb{R}^n ). But it is needed in order to maintain the linear independence of the states in an infinite-dimensional Hilbert space. We have here, then, an important example of the limits of mathematical modeling of nature. EXCERPT #Y7PGTT p. 21 For a finite Hilbert space \mathcal{H} , an orthonormal state basis of non-zero kets \{|\phi_1\rangle, |\phi_2\rangle, \dots\} can be defined so that \forall j, k with j \neq k \langle \phi_j | \phi_k \rangle = 0 (all kets are orthogonal) so that EXCERPT #36ASV7 p. 21 \langle \phi_j | \phi_k \rangle = \delta_{jk}. EXCERPT #B4J89N p. 21 Given that the edges in the mathematics ontology are subset-relations, \mathcal{H} can be represented as shown in Fig. 6. EXCERPT #5SGSZA p. 21 Note that in quantum physics the Hilbert space is the phase space of the modelled system. EXCERPT #4PJUKA p. 21 50 Proportionality is ubiquitous in physics, for example the harmonic oscillator shown in eqn. (2) has the proportionality relation F \propto -x . EXCERPT #EMD274 p. 21 51 A sequence in which the distance of its elements shrinks arbitrarily as the sequence progresses. EXCERPT #CPPBZC p. 21 52 There is a derivation of the Dirac function as a linear form acting on functions in the theory of distributions (L. Schwartz). EXCERPT #27DB3N p. 21 EXCERPT #V62N6Z p. 22 graph TD A[structure] --> B[algebraic structure] B --> C[vector space (C)] C --> D[metric space] D --> E[H] A vertical hierarchy diagram showing the ontological context of a Hilbert space. The boxes are stacked from top to bottom: 'structure', 'algebraic structure', 'vector space (C)', 'metric space', and 'H'. EXCERPT #BKFR6M p. 22 Figure 6: Hilbert space in ontological context. EXCERPT #Z4HR58 p. 22 4.2.1.2 State vector An element of a finite \mathcal{H} is a quantum state vector |\psi\rangle which can be expressed as a linear combination of the basis vectors j of \mathcal{H} as indicated in equation (5). One or more state vectors can form a subspace \mathcal{A} \subseteq \mathcal{H} . Ontologically we have the structure shown in Fig 7. EXCERPT #K4B2EY p. 22 graph TD A[H] --> B[A] B --> C[psi] A vertical hierarchy diagram showing the ontological context of a quantum state vector. The boxes are stacked from top to bottom: 'H', 'A', and 'psi'. EXCERPT #CM6VKZ p. 22 Figure 7: Quantum state vector in ontological context. EXCERPT #48NZDB p. 22 4.2.1.3 Inner product An inner product (the operator is written as \cdot ) is a relation which assigns a scalar to two vectors, for example: \mathbf{W} = \mathbf{F} \cdot \mathbf{s} = |\mathbf{F}| |\mathbf{s}| \cos \phi , where \mathbf{W} , \mathbf{F} and \mathbf{s} are the work, force and distance vectors, respectively, and \phi is the angle between the force and the distance vector. EXCERPT #9JMZR9 p. 22 In a quantum Hilbert space, the inner product \mathcal{I}(|\phi\rangle, |\psi\rangle) defined in equation (6) yields a complex number indicating the amplitude of a particle to change from one state into another. EXCERPT #H389Y6 p. 22 Ontologically, we have the relationships shown in Fig. 8. EXCERPT #8MBEEP p. 22 EXCERPT #75ULYU p. 23 A vertical flow diagram showing the ontological context of the inner product. It consists of four rectangular boxes connected by vertical lines. The top box contains the expression 'R ⊆ (A × B)'. A vertical line connects it to a box labeled 'operator'. Another vertical line connects 'operator' to a box labeled 'inner product'. A final vertical line connects 'inner product' to the bottom box, which contains the expression 'I(|ϕ⟩, |ψ⟩)'. EXCERPT #C2L5RH p. 23 Figure 8: Inner product in ontological context. EXCERPT #AZUGY6 p. 23 4.2.1.4 Projector The spin quantities of the particles a and b in equation (4) measured in the z -direction are S_{az} and S_{bz} . In the mathematics of quantum mechanics they are modelled as Hilbert projectors, which are operators with properties defined as follows. EXCERPT #RF7LRM p. 23 We first define an operator A in quantum mechanics. It is a binary relation which maps a ket (a vector subset of \mathcal{H} used to model quantum mechanical states, see [12, ch. 3]) to another ket: EXCERPT #P2PNXV p. 23 A : \mathcal{H} \mapsto \mathcal{H}, A(\alpha |\phi\rangle + \beta |\psi\rangle) = \alpha A(|\phi\rangle) + \beta A(|\psi\rangle) EXCERPT #24NVXD p. 23 with |\phi\rangle, |\psi\rangle \in \mathcal{H} and \alpha, \beta \in \mathbb{C} . Then a projector is an operator P with the following additional properties: EXCERPT #QSGFGF p. 23 P^2 = P, \quad P^\dagger = P, EXCERPT #8U3CL4 p. 23 where P^\dagger indicates the complex conjugate transpose 53 of P . The |j\rangle\langle j| referred to in equation (5) is also called a dyadic projector. EXCERPT #7AMSXQ p. 23 Ontologically, we have the structure shown in Fig. 9. EXCERPT #CMBU24 p. 23 4.2.1.5 Tensor product To understand the ontology of the entangled photon system we need to understand the tensor product \otimes . In the quantum mechanics formalism of Griffiths, a tensor product of two Hilbert spaces is used to describe two related systems which together form a new system, such as the two particles described in equation (4). This tensor product is defined as follows: Let |a_j\rangle and |b_p\rangle with j = 1 \dots m, p = 1 \dots n be EXCERPT #HWZ2A6 p. 23 53 This is also known as the Hermitian transpose: For any m \times n complex matrix A , this is an n \times m matrix obtained by transposing A and applying the complex conjugate on each entry. EXCERPT #W5UFBG p. 23 EXCERPT #2KVJVG p. 24 graph TD A["R ⊆ (A × B)"] --- B["operator"] B --- C["H – operator"] C --- D["H – projector"] A vertical flowchart showing the hierarchy of quantum projectors in an ontological context. It starts with a box labeled 'R ⊆ (A × B)' at the top, followed by 'operator', then 'H – operator', and finally 'H – projector' at the bottom, connected by vertical lines. EXCERPT #QXL5GW p. 24 Figure 9: Quantum projectors in ontological context. EXCERPT #WNRLEP p. 24 orthonormal bases 54 for the Hilbert spaces \mathcal{A}^m \subset \mathcal{H} and \mathcal{B}^n \subset \mathcal{H} , respectively. Then the collection of mn elements |a_j\rangle \otimes |b_p\rangle forms an orthonormal basis of the tensor product \mathcal{A}^m \otimes \mathcal{B}^n . This is the set of all linear combinations EXCERPT #7E6G9Q p. 24 |\psi\rangle = \sum_j \sum_p \gamma_{jp} (|a_j\rangle \otimes |b_p\rangle), \quad (8) EXCERPT #H4UKKF p. 24 where the \gamma_{jp} are complex numbers [12, sect. 6.2]. SECTION #MDGYDQ 4.2.2 Magnitudes in the entangled photons model EXCERPT #X27P5A p. 24 The model defined by equation (4) contains only one magnitude, the spin, which can take the quantities w^+ = \frac{\hbar}{2} and w^- = -\frac{\hbar}{2} , where w is an arbitrary axis in \mathbb{R}^3 , usually x, y or z . EXCERPT #FE4EFJ p. 24 Here ‘+’ and ‘−’ represent spin qualities referred to as ‘up’ and ‘down’, according to the way that they are measured. EXCERPT #QBAJCQ p. 24 The spin of a particle is the quantum mechanical equivalent of the angular momentum of classical physics, but with the restrictions made at the beginning of this section. EXCERPT #KRVWPZ p. 24 But where the latter can be imagined using our common sense knowledge of the world, this is not so of the former, which can only be understood as a property that leads to certain indirect measurements results (see above page 19). We model this property mathematically. It is a vector magnitude because it has three spatial components \mathbf{s} = (s_x, s_y, s_z) . It is a vector modelling a conserved quality of the particle and is therefore a continuant quality like mass and not a process characteristic like acceleration. EXCERPT #J48JPR p. 24 54 An orthonormal basis is a set of linearly independent basis vectors of a vector space which are all normed to length one. EXCERPT #MEKB2Y p. 24 SECTION #W7M2FJ 4.2.3 The meaning of the model EXCERPT #E5LHTR p. 25 From the ontological analysis of the entangled photon model, we can derive the following interpretation of equation (4), which we repeat for convenience: EXCERPT #8UX6KZ p. 25 |\phi_0\rangle = \frac{1}{\sqrt{2}} (|z_a^+ z_b^-\rangle - |z_a^- z_b^+\rangle). EXCERPT #PNYW3L p. 25 In the notation of Griffiths |z_a^+ z_b^-\rangle = |z^+\rangle_a \otimes |z^-\rangle_b is “an eigenstate of both S_{az} [z-direction spin] for the a particle, eigenvalue +1/2 , and of S_{bz} for the b particle, eigenvalue -1/2 ; the state |z_a^- z_b^+\rangle has a similar interpretation with eigenvalues of the opposite sign.” [13, sect. V A]. 55 That the projectors S_{az} and S_{bz} have the tensor product |z^+\rangle_a \otimes |z^-\rangle_b as eigenstate means that they have a physical interpretation in the model of the spin states of the photons with phase space \mathcal{A}^m \otimes \mathcal{B}^n . One can also show how to add the experimental process of the measurement of the spin to the phase space of the model [13]. EXCERPT #VZWJ3W p. 25 But the superposition expressed as |\phi_0\rangle in equation (4) is not an eigenstate of S_{az} or S_{bz} , and therefore the state cannot be interpreted as a property of the system because the measurement operators cannot be related to it. EXCERPT #NFDGW4 p. 25 Therefore equation (4) and the state it describes have no physical interpretation, and it makes no sense to say that a quantum system with property \phi_0 has any non-trivial property corresponding to a subspace of the Hilbert spaces \mathcal{A} or \mathcal{B} or their tensor product (see 4.2.1.5). From this perspective, the entanglement per se does not mean that there are non-local effects. The experiments performed on entangled systems do, certainly, seem to imply this. 56 But we think that the ontological character of entanglement is in this sense void – that there is nothing that we can learn from this entanglement model about the mode of existence of the world. We can, however, describe the ontological significance of various parts of the entanglement model. EXCERPT #3FRNNP p. 25 Each of the ket-terms on the right side of equation (4) has an ontological meaning in physics, namely opposite spin directions of the two particles. But |\phi_0\rangle is ontologically void, we cannot imagine or understand it. EXCERPT #R78Z7U p. 25 The equation describes a state superposition that is generated using a machine and that can be dissolved using a second (measurement) machine. When the particles are prepared by the first machine, which creates the spontaneous parametric down-conversion of the high energy photon, we actively prepare a state which has no ontological significance in particle physics, but only the state of being engineered (a technical entity – \tau\epsilon\chi\nu\eta\acute{\omega}\nu , Aristotle, Physics Γ). The entangled particles form a pattern (in information theory: contain information) that we can model mathematically using the singlet model EXCERPT #JZT8E6 p. 25 55 In the linear algebra of the Hilbert space describing the quantum system under consideration here, an eigenstate and its eigenvalue are the results of a linear mapping of \mathcal{H} on itself (an endomorphism). EXCERPT #VGTJRB p. 25 56 The consistent histories school [12, 20] has argued that there is no such implication, but at the price of losing world unity, i.e. the view that we can understand the world as a coherent whole. EXCERPT #XAP3Y7 p. 25 EXCERPT #NG2H7B p. 26 of equation (4); but we cannot give it any ontological significance other than that these are artificial particles that show a characteristic pattern. EXCERPT #6PQMBB p. 26 When we measure the particles at different points in space, we do not get any non-local effects, but we just use another machine to recover the information we had earlier introduced. While doing this, we learn about what we can achieve by applying sophisticated machines to natural particles. We therefore learn something that is ontologically valid, namely about our machines. SECTION #74LR6W 5 The ontological relation of common sense, physics and mathematics EXCERPT #FDVHRT p. 26 What can we learn about the relationship of the ontologies of common sense, physics and mathematics from the examples we analysed? This becomes clear when we use the subdivision of the ontology of physics into system entities, magnitudes and models. In classical physics, which is tightly linked to common sense and in which we understand what we are observing using common-sense thinking, all system entities are real-world entities understandable by common sense. Thus they are either universals in the sense of BFO (see figure 10, top of left panel) or they are instances of such universals. EXCERPT #BPDW7V p. 26 The magnitudes of classical physics, however, have a dual character. As universals, they are again parts of the coverage domain of BFO (middle of left panel). When viewed from the mathematical point of view, on the other hand, in other words viewed as parts of mathematical models, they are not. Therefore \text{BFO} \cap \text{PhysO} = \text{PhysO} \setminus \text{MathO} . The models have no BFO part because they do not have instances in reality (bottom of left panel). EXCERPT #HPNU3N p. 26 To see how this works consider the mass used in the forced harmonic oscillator model with dampening (a realistic model). This is the mass of BFO, a quality of a material entity. The magnitude mass used to measure the weight of the entity is this same BFO entity. But the magnitude viewed mathematically is something quite different. Of course, the harmonic oscillator equation cannot be viewed as we can view the real system with a common sense stance, because it exists only as a mathematical entity, and mathematical entities, as we saw, do not have instances in the real world. EXCERPT #LYXGNG p. 26 With modern physics (since 1905), the relationship between common sense, physics and mathematics looks very different (see right hand panel of figure 10). Modern physics has no entities in the sense of BFO , because it relies on mathematics to define system elements, magnitudes and models. The latter are one and all mathematical entities, which have nothing corresponding to them in common-sense reality, the tuple (mathematical entity, measurement) is not discovered but rather declared. EXCERPT #6AEN9X p. 26 For example, an electron is a BFO:material entity which falls within the coverage domain of classical physics. But in quantum field theory (QFT), an electron is a probabilistic field quant and not a BFO entity of any sort. Because all system elements are made of matter, which is described in QFT using mathematical entities (non-universals), there are no QFT system elements that are BFO entities. All magnitudes in modern physics are grounded either in QFT or in the general theory of relativity (for phenomena which QFT cannot model). Yet all their magnitudes are mathematical entities which do not correspond to any natural universals, but are rather only a matter of entity-measurement tuples which we declare in the way described in section 2.2. Only in the model branch of the physics ontology is the relationship to mathematics and BFO the same in classical as in modern physics: there are no BFO entities in either case. EXCERPT #GPGHPL p. 26 EXCERPT #G5T5WQ p. 27 Classical physics Modern physics BFO = PhysO BFO \cap PhysO = \emptyset BFO \cap PhysO = PhysO \setminus MathO BFO \cap PhysO = \emptyset BFO \cap PhysO = \emptyset BFO \cap PhysO = \emptyset Venn diagram for Classical physics System element: a single white oval labeled 'System element'. Venn diagram for Modern physics System element: two overlapping ovals, the left one is white and labeled 'System element', the right one is grey. Venn diagram for Classical physics Magnitude: two overlapping ovals, the left one is white and labeled 'Magnitude', the right one is grey. Venn diagram for Modern physics Magnitude: two overlapping ovals, the left one is