# Assistant conversation HR32SY

- Canonical URL: <https://m.sheaf.less.rest/HR32SY>
- Resource IRI: <https://m.sheaf.less.rest/HR32SY>
- Type: assistant conversation
- Mode: quick

## Transcript

### 1. User

let's think about RDF Surfaces and Ellerman's partition logic

### 2. Assistant tool calls

#### Tool call: `list_documents`

```json
{}
```

#### Tool call: `search_text`

```json
{
  "limit": 10,
  "query": "RDF Surfaces partition logic Ellerman distinctions dit sets"
}
```

#### Tool call: `search_text`

```json
{
  "limit": 10,
  "query": "RDF Surfaces negation graph logic surfaces"
}
```

### 3. Tool result: list_documents

FOLDERS
- Accounting Algebra & Property Theory
- Living Structure, Wholeness & Beauty
- Living Structure, Wholeness &amp; Beauty
- Procedural Generation & Co-Creation
- Roads, Trails & Movement
- 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.

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.

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.

### 4. Tool result: search_text

Exact matches

1. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 7
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Matching excerpt #LFDVDT:
      Our purpose has been to develop the notions of negation and implication (relative negation) in the logic of partitions. Since partition relations (ditsets) and equivalence relations (indit sets) are complementary in U \times U , every result in the logic of partitions has a complementary-dual result in the logic of equivalence relations so the latter is not really a different logic but a complementary way to view partition logic. There is a similar complementary-duality in intuitionistic logic between Heyting algebras (modelled by the open subsets of a topological space) and co-Heyting algebras [13] (modelled by the closed subsets). 4 Intuitionistic logic makes the symmetry-breaking choice to deal with Heyting algebras rather than co-Heyting algebras, and we have made the similar choice to develop the logic of partitions rather than the (‘anti-isomorphic’) logic of equivalence relations. For instance, the complementary-dual to the implication operation on partitions is the difference or subtraction operation on equivalence relations. The partition logic tautology of modus ponens has the customary form: (\sigma \wedge (\sigma \Rightarrow \pi)) \Rightarrow \pi , whereas the corresponding formula in the dual logic of

2. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 0
  Context:
    #L8JZAG On implication and negation in partition logic
      #ZZPBEJ 1. Introduction: The Boolean logic of subsets and the logic of partitions
  Matching excerpt #BLF82B:
      T oday, the ‘classical’ form of logic is seen as the Boolean logic of subsets usually presented as the special case of propositional logic (i.e., the logic of subsets 0 and 1 of the one element set 1). Other related logics, such as intuitionistic logic (e.g., the logic of the open subsets of a topological space) are considered as non-classical. But there is another recently developed and little-known logic that is at the same mathematical level of fundamentality as subset logic and is thus ‘classical’ in that nontemporal sense. Since the development of category theory starting in the middle of the twentieth century, it has been known that the concept of a subset has a category-theoretic dual in the notion of a quotient set (or, equivalently, a partition or equivalence relation). F. William Lawvere calls the generalization of a subset a “part” and “The dual notion (obtained by reversing the arrows) of ‘part’ is the notion of partition.” [1, p. 85] The simplest illustration of this is the fact that given a set function f : X \rightarrow Y , the image of f is a subset f(X) \subseteq Y of the codomain Y and the inverse-image \{f^{-1}(y) \neq \emptyset : y \in Y\} is a partition on the domain X . Hence, it should be expected that there is a logic of partitions ([2,3]) dual to the Boolean logic of subsets. And since subsets and quotient sets are at the same basic level from the mathematical point of view, partition logic is more of a dual sibling to subset logic rather than being another ‘non-classical’ off-shoot of the classical subset logic.

3. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 2
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Matching excerpt #ZTPFTV:
      The equivalence relation corresponding to the indiscrete partition 0 is the universal relation U \times U . For any two equivalence relations E, E' \subseteq U \times U , if E \cup E' = U \times U , then E = U \times U or E' = U \times U . This is essentially the standard result of graph theory that the complement of any disconnected graph is connected [8, p. 30]. Since the indiscrete partition has no distinctions, i.e., \text{dit}(\mathbf{0}) = \emptyset , the complementary form of that result is that for any two partitions \sigma, \pi , if \text{dit}(\sigma) \cap \text{dit}(\pi) = \emptyset , then \text{dit}(\sigma) = \emptyset or \text{dit}(\pi) = \emptyset , i.e., \sigma = \mathbf{0} or \pi = \mathbf{0} . An alternative form of the result is useful to understand the negation \sigma \Rightarrow \mathbf{0} .

4. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 3
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Matching excerpt #2TNJYT:
      Since intuitionistic logic is the most developed logic aside from Boolean logic, it is often suggestive to compare the ditsets of partition logic with the open sets in the topological representation of intuitionistic logic, i.e., of a Heyting algebra (also called a pseudo-Boolean algebra or Brouwer algebra). The negation of an open set is the largest open set disjoint from the given set. But now we see that there is no non-empty ditsets disjoint from any given non-empty ditset. Hence intuitively the negation of any partition \sigma \neq \mathbf{0} , is the partition \mathbf{0} with an empty ditset. The definition \neg\sigma := \sigma \Rightarrow \mathbf{0} gives the same result since the only block U in \mathbf{0} = \{U\} is not contained in any block of \sigma \neq \mathbf{0} . And when \sigma = \mathbf{0} , then \neg\mathbf{0} = \mathbf{0} \Rightarrow \mathbf{0} = \mathbf{1} since U \subseteq U so it is discretized in the implication. That is why the absolute negation \neg\sigma is of less interest than the relative negation \frac{\pi}{\sigma}\sigma = \sigma \Rightarrow \pi which is simply the partition implication.

5. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 8
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Matching excerpt #T6E7D9:
      No new logical operations on partitions, aside from join and meet, were defined throughout the twentieth century. The definition of the partition implication (or relative negation) in any of the many equivalent ways was the key to the development of the full logic of partitions. Why the delay? One reason is perhaps the fact that partition lattices are so general that any partition tautology or identity involving just the lattice operations and top and bottom, e.g., \mathbf{1} \wedge \pi = \pi or \mathbf{0} \vee \pi = \pi , are in fact identities that hold on all such lattices [15]. Thus the logic of general set partitions only becomes of independent interest by moving beyond the lattice operations on partitions. Alternatively, one could develop a 'logic' of equivalence relations sticking with only the lattice operations but specializing to certain types of equivalence relations [16]. But the main reason for the delay seems to be that the Boolean logic of subsets was and is presented in only the special case of the logic of propositions. And propositions, unlike subsets, do not have a category-theoretic dual concept. Hence twentieth century mathematical logicians were not even looking for the dual logic of quotient sets, equivalence relations, or partitions.

6. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 29
  Context:
    #DGJK69 3 Results
      #MUCGHX 3.6 Quantum Mechanics Math as the Hilbert Space Version of Partition Math
        #J4NVLA 3.6.7 Group Representation Theory
  Matching excerpt #P2CSVL:
      In the development of the math of partitions, we have seen that a (non-discrete) partition can be refined by adding more distinctions, e.g., \pi can be refined to \pi \vee \sigma by adding the new distinctions of \sigma since \text{dit}(\pi \vee \sigma) = \text{dit}(\pi) \cup \text{dit}(\sigma) so the new distinctions are \text{dit}(\sigma) - \text{dit}(\pi) . If H is a subgroup of G , then \{R_g : U \rightarrow U\}_{g \in H} is a group representation of H on U and since it has no indistinctions R_g(u) = u' for g \in G - H , its orbit partition will refine the orbit partition of the G -representation.

7. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 46
  Context:
    #L62FZR 4 Discussion and Conclusions
  Matching excerpt #QKEJGK:
      While the fundamental duality finds its most mathematical formulation as category theory's reverse-the-arrows duality, that is far too abstract to elicit the multitude of applications throughout the sciences. The more specific formulation between subsets and partitions at the logical level, and the even more granular formulation as the elements-and-distinctions (or Its & Dits) duality, brought out many applications—including the 'origin' of the category-theoretic duality in the ur-category of Sets . Outside of category theory, the historical development has been largely on the subset side of the duality so it was the new developments of the partition side, starting with partition logic and running through logical information theory, quantum theory, and finally to the biological notion of a generative mechanism, that revealed the wide range of applications of the fundamental duality throughout the mathematical and natural sciences—from logic to biology.

8. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 5
  Context:
    #ZJN679 2 Methods: The Dual Logics of Subsets and Partitions
  Matching excerpt #3AJ2XX:
      A partition \pi on U is a set of non-empty blocks \pi = \{B_1, \dots, B_m\} such that the blocks are disjoint and their union is all of U . The corresponding equivalence relation is \text{indit}(\pi) = \bigcup_{j=1}^m B_j \times B_j \subseteq U \times U is the set of ordered pairs of elements that are in the same block of the partition which are called the indistinctions of \pi . A distinction of \pi is an ordered pair of elements in different blocks and the set of all distinctions is \text{dit}(\pi) = U \times U - \text{indit}(\pi) . The set of all partitions on U is denoted \Pi(U) and the partial order on it is defined by refinement, i.e., for another partition \sigma = \{C_1, \dots, C_{m'}\} , the partition \sigma is refined by \pi , written \sigma \preceq \pi , if for every block B_j \in \pi , there is a block C_{j'} \in \sigma such that B_j \subseteq C_{j'} . Note that when \sigma \preceq \pi , then there is a canonical surjective set function \pi \rightarrow \sigma taking each block B_j \in \pi to the block C_{j'} that it is contained in. In terms of distinctions, refinement is equivalent to inclusion of ditsets, i.e., \sigma \preceq \pi iff \text{dit}(\sigma) \subseteq \text{dit}(\pi) .

9. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 45
  Context:
    #L62FZR 4 Discussion and Conclusions
  Matching excerpt #2KDWDJ:
      We have argued that there is a fundamental or foundational duality that runs through logic, mathematics, probability and information theory, physics, and even the life sciences. At the logical level, it is the duality between subsets (or subobjects or ‘parts’) and partitions (or equivalence relations or quotient objects). At a more granular level, it is the duality between elements (of a subset) and distinctions (of a partition) or “Its & Dits.” In most cases, there has been a fulsome development of the subset-side of the duality to the neglect of the partition-side.

10. Source: Partition lens across the corpus (#TUCFMG)
  Matching note #TUCFMG:
      Ellerman’s partition logic suggests a useful cross-corpus lens, but only where blocks are mutually exclusive and jointly exhaustive (#ZBQE96). The clearest exact case is hydrological catchments: terrain cells are equivalent when they drain to the same outlet, and watershed labeling assigns one common label to every such equivalence class (#NNC37P, #477DMT). FastFlow similarly defines a basin as cells sharing a stream tree/root (#A6G5PJ) and propagates basin identifiers upstream (#C9XNSJ); saddle crossings then connect or merge basin blocks (#L94DMG), suggesting dynamic coarsening of a catchment partition. Hydrological terrain generation also constructs Voronoi cells and hierarchical watersheds/subwatersheds (#KAMGFF, #WL3SCW), giving nested partitions at multiple scales. Other exact or near-exact corpus examples include planar regions cut by major streets/topographic boundaries (#PUZXXV); recursive figure/ground and connected-pixel segmentation (#FXLSNG); head/tail classes (#WQW77N); MAP-Elites bins that partition behavior/search space (#DJQRBX, #964T2S); fluid particles classified into rendering layers (#48RCD2); and walker populations grouped by entry–destination pair (#RRJ2BJ). Alexanderian centers should not be treated as a partition without qualification because their local symmetries and centers overlap across scales (#LNKPPL). Conceptually, a deterministic map from each terrain cell to its terminal outlet realizes Ellerman’s function-to-partition idea (#BLF82B): catchments are the inverse-image fibers of the outlet map.

Approximate matches

1. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 4
  Context:
    #L8JZAG On implication and negation in partition logic
      #FEMQRA 4. Three more equivalent ways to define implication for partitions
        #H2YZ4E 4.3. The ditset definition
  Score: 0.029
  Related excerpt #LTGSSD:
      Another equivalent way to define partition implication is to mimic the subset definition using ditsets except for the fact that Boolean subset operations on ditsets do not necessarily lead to ditsets. By the analogy with the interior operation on subsets of a topological space, the interior \text{int}(S) of an arbitrary subset S \subseteq U , where U is a topological space, is the largest open set contained in S . The interior \text{int}(S) can also be defined as the complement of the (topological) closure of the complement, i.e., (\overline{S^c})^c . Similarly, we could start with any subset S \subseteq U \times U , and define the interior of S as the largest ditset contained in S . It could be constructed by first taking the complement S^c in U \times U and then its reflexive, symmetric, and transitive closure \overline{S^c} which is just the intersection of all the equivalence relations containing S^c . Then the complement is the interior : \text{int}(S) := (\overline{S^c})^c . This partition-theoretic closure operation is not a topological closure operation, e.g., since the intersection of two ditsets is not necessarily a ditset, whereas the intersection of two open sets is open.

2. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 1
  Context:
    #L8JZAG On implication and negation in partition logic
      #7WB5RZ 2. The logic of partitions
  Score: 0.028
  Related excerpt #B7W87A:
      A distinction of a partition is an ordered pair (u, u') of elements of U in distinct blocks of the partition. The set of distinctions (abbreviated "dits") of a partition is the ditset

3. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 1
  Context:
    #L8JZAG On implication and negation in partition logic
      #7WB5RZ 2. The logic of partitions
  Score: 0.027
  Related excerpt #HTEQ5J:
      The indit set of a partition is the equivalence relation defined by the partition, and the ditset of a partition is the complementary partition relation defined by the partition.

4. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 2
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Score: 0.023
  Related excerpt #3K4JXL:
      Proof. Let \pi and \sigma be any two partitions on U with non-empty dit sets, i.e., \pi \neq \mathbf{0} \neq \sigma . We need to show that \text{dit}(\pi) \cap \text{dit}(\sigma) \neq \emptyset . Since \sigma is not the blob 0 , consider two elements u and u' distinguished by \sigma but identified by \pi [otherwise (u, u') \in \text{dit}(\pi) \cap \text{dit}(\sigma) and we are finished]. Since \pi is also not the blob, there must be a third element u'' not in the same block of \pi as u and u' .

5. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 2
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Score: 0.023
  Related excerpt #ZTPFTV:
      The equivalence relation corresponding to the indiscrete partition 0 is the universal relation U \times U . For any two equivalence relations E, E' \subseteq U \times U , if E \cup E' = U \times U , then E = U \times U or E' = U \times U . This is essentially the standard result of graph theory that the complement of any disconnected graph is connected [8, p. 30]. Since the indiscrete partition has no distinctions, i.e., \text{dit}(\mathbf{0}) = \emptyset , the complementary form of that result is that for any two partitions \sigma, \pi , if \text{dit}(\sigma) \cap \text{dit}(\pi) = \emptyset , then \text{dit}(\sigma) = \emptyset or \text{dit}(\pi) = \emptyset , i.e., \sigma = \mathbf{0} or \pi = \mathbf{0} . An alternative form of the result is useful to understand the negation \sigma \Rightarrow \mathbf{0} .

6. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 1
  Context:
    #L8JZAG On implication and negation in partition logic
      #7WB5RZ 2. The logic of partitions
  Score: 0.021
  Related excerpt #HG4C5G:
      To define the meet \pi \wedge \sigma (greatest lower bound) of \pi and \sigma , we define an equivalence relation on U that is generated by u \sim u' if u and u' are in the same block of \pi or \sigma . Thus if two blocks of \pi and \sigma overlap (non-empty intersection) then all the elements of the two blocks are equated and so forth for any finite sequence of overlapping blocks. Hence a block of the meet partition, i.e., an equivalence class of that equivalence relation, is a precise union of blocks of \pi and a union of blocks of \sigma , and is the smallest such union. These definitions of refinement, join, and meet turn the set \Pi(U) of partitions on U into a lattice. The notion of refinement between partitions is equivalent to inclusion between their corresponding ditsets or partition relations, i.e., \sigma \preceq \pi iff \text{dit}(\sigma) \subseteq \text{dit}(\pi) , so the lattice of partitions on U can be represented as the isomorphic lattice of partition relations on U \times U . But it should be carefully noted that what many textbooks call the "lattice of partitions" is really the opposite lattice of equivalence relations, e.g., Birkhoff [4] or Grätzer [5], where the join and meet are interchanged.

7. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 5
  Context:
    #ZJN679 2 Methods: The Dual Logics of Subsets and Partitions
  Score: 0.028
  Related excerpt #3AJ2XX:
      A partition \pi on U is a set of non-empty blocks \pi = \{B_1, \dots, B_m\} such that the blocks are disjoint and their union is all of U . The corresponding equivalence relation is \text{indit}(\pi) = \bigcup_{j=1}^m B_j \times B_j \subseteq U \times U is the set of ordered pairs of elements that are in the same block of the partition which are called the indistinctions of \pi . A distinction of \pi is an ordered pair of elements in different blocks and the set of all distinctions is \text{dit}(\pi) = U \times U - \text{indit}(\pi) . The set of all partitions on U is denoted \Pi(U) and the partial order on it is defined by refinement, i.e., for another partition \sigma = \{C_1, \dots, C_{m'}\} , the partition \sigma is refined by \pi , written \sigma \preceq \pi , if for every block B_j \in \pi , there is a block C_{j'} \in \sigma such that B_j \subseteq C_{j'} . Note that when \sigma \preceq \pi , then there is a canonical surjective set function \pi \rightarrow \sigma taking each block B_j \in \pi to the block C_{j'} that it is contained in. In terms of distinctions, refinement is equivalent to inclusion of ditsets, i.e., \sigma \preceq \pi iff \text{dit}(\sigma) \subseteq \text{dit}(\pi) .

8. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 2
  Context:
    #MEVK4Z 1 Introduction: A Fundamental Duality in the Sciences
  Score: 0.028
  Related excerpt #3PDWZY:
      The duality between subsets and partitions can also be expressed, in a more elementary or granular form, as the duality between the elements or ‘ Its ’ of a subset and the distinctions or ‘ Dits ’ of a partition—where a distinction of a partition is an ordered pairs of elements from the underlying set that are in different blocks of the partition (or different equivalence classes of the equivalence relation).

9. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 6
  Context:
    #ZJN679 2 Methods: The Dual Logics of Subsets and Partitions
  Score: 0.026
  Related excerpt #YUEKZJ:
      In the refinement partial order, the join \pi \vee \sigma is the partition whose blocks are all the nonempty intersections B_j \cap C_{j'} for j = 1, \dots, m and j' = 1, \dots, m' . The ditset of the join is just the union of the ditsets, i.e., \text{dit}(\pi \vee \sigma) = \text{dit}(\pi) \cup \text{dit}(\sigma) . To form the meet \pi \wedge \sigma , take the intersection of all equivalence relations E \subseteq U \times U such that \text{indit}(\pi), \text{indit}(\sigma) \subseteq E . The intersection of equivalence relations is always an equivalence relation, and the meet \pi \wedge \sigma is the partition whose blocks are the equivalence classes of the intersection of those equivalence relations. The ditset of the meet \pi \wedge \sigma is the largest ditset contained in the ditsets of \pi and \sigma . The join and meet operations turn \Pi(U) into the lattice of partitions on U —which was known in the nineteenth century (e.g., Richard Dedekind and Ernst Schröder). The lattice of partitions has a top which is the discrete partition \mathbf{1}_U = \{\{u_1\}, \dots, \{u_n\}\} where all the blocks are singletons. The bottom is the indiscrete partition \mathbf{0}_U = \{U\} with only one block U . There is an implication \sigma \Rightarrow \pi which is such that: \sigma \Rightarrow \pi = \mathbf{1}_U iff \sigma \preceq \pi . The partition \sigma \Rightarrow \pi which has that property is like \pi except that for any B_j \in \pi , if there is a C_{j'} \in \sigma such that B_j \subseteq C_{j'} , then the block B_j is discretized, i.e., replaced by singletons of all the elements of B_j . Thus \sigma \Rightarrow \pi is an indicator or characteristic function for refinement in the sense that if there is a C_{j'} such that B_j \subseteq C_{j'} , then B_j is replaced by its discrete version \mathbf{1}_{B_j} , and otherwise B_j remains in its indiscrete version \mathbf{0}_{B_j} . That is why it satisfies the property: \sigma \Rightarrow \pi = \mathbf{1}_U iff \sigma \preceq \pi . The partition lattice structure of joins and meets enriched with the partition implication operation makes \Pi(U) in an algebra of partitions.