white and labeled 'Magnitude', the right one is grey. Venn diagram for Classical physics Model: two overlapping ovals, the left one is white and labeled 'Model', the right one is grey. Venn diagram for Modern physics Model: two overlapping ovals, the left one is white and labeled 'Model', the right one is grey. EXCERPT #UMHURW p. 27 Figure 10: Ontological relationships between common-sense ontology (BFO) and the physics and mathematics ontologies (PhysO and MathO) in classical and modern physics. PhysO: open set, MathO: filled set (grey). Relations between system element, magnitude and model not shown. EXCERPT #T3MA82 p. 27 SECTION #VPLR2G 6 Discussion EXCERPT #JPA9DL p. 28 The approach presented here differs in several ways from the current attempts to formulate an ontology of physics. The epistemological foundation of our view is that modern, mathematical physics as a science mediates between our perceptual experience of the world and mathematical a priori entities which are mind-dependent and do not exist outside the collective consisting of the minds of mathematicians. It is crucial to understand that such entities exist and to grasp their fundamental role as objects of human thinking as well as the ways we think of them and how we relate them to each other ontologically: not via Aristotelian genus-species-subsumption hierarchies, but using set-theoretical relations to describe their taxonomic relations (and otherwise using the full arsenal of mathematical relations). Because they essentially use the abstract entities of mathematics, the systems modelled by physics are abstractions of reality in mathematical form. When a physicist creates a model, he gives a description of a system which uses simplified system properties in order to enable the construction of a mathematical model. Once the mathematical model is available, purely mathematical reasoning processes such as variable substitution or term approximation can be used to refine it. EXCERPT #5F7LKE p. 28 In classical physics the model then has two roles. As a mathematical equation, it is an essential structure which the mathematician uses to manipulate the equation algebraically. But at the same time, it relates real-world magnitudes to each other inside systems. The fact that variables used in mathematical equations are both mathematical entities and represent real magnitudes enables classical physics to mediate between reality and mathematics. Modern theoretical physics, in contrast, cannot achieve this, as there are no available universals. Applied physics can be used to engineer technical systems that can be used to demonstrate the ability of the models to explain and predict aspects of nature. They thereby combine classical and modern physics. But even here the system elements from modern physics are not universals either, as we have seen. When engineers use modern physics for engineering, for example when building a laser, they combine instances of universals together to create technical devices using their knowledge of mathematical entities. EXCERPT #Z598KL p. 28 Our approach to physics ontology with the division of physics entities into systems, magnitudes and models reflects our understanding of the physicist's knowledge generation process. In classical physics, systems are selections from reality which are chosen and delimited as objects of scientific inquiry. Here, the magnitudes are universals. They mediate between the reality of the system and the idealised nature of the model. We saw that the harmonic oscillator model illustrates in which way we abstract from reality to create idealised models in physics, but the model can be refined to an extent that it can model reality very closely, for example, to yield the forced oscillator with damping that models a real electric circuit. Because the model uses Euclidean space ( \mathbb{R}^3 ) as phase space, we can easily imagine the relationship of the model to the sensory reality we perceive in our common-sense view of the world as an environment extending in three spatial and one temporal dimension. The magnitudes we use to link the model to reality can not only be imagined quite well, they can also be experienced: we can feel our own mass and the acceleration and force acting on our bodies, and we can feel the retraction of a spring and also the passing of time as our heart beats and as we breathe. EXCERPT #AYRYAE p. 28 EXCERPT #RB4EMC p. 29 With the second system, the entangled photons , we have again a system that really exists . But the model is to a much greater degree a creature of the mind, and though it models our measurements, it does not represent the real system. As we have shown, though the model defined by equation (4) has a physical state to which it relates, we do not understand it; in this special case, the model merely expresses the artificial information distribution (what we call ‘photon entanglement’) that we have created using parametric down-conversion with a machine . EXCERPT #DU4DCU p. 29 More generally speaking, in modern physics, models merely mediate between the mostly indirect measurement of system entities on one side and the highly abstract essential structures of mathematics on the other. The real part that remains are the measurements to which we have attributed mathematical magnitudes. But we have no universals. EXCERPT #J9PMGD p. 29 The mathematical knowledge stack that is needed to conceive and understand such models is quite deep and broad, and talented students usually need three to four years to acquire it. The model, as we have seen, can be understood only through the view of the mathematical entities it is made of, and even the one magnitude that it describes – the spin – cannot be imagined in the way that we think of magnitudes in classical physics using our natural, common-sense attitude. We cannot think that the particle turns on its own axis, though sometimes the spin is shown like this in textbooks as a pseudo-illustration. Rather, spin is really only a property that we can measure with complicated machines and which we model using quantum projectors (see 4.2.1.4). We cannot imagine this property: we can only think of it as a \mathcal{H} -space projector. EXCERPT #AHFFE5 p. 29 Therefore, post-classical physics cannot be ontologically represented using common-sense-based ontologies, but requires an ontology of mathematics which gives us a possibility of thinking about the ideal entities of it postulates. It is nevertheless a miracle of the human mind that we can use these entities to model and engineer useful machines such as the MRI, the laser, or quantum sensors. SECTION #86KNF3 References EXCERPT #QAXMTL p. 29 [1] ISO/IEC 21838-2. Information technology — Top-level ontologies (TLO) — Part 2: Basic Formal Ontology (BFO) . New York: International Standardization Organization, 2020. [2] Robert Arp, Barry Smith, and Andrew D. Spear. Building Ontologies with Basic Formal Ontology . Cambridge, MA: MIT Press, 2015. [3] Robert W Boyd. Nonlinear optics . Academic press, 2020. [4] Jeremy Butterfield and John Earman, eds. Handbook of the Philosophy of Science: Philosophy of Physics . Amsterdam: Elsevier, 2007. EXCERPT #YUXVS9 p. 29 EXCERPT #BYQWSW p. 30 [5] Nancy Cartwright. How the Laws of Physics Lie . Oxford, England: Oxford University Press, 1983. [6] Nancy Cartwright et al. The dappled world: A study of the boundaries of science . Cambridge University Press, 1999. [7] Gene Ontology Consortium. “The Gene Ontology resource: enriching a GOLD mine”. In: Nucleic acids research 49.D1 (2021), pp. D325–D334. [8] H.-D. Ebbinghaus. “Mengenlehre und Mathematik”. In: Zahlen . Ed. by H.-D. Ebbinghaus. Berlin: Springer, 1992, pp. 300–317. [9] Michael Esfeld, ed. Philosophie der Physik . Berlin: Suhrkamp, 2012. [10] Michael Esfeld and Dirk-André Deckert. A minimalist ontology of the natural world . Routledge, 2017. [11] Richard P. Feynman, Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics (1964) . Boston, MA: Addison-Wesley, 2010. [12] Robert B Griffiths. Consistent quantum theory . Cambridge University Press, 2002. [13] Robert B Griffiths. “EPR, Bell, and quantum locality”. In: American Journal of Physics 79.9 (2011), pp. 954–965. [14] Janna Hastings et al. “The ChEBI reference database and ontology for biologically relevant chemistry: enhancements for 2013”. In: Nucleic acids research 41.D1 (2012), pp. D456–D463. [15] Ingvar Johansson. “The mole is not an ordinary measurement unit”. In: Accreditation and Quality Assurance 16.8 (2011), pp. 467–470. [16] Jobst Landgrebe and Barry Smith. Why machines will never rule the world. AI without fear . London: Routledge, 2022. [17] Peter J Lewis. Quantum ontology: A guide to the metaphysics of quantum mechanics . Oxford University Press, 2016. [18] Tim Maudlin. “Distilling metaphysics from quantum physics”. In: The Oxford handbook of metaphysics . Ed. by MJ Loux and DW Zimmerman. Oxford, England: Oxford University Press, 2003, pp. 461–487. [19] Elliot Mendelson. Introduction to Mathematical Logic . Boca Raton, FL: CRC Press, 1997. [20] Roland Omnès. The interpretation of quantum mechanics . Princeton University Press, 1994. [21] Erich Reck and Georg Schiemer. “Structuralism in the Philosophy of Mathematics”. In: The Stanford Encyclopedia of Philosophy . Ed. by Edward N. Zalta. Spring 2020. Metaphysics Research Lab, Stanford University, 2020. [22] Adolf Reinach. Sämtliche Werke: Textkritische Ausgabe in 2 Bänden . Ed. by Karl Schuhmann and Barry Smith. Philosophia, 1989. EXCERPT #E74C9F p. 30 EXCERPT #YJ95Q4 p. 31 [23] Barry Smith. “An Essay on Material Necessity”. In: Canadian Journal of Philosophy (1992), pp. 301–322. [24] Barry Smith. “Biomedical Ontologies”. In: Terminology, Ontology and their Implementations . Ed. by Peter Elkin. Springer Nature Switzerland AG, forthcoming. [25] Barry Smith. “Classifying processes: an essay in applied ontology”. In: Ratio 25.4 (2012), pp. 463–488. [26] Barry Smith. “Ontology”. In: Blackwell Guide to the Philosophy of Computing and Information . Blackwell, 2003, pp. 155–166. [27] Stefan Thurner, Peter Klimek, and Rudolf Hanel. Introduction to the Theory of Complex Systems . Oxford: Oxford University Press, 2018. [28] Steven Weinberg. The quantum theory of fields . Vol. 1. Cambridge University Press, 1995. EXCERPT #ZG6QY9 p. 31 DOCUMENT #GN66WW Ontologies of Common Sense, Physics and Mathematics SECTION #74LR6W 5 The ontological relation of common sense, physics and mathematics EXCERPT #FDVHRT p. 26 What can we learn about the relationship of the ontologies of common sense, physics and mathematics from the examples we analysed? This becomes clear when we use the subdivision of the ontology of physics into system entities, magnitudes and models. In classical physics, which is tightly linked to common sense and in which we understand what we are observing using common-sense thinking, all system entities are real-world entities understandable by common sense. Thus they are either universals in the sense of BFO (see figure 10, top of left panel) or they are instances of such universals. EXCERPT #BPDW7V p. 26 The magnitudes of classical physics, however, have a dual character. As universals, they are again parts of the coverage domain of BFO (middle of left panel). When viewed from the mathematical point of view, on the other hand, in other words viewed as parts of mathematical models, they are not. Therefore \text{BFO} \cap \text{PhysO} = \text{PhysO} \setminus \text{MathO} . The models have no BFO part because they do not have instances in reality (bottom of left panel). EXCERPT #HPNU3N p. 26 To see how this works consider the mass used in the forced harmonic oscillator model with dampening (a realistic model). This is the mass of BFO, a quality of a material entity. The magnitude mass used to measure the weight of the entity is this same BFO entity. But the magnitude viewed mathematically is something quite different. Of course, the harmonic oscillator equation cannot be viewed as we can view the real system with a common sense stance, because it exists only as a mathematical entity, and mathematical entities, as we saw, do not have instances in the real world. EXCERPT #LYXGNG p. 26 With modern physics (since 1905), the relationship between common sense, physics and mathematics looks very different (see right hand panel of figure 10). Modern physics has no entities in the sense of BFO , because it relies on mathematics to define system elements, magnitudes and models. The latter are one and all mathematical entities, which have nothing corresponding to them in common-sense reality, the tuple (mathematical entity, measurement) is not discovered but rather declared. EXCERPT #6AEN9X p. 26 For example, an electron is a BFO:material entity which falls within the coverage domain of classical physics. But in quantum field theory (QFT), an electron is a probabilistic field quant and not a BFO entity of any sort. Because all system elements are made of matter, which is described in QFT using mathematical entities (non-universals), there are no QFT system elements that are BFO entities. All magnitudes in modern physics are grounded either in QFT or in the general theory of relativity (for phenomena which QFT cannot model). Yet all their magnitudes are mathematical entities which do not correspond to any natural universals, but are rather only a matter of entity-measurement tuples which we declare in the way described in section 2.2. Only in the model branch of the physics ontology is the relationship to mathematics and BFO the same in classical as in modern physics: there are no BFO entities in either case. EXCERPT #GPGHPL p. 26 EXCERPT #G5T5WQ p. 27 Classical physics Modern physics BFO = PhysO BFO \cap PhysO = \emptyset BFO \cap PhysO = PhysO \setminus MathO BFO \cap PhysO = \emptyset BFO \cap PhysO = \emptyset BFO \cap PhysO = \emptyset Venn diagram for Classical physics System element: a single white oval labeled 'System element'. Venn diagram for Modern physics System element: two overlapping ovals, the left one is white and labeled 'System element', the right one is grey. Venn diagram for Classical physics Magnitude: two overlapping ovals, the left one is white and labeled 'Magnitude', the right one is grey. Venn diagram for Modern physics Magnitude: two overlapping ovals, the left one is white and labeled 'Magnitude', the right one is grey. Venn diagram for Classical physics Model: two overlapping ovals, the left one is white and labeled 'Model', the right one is grey. Venn diagram for Modern physics Model: two overlapping ovals, the left one is white and labeled 'Model', the right one is grey. EXCERPT #UMHURW p. 27 Figure 10: Ontological relationships between common-sense ontology (BFO) and the physics and mathematics ontologies (PhysO and MathO) in classical and modern physics. PhysO: open set, MathO: filled set (grey). Relations between system element, magnitude and model not shown. EXCERPT #T3MA82 p. 27 DOCUMENT #GN66WW Ontologies of Common Sense, Physics and Mathematics SECTION #VPLR2G 6 Discussion EXCERPT #JPA9DL p. 28 The approach presented here differs in several ways from the current attempts to formulate an ontology of physics. The epistemological foundation of our view is that modern, mathematical physics as a science mediates between our perceptual experience of the world and mathematical a priori entities which are mind-dependent and do not exist outside the collective consisting of the minds of mathematicians. It is crucial to understand that such entities exist and to grasp their fundamental role as objects of human thinking as well as the ways we think of them and how we relate them to each other ontologically: not via Aristotelian genus-species-subsumption hierarchies, but using set-theoretical relations to describe their taxonomic relations (and otherwise using the full arsenal of mathematical relations). Because they essentially use the abstract entities of mathematics, the systems modelled by physics are abstractions of reality in mathematical form. When a physicist creates a model, he gives a description of a system which uses simplified system properties in order to enable the construction of a mathematical model. Once the mathematical model is available, purely mathematical reasoning processes such as variable substitution or term approximation can be used to refine it. EXCERPT #5F7LKE p. 28 In classical physics the model then has two roles. As a mathematical equation, it is an essential structure which the mathematician uses to manipulate the equation algebraically. But at the same time, it relates real-world magnitudes to each other inside systems. The fact that variables used in mathematical equations are both mathematical entities and represent real magnitudes enables classical physics to mediate between reality and mathematics. Modern theoretical physics, in contrast, cannot achieve this, as there are no available universals. Applied physics can be used to engineer technical systems that can be used to demonstrate the ability of the models to explain and predict aspects of nature. They thereby combine classical and modern physics. But even here the system elements from modern physics are not universals either, as we have seen. When engineers use modern physics for engineering, for example when building a laser, they combine instances of universals together to create technical devices using their knowledge of mathematical entities. EXCERPT #Z598KL p. 28 Our approach to physics ontology with the division of physics entities into systems, magnitudes and models reflects our understanding of the physicist's knowledge generation process. In classical physics, systems are selections from reality which are chosen and delimited as objects of scientific inquiry. Here, the magnitudes are universals. They mediate between the reality of the system and the idealised nature of the model. We saw that the harmonic oscillator model illustrates in which way we abstract from reality to create idealised models in physics, but the model can be refined to an extent that it can model reality very closely, for example, to yield the forced oscillator with damping that models a real electric circuit. Because the model uses Euclidean space ( \mathbb{R}^3 ) as phase space, we can easily imagine the relationship of the model to the sensory reality we perceive in our common-sense view of the world as an environment extending in three spatial and one temporal dimension. The magnitudes we use to link the model to reality can not only be imagined quite well, they can also be experienced: we can feel our own mass and the acceleration and force acting on our bodies, and we can feel the retraction of a spring and also the passing of time as our heart beats and as we breathe. EXCERPT #AYRYAE p. 28 EXCERPT #RB4EMC p. 29 With the second system, the entangled photons , we have again a system that really exists . But the model is to a much greater degree a creature of the mind, and though it models our measurements, it does not represent the real system. As we have shown, though the model defined by equation (4) has a physical state to which it relates, we do not understand it; in this special case, the model merely expresses the artificial information distribution (what we call ‘photon entanglement’) that we have created using parametric down-conversion with a machine . EXCERPT #DU4DCU p. 29 More generally speaking, in modern physics, models merely mediate between the mostly indirect measurement of system entities on one side and the highly abstract essential structures of mathematics on the other. The real part that remains are the measurements to which we have attributed mathematical magnitudes. But we have no universals. EXCERPT #J9PMGD p. 29 The mathematical knowledge stack that is needed to conceive and understand such models is quite deep and broad, and talented students usually need three to four years to acquire it. The model, as we have seen, can be understood only through the view of the mathematical entities it is made of, and even the one magnitude that it describes – the spin – cannot be imagined in the way that we think of magnitudes in classical physics using our natural, common-sense attitude. We cannot think that the particle turns on its own axis, though sometimes the spin is shown like this in textbooks as a pseudo-illustration. Rather, spin is really only a property that we can measure with complicated machines and which we model using quantum projectors (see 4.2.1.4). We cannot imagine this property: we can only think of it as a \mathcal{H} -space projector. EXCERPT #AHFFE5 p. 29 Therefore, post-classical physics cannot be ontologically represented using common-sense-based ontologies, but requires an ontology of mathematics which gives us a possibility of thinking about the ideal entities of it postulates. It is nevertheless a miracle of the human mind that we can use these entities to model and engineer useful machines such as the MRI, the laser, or quantum sensors. ### 54. Tool result: read DOCUMENT #KYQGNH On Credentials SECTION #XBYXRY Barry Smith, Olimpia Giuliana Loddo and Giuseppe Lorini* On Credentials EXCERPT #TP3MHS p. 0 https://doi.org/10.1515/jso-2019-0034 EXCERPT #DNZWWT p. 0 published online August 7, 2020 EXCERPT #2WJEE9 p. 0 Abstract: Credentials play an important role in all modern societies, but the analysis of their nature and function has thus far been neglected by social philosophers. We present a view according to which the defining function of credentials is to certify the identity and the institutional status (including certain rights) of individuals. More importantly, credentials enable rights-holders to exercise their rights, so that for a particular right to be exercisable the right-holder should possess, carry and sometimes show to an authority a document of a specific kind. Driving licenses, identity cards, passports, boarding passes, library passes, credit cards, ATM cards, health insurance cards are all examples of credentials in this sense. Credentials have in every case a bearer, and the bearer should be able to carry them easily on his or her person. Credentials should also be inspectable – not least because credentials can be forged. The authors analyze several historical and contemporary examples of credentials, focusing on the credentials carried by the pilgrims of the Way of Saint James. EXCERPT #G75N9Q p. 0 Keywords: credentials, document acts, deontic powers, Way of Saint James, institutional objects, status indicators DOCUMENT #KYQGNH On Credentials SECTION #WKUZRV 2 What are Credentials? EXCERPT #98MHDL p. 2 The term “credential” derives from the medieval Latin “ credentialis ”, meaning “giving credence to” (from the Latin “ credo ” – “I believe”). 5 The Oxford English Dictionary defines “credential” as a “document or certificate proving a person’s identity or qualifications”. EXCERPT #MKE8EU p. 2 Credentials are not solely linguistic artifacts (indeed we shall see that they are in some cases wordless, see section 3.3). In addition to the information they convey, they also have a material substratum that plays an essential role in the performance of the document acts in which they are involved. This is because, as we shall see, the possession of and control over a material object is a felicity condition (in Austin’s sense) of the associated document acts. 6 The material object may serve in this capacity either through being offered for inspection, as when you hand your passport to the border control officer, or by serving as the channel for signals, as when you hold your watch over the QR code scanner at the gate when entering a secure area. 7 EXCERPT #TDJFGL p. 2 Yet although we need to prove our status as rights-holders using credentials in order to enjoy the associated rights, it is not the credentials themselves that provide us with those rights. Credentials are not “status-making” or “rights-making”. Indeed, to follow the terminology proposed by Hohfeld (1919), credentials provide us with certain legal advantages ; for example, your passport confers new advantages when a visa stamp has been entered. EXCERPT #DLTKDH p. 2 Because credentials typically have nothing to do with the genesis of the rights they grant to their possessor, they are connected not with the having of a right but rather with the exercise thereof. It is generally a necessary condition of acquiring a credential that one possess already the right that the credential certifies. It is important here to distinguish between “being the holder of a right” and “having the deontic EXCERPT #UMRXMY p. 2 5 Curiously, the term ‘document’ has a very similar etymology, since it comes from the Latin “ documentum ”, meaning proof or demonstration. The expression “ documentum dare ” means “to provide proof” or “to demonstrate”. On documents as instruments of verification, see Immorlica et al. (2019). EXCERPT #5GZDPC p. 2 6 By “possession” we mean the material possession of something, which is to say: the state of having something with you or under your control. “Material possession” connotes a simple factual relation between a person and a thing, as contrasted with the legal meaning of “possession”, which connotes a person’s having a legal power over a thing. EXCERPT #KQGHFG p. 2 7 Issuing bus or theater or plane tickets is not just a way of allowing the exercise of rights, but also a way of coordinating the plans of many people, for example, by ensuring that seats are still available when they arrive. EXCERPT #VYQ8EV p. 3 EXCERPT #95XG34 p. 3 EXCERPT #7EXKDA p. 3 power to exercise that right concretely ”. 8 A right such as freedom of movement may long pre-exist the emergence of credentials that enable the exercise of that right. EXCERPT #R98LC3 p. 3 The right to travel, for example, is commonly guaranteed by both national and international legislation. For instance, Article 16.2 of the Italian Constitution provides that “Every citizen is free to leave the territory of the Republic and return to it, notwithstanding any legal obligations”. 9 Article 13 of the Universal Declaration of Human Rights provides that “(1) Everyone has the right to freedom of movement and residence within the borders of each State. (2) Everyone has the right to leave any country, including his own, and to return to his country”. But it is nowadays only the possession of a passport or other similar credentials that gives the possibility to exercise these rights concretely. 10 If you temporarily lose possession of your passport, then certainly your constitutional right to travel will not, for that reason, disappear. Rather, you will only temporarily not be able to exercise it. Even if your passport is lost or stolen while you are abroad, this does not mean that you are trapped in a foreign country forever because it is no longer possible for you to cross national borders. This is because it is possible to request and obtain a replacement credential. Credentials are documents that can be replaced after a process of legitimation. SECTION #K9YZET 2.1 Credentials as Status Indicators EXCERPT #9QAL6N p. 3 As the OED definition of credential makes clear, since a credential is a “document or certificate proving a person’s identity or qualifications” they mainly perform an “epistemic function”. In this sense, if we use Searle’s lexicon, they can be considered “status indicators”. 11 Moreover Searle (1995, 210) explicitly mentions EXCERPT #XFGTZS p. 3 8 Following Searle (1995, 100; 2010, 8f), deontic powers are conventional powers that regulate relations between people. Searle includes in this category rights, responsibilities, obligations, duties, privileges, entitlements, penalties, authorizations, permissions, “and other such deontic phenomena”. EXCERPT #YF89Q4 p. 3 9 See Article 23 of the Treaty of Rome on “The Functioning of the European Union” (1957). On this topic, see Muchmore (2004, 328). EXCERPT #ACK4YL p. 3 10 “Every citizen is free to enter the territory of the Italian Republic and to go out of the territory of the Italian Republic, without prejudice to legally imposed obligation, using a valid passport or a passport substitute document ” (from Law no. 1185, 1967, article 1; our translation. Italics added). EXCERPT #FULS2T p. 3 11 This feature distinguishes credentials from money. Indeed, money is an object of exchange and not an instrument that certifies already existing deontic powers. It is not a status indicator: it is (according to Searle, at least) a status function (Searle 2017). In ontologically more careful terms, an amount of physical money is a deontic artifact that confers powers. Money is not a credential mainly for two reasons: (i) banknotes do not certify anything, they do not say anything about their bearer; (ii) it is true that, as Searle (1995) claims, money gives us a deontic power, but, unlike credentials, it is impossible to exercise the deontic power only by showing a banknote or a coin. If you want to buy something you need to deliver/hand over the money to the seller. EXCERPT #4Z7EP7 p. 4 EXCERPT #6WEH6N p. 4 EXCERPT #DBB7HJ p. 4 some kinds of credential as examples of status indicators (uniforms, passports, driver's licenses). EXCERPT #HCMAKY p. 4 According to Searle (1995, 119), the status indicators are "official representations" required by institutional facts, "because the existence of institutional facts cannot in general be read off from brute physical facts of the situation". As stated by Searle (2006, 21–22): "in order for status functions to be recognized, there typically have to be some sort of status indicators , because there is nothing in the man or the object itself that will indicate its status, since the status is only there by collective acceptance or recognition. Thus, we have policemen's uniforms, wedding rings, marriage certificates and passports, all of which are status indicators." EXCERPT #PTTKWS p. 4 Status indicators document institutional realities. They make them evident and recognizable, without creating by themselves the institutional reality they represent. Nevertheless, they perform an important role also from the ontological point of view: for instance, according to Searle, "the uniform does not constitute being a policeman, but it does symbolize a status-function; and that symbolization, in some form or other, is essential to the existence of the status-function". 12 If institutional realities exist in virtue of social recognition, then the tools/instruments that make it possible for us to recognize these realities also play a role in maintaining them in existence. In this sense, "collective recognition is not enough. There has to be official recognition by some agency, itself supported by collective recognition, and there have to be status indicators issued by the official agency" (Searle 2005, 15). EXCERPT #KDV39B p. 4 Searle's analysis of status indicators neglects two further important features of this epistemic function. First, that status indicators are also physical entities. And second, that many status indicators play an important role also in the realm of deontic powers. Searle does not realize that those status indicators which are credentials do not only perform an epistemic function (a "status-indicating function") and the indirect ontological functions mentioned above. They also have the deontic function identified in this paper, namely of making it "possible to exercise a certain right concretely". 