10. Source: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology (#GANFP9), David Ellerman, p. 6
  Context:
    #ZJN679 2 Methods: The Dual Logics of Subsets and Partitions
  Score: 0.023
  Related excerpt #E6KNTZ:
      The Boolean algebra of subsets and the algebra of partitions have been developed in a way to emphasize the underlying duality of elements of a subset and distinctions of a partition, i.e., its and dits. The canonical injections and surjections defined just by the dual logical partial orders are the “ur-morphisms” that define the ‘canonical’ morphisms in the universal constructions in the category of

### 5. Tool result: search_text

Exact matches

1. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 7
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Matching excerpt #LFDVDT:
      Our purpose has been to develop the notions of negation and implication (relative negation) in the logic of partitions. Since partition relations (ditsets) and equivalence relations (indit sets) are complementary in U \times U , every result in the logic of partitions has a complementary-dual result in the logic of equivalence relations so the latter is not really a different logic but a complementary way to view partition logic. There is a similar complementary-duality in intuitionistic logic between Heyting algebras (modelled by the open subsets of a topological space) and co-Heyting algebras [13] (modelled by the closed subsets). 4 Intuitionistic logic makes the symmetry-breaking choice to deal with Heyting algebras rather than co-Heyting algebras, and we have made the similar choice to develop the logic of partitions rather than the (‘anti-isomorphic’) logic of equivalence relations. For instance, the complementary-dual to the implication operation on partitions is the difference or subtraction operation on equivalence relations. The partition logic tautology of modus ponens has the customary form: (\sigma \wedge (\sigma \Rightarrow \pi)) \Rightarrow \pi , whereas the corresponding formula in the dual logic of

2. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 0
  Context:
    #L8JZAG On implication and negation in partition logic
  Matching excerpt #KS4M7Q:
      Keywords: Boolean logic of subsets, logic of partitions, duality of subsets and partitions, implication, negation

3. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 3
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Matching excerpt #2TNJYT:
      Since intuitionistic logic is the most developed logic aside from Boolean logic, it is often suggestive to compare the ditsets of partition logic with the open sets in the topological representation of intuitionistic logic, i.e., of a Heyting algebra (also called a pseudo-Boolean algebra or Brouwer algebra). The negation of an open set is the largest open set disjoint from the given set. But now we see that there is no non-empty ditsets disjoint from any given non-empty ditset. Hence intuitively the negation of any partition \sigma \neq \mathbf{0} , is the partition \mathbf{0} with an empty ditset. The definition \neg\sigma := \sigma \Rightarrow \mathbf{0} gives the same result since the only block U in \mathbf{0} = \{U\} is not contained in any block of \sigma \neq \mathbf{0} . And when \sigma = \mathbf{0} , then \neg\mathbf{0} = \mathbf{0} \Rightarrow \mathbf{0} = \mathbf{1} since U \subseteq U so it is discretized in the implication. That is why the absolute negation \neg\sigma is of less interest than the relative negation \frac{\pi}{\sigma}\sigma = \sigma \Rightarrow \pi which is simply the partition implication.

4. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 0
  Context:
    #L8JZAG On implication and negation in partition logic
  Matching excerpt #AHAPPT:
      Abstract: The Boolean logic of subsets, usually presented as ‘propositional logic,’ is considered as being “classical” while intuitionistic logic and the many sublogics and off-shoots are “non-classical.” But there is another mathematical logic, the logic of partitions, that is at the same mathematical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of implication and negation in that other mathematical logic of partitions.

5. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 8
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Matching excerpt #XPGYMY:
      Much of the modern development in "logic" is focused on the propositional interpretation of the variables which is a special case of the general logic of subsets of some given universe U (where U = 2 = \{0,1\} or \{F,T\} in the propositional case). Then attention has been focused on the many variations and sublogics of the Boolean logic of subsets. Partition logic is not about propositions at all and it is not a variation or sublogic of the Boolean logic of subsets. Partition logic is the category-theoretic dual to subset logic, and it is coequal to subset logic from the purely mathematical point of view (see, for instance, [17]). Hence our focus has been to address the topic of implication and negation in this little-known form of logic that is the mathematical coequal twin or dual to the Boolean logic of subsets.

6. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 4
  Context:
    #L8JZAG On implication and negation in partition logic
      #KS4RBJ 5. Relative negation in partition logic
  Matching excerpt #TTFKUQ:
      To study relative negation, we take the ‘consequence’ \pi as fixed and then let the ‘antecedent’ \sigma vary in the \pi -negation \neg_\pi \sigma := \sigma \Rightarrow \pi . Another suggestion from intuitionistic logic is that the negated elements in a Heyting algebra form a Boolean algebra. In partition logic, this is trivially true for absolute negation since the negated elements form the two-element Boolean algebra. And it is also true for the relative \pi -negation as was suggested by viewing the implication \sigma \Rightarrow \pi as an indicator or characteristic function for the inclusion of the blocks of \pi in the blocks of \sigma . And the double \pi -negation \neg_\pi \neg_\pi \sigma = (\sigma \Rightarrow \pi) \Rightarrow \pi just interchanges the 0_B and 1_B so \pi -negation is like the usual negation of a subset represented by its indicator function (i.e., negation interchanges the zero-one values). 1 Thus the triple \pi -negation is the same as the single \pi -negation. For another partition \tau = \{D, D', \dots\} , the join \overset{\pi}{\neg}\sigma \vee \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \vee (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B , iff B is contained in a block C \in \sigma or B is contained in a block D \in \tau , so it acts like the Boolean join or disjunction: \mathbf{0}_B \vee \mathbf{1}_B = \mathbf{1}_B \vee \mathbf{0}_B = \mathbf{1}_B \vee \mathbf{1}_B = \mathbf{1}_B and \mathbf{0}_B \vee \mathbf{0}_B = \mathbf{0}_B . Similarly, the meet \overset{\pi}{\neg}\sigma \wedge \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \wedge (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B iff B is contained in a block C \in \sigma and B is contained in a block D \in \tau , so it acts like the Boolean conjunction. Thus all the partitions over U in the form of a \pi -negation \overset{\pi}{\neg}\sigma = \sigma \Rightarrow \pi form a Boolean algebra \mathcal{B}_\pi with \pi = \overset{\pi}{\neg}\mathbf{1} as the bottom element and \mathbf{1} = \pi \Rightarrow \pi as the top element. Since all the \pi -negated partitions (also called \pi -regular partitions) refine \pi , the Boolean algebra \mathcal{B}_\pi is contained in the upper segment [\pi, \mathbf{1}] and might be called the Boolean core \mathcal{B}_\pi of [\pi, \mathbf{1}] .

7. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 7
  Context:
    #L8JZAG On implication and negation in partition logic
      #LEZCCF 6. Valid formulas
  Matching excerpt #L4CR23:
      Then the single \pi -negation transform of any classical tautology will still be a tautology but now expressed in \mathcal{B}_\pi and thus it is also a partition tautology. Thus all partition tautologies are ordinary Boolean logic tautologies, and any ordinary subset tautology transforms into a partition tautology via the single \pi -negation transform.

8. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 8
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Matching excerpt #T6E7D9:
      No new logical operations on partitions, aside from join and meet, were defined throughout the twentieth century. The definition of the partition implication (or relative negation) in any of the many equivalent ways was the key to the development of the full logic of partitions. Why the delay? One reason is perhaps the fact that partition lattices are so general that any partition tautology or identity involving just the lattice operations and top and bottom, e.g., \mathbf{1} \wedge \pi = \pi or \mathbf{0} \vee \pi = \pi , are in fact identities that hold on all such lattices [15]. Thus the logic of general set partitions only becomes of independent interest by moving beyond the lattice operations on partitions. Alternatively, one could develop a 'logic' of equivalence relations sticking with only the lattice operations but specializing to certain types of equivalence relations [16]. But the main reason for the delay seems to be that the Boolean logic of subsets was and is presented in only the special case of the logic of propositions. And propositions, unlike subsets, do not have a category-theoretic dual concept. Hence twentieth century mathematical logicians were not even looking for the dual logic of quotient sets, equivalence relations, or partitions.

9. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 2
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Matching excerpt #ZTPFTV:
      The equivalence relation corresponding to the indiscrete partition 0 is the universal relation U \times U . For any two equivalence relations E, E' \subseteq U \times U , if E \cup E' = U \times U , then E = U \times U or E' = U \times U . This is essentially the standard result of graph theory that the complement of any disconnected graph is connected [8, p. 30]. Since the indiscrete partition has no distinctions, i.e., \text{dit}(\mathbf{0}) = \emptyset , the complementary form of that result is that for any two partitions \sigma, \pi , if \text{dit}(\sigma) \cap \text{dit}(\pi) = \emptyset , then \text{dit}(\sigma) = \emptyset or \text{dit}(\pi) = \emptyset , i.e., \sigma = \mathbf{0} or \pi = \mathbf{0} . An alternative form of the result is useful to understand the negation \sigma \Rightarrow \mathbf{0} .

10. Source: Algebraic Models for Accounting Systems (#NBH3BE), Derek J. S. Robinson, José García Pérez, Robert A. Nehmer, Salvador Cruz Rambaud, p. 21
  Context:
    #YRM968 Chapter OneApproaches to Accounting Theory
      #SWTSUG 1.4. A Formal Grammar Approach
  Matching excerpt #KJP4VC:
      The second criterion, consistency, refers to the cross-statement compatibility of the grammars. This compatibility can perhaps best be addressed in terms of first order logic, rather than the formal grammar approach used in the article. The two systems are equivalent, so the change in approach is warranted. In first order logic consistency is defined in terms of the sentences which can be proved from the axioms. If both a sentence and its negation are provable from the axioms, then the system is inconsistent and in fact any sentence is then provable from the axiom system. In terms of the article, in order for a formal grammatical analysis to succeed, a single formal grammar containing all accounting standards must be demonstrated. Then any proposed new standards could be appraised in terms of their consistency with the current formal grammar.

Approximate matches

1. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 3
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Score: 0.029
  Related excerpt #2TNJYT:
      Since intuitionistic logic is the most developed logic aside from Boolean logic, it is often suggestive to compare the ditsets of partition logic with the open sets in the topological representation of intuitionistic logic, i.e., of a Heyting algebra (also called a pseudo-Boolean algebra or Brouwer algebra). The negation of an open set is the largest open set disjoint from the given set. But now we see that there is no non-empty ditsets disjoint from any given non-empty ditset. Hence intuitively the negation of any partition \sigma \neq \mathbf{0} , is the partition \mathbf{0} with an empty ditset. The definition \neg\sigma := \sigma \Rightarrow \mathbf{0} gives the same result since the only block U in \mathbf{0} = \{U\} is not contained in any block of \sigma \neq \mathbf{0} . And when \sigma = \mathbf{0} , then \neg\mathbf{0} = \mathbf{0} \Rightarrow \mathbf{0} = \mathbf{1} since U \subseteq U so it is discretized in the implication. That is why the absolute negation \neg\sigma is of less interest than the relative negation \frac{\pi}{\sigma}\sigma = \sigma \Rightarrow \pi which is simply the partition implication.

2. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 4
  Context:
    #L8JZAG On implication and negation in partition logic
      #KS4RBJ 5. Relative negation in partition logic
  Score: 0.028
  Related excerpt #TTFKUQ:
      To study relative negation, we take the ‘consequence’ \pi as fixed and then let the ‘antecedent’ \sigma vary in the \pi -negation \neg_\pi \sigma := \sigma \Rightarrow \pi . Another suggestion from intuitionistic logic is that the negated elements in a Heyting algebra form a Boolean algebra. In partition logic, this is trivially true for absolute negation since the negated elements form the two-element Boolean algebra. And it is also true for the relative \pi -negation as was suggested by viewing the implication \sigma \Rightarrow \pi as an indicator or characteristic function for the inclusion of the blocks of \pi in the blocks of \sigma . And the double \pi -negation \neg_\pi \neg_\pi \sigma = (\sigma \Rightarrow \pi) \Rightarrow \pi just interchanges the 0_B and 1_B so \pi -negation is like the usual negation of a subset represented by its indicator function (i.e., negation interchanges the zero-one values). 1 Thus the triple \pi -negation is the same as the single \pi -negation. For another partition \tau = \{D, D', \dots\} , the join \overset{\pi}{\neg}\sigma \vee \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \vee (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B , iff B is contained in a block C \in \sigma or B is contained in a block D \in \tau , so it acts like the Boolean join or disjunction: \mathbf{0}_B \vee \mathbf{1}_B = \mathbf{1}_B \vee \mathbf{0}_B = \mathbf{1}_B \vee \mathbf{1}_B = \mathbf{1}_B and \mathbf{0}_B \vee \mathbf{0}_B = \mathbf{0}_B . Similarly, the meet \overset{\pi}{\neg}\sigma \wedge \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \wedge (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B iff B is contained in a block C \in \sigma and B is contained in a block D \in \tau , so it acts like the Boolean conjunction. Thus all the partitions over U in the form of a \pi -negation \overset{\pi}{\neg}\sigma = \sigma \Rightarrow \pi form a Boolean algebra \mathcal{B}_\pi with \pi = \overset{\pi}{\neg}\mathbf{1} as the bottom element and \mathbf{1} = \pi \Rightarrow \pi as the top element. Since all the \pi -negated partitions (also called \pi -regular partitions) refine \pi , the Boolean algebra \mathcal{B}_\pi is contained in the upper segment [\pi, \mathbf{1}] and might be called the Boolean core \mathcal{B}_\pi of [\pi, \mathbf{1}] .

3. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 3
  Context:
    #L8JZAG On implication and negation in partition logic
      #FEMQRA 4. Three more equivalent ways to define implication for partitions
        #WUWW5V 4.2. The graph-theoretic definition
  Score: 0.027
  Related excerpt #WR8JA4:
      Specializing to the implication operation \sigma \Rightarrow \pi , the links retained in G(\sigma \Rightarrow \pi) are the links labelled with T_\sigma and F_\pi since that combination is the only one to be evaluated to F_{\sigma \Rightarrow \pi} in the truth table for implication (or conditional). The connected components in that graph G(\sigma \Rightarrow \pi) are the blocks in the partition implication \sigma \Rightarrow \pi .

4. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 2
  Context:
    #L8JZAG On implication and negation in partition logic
      #9DDUBK 3. Implication and negation in partition logic
  Score: 0.024
  Related excerpt #FFVCXY:
      With the implication operation, the (absolute) negation of \sigma can be defined as \neg\sigma := \sigma \Rightarrow \mathbf{0} . But the more interesting (relative) \pi -negation of \sigma is defined as: {}^\pi\neg\sigma := \sigma \Rightarrow \pi , so the \pi -negation of \sigma is just another way of considering the implication \sigma \Rightarrow \pi .

5. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 5
  Context:
    #L8JZAG On implication and negation in partition logic
      #KS4RBJ 5. Relative negation in partition logic
  Score: 0.024
  Related excerpt #MMGG3Y:
      The single \pi -negation \overset{\pi}{\neg}\sigma , the double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma , and the excluded middle \sigma \vee \overset{\pi}{\neg}\sigma are all partitions of special interest. The non-singleton blocks of \overset{\pi}{\neg}\sigma are the blocks B \in \pi that intersect two or more blocks of \sigma . Thus the non-singleton blocks of the double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma are the blocks B \in \pi that are contained in blocks of \sigma so \sigma \Rightarrow \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma is a partition tautology and \sigma \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma . The double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma can be thought of as the \pi -closure of any \sigma inside of \mathcal{B}_\pi . Since \pi \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma , we also have that \sigma \vee \pi \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma .

6. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 5
  Context:
    #L8JZAG On implication and negation in partition logic
      #KS4RBJ 5. Relative negation in partition logic
  Score: 0.023
  Related excerpt #VQ5SJ2:
      Moreover, since the non-singleton blocks of \overset{\pi}{\neg}\sigma intersect two or more blocks of \sigma , the blocks (always non-singleton unless otherwise specified) in the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma are all (strictly) smaller than the blocks of \pi so \pi \lesssim \sigma \vee \overset{\pi}{\neg}\sigma and thus \sigma \vee \pi \lesssim \sigma \vee \overset{\pi}{\neg}\sigma . And since the blocks of \sigma \vee \overset{\pi}{\neg}\sigma are strictly smaller than the blocks of \pi , no blocks of \pi are discretized in its \pi -negation, i.e., \overset{\pi}{\neg}(\sigma \vee \overset{\pi}{\neg}\sigma) = \pi . Thus the double \pi -negation of the excluded-middle partition is \mathbf{1} , i.e., \overset{\pi}{\neg}\overset{\pi}{\neg}(\sigma \vee \overset{\pi}{\neg}\sigma) is a partition tautology. While the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma is not (in general) equal to \mathbf{1} (i.e., is not in general a partition tautology) and is not even in \mathcal{B}_\pi , it could be said to be \pi -dense in \mathbf{1} since its \pi -closure is \mathbf{1} .

7. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 7
  Context:
    #L8JZAG On implication and negation in partition logic
      #GNBDTD 7. Concluding remarks
  Score: 0.023
  Related excerpt #LFDVDT:
      Our purpose has been to develop the notions of negation and implication (relative negation) in the logic of partitions. Since partition relations (ditsets) and equivalence relations (indit sets) are complementary in U \times U , every result in the logic of partitions has a complementary-dual result in the logic of equivalence relations so the latter is not really a different logic but a complementary way to view partition logic. There is a similar complementary-duality in intuitionistic logic between Heyting algebras (modelled by the open subsets of a topological space) and co-Heyting algebras [13] (modelled by the closed subsets). 4 Intuitionistic logic makes the symmetry-breaking choice to deal with Heyting algebras rather than co-Heyting algebras, and we have made the similar choice to develop the logic of partitions rather than the (‘anti-isomorphic’) logic of equivalence relations. For instance, the complementary-dual to the implication operation on partitions is the difference or subtraction operation on equivalence relations. The partition logic tautology of modus ponens has the customary form: (\sigma \wedge (\sigma \Rightarrow \pi)) \Rightarrow \pi , whereas the corresponding formula in the dual logic of

8. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 6
  Context:
    #L8JZAG On implication and negation in partition logic
      #LEZCCF 6. Valid formulas
  Score: 0.021
  Related excerpt #J8KTXB:
      Conversely, given any formula using the connectives of \vee, \wedge, \Rightarrow , and the constants of \mathbf{0} and \mathbf{1} , its single \pi -negation transform is obtained by replacing each atomic variable \sigma by its single \pi -negation \frac{\pi}{\neg\sigma} = \sigma \Rightarrow \pi and by replacing the constant \mathbf{0} by \pi . The binary operations \vee, \wedge , and \Rightarrow as well as the constant \mathbf{1} all remain the same. For instance, the single \pi -negation transform of the excluded middle formula \sigma \vee \neg\sigma = \sigma \vee (\sigma \Rightarrow \mathbf{0}) is the partition tautology of the weak law of excluded middle for \pi -negation:

9. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 5
  Context:
    #L8JZAG On implication and negation in partition logic
      #KS4RBJ 5. Relative negation in partition logic
  Score: 0.021
  Related excerpt #7FLLBB:
      Since both the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma and the double \pi -negation partition \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma refine \sigma \vee \pi , they meet (greatest lower bound) (\sigma \vee \overset{\pi}{\neg}\sigma) \wedge \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma must also refine \sigma \vee \pi . Moreover, that is an equality since the blocks of \sigma \vee \overset{\pi}{\neg}\sigma are the non-empty intersections C \cap B for C \in \sigma and B \in \pi where B is not contained in any C \in \sigma , and the blocks of \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma are the blocks B contained in some C \in \sigma . Those non-singleton blocks are all disjoint, so there are no overlaps in the meet operation. Hence those blocks remain the same in the meet and they are precisely the blocks of the join \sigma \vee \pi , i.e.,

10. Source: On implication and negation in partition logic (#C8FHDZ),  , David Ellerman, p. 0
  Context:
    #L8JZAG On implication and negation in partition logic
      #7WB5RZ 2. The logic of partitions
  Score: 0.019
  Related excerpt #ZBQE96:
      Our purpose here is briefly present the basics of partition logic that suffice to explore the role of negation and implication in that logic. A partition \pi = \{B, B', \dots\} on a set U is a set of non-empty subsets B, B', \dots (“blocks”) of U where the blocks are mutually exclusive (the intersection of distinct blocks is empty) and jointly exhaustive (the union of the blocks is U ). An equivalence relation is a binary relation E \subseteq U \times U that is reflexive, symmetric, and transitive. Every equivalence relation on a set U determines a partition on U where the equivalence classes are the mutually exclusive and jointly exhaustive blocks of the partition. Conversely, every partition on a set determines an equivalence relation on the set; two elements are equivalent if they are in the same block of the partition. The notions of a partition on a set and an equivalence relation on a set are thus interdefinable. Indeed, equivalence relations and partitions are often considered as the “same” as in the conventional practice (not used here) of defining the “lattice of partitions” as the lattice of equivalence relations [4].

### 6. Assistant tool calls

#### Tool call: `web_search`

```json
{
  "query": "RDF Surfaces specification logic negation graphs Hayes Harmelen 2024"
}
```

#### Tool call: `web_search`

```json
{
  "query": "RDF Surfaces paper semantics graffiti surface negation existential universal variables"
}
```

#### Tool call: `get_document`

```json
{
  "id": "C8FHDZ"
}
```

#### Tool call: `get_document`

```json
{
  "id": "GANFP9"
}
```

### 7. Tool result: web_search

WEB SEARCH RESULTS
Query: RDF Surfaces specification logic negation graphs Hayes Harmelen 2024

You’re likely looking for the **2024 RDF Surfaces work on adding classical negation and first-order logic to RDF**.

### Main 2024 publications

1. **Hochstenbach et al., “RDF Surfaces: Enabling Classical Negation on the Semantic Web”**  
   Authors: Patrick Hochstenbach, Mathijs van Noort, Dörthe Arndt, Rebekka Martens, Jos De Roo, Ruben Verborgh, Pieter Bonte, and Femke Ongenae.  
   The paper introduces:
   - positive and negative RDF “surfaces”;
   - **Hayes triples/graphs**;
   - scoped variables called *graffiti*;
   - classical—not failure-based—negation;
   - encodings of full first-order logic through conjunction, negation, nesting, and quantification. ([arxiv.org](https://arxiv.org/abs/2406.10659?utm_source=openai))

2. **Arndt et al., “RDF Surfaces as a First-Order Language for the Semantic Web”**  
   Published at **RuleML+RR 2024**, pp. 200–216, DOI `10.1007/978-3-031-72407-7_15`. It supplies a more formal semantics by mapping RDF Surfaces to first-order formulas and implements the `rs2fol` translation to TPTP. ([imec-publications.be](https://imec-publications.be/entities/publication/bcac214e-b80b-4f78-abfc-49cb6692c49d?utm_source=openai))

### Specification

The **RDF Surfaces Primer**, dated **May 16, 2024**, is a living W3C Community Group document—not a W3C Recommendation. It defines an N3 sublanguage using:

```n3
() log:onNegativeSurface {
    :WOS :indexed :JournalABC .
}.
```

This means:

\[
\neg indexed(WOS, JournalABC)
\]

Multiple triples inside the surface are interpreted conjunctively, so the surface denotes the negation of their conjunction. Nested negative surfaces provide disjunction and implication; scoped blank nodes provide existential and universal quantification. ([w3c-cg.github.io](https://w3c-cg.github.io/rdfsurfaces/?utm_source=openai))

### Hayes versus van Harmelen

- **Pat Hayes** is the intellectual source: RDF Surfaces develops his **BLOGIC/RDF Redux** proposal from his 2009 ISWC invited talk, itself inspired by Charles Sanders Peirce’s existential graphs. “Hayes graph” and “Hayes triple” acknowledge this origin. ([backoffice.biblio.ugent.be](https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ))
- The 2024 author is **Mathijs van Noort**, not Frank van Harmelen.
- **Frank van Harmelen is not an author** of either principal 2024 RDF Surfaces paper; his name appears in related Semantic Web/OWL literature and references. ([imec-publications.be](https://imec-publications.be/entities/publication/bcac214e-b80b-4f78-abfc-49cb6692c49d?utm_source=openai))

A suitable citation is:

> Arndt, D., De Roo, J., Hochstenbach, P., Martens, R., Ongenae, F., & van Noort, M. (2024). *RDF Surfaces as a First-Order Language for the Semantic Web*. RuleML+RR 2024, 200–216. DOI: `10.1007/978-3-031-72407-7_15`.

Cited sources:
- RDF Surfaces: Enabling Classical Negation on the Semantic Web: https://arxiv.org/abs/2406.10659?utm_source=openai
- RDF Surfaces as a First-Order Language for the Semantic Web: https://imec-publications.be/entities/publication/bcac214e-b80b-4f78-abfc-49cb6692c49d?utm_source=openai
- RDF Surfaces Primer: https://w3c-cg.github.io/rdfsurfaces/?utm_source=openai
- : https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ

### 8. Tool result: web_search

WEB SEARCH RESULTS
Query: RDF Surfaces paper semantics graffiti surface negation existential universal variables

## RDF Surfaces: semantics of surfaces, graffiti, and variables

**RDF Surfaces** extends RDF with two main constructs:

1. **Surfaces**, which group RDF triples and may negate their contents.
2. **Graffiti**, which explicitly bind blank nodes as quantified variables.