13 EXCERPT #2ELRBT p. 4 12 Searle (2006, 22) writes: "Many societies find that they [the status functions] cannot exist without status indicators, as, for example, the proliferation of identity cards and driver's licenses will attest." EXCERPT #CN2TP2 p. 4 13 Credentials do not give you a new (second-order) legal right to exercise a (first-order) right. Rather, what they give you is just a de facto deontic power to exercise a (first-order) right. Searle comes closest to this insight, when, speaking of Hernando de Soto's highly influential exploitation of the powers of those particular status indicators that are property deeds in developing countries, he asserts that "sometimes the status indicators acquire a kind of life of their own" (Searle 2006, 22). See de Soto (2000). EXCERPT #9LW3FW p. 5 EXCERPT #6V8N7B p. 5 EXCERPT #SVS4UN p. 5 In conclusion: we can say that all credentials are also status indicators, but not all status indicators are also credentials (for example, wedding rings are status indicators but not also credentials). SECTION #83D5QY 2.2 Credentials as Institutional Objects EXCERPT #D7FXY5 p. 5 Credentials can be either valid or invalid. This is because credentials are not mere brute objects, but institutional objects . By institutional objects we mean material things with features that exist only in virtue of a system of constitutive rules (Searle 1969 and Znamierowski 1924) and thus only in the context of institutions (Anscombe 1958 and Searle 1969). 14 The type of each institutional object is determined by its associated constitutive rules. Examples of institutional objects are: a banknote, a check, a chess piece, playing cards, a traffic sign, a testament (document), a contract (document), a ballot paper, a commercial invoice, a receipt, a graduation diploma, a promissory note, a stone border, identity cards, passports, driving licenses, traffic lights, and so on. EXCERPT #KZWNJJ p. 5 Within the system of constitutive rules that make credentials possible, there are constitutive rules of a particular type, that establish conditions of validity for the entities in question (for example, “The ticket must be validated (a) by being punched in the machine (b) before boarding the train”). 15 Conte (1997) introduced the term “anankastic-constitutive rules” for rules of this type. 16 EXCERPT #AGK278 p. 5 Some anankastic-constitutive rules impose limits on activities that require credentials, for instance, rules establishing credential-related conditions on certain acts, for example of crossing a border (crosser must carry a passport), traveling by car (driver must carry a driver’s license), and so on. As we have seen, the possession of a credential gives us the deontic power to exercise one or more associated rights. However, at the same time, the anankastic-constitutive rules associated with that right set limits to its exercise. EXCERPT #77JJS3 p. 5 14 On Znamierowski’s social ontology, see Lorini and Żelaniec (2016). EXCERPT #P98U5Y p. 5 15 See Searle (1969); Gizbert-Studnicki (1975); Lorini (2000). EXCERPT #YCSSQV p. 5 16 More precisely, by anankastic-constitutive rules Conte means rules that set a necessary condition for the validity of what they regulate (for example, the rule that establishes that a signature is a necessary condition of validity for a will). Anankastic-constitutive rules are different in kind from Searle’s constitutive rules. Indeed, whereas the latter create new types of institutional entities and therefore make possible new institutional phenomena, the former presuppose the institutional types created by Searle’s constitutive rules and state validity’s conditions for the institutional tokens of these types. Anankastic-constitutive rules work at the level of felicity of acts, to use J.L. Austin’s terminology. For an investigation on anankastic-constitutive rules see Lorini (2017). EXCERPT #DF9TXQ p. 6 EXCERPT #RZE3QY p. 6 EXCERPT #UFEB8Y p. 6 In addition, credentials themselves can exercise their function only if they fulfill certain conditions of validity set by the anankastic-constitutive rules. For example, where credentials have an expiration date they cannot exercise their function if they have expired. 17 SECTION #29CHSE 2.3 Counterfeiting Credentials EXCERPT #RGDNLB p. 6 As we have seen, each type of credential is associated with a specific type of institutional state of affairs, for example the granting of a right of the corresponding type. You are awarded a degree certificate only after you have acted in accordance with all the appropriate procedures, including passing all necessary examinations and receiving the degree. One way in which a (putative) credential may be invalid is because it was not issued in accordance with the appropriate procedures because of departures from these procedures on the side of either the issuer, or the receiver, or both. EXCERPT #83SWZ8 p. 6 In the case of forged credentials the associated institutional state of affairs typically does not obtain. A forged credential is either a new or a modified artifact designed to look like an authentic credential. Authentic credentials can also be used in illegal ways, for example when a twin uses the driving license of his brother. 18 EXCERPT #VBDS2S p. 6 Many techniques have been devised to prevent the forgery of credentials through the use of signatures, through complex and multiply redundant sets of identifying numbers, bar codes or RFID chips, through graphics and photographic and holographic images, and through stamps or seals (which are institutional objects imprinted upon or attached to other institutional objects). Graphic devices can also be used to confer new deontic powers on the possessor of a credential or to EXCERPT #SGKJ4L p. 6 17 Certain credentials can have some of their functions exercised even if they are no longer formally valid. A first example is the European Health Insurance Card (EHIC), which allows its possessor to receive medical treatment even in another EU member state if this becomes medically necessary. But it can be used even after its expiry date to certify personal data, for instance when using cigarette- or alcohol-vending machines. A second example is documented in the Italian Presidential Decree no. 445, 28 December 2000 (D.P.R. 445/2000), according to which it is not necessary to produce certifications to attest to one's citizenship, personal data, or marital status; it will be sufficient to exhibit a valid ID document. The data will be recorded by acquiring a photocopy of the ID document. It is possible to exhibit an expired document if the subject declares in a note on the photocopy that the data certified by the document are unchanged. EXCERPT #SVZQB8 p. 6 18 A famous example of this illegal use of credentials is the case of the Berlin shoemaker, documented in Carl Zuckmeyer's play The Captain of Köpenick , who "confiscated" more than 4000 marks from a municipal treasury while wearing a Prussian military uniform with a captain's insignia. See Kelsen, ([1934]1992, 9). EXCERPT #CXYLEV p. 7 EXCERPT #29UZJT p. 7 EXCERPT #E677CF p. 7 provide a historical or forensic trail of the functions fulfilled by the credential through time (for example, the hole punched in a bus ticket). SECTION #BPKCKS 2.4 Credentials as Bearer-dependent, Portable, Inspectable Documents EXCERPT #QCH4Z6 p. 7 Connection to a bearer, portability and inspectability are three fundamental characteristics of a credential. Connection means that each credential is connected to its own specific bearer; 19 portability that the bearer should be able to carry the credential easily on his or her person 20 (thus that it be small and light); 21,22 inspectability requires that the credential can be made publicly visible (to a human being, barcode scanner, chip reader). All three characteristics can be realized by wearing the credential, for example as a badge or as attached to a lanyard around one's neck. Generally, credentials should also be immediately recognizable and identifiable as being a credential of the relevant type. Typographical and graphical features of the document are designed to make it easily and immediately identifiable in a single glance, even by someone who does not understand the language of the text written on them. DOCUMENT #KYQGNH On Credentials SECTION #89CNFV 3 Toward a Typology of Credentials SECTION #2ACGZB 3.1 Bearer Credentials versus Identifying Credentials EXCERPT #RNNJBA p. 7 In the typology of credentials we can distinguish, first, between bearer credentials and identifying credentials . The former does not carry any specific information about the possessor. Examples are a bearer bond, a bus ticket, an (unintelligent) EXCERPT #SAHKQB p. 7 19 Indeed, some credential (such as credit cards and bus tickets) can be shared, for example between family members. EXCERPT #RERMJQ p. 7 20 We ignore here artifacts such as car registration plates and mussel stickers, which serve as credentials not for persons but for inanimate objects such as cars or boats. EXCERPT #8PFK6A p. 7 21 Portability (the ability to carry) goes hand in hand with the requirement to carry; the latter, however, may be subject to qualified exceptions. In the US, for example, drivers are required to carry their driving licenses whenever they drive; but failure to carry is at worst a minor offense if a valid license is produced within a reasonable period. EXCERPT #PGF3XR p. 7 22 Portability is commonly considered to be an intrinsic characteristic also of money. An interesting counterexample, documented in Furness, ([1903]2005), is the Rai, or stone money, which are large, circular stone disks used as money in the island of Yap. These stones are too large to move; buying an item with one simply involves agreeing that the ownership of the Rai has changed. EXCERPT #3Y27LL p. 8 EXCERPT #GNBEJ5 p. 8 EXCERPT #HKJKVB p. 8 hotel keycard that allows entrance into a specific room or area of the hotel (a keycard of this type plays the same role as a password). 23 EXCERPT #X2PQQT p. 8 Identifying credentials carry specific data concerning their possessor. Thus where generally, for a bearer credential, simple possession is sufficient to prove entitlement, 24 for an identifying credential possession is necessary but not sufficient, since the bearer must in this case be identifiable, on the basis of what is indicated on the credential, as the person whose credential it is. EXCERPT #WRUCLS p. 8 Identifying credentials often include graphical elements, some of which – including fingerprints, photographs, or signatures – serve alongside descriptions of identifying marks to anchor the credential to some specific person and thereby allow the validation of the legitimacy of the bearer. A signature is part of both the textual and graphical content of an identifying credential because it is composed of signs representing letters of the alphabet that also have a peculiar graphic form that identifies its author (Harris 2000). EXCERPT #Z3PTQH p. 8 Legitimation can consist merely in determining the authenticity of the credential (through a set of techniques that are connected to the design of its material substratum). Or it can involve comparison of the data appearing on the credential with the corresponding features of its bearer (this process may also involve calling up data from a central registry. Where some talk of knowledge by acquaintance and knowledge by description (Russell 1910), the type of knowledge achieved in the latter case, which combines aspects of each of these, might best be called knowledge by comparison . The type of knowledge achieved purely through appeal to the fact that the credential is in the immediate physical possession of the bearer might be called knowledge by co-localization . 25 SECTION #M88RCU 3.2 Credential Templates versus Credentials Proper EXCERPT #7RM7DD p. 8 Second, we can distinguish between credential templates (which need to be filled in ), and complete credentials (credentials proper), which are the result of such filling in. 26 This distinction turns on the fact that many sorts of credentials consist of two sorts of (textual or graphical) content: content that is compulsorily provided for all instances of credentials of the given type; and content that is added in each specific instance. The latter is used to fill in the cells left blank in the former. For EXCERPT #H6WF68 p. 8 23 Both usernames and passwords are nowadays commonly referred to as “credentials”. EXCERPT #WFJEHG p. 8 24 A keycard or a bus ticket can of course be stolen. In that case, it is being used in order to prove something that is not true. The reason for stealing a bearer credential is that it gives the possibility to perform illegally the act that the lawful holder of the credential could perform legally. EXCERPT #HLZ8R9 p. 8 25 The idea of co-localization is elaborated by Stjernfelt (2019), who draws in turn on Peirce. EXCERPT #BRZGPC p. 8 26 The template is ‘unsaturated’ in the terminology of Frege (1952). EXCERPT #NP6QEN p. 9 EXCERPT #ZFUS5Z p. 9 EXCERPT #RMA8G3 p. 9 traditional types of credentials, the template 27 is typically printed; for electronic credentials it is hard-wired into the system through which credentials are generated. SECTION #Q73TWT 3.3 Credentials with Displayed Text versus Credentials without Displayed Text EXCERPT #5ZSCVA p. 9 Third, we distinguish between credentials with displayed text , and credentials without displayed text . The root of this distinction turns on the fact that credential documents are still, in our common thinking, assumed to be a textual, linguistic documents, i. e., to include some visually displayed text that specifies the document's function and the bearer's identity. However, some credentials do not show any textual content of this sort. The ink patterns created by the re-entry hand stamps used by organizers of festivals and other special events are examples of credentials without displayed text – wordless credentials – of this sort. EXCERPT #5TTY57 p. 9 Further examples are badges allowing entry to a building, which may show just a picture representing the bearer. There are also wordless credentials whose specific design contains elements taken from a generally acknowledged symbolism, for example the insignia of rank used on uniforms. Others contain coded content which enables them to activate some specific devices, for example a QR code on a dog collar or an electronic access badge, where the badge reader checks that the badge information is included in the table of authorized badges in the application program and returns a command to open the electronic access badge lock (Champlain 2003, 111–112). Here the credential performs not only a deontic function, but also a forcing function (Norman 2013) or what Searle calls a causal function (Searle 1995, 2010). 28 EXCERPT #FHXZFJ p. 9 27 The term 'template' possibly derives from the French templet , meaning: weaver's stretcher. The more general meaning "pattern or gauge for shaping a piece of work" is first recorded in 1819 (Oxford English Dictionary). EXCERPT #3C63SS p. 9 28 By "forcing functions" Norman (2013, 141) means "strong constraints that can prevent inappropriate behavior". Forcing functions work on the basis of the mere physical structure of the associated artifact. For example, "[s]tarting a car has a forcing function associated with it – the driver must have some physical object that signifies permission to use the car. In the past, it was a physical key to unlock the car doors and also to be placed into the ignition switch, which allowed the key to turn on the electrical system and, if rotated to its extreme position, to activate the engine". A similar definition is that of causal function proposed by Searle. By causal function Searle (1995, 41) means a function that can be performed by an object (for example, a nutcracker) solely by virtue of its physical structure. EXCERPT #J8Q8DB p. 10 EXCERPT #PEWN7L p. 10 EXCERPT #78A27X p. 10 Table 1: Four Types of Credentials. EXCERPT #2H68RJ p. 10 Credentials Bearer Identifying with displayed text bearer bonds passports bearer shares ID cards share warrants credit cards without displayed text military rank insignia a credential without text, but with hand stamps (to a picture identifying its regain access to an event) bearer and a graphical logo of the issuing institution EXCERPT #GVBZWK p. 10 Table 1 combines the first and last of these distinctions. DOCUMENT #KYQGNH On Credentials SECTION #8Y4CDY 4 The Functions of Credentials EXCERPT #3N9BXY p. 10 Credentials serve to make someone (or some machine) believe that some proposition is true (this is their epistemic function). But they also fulfill a range of different deontic functions, which makes them in some ways analogous to performatives in the realm of speech acts. 29 In the simplest case they enable their possessor to exercise certain deontic powers – for example gaining access to a certain area. In a more complex case a credential may have the function to prevent the exercise of rights by enabling discrimination between different classes of persons. The possession of these negative credentials is not a condition for the exercise of a right but eases the imposition of burdens and obligations by an authority (recall the Nazi Judenstern , or the yellow ticket used as substitute passport for prostitutes in the Russian Empire). EXCERPT #LDMDK3 p. 10 In a way, credentials are “deontic artifacts” (like, for instance, traffic signs, roundabouts and traffic lights) since they are material objects that play a deontic role in a way that is intelligible to their (human or machine) addressees (see Lorini and Moroni 2020b). SECTION #QXC782 4.1 Performing Document Acts with Credentials EXCERPT #DFURM5 p. 10 We can now classify the different kinds of document acts performed with the aid of credentials according to the way they are performed: EXCERPT #58VDS8 p. 10 29 A credential itself may perhaps be considered as a kind of performative object. See Strother (2000). EXCERPT #HQQXSC p. 11 EXCERPT #W6VSBX p. 11 EXCERPT #UQ8RNK p. 11 i. thetic document acts (from the Greek thésis /θέσις, meaning: position-creation), which are performed by creating new documents, for instance by issuing a passport or ID card; 30 ii. epideictic document acts (from the Greek epideiknumi /ἐπιδεικνύμι, meaning: to exhibit), which are performed by exhibiting already existing documents, for instance producing a passport to cross a border legally; iii. anaretic document acts (from the Greek anairesis /ἀνάλειψις, meaning: destruction), which are performed through the destruction of the material substratum of a document, 31 for instance when an official destroys a passport to prevent a person from leaving the country. 32 EXCERPT #XK2J2V p. 11 Where thetic document acts have acts such as promising as counterparts in the speech act realm, epideictic document acts correspond to acts such as entering a passport, and anaretic document acts correspond to acts of forgiving a debt or waiving a claim. SECTION #M9UNER 4.2 Deontic Functions and Causal Functions of Credentials EXCERPT #8N9R33 p. 11 In general, credentials have a primary deontic function that typically emerges already in the name of the corresponding credential type. We note that a function that is primary for credentials of one type can be secondary for those of another type. For instance, the primary deontic function of an ID card is to allow its bearer to be identified. However, an Italian driving license, because it includes a photograph of the citizen and a stamp or signature from the issuing state administration, can also be used (in Italy) as a substitute ID Card. 33 EXCERPT #7XTCAX p. 11 30 Sometimes the creation of a new document goes hand in hand with the creation of a new “social object such as an easement or an award of damages. Some documents play both a recording and an object-generating role. The issuance of a paper guarantee, for instance, brings a certain sort of right into existence and provides evidence of this right in the future” (Koeppell and Smith 2014, 222). EXCERPT #SQ9MPN p. 11 31 The distinction between thetic and anaretic performative acts is due to A. G. Conte (1994). See also M.-E. Conte (1985). EXCERPT #8M55G9 p. 11 32 Destroying a document is in this case not a merely physical act (like destroying a physical object out of rage or by accident), for it is an act performed because of its legal consequence (for instance in Italy destruction of one's ATM card is part of the procedure to close a bank account). Another example: in 1520, Luther launched his revolt against Catholicism by burning his bull of excommunication, the Decretals of Clement VI, and other Church documents with which he disagreed. EXCERPT #DRYJS2 p. 11 33 Circular 300/A/744/13/101/3/3/9 of the Ministry of Interior, Department of Public Security on Title IV of the Highway Code (dated January 25, 2013). EXCERPT #ATGB6H p. 12 EXCERPT #BCRUED p. 12 EXCERPT #5G5H2W p. 12 As will be clear from such examples, credentials can simultaneously fulfill both causal and deontic functions. The same goes for a paper bus ticket, whose typographic or graphical features as identified by a physical (human or electronic) scanner allow you (in both the deontic and causal senses of ‘allow’) to board the bus. DOCUMENT #KYQGNH On Credentials SECTION #G6S8LB 6 Conclusions EXCERPT #QKDU5D p. 17 Curiously, the analysis of credentials has been neglected by social philosophers, including those working in the field of social ontology, and we here provide a first systematic study of a phenomenon that plays an important role in our everyday life. 41 EXCERPT #7FJCRC p. 17 Yet as we have seen, credentials provide a series of clear examples of the way a single document act – for example entering a certain stamp into a hostel passbook – can serve a plurality of performative ends. There emerges from our analysis of credentials a distinction between two fundamental categories of our everyday deontology, namely between “holding a right” and “having the deontic power to exercise that right concretely”. EXCERPT #SSK6VR p. 17 We believe that our analysis is significant also because, as paper and plastic credentials give way to electronic credentials in your phone, or in an RFID chip implanted in your earlobe, new types of uses for credentials are being invented that are transforming modern societies, uses which involve execution by software. Already credentials stored on your phone can enable you to get a loan, refill a prescription, disclose your medical data, request a car and driver to be sent to your current location, or take control over a car or bicycle or scooter that is left on the street. At the same time, when you use these services the stored credentials will see to it that your credit record/medical record/credit card account are automatically updated accordingly. EXCERPT #QC6XFD p. 17 As we saw in section 5.1 above, credentials were used already in Biblical times as instruments to control migration flows. Reflecting on the Chinese invasion of Tibet in the 1950s, Pallis (1985, 7) writes: EXCERPT #QVH9TC p. 17 One of the side effects of modern technology has been to place in the hands of those who control the machinery of government a range of coercive apparatus undreamed of by any ancient despotism. It is not only such obvious means of intimidation as machine guns or concentration camps that count; such a petty product of the printing press as an identity card, by making it easy for the authorities to keep constant watch on everybody's movements, represents in the long run a still more effective curb on liberty. In Tibet, for instance, the introduction of such a system by the Chinese Communists, following the abortive rising of 1959, and its application to food rationing has been one of the principal means of keeping the EXCERPT #3YJEZ5 p. 17 41 As we mentioned in section 2.1, an important contribution to the ontology of credentials derives from Searle's research on “status indicators”. The present research on credentials can accordingly also be viewed as a contribution to the ontological investigation of status indicators. EXCERPT #M5FT4W p. 18 EXCERPT #BUHUXC p. 18 EXCERPT #SLR5TX p. 18 whole population in subjection and compelling them to do the work decreed by their foreign overlords. EXCERPT #9TJD97 p. 18 With the growth and ubiquity of electronic credentials, the possibilities for such control have of course increased by orders of magnitude. They will manifest themselves both in good ways – for example in allowing more effective means of controlling traffic flows by exploiting the credentials built into our car Wi-Fi and autonomous driving systems – but also in bad ways, by enhancing the ability of the state to control its citizens' thought and behavior. ### 55. Tool result: read DOCUMENT #PHAFYA Truth and the Visual Field SECTION #JMETCH Truth and the Visual Field EXCERPT #AZ9ZZN p. 1 BARRY SMITH EXCERPT #CMRJAU p. 1 In this study I use the tools of mereotopology (the theory of parts, wholes, and boundaries) to work out the implications of certain analogies between the “ecological psychology” of J. J. Gibson and the phenomenology of Edmund Husserl. I present an ontological theory of spatial boundaries and spatially extended entities. By reference to examples from geography I show that both boundaries and extended entities fall into two broad categories: those which exist independently of our cognitive acts (for example, the planet Earth, its exterior surface); and those which exist only by virtue of such acts (for example: the International Date Line, the state of Wyoming). The visual field, too, can be conceived as an example of an extended entity that is dependent in the sense at issue. I here argue that we can extend this analogy by postulating entities that would stand to true judgments as the visual field stands to acts of visual perception. Such a “judgment field” can then be defined as that complex extended entity which comprehends all entities that are relevant to the truth of a given (true) judgment. The work of cognitive linguists such as Leonard Talmy and Ronald Langacker, when properly interpreted, can be shown to yield a detailed account of the structures of the judgment fields corresponding to sentences of different sorts. Such an account can serve as the basis for a new sort of correspondence-theoretic definition of truth for sentences in a natural language. DOCUMENT #PHAFYA Truth and the Visual Field SECTION #BZ6CLE PREAMBLE: GIBSON AND PHENOMENOLOGY EXCERPT #VKQDLN p. 1 This study is part of a larger project designed to exploit the ecological psychology of J. J. Gibson to yield a new, naturalized interpretation of Husserlian phenomenology. The world, as Gibson points out, is a complex hierarchy of internested levels: molecules are nested within cells, cells are nested within leaves, leaves are nested within trees, trees are nested within forests (Gibson 1979: 101). Each type of organism is tuned in its behavior to entities on a specific level of granularity within this complex hierarchy, to entities which together form what Gibson calls an “ecological niche.” A niche is that into which an animal fits; it is that in relation to which the animal is habituated in its behavior (Gibson 1979: 129). A niche embraces not only objects of different sorts, but also shapes, colors, textures, tendencies, and boundaries (surfaces, edges, and contours), all of which are organized in such a way that they enjoy affordance-character for the animal in question. That is, the given features of the entities in the niche motivate the organism; they intrude upon its life; they stimulate it in a wide range of different though characteristically understandable and familiar ways. The niche shared by all human beings—called by Husserl the “life-world”—is thus such that its basic organizing features are intrinsically comprehensible to the human organism (yielding what Husserl calls the “a priori of the life-world”). These basic organizing features include simple geometrical and topological relations and relations of identity, part, and whole, as well as relations between qualities of different sorts (B. Smith and Varzi, in press [a]). EXCERPT #VV6LNS p. 1 EXCERPT #MFLQ2K p. 2 EXCERPT #LLWYN2 p. 2 According to Gibson, human beings, like other animals, are integrated into the world order via their perceptions and actions in virtue of the fact that these perceptions and actions are pre-tuned to the characteristic shapes and qualities and patterns of behavior of the respective environments. In the case of human beings this mutual entanglement is extended further through cultural phenomena, above all through language and its associated institutions. To learn a language is in part also to extend the range of objects in relation to which we are able spontaneously to adjust our behavior. Just as our experiences of objects of perception in our everyday environment are characteristically and for the most part not subject to deliberation, so our experience of the words of a language we thoroughly understand is spontaneously bound together completely with our grasping of the associated meanings and thereby also with our being spontaneously directed toward corresponding objects in the world. EXCERPT #P3KJKC p. 2 The concept of niche can be extended and generalized beyond the basic level of the life-world of common sense in other ways as well. A humanly extended niche might include, for example, the interior of a cockpit, the floor of a stock exchange, or the environment of a keyboard and computer screen; it might include a library or a highway system, or it might include the world of a scientific theory or of some other specialist activity (for example, of measuring or legislating) in which a human being feels at home. For as Gibson himself intimated, and as Husserl argued in detail in the second book of his Ideas (see also the extremely provocative Katz 1987), the activity of scientific theorizing on the part of different specialist sciences can be compared in important respects to the behavior of animals and humans in their respective natural environments. There is a deep-rooted analogy between the relationship of animal or human behavior to niche or life-world on the one EXCERPT #AMLVJJ p. 2 EXCERPT #CK5HGJ p. 2 hand, and the relationship of the scientist (or a specialist community of scientists) to the corresponding scientific subject matter on the other. EXCERPT #XJ6DQ9 p. 2 The basic axiom of Husserl's constitutive phenomenology is this: that all objects refer back to corresponding acts in which they are (or can be) given. All entities, on whatever level, are correlates of corresponding acts, and each subject is directed in its acts toward a corresponding world of correlates: “As person I am what I am (and each other person is what he is) as subject of a surrounding world. The concepts of ego and surrounding world are related to one another inseparably” ( Ideen II, §50). The world of common sense is the accomplishment of a community of persons recognizing one another (or better: taking one another for granted) as being in agreement. The things of the commonsense world are direct correlates not of abstract, theoretical experiences, but of intuitive experiences; they are “things we see, grasp, and touch, just as we, and other people, see them, grasp them, etc.” ( Ideen II §62; see also B. Smith 1995). EXCERPT #KEPZP8 p. 2 From the basic axiom it follows that physical things, too, can be nothing other than the correlates of certain acts, namely of the theoretical acts of physicists. Physical nature is then for Husserl the common “surrounding world” of physicists, precisely as they know of it in their theories and conceived as infinitely extended in perfect regularity. Other such special “surrounding worlds” can be distinguished also. Thus, for example, there are the worlds of mathematical or legal objects, of financial instruments, of chess, and so on. Each such realm of objects is, from Husserl's point of view, an interpersonal, cultural accomplishment, presupposing a certain association of human beings. It is a product of “constitution.” EXCERPT #ETP2ED p. 2 The Gibsonian perspective has obvious implications for our understanding of the theories of the life-world (or of Umwelt or “bodily space”) put forward, not only by Husserl, but also by Scheler, Heidegger, Merleau-Ponty, and other phenomenologists in their various writings. This same perspective yields also, however, a radically new, realist interpretation of Husserl's “constitutive phenomenology”: for constitution is not, from the Gibsonian point of view, the creation of a new domain of entities in some spurious “transcendent” realm: rather, it is the carving out of a new sort of niche from within the already existing surrounding world of the relevant subject or specialist community (B. Smith, forthcoming). EXCERPT #Y3SYLS p. 2 The Gibsonian perspective has implications also for our understanding of the relation of individual acts to their corresponding objective correlates. Thus consider once again the analogy between the relationship of animal or human behavior to niche on the one hand, and the relationship of the specialist community of scientists to its corresponding scientific subject matter on the other. This same analogy can be applied not merely to global behavior-patterns but also to specific acts: an act of visual perception stands to a visual field as an act of (true) judgment stands to a fact or state of affairs. I devote the bulk of what follows to working out some of the implications of this latter analogy. EXCERPT #S3CMWM p. 3 SECTION #D6U7A6 1. TYPES OF BOUNDARIES EXCERPT #6J3AVD p. 3 We most commonly demarcate reality along what we might call natural or bona fide boundaries. The most prominent (and most salient) examples of such natural boundaries are the outer boundaries of objects in space and processes in time. Such natural boundaries are boundaries in the things themselves. They would exist even in the absence of all articulating activity on our part. The natural boundary of you is (roughly speaking) the surface of your skin. EXCERPT #CBLSFV p. 3 We can also recognize internal natural boundaries—for example, the boundaries around your heart, lungs, and other organs. But we can recognize unnatural boundaries as well, that is to say, boundaries, both internal and external, which correspond to no genuine heterogeneity (natural articulations) on the side of the bounded entities themselves. The boundary of Utah corresponds to no local physical discontinuity, and to no qualitative heterogeneity (of material constitution, color, texture, etc.) in the world itself. EXCERPT #P3SG2Y p. 3 Let us call inner and outer boundaries of this second sort fiat boundaries, a terminology that is designed to draw attention to the sense in which the latter owe their existence to acts of human decision or fiat or to cognitive phenomena of associated sorts (B. Smith 1994; B. Smith and Varzi, in press [b]). The plausibility of extending our ontology by acknowledging fiat boundaries in this way lies first of all in the fact that all of the standard distinctions we can make between types of natural boundaries can be straightforwardly applied to their fiat counterparts as well. Thus we can distinguish between natural and fiat boundaries of different numbers of dimensions: the equator, like the edge of this table, is a one-dimensional boundary; the North Pole, like the corner of this table, is a zero-dimensional boundary. We can distinguish between complete and incomplete boundaries, whether natural or fiat: the Western Front (anno 1916) and the boundary between France and Germany are examples of incomplete fiat boundaries, in the sense that they do not of themselves serve to demarcate any object in the way in which this is done, for example, by the equator (which demarcates the two hemispherical surfaces of the Earth) and by the boundary of my body (which demarcates the corporeal me). We can similarly distinguish between endur- EXCERPT #KVUMWC p. 3 EXCERPT #HAAEMW p. 3 ing and transient natural and fiat boundaries: the Western Front, again, is an example of a transient fiat boundary, the boundary of Iceland (modulo the movement of tides) is an example of a (relatively) enduring fiat boundary, and the boundaries of this cloud and of that stone are transient and enduring natural boundaries, respectively. We can distinguish equally between crisp and fuzzy natural and fiat boundaries: the equator is crisp and the product of fiat; the boundary of this cone of light is crisp but exists as part of the natural world; the boundary of Asia is fuzzy but it is still (we can suppose) a product of fiat; the boundary of the polar ice cap is likewise fuzzy but a product of nature. Deserts, valleys, dunes, and so on, are delineated not by crisp outer boundaries but rather by boundary-like regions that are to some degree indeterminate (Cohn and Gotts 1994). Most peninsular objects (including fingers, hands, arms) are characterized likewise by the possession of indeterminate boundaries in the area where they abut their larger hosts. (We leave to one side here the question whether, as quantum physics seems to suggest, there is an additional type of boundary indeterminacy that pertains to all material objects given in our normal experience.) SECTION #6AFHK7 2. FIAT OBJECTS EXCERPT #VTLRVJ p. 3 Once fiat boundaries have been recognized, then it becomes clear that the opposition between bona fide and fiat can be drawn in relation to objects also (B. Smith 1994). Fiat objects are those objects which exist only by virtue of the fact that some corresponding (complete) fiat boundary has come to be drawn. Examples of genuine objects are you and me, the planet Earth. Examples of fiat objects are all geographical entities—Dade County, Florida, the United States, the Northern Hemisphere—which are demarcated in ways that do not, or do not everywhere, respect qualitative differentiations or spatiotemporal discontinuities in the underlying territory. And then, not the least important reason for admitting fiat objects into our general ontology turns on the fact that most of us live in one (or in what turns out to be a nested hierarchy of such objects). EXCERPT #CDL2NC p. 3 Clearly, most geographical fiat objects will have boundaries that involve a combination of bona fide and fiat elements: the shores of the North Sea are bona fide boundaries, not, however, its boundaries at those points where it abuts the Atlantic. The Western Front was built out of bona fide stretches, where opposing armies faced off against each other in more or less linear fashion, knitted together by interspersed fiat stretches, generated algorithmically, by joining up the dots (roughly: a front line is the shortest distance transecting the region separating two neighboring but opposed infantry companies). We might in light of this example distinguish between the following: EXCERPT #ECYKNB p. 4 EXCERPT #L9P46W p. 4 EXCERPT #NZRZFG p. 4 1. Fiat boundaries, every portion of which is laid down by explicit human fiat (for example, by treaty, or by drawing lines on a map) 2. Fiat boundaries, stretches of which are determined in whole or in part in relation to natural boundaries (or to preexisting fiat boundaries) on the basis of geometrical algorithms (most determinations effected by boundary commissions are of this sort, for example, when a boundary is specified as lying in the middle of a river bed) 3. Fiat boundaries determined algorithmically not in relation to boundaries but in relation to other, real properties of the underlying subject matter: the boundaries depicted in dialect and electoral atlases are of this sort, as are the transient boundaries depicted in weather maps SECTION #EMC9A9 3. FIAT BOUNDARIES AS CREATED ENTITIES EXCERPT #DXLNG2 p. 4 What begins as a fiat geographical boundary may evolve over time into a natural boundary, reflecting not merely new features of the landscape but also differences in the language or dialect or trading habits of those who live on either side—all of which suggests that we develop a view of geographical boundaries as created entities, entities subject to the vagaries of history. Thus fiat boundaries seem to have a beginning in time, and geographical boundaries in general are such as to instantiate one of a number of characteristic patterns of boundary evolution (Prescott 1978). EXCERPT #UT225T p. 4 Against this, however, is an alternative view according to which spatial boundaries are merely abstract mathematical constructions and are thus not the sorts of things that can be subject to historical change. Boundaries are not created, on the given view, but discovered or picked out from the infinite totality of all geometrically possible alternative ways of dividing up (say) the surface of the earth. Utah, on the given reading, existed long before its boundaries were first picked out by the responsible administrators, and it may similarly continue to exist for long after human beings have ceased to occupy this planet. EXCERPT #FQG532 p. 4 Are fiat boundaries, and the fiat objects they circumscribe, discovered or created? The former view has in its favor the virtue of ontological parsimony: only one sort of boundary needs to be admitted into our ontology, where on the latter view we should have to admit in addition to purely geometrical boundaries also certain historically determined boundaries that coincide with these. An argument in favor of the existence of historically created boundaries can however be formulated as follows. We note, first of all, EXCERPT #7VCU8W p. 4 that “Hamburg” is an ambiguous term, standing on the one hand for a certain city (Hamburg-Stadt) and on the other hand for a certain administrative entity (Hamburg-Land), which is one of the constituent Länder (states, cantons) of the German Federal Republic. Hamburg-Stadt and Hamburg-Land are distinct entities, which happen to coincide spatially. On the geometrical account of boundaries (boundaries are discovered, not created) Hamburg-Stadt and Hamburg-Land have identical boundaries; on the alternative, historical reading, they have boundaries which are distinct from each other and from the underlying geometrical boundaries, even though all three sets of boundaries happen to coincide spatially. EXCERPT #DQYPQX p. 4 Why on earth, now, should we not embrace the more parsimonious reading and save ourselves the embarrassment of, in this case, three complete sets of boundaries in the very same place? The answer to this question turns on the possibility of divergent histories. The boundary of Hamburg-Stadt might, after all, have lain elsewhere. Each geometrically determined boundary is, however, as a matter of necessity exactly where it is. If, therefore, the boundary b of Hamburg-Stadt were identical to (and not merely contingently such as to coincide with) a certain geometrical boundary, then we should have to swallow the simultaneous truth of (1) b could have lain elsewhere and (2) b is as a matter of necessity exactly where it is. EXCERPT #RB9SSR p. 4 One must reject the temptation to suppose that we are confronted here with a mere verbal dispute, which could be resolved by some alternative choice of words. For consider the in-many-ways-analogous case of Bremen. “Bremen,” too, is ambiguous; it refers on the one hand to a certain city, and on the other to a certain Land. In this case, however, the boundaries of Bremen-Stadt and -Land do not coincide. And of course something analogous might hold in the case of Hamburg, too: it would be an administrative act of no great difficulty to bring it about that, as of tomorrow, the boundaries of Hamburg-Stadt and -Land should likewise be distinct or be, in however subtle a fashion, differently defined. This implies, however, that already today we are dealing with entities that could have distinct histories, and this is possible only if the entities themselves are already distinct. SECTION #ADGTJ6 4. FIAT OBJECTS IN PERCEPTION EXCERPT #PF95T8 p. 4 Geographical boundaries such as those of Hamburg-Stadt and -Land are, if the argument above can be accepted, human creations that are subject to the vagaries of history. It is as if, through the evolution of our political and administrative and legal practices and through practices relating to property law, new boundaries come to be inscribed in reality in addition to the natural boundaries in relation to which these supernumerary fiat boundaries are constructed and in terms of which they are defined. As will already have become clear from some of the examples mentioned above, however, we are confronted in our everyday experience also with a great wealth of such supernumerary boundaries of a more transient sort, boundaries created by our acts of perception and by human cognitive processes of other sorts. Imagine, for example, that I am outdoors on a clear day looking out over the landscape. One prominent object in the visual field hereby determined is my present horizon, a transient and incomplete and roughly linear boundary between earth and sky, whose existence and nature are determined not by any simple act of decision or fiat on my part but by my very existence as a visually perceiving subject in a given location at a given time, as also by the perimetric properties of my visual system, by topographical features of the location, and by the laws of optics. Note, however, that even in this case there is a residual element of human decision at work, namely, the decision on my part to turn my head in a given direction at a given moment. EXCERPT #7RPDNV p. 5 EXCERPT #7MX4S2 p. 5 EXCERPT #FUMALN p. 5 The horizon is a component object of the visual field, and the latter may be defined, with Ewald Hering, “as the totality of real objects imaged at a given moment on the retina of the right or left eye” (1964: 226). Let us assume that the eye sees in normal fashion, that it is not momentarily startled, and that there are no tricks, mirrors, or special equipment, and no clouds, fog, stained glass, or the like, in its way of seeing given objects. The depictions of the visual field provided by Ernst Mach (1959: 19; see Figure 10.1) and by Gibson (1979: 118f.) tell us that the objects making up the visual field according to Hering’s definition are primarily the surfaces of three-dimensional entities (the surfaces of walls, trousers, bookends, etc.). In fact we can distinguish three sorts of component object: (1) two-dimensional surfaces (with their own intrinsic curvature in three-dimensional space); (2) the boundaries of these two-dimensional surfaces (both one-dimensional edges and zero-dimensional vertices; both fiat and natural boundaries: the horizon is an example of a one-dimensional fiat boundary in the interior of the visual field); and (3) the one-dimensional psychologically induced fiat outer boundary of the visual field itself. The boundary of the visual field is a complex, subtle, ever-changing and gappy patchwork of physical surfaces and other components. The patchwork is “open,” topologically speaking: its external boundary is not a part of the visual field itself (as death is not an event in life). The patchwork is organized further in terms of an opposition between entities (“figures”) in the focus of attention, which characteristically manifest determinate boundaries, and entities which have indeterminate boundaries and which are experienced as running on (as “ground”) behind them. EXCERPT #VVG4DD p. 5 A black and white line drawing by Ernst Mach (1959) titled 'The visual field'. It depicts a person lying on their back on a wooden floor, looking up at a ceiling with a grid of square panels. The person's head is tilted back, and their arms are resting on their knees. The drawing illustrates the concept of the visual field as a totality of objects imaged on the retina, showing how the person's position and orientation determine what is visible. EXCERPT #AAJUN5 p. 5 FIGURE 10.1. The visual field. (Mach 1959) SECTION #NGNLAC 5. LANGUAGE-GENERATED FIAT OBJECTS EXCERPT #RT5BNY p. 5 A further important class of transient fiat boundaries are those effected through our everyday use of natural language. As Talmy puts it, drawing attention to a hitherto insufficiently studied analogy between the articulations effected by the descriptive use of language and those effected by acts of visual perception: “Linguistic forms can direct the distribution of one’s attention over a referent scene in a certain type of pattern, the placement of one or more windows of greatest attention over the scene, in a process that can be termed the windowing of attention” (1996: 236). Common to all such processes is the determination of a boundary, which might be a sharp line or a gradient zone, and whose particular scope and contour—hence, the particular quantity and portions of material that it encloses—can be seen to vary from context to context. EXCERPT #2ERFWV p. 6 EXCERPT #V98XHK p. 6 The characteristics of such boundaries are described by Talmy as follows: EXCERPT #DU6GRY p. 6 First, the material enclosed within the boundary is felt to constitute a unitary, coherent conceptual entity distinct from the material outside the boundary. Second, there seems to be some sense of connectivity throughout the material enclosed within the boundary and, contrariwise, some sense of discontinuity or disjuncture across the boundary between the enclosed and external material. Third, the various portions of the material within the boundary are felt to be co-relevant to each other, whereas the material outside the boundary is not relevant to that within. (Talmy 1996: 240; compare the characteristics of the ecological niche as set forth in B. Smith and Varzi, in press [b]) EXCERPT #6SB4QU p. 6 As Talmy and Langacker have shown in great detail, and as the phenomenologist Johannes Daubert emphasized in the "delineationist" ontology of states of affairs he developed in the early years of the twentieth century (Schuhmann and Smith 1987), the very same material can be subject to such windowing or profiling in different ways, amounting, in our terms, to the inscription within one and the same whole of internal fiat boundary-structures of different sorts. Thus to take one very simple example, the very same totality of objects and processes is windowed in different ways by "Blood flowed from his nose" and "He was bleeding from the nose." EXCERPT #LGXRVL p. 6 The thesis that the windowing effected through a complete linguistic act is a matter of the drawing of a topologically complete fiat boundary around a given portion of worldly material then allows us to develop a sort of topological grammar, a grammar that exploits the formal tools of the topologist (more precisely: of the mereotopologist; see Simons 1987, Varzi 1994, B. Smith 1996, and B. Smith and Varzi, in press [a]), in giving an account of the ways in which, through language, we effect systematically different sorts of windowing or profiling of reality (or fail in the attempt). Thus, for example, we can associate different sorts of incomplete or syncategorematic expressions ("John caused . . .," "John closed . . .," "John . . . quickly," and so on) with different sorts of incompleteness on the side of the corresponding fiat boundaries. There are then incomplete boundaries, analogous to the geographical cases of incompleteness previously referred to, in the linguistic sphere as well. EXCERPT #XADXPC p. 6 A further type of articulation, in some sense complementary to the addition of fiat boundaries within the interiors of objects, arises where bona fide interior part-structure is as it were stripped away, as occurs, for example, when an extended entity with genuine interior boundaries is treated as if it were a homogeneous whole. One variety of this phenomenon in the linguistic sphere might be called fiat continuity, which occurs where natural language sanctions the use of mass terms ("water," "sugar," "luggage") to re- EXCERPT #7CPETY p. 6 EXCERPT #VNDR9E p. 6 fer to entities that are in fact made up of discrete units in such a way that they come to be treated as continuous. It is here that we encounter the granularity that is characteristic of all phenomena of natural cognition: only those extended parts of objects and processes which enjoy a certain minimal extent come to be counted as parts within natural language fiat articulations. (See Ojeda 1993 and Habel 1994.) EXCERPT #M49H28 p. 6 There is one important difference between the views of Daubert on the windowing of language, and those of Talmy and Langacker, however. Daubert very clearly saw the boundaries in question—by analogy with the geographical case—as boundaries in reality, although generated by human fiat. In this he was struggling against the "constitutive phenomenology" of his master Husserl (Schuhmann and Smith 1985). Talmy and Langacker, in contrast, with their talk of "conceptual boundaries," of "boundaries in conceptual reality," of boundaries in "our concept of reality," and so on, seem unclear whether language-induced boundaries would be drawn within the mind or in exterior reality or in some other not clearly specified "conceptual realm." The motivation for this unclarity is understandable: it derives from the desire to develop a theory of linguistic usage that would apply equally to all the myriad different sorts of objects to which our sentences relate. Thus as Langacker points out: EXCERPT #X6FXAS p. 6 We are capable of constructing conceptual worlds of arbitrary complexity involving entities and phenomena that have no direct counterpart in peripherally connected experience. Such are the worlds of dreams, stories, mythology, mathematics, predictions about the future, flights of the imagination, and linguistic theories. All of us have constructed many conceptual worlds that differ in genre, complexity, conventionality, abstractness, degree of entrenchment, and so on. For many linguistic purposes all of these worlds are on a par with the one we distinguish as "reality." (1987/1991, 1: 113) EXCERPT #EL2PZY p. 6 Note, however, that constructing these worlds is not comparable to what some might argue is the most important of all linguistic purposes, namely that of giving an account of how, through language, human beings are able to become related to peripherally connected reality at all. Note, further, that if reality (or what Langacker calls "reality") is regarded as a mere constructed world, then one runs the risk of flouting our normal distinction between objects and concepts (for example between rabbits and our concepts of rabbits), with much confusion as its consequence: EXCERPT #R3H7VX p. 6 A person's conception of reality is itself a conceptual world that is built up from peripherally connected experience through complex sequences of mental operations. We construct our conception of the "real world" bit by bit, stage by stage, from myriad and multifarious sensory and motor experiences. . . . It is our conception of reality (not the real world per se ) that is relevant to linguistic semantics. (1987/1991, 1: 114) EXCERPT #HNFHDA p. 7 EXCERPT #5WLJ8N p. 7 EXCERPT #4KQ8CC p. 7 In the eyes of the cognitive linguist, it would seem, our natural language sentences about rabbits are not about rabbits ( per se ) at all; rather they are about conceptual rabbits that we ourselves have constructed bit by bit. The whole thrust of cognitive grammar à la Talmy and Langacker is unfortunately to minimize in this fashion the ontologically crucial differences between human concepts and reality. DOCUMENT #PHAFYA Truth and the Visual Field SECTION #EMC9A9 3. FIAT BOUNDARIES AS CREATED ENTITIES EXCERPT #DXLNG2 p. 4 What begins as a fiat geographical boundary may evolve over time into a natural boundary, reflecting not merely new features of the landscape but also differences in the language or dialect or trading habits of those who live on either side—all of which suggests that we develop a view of geographical boundaries as created entities, entities subject to the vagaries of history. Thus fiat boundaries seem to have a beginning in time, and geographical boundaries in general are such as to instantiate one of a number of characteristic patterns of boundary evolution (Prescott 1978). EXCERPT #UT225T p. 4 Against this, however, is an alternative view according to which spatial boundaries are merely abstract mathematical constructions and are thus not the sorts of things that can be subject to historical change. Boundaries are not created, on the given view, but discovered or picked out from the infinite totality of all geometrically possible alternative ways of dividing up (say) the surface of the earth. Utah, on the given reading, existed long before its boundaries were first picked out by the responsible administrators, and it may similarly continue to exist for long after human beings have ceased to occupy this planet. EXCERPT #FQG532 p. 4 Are fiat boundaries, and the fiat objects they circumscribe, discovered or created? The former view has in its favor the virtue of ontological parsimony: only one sort of boundary needs to be admitted into our ontology, where on the latter view we should have to admit in addition to purely geometrical boundaries also certain historically determined boundaries that coincide with these. An argument in favor of the existence of historically created boundaries can however be formulated as follows. We note, first of all, EXCERPT #7VCU8W p. 4 that “Hamburg” is an ambiguous term, standing on the one hand for a certain city (Hamburg-Stadt) and on the other hand for a certain administrative entity (Hamburg-Land), which is one of the constituent Länder (states, cantons) of the German Federal Republic. Hamburg-Stadt and Hamburg-Land are distinct entities, which happen to coincide spatially. On the geometrical account of boundaries (boundaries are discovered, not created) Hamburg-Stadt and Hamburg-Land have identical boundaries; on the alternative, historical reading, they have boundaries which are distinct from each other and from the underlying geometrical boundaries, even though all three sets of boundaries happen to coincide spatially. EXCERPT #DQYPQX p. 4 Why on earth, now, should we not embrace the more parsimonious reading and save ourselves the embarrassment of, in this case, three complete sets of boundaries in the very same place? The answer to this question turns on the possibility of divergent histories. The boundary of Hamburg-Stadt might, after all, have lain elsewhere. Each geometrically determined boundary is, however, as a matter of necessity exactly where it is. If, therefore, the boundary b of Hamburg-Stadt were identical to (and not merely contingently such as to coincide with) a certain geometrical boundary, then we should have to swallow the simultaneous truth of (1) b could have lain elsewhere and (2) b is as a matter of necessity exactly where it is. EXCERPT #RB9SSR p. 4 One must reject the temptation to suppose that we are confronted here with a mere verbal dispute, which could be resolved by some alternative choice of words. For consider the in-many-ways-analogous case of Bremen. “Bremen,” too, is ambiguous; it refers on the one hand to a certain city, and on the other to a certain Land. In this case, however, the boundaries of Bremen-Stadt and -Land do not coincide. And of course something analogous might hold in the case of Hamburg, too: it would be an administrative act of no great difficulty to bring it about that, as of tomorrow, the boundaries of Hamburg-Stadt and -Land should likewise be distinct or be, in however subtle a fashion, differently defined. This implies, however, that already today we are dealing with entities that could have distinct histories, and this is possible only if the entities themselves are already distinct. DOCUMENT #PHAFYA Truth and the Visual Field SECTION #ADGTJ6 4. FIAT OBJECTS IN PERCEPTION EXCERPT #PF95T8 p. 4 Geographical boundaries such as those of Hamburg-Stadt and -Land are, if the argument above can be accepted, human creations that are subject to the vagaries of history. It is as if, through the evolution of our political and administrative and legal practices and through practices relating to property law, new boundaries come to be inscribed in reality in addition to the natural boundaries in relation to which these supernumerary fiat boundaries are constructed and in terms of which they are defined. As will already have become clear from some of the examples mentioned above, however, we are confronted in our everyday experience also with a great wealth of such supernumerary boundaries of a more transient sort, boundaries created by our acts of perception and by human cognitive processes of other sorts. Imagine, for example, that I am outdoors on a clear day looking out over the landscape. One prominent object in the visual field hereby determined is my present horizon, a transient and incomplete and roughly linear boundary between earth and sky, whose existence and nature are determined not by any simple act of decision or fiat on my part but by my very existence as a visually perceiving subject in a given location at a given time, as also by the perimetric properties of my visual system, by topographical features of the location, and by the laws of optics. Note, however, that even in this case there is a residual element of human decision at work, namely, the decision on my part to turn my head in a given direction at a given moment. EXCERPT #7RPDNV p. 5 EXCERPT #7MX4S2 p. 5 EXCERPT #FUMALN p. 5 The horizon is a component object of the visual field, and the latter may be defined, with Ewald Hering, “as the totality of real objects imaged at a given moment on the retina of the right or left eye” (1964: 226). Let us assume that the eye sees in normal fashion, that it is not momentarily startled, and that there are no tricks, mirrors, or special equipment, and no clouds, fog, stained glass, or the like, in its way of seeing given objects. The depictions of the visual field provided by Ernst Mach (1959: 19; see Figure 10.1) and by Gibson (1979: 118f.) tell us that the objects making up the visual field according to Hering’s definition are primarily the surfaces of three-dimensional entities (the surfaces of walls, trousers, bookends, etc.). In fact we can distinguish three sorts of component object: (1) two-dimensional surfaces (with their own intrinsic curvature in three-dimensional space); (2) the boundaries of these two-dimensional surfaces (both one-dimensional edges and zero-dimensional vertices; both fiat and natural boundaries: the horizon is an example of a one-dimensional fiat boundary in the interior of the visual field); and (3) the one-dimensional psychologically induced fiat outer boundary of the visual field itself. The boundary of the visual field is a complex, subtle, ever-changing and gappy patchwork of physical surfaces and other components. The patchwork is “open,” topologically speaking: its external boundary is not a part of the visual field itself (as death is not an event in life). The patchwork is organized further in terms of an opposition between entities (“figures”) in the focus of attention, which characteristically manifest determinate boundaries, and entities which have indeterminate boundaries and which are experienced as running on (as “ground”) behind them. EXCERPT #VVG4DD p. 5 A black and white line drawing by Ernst Mach (1959) titled 'The visual field'. It depicts a person lying on their back on a wooden floor, looking up at a ceiling with a grid of square panels. The person's head is tilted back, and their arms are resting on their knees. The drawing illustrates the concept of the visual field as a totality of objects imaged on the retina, showing how the person's position and orientation determine what is visible. EXCERPT #AAJUN5 p. 5 FIGURE 10.1. The visual field. (Mach 1959) DOCUMENT #PHAFYA Truth and the Visual Field SECTION #NGNLAC 5. LANGUAGE-GENERATED FIAT OBJECTS EXCERPT #RT5BNY p. 5 A further important class of transient fiat boundaries are those effected through our everyday use of natural language. As Talmy puts it, drawing attention to a hitherto insufficiently studied analogy between the articulations effected by the descriptive use of language and those effected by acts of visual perception: “Linguistic forms can direct the distribution of one’s attention over a referent scene in a certain type of pattern, the placement of one or more windows of greatest attention over the scene, in a process that can be termed the windowing of attention” (1996: 236). Common to all such processes is the determination of a boundary, which might be a sharp line or a gradient zone, and whose particular scope and contour—hence, the particular quantity and portions of material that it encloses—can be seen to vary from context to context. EXCERPT #2ERFWV p. 6 EXCERPT #V98XHK p. 6 The characteristics of such boundaries are described by Talmy as follows: EXCERPT #DU6GRY p. 6 First, the material enclosed within the boundary is felt to constitute a unitary, coherent conceptual entity distinct from the material outside the boundary. Second, there seems to be some sense of connectivity throughout the material enclosed within the boundary and, contrariwise, some sense of discontinuity or disjuncture across the boundary between the enclosed and external material. Third, the various portions of the material within the boundary are felt to be co-relevant to each other, whereas the material outside the boundary is not relevant to that within. (Talmy 1996: 240; compare the characteristics of the ecological niche as set forth in B. Smith and Varzi, in press [b]) EXCERPT #6SB4QU p. 6 As Talmy and Langacker have shown in great detail, and as the phenomenologist Johannes Daubert emphasized in the "delineationist" ontology of states of affairs he developed in the early years of the twentieth century (Schuhmann and Smith 1987), the very same material can be subject to such windowing or profiling in different ways, amounting, in our terms, to the inscription within one and the same whole of internal fiat boundary-structures of different sorts. Thus to take one very simple example, the very same totality of objects and processes is windowed in different ways by "Blood flowed from his nose" and "He was bleeding from the nose." EXCERPT #LGXRVL p. 6 The thesis that the windowing effected through a complete linguistic act is a matter of the drawing of a topologically complete fiat boundary around a given portion of worldly material then allows us to develop a sort of topological grammar, a grammar that exploits the formal tools of the topologist (more precisely: of the mereotopologist; see Simons 1987, Varzi 1994, B. Smith 1996, and B. Smith and Varzi, in press [a]), in giving an account of the ways in which, through language, we effect systematically different sorts of windowing or profiling of reality (or fail in the attempt). Thus, for example, we can associate different sorts of incomplete or syncategorematic expressions ("John caused . . .," "John closed . . .," "John . . . quickly," and so on) with different sorts of incompleteness on the side of the corresponding fiat boundaries. There are then incomplete boundaries, analogous to the geographical cases of incompleteness previously referred to, in the linguistic sphere as well. EXCERPT #XADXPC p. 6 A further type of articulation, in some sense complementary to the addition of fiat boundaries within the interiors of objects, arises where bona fide interior part-structure is as it were stripped away, as occurs, for example, when an extended entity with genuine interior boundaries is treated as if it were a homogeneous whole. One variety of this phenomenon in the linguistic sphere might be called fiat continuity, which occurs where natural language sanctions the use of mass terms ("water," "sugar," "luggage") to re- EXCERPT #7CPETY p. 6 EXCERPT #VNDR9E p. 6 fer to entities that are in fact made up of discrete units in such a way that they come to be treated as continuous. It is here that we encounter the granularity that is characteristic of all phenomena of natural cognition: only those extended parts of objects and processes which enjoy a certain minimal extent come to be counted as parts within natural language fiat articulations. (See Ojeda 1993 and Habel 1994.) EXCERPT #M49H28 p. 6 There is one important difference between the views of Daubert on the windowing of language, and those of Talmy and Langacker, however. Daubert very clearly saw the boundaries in question—by analogy with the geographical case—as boundaries in reality, although generated by human fiat. In this he was struggling against the "constitutive phenomenology" of his master Husserl (Schuhmann and Smith 1985). Talmy and Langacker, in contrast, with their talk of "conceptual boundaries," of "boundaries in conceptual reality," of boundaries in "our concept of reality," and so on, seem unclear whether language-induced boundaries would be drawn within the mind or in exterior reality or in some other not clearly specified "conceptual realm." The motivation for this unclarity is understandable: it derives from the desire to develop a theory of linguistic usage that would apply equally to all the myriad different sorts of objects to which our sentences relate. Thus as Langacker points out: EXCERPT #X6FXAS p. 6 We are capable of constructing conceptual worlds of arbitrary complexity involving entities and phenomena that have no direct counterpart in peripherally connected experience. Such are the worlds of dreams, stories, mythology, mathematics, predictions about the future, flights of the imagination, and linguistic theories. All of us have constructed many conceptual worlds that differ in genre, complexity, conventionality, abstractness, degree of entrenchment, and so on. For many linguistic purposes all of these worlds are on a par with the one we distinguish as "reality." (1987/1991, 1: 113) EXCERPT #EL2PZY p. 6 Note, however, that constructing these worlds is not comparable to what some might argue is the most important of all linguistic purposes, namely that of giving an account of how, through language, human beings are able to become related to peripherally connected reality at all. Note, further, that if reality (or what Langacker calls "reality") is regarded as a mere constructed world, then one runs the risk of flouting our normal distinction between objects and concepts (for example between rabbits and our concepts of rabbits), with much confusion as its consequence: EXCERPT #R3H7VX p. 6 A person's conception of reality is itself a conceptual world that is built up from peripherally connected experience through complex sequences of mental operations. We construct our conception of the "real world" bit by bit, stage by stage, from myriad and multifarious sensory and motor experiences. . . . It is our conception of reality (not the real world per se ) that is relevant to linguistic semantics. (1987/1991, 1: 114) EXCERPT #HNFHDA p. 7 EXCERPT #5WLJ8N p. 7 EXCERPT #4KQ8CC p. 7 In the eyes of the cognitive linguist, it would seem, our natural language sentences about rabbits are not about rabbits ( per se ) at all; rather they are about conceptual rabbits that we ourselves have constructed bit by bit. The whole thrust of cognitive grammar à la Talmy and Langacker is unfortunately to minimize in this fashion the ontologically crucial differences between human concepts and reality. DOCUMENT #PHAFYA Truth and the Visual Field SECTION #6LAAZD 6. TRUTH EXCERPT #GN3L33 p. 7 What I now want to claim is that the construction of transient sentence-generated fiat boundaries of the sort described by Daubert, boundaries in reality , is pervasively involved in all descriptive statement-making uses of language, that there are transient fiat boundaries in the judgmental sphere analogous to the transient boundaries of visual fields associated with acts of visual perception. In this connection it is important to bear in mind that truth for empirical sentences has classically been understood in terms of a correspondence relation (that is to say, of some sort of isomorphism) between a judgment or an assertion on the one hand and a certain portion of reality on the other. The central difficulty standing in the way of this classical theory turned always on the fact that reality evidently does not come ready-parceled into judgment-shaped portions of the sort that would be predisposed to stand in relations of correspondence of the suggested sort. The theory of language-induced fiat boundaries can, however, allow us to treat judgment itself as a way to draw fiat boundaries around entities in reality of the appropriate (truth-making) sort. In this fashion it yields a way of putting the world back into semantics, or of anchoring true judgment to a reality of exactly the sort required by the correspondence theory (Smith 1993). EXCERPT #WNNYK5 p. 7 Let us define the judgment field as a portion of reality, a fiat object, that is demarcated by the transient fiat boundary associated with a given true empirical judgment. A judgment field is then a certain region of reality through and around which the relevant judgmental fiat boundary is drawn. As such it exists in and of itself, regardless of our judging activity. The judgment field—called by Daubert the state of affairs or Sachverhalt —is, however, also in a certain sense dependent on our judgment. For in the absence of the judging activity, an entity of the given sort would in no way be demarcated from its surroundings, nor would it have the internal demarcation-structure which it comes to have by virtue of the sentence forms employed. In this way, EXCERPT #FGXQLV p. 7 then, cognitive linguistics can replace its confused notion of conceptual reality with the geographer's notion of reality as subject to fiat articulations. It will then be in a position to exploit its remarkably sophisticated resources for the analysis of the grammatical structures at work in natural language in order to produce a truly adequate account of truth for natural language in correspondence-theoretic terms. ### 56. Assistant tool calls #### Tool call: `update_document_metadata` ```json { "document_id": "2F8T3H", "folder": "ontology", "micro_abstract": "Defends Gibsonian ecological realism: organisms directly perceive affordances in physically real niches, while granular partitions explain how different species inhabit distinct perspectives on one world rather than separate constructed worlds." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "DT9Y7X", "folder": "ontology", "micro_abstract": "Generalizes Alberti’s perspectival grid into a realist theory of projection: pictures, maps, names, concepts, and databases are “true grids” when their cells preserve relevant structure and refer transparently to reality." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "KY3Y9U", "folder": "ontology", "micro_abstract": "Reconstructs naïve physics as a realist ontology of the common-sense world—objects, processes, stuffs, boundaries, media, and values—drawing on Gestalt psychology and phenomenology to broaden AI’s set-theoretic models." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "FJ5KCA", "folder": "ontology", "micro_abstract": "Develops an ontology of spatial regions and boundaries, arguing that political territories are historically created fiat objects through performative maps while also recognizing vague, overlapping, and incomplete geographic objects." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "TQPVBD", "folder": "ontology", "micro_abstract": "Argues for a scientific ontology of the qualitative common-sense world, using morphological discontinuities to connect physical substrates, sensible qualities, Aristotelian categories, and ecologically constrained cognition." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "XZX6PE", "folder": "ontology", "micro_abstract": "Formalizes vague reference as multiple crisp candidate referents within granular partitions, then explains approximation as using familiar spatial or temporal reference grids to constrain vagueness without truth-value indeterminacy." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "88BVY3", "folder": "ontology", "micro_abstract": "Reconstructs Aristotle’s categories as the philosophical basis of BFO, extending the ontological square with processes into a six-category framework for continuants, occurrents, dependence, and multiple scientific granularities." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "GN66WW", "folder": "ontology", "micro_abstract": "Proposes linked upper ontologies for common sense, physics, and mathematics, arguing that classical models connect real magnitudes to mathematics whereas modern physics relates measurements to entities lacking commonsense universals." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "KYQGNH", "folder": "ontology", "micro_abstract": "Provides a social ontology of credentials as portable, inspectable institutional documents that certify identity or status and give bearers the practical deontic power to exercise rights, with a typology of their forms and functions." } ``` #### Tool call: `update_document_metadata` ```json { "document_id": "PHAFYA", "folder": "ontology", "micro_abstract": "Uses mereotopology and Gibsonian ecology to treat perception and language as carving transient fiat boundaries in reality, defining a judgment field as the truth-making portion of the world selected by a true sentence." } ``` ### 57. Tool result: update_document_metadata {"error":"\"\\\"micro_abstract must be no more than 240 characters\\\"\""} ### 58. Tool result: update_document_metadata Updated metadata for #DT9Y7X: folder, micro_abstract ### 59. Tool result: update_document_metadata Updated metadata for #KY3Y9U: folder, micro_abstract ### 60. Tool result: update_document_metadata Updated metadata for #FJ5KCA: folder, micro_abstract ### 61. Tool result: update_document_metadata Updated metadata for #TQPVBD: folder, micro_abstract ### 62. Tool result: update_document_metadata Updated metadata for #XZX6PE: folder, micro_abstract ### 63. Tool result: update_document_metadata Updated metadata for #88BVY3: folder, micro_abstract ### 64. Tool result: update_document_metadata Updated metadata for #GN66WW: folder, micro_abstract ### 65. Tool result: update_document_metadata Updated metadata for #KYQGNH: folder, micro_abstract ### 66. Tool result: update_document_metadata Updated metadata for #PHAFYA: folder, micro_abstract ### 67. Assistant tool calls #### Tool call: `update_document_metadata` ```json { "document_id": "2F8T3H", "folder": "ontology", "micro_abstract": "Defends Gibsonian ecological realism: organisms directly perceive affordances in physically real niches, while granular partitions show how different species inhabit perspectives on one world, not separate constructed worlds." } ``` ### 68. Tool result: update_document_metadata Updated metadata for #2F8T3H: folder, micro_abstract ### 69. Assistant Moved all 10 documents into the **ontology** folder and added new, argument-focused micro-abstracts: #2F8T3H, #DT9Y7X, #KY3Y9U, #FJ5KCA, #TQPVBD, #XZX6PE, #88BVY3, #GN66WW, #KYQGNH, and #PHAFYA.