Together with RDF’s conjunction of triples, these constructs can represent first-order logic. ([arxiv.org](https://arxiv.org/abs/2406.10659))

### 1. Surfaces

Triples on the default positive surface are asserted and combined by conjunction:

```n3
:Alice a :Person .
:Alice :knows :Bob .
```

means:

\[
Person(Alice)\land knows(Alice,Bob)
\]

A negative surface negates the conjunction of everything inside it:

```n3
() log:onNegativeSurface {
    :Alice a :Cat .
    :Alice :likes :Fish .
} .
```

means:

\[
\neg\bigl(Cat(Alice)\land likes(Alice,Fish)\bigr)
\]

It does **not** mean that each triple is independently negated. ([w3c-cg.github.io](https://w3c-cg.github.io/rdfsurfaces/?utm_source=openai))

### 2. Graffiti

The list in the subject position of a surface declares its **graffiti variables**:

```n3
(_:x) log:onNegativeSurface {
    _:x a :Human .
} .
```

Here `_:x` in the list is the variable declaration; occurrences of `_:x` inside the graph are references to that declaration. A nested declaration with the same label shadows the outer one. ([w3c-cg.github.io](https://w3c-cg.github.io/rdfsurfaces/))

Formally, a positive surface is translated as:

\[
\exists x_1\ldots \exists x_n\,H
\]

and a negative surface as:

\[
\forall x_1\ldots \forall x_n\,\neg H
\]

where \(H\) is the conjunction of the surface’s contents. ([backoffice.biblio.ugent.be](https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ))

### 3. Existential variables

A graffiti variable on a positive—or even-parity—surface is existential:

```n3
_:x a :Human .
```

means:

\[
\exists x\,Human(x)
\]

Plain RDF blank nodes remain compatible with this interpretation because undeclared/free blank nodes receive existential closure at the default positive surface. ([backoffice.biblio.ugent.be](https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ))

### 4. Universal variables

A graffiti declaration inside one negation has odd parity and consequently behaves universally through De Morgan duality:

\[
\neg\exists x\,P(x)\equiv \forall x\,\neg P(x)
\]

Thus:

```n3
(_:x) log:onNegativeSurface {
    _:x a :Human .
} .
```

means:

\[
\neg\exists x\,Human(x)
\equiv
\forall x\,\neg Human(x)
\]

In words: **nothing is human**. It does not mean “everything is human.” ([arxiv.org](https://arxiv.org/abs/2406.10659))

To express **everything is human**, another negative surface is needed:

```n3
(_:x) log:onNegativeSurface {
    () log:onNegativeSurface {
        _:x a :Human .
    } .
} .
```

This translates as:

\[
\neg\exists x\,\neg Human(x)
\equiv
\forall x\,Human(x)
\]

### 5. Implication pattern

The standard RDF Surfaces implication pattern is:

```n3
(_:x) log:onNegativeSurface {
    _:x a :Human .

    () log:onNegativeSurface {
        _:x a :Mortal .
    } .
} .
```

Its logical form is:

\[
\neg\exists x\bigl(Human(x)\land\neg Mortal(x)\bigr)
\]

which is equivalent to:

\[
\forall x\bigl(Human(x)\rightarrow Mortal(x)\bigr)
\]

So the outer graffiti variable is universally interpreted, while the nested negative surface constructs the implication’s consequent. ([backoffice.biblio.ugent.be](https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ))

### Important distinction

These two forms have very different meanings:

```n3
# x is explicitly graffiti on the negative surface
(_:x) log:onNegativeSurface {
    _:x a :Human .
} .
```

\[
\forall x\,\neg Human(x)
\]

versus:

```n3
# x is free here and therefore existentially closed outside
() log:onNegativeSurface {
    _:x a :Human .
} .
```

\[
\exists x\,\neg Human(x)
\]

The first says **nothing is human**; the second says **something is not human**. ([backoffice.biblio.ugent.be](https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ))

### Summary table

| Construction | Logical interpretation |
|---|---|
| Multiple triples on a surface | Conjunction |
| Negative surface around \(H\) | \(\neg H\) |
| Graffiti on positive/even-parity surface | Existential variable |
| Graffiti on negative/odd-parity surface | Universal effect through negated existential |
| Undeclared/free blank node | Existentially closed at the outer level |
| Nested negative surfaces | Used to construct universal claims, implication, disjunction and double negation |

The most precise interpretation is therefore: **graffiti provide explicit variable binding and scope; negation parity determines how those existential bindings appear in the equivalent normalized FOL formula.**

Cited sources:
- RDF Surfaces: Enabling Classical Negation on the Semantic Web: https://arxiv.org/abs/2406.10659
- RDF Surfaces Primer: https://w3c-cg.github.io/rdfsurfaces/?utm_source=openai
- RDF Surfaces Primer: https://w3c-cg.github.io/rdfsurfaces/
- : https://backoffice.biblio.ugent.be/download/01J8SDWH9HEVQMZR0R4NVF8F76/01J8SDYCSHMHC3YQNZ2Q5AA3EJ

### 9. Tool result: get_document

DOCUMENT #C8FHDZ
Title: On implication and negation in partition logic

Outline:
  - 1 #L8JZAG On implication and negation in partition logic
    - 1.1 #ZZPBEJ 1. Introduction: The Boolean logic of subsets and the logic of partitions
    - 1.2 #7WB5RZ 2. The logic of partitions
    - 1.3 #9DDUBK 3. Implication and negation in partition logic
    - 1.4 #FEMQRA 4. Three more equivalent ways to define implication for partitions
      - 1.4.1 #8P96YU 4.1. The adjunctive definition
      - 1.4.2 #WUWW5V 4.2. The graph-theoretic definition
      - 1.4.3 #H2YZ4E 4.3. The ditset definition
    - 1.5 #KS4RBJ 5. Relative negation in partition logic
    - 1.6 #LEZCCF 6. Valid formulas
    - 1.7 #GNBDTD 7. Concluding remarks
    - 1.8 #87DAFA References

### 10. Tool result: get_document

DOCUMENT #GANFP9
Title: A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology

Outline:
  - 1 #HY74XE A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology
    - 1.1 #ST2HVQ Abstract
  - 2 #FLB78P Contents
  - 3 #MEVK4Z 1 Introduction: A Fundamental Duality in the Sciences
  - 4 #ZJN679 2 Methods: The Dual Logics of Subsets and Partitions
  - 5 #DGJK69 3 Results
    - 5.1 #3EY9FC 3.1 The Fundamental Duality as the Reverse-the-Arrows in Category Theory
      - 5.1.1 #8FLPXH 3.1.1 The Elements-and-Distinctions Definition of Functions
      - 5.1.2 #4EJ7VK 3.1.2 Subsets and Partitions as Morphisms
    - 5.2 #4V3BVP 3.2 The Canonical Morphisms in Universal Mapping Properties in Sets
      - 5.2.1 #BB55N6 3.2.1 Coproduct in Sets
      - 5.2.2 #TPPZXR 3.2.2 Product in Sets
      - 5.2.3 #QHXNHR 3.2.3 The Duality in Sets
    - 5.3 #6UWNUS 3.3 Probability and Information: The Quantitative Versions of the Dual Logics
      - 5.3.1 #BCJ8ZC 3.3.1 Probability Theory
      - 5.3.2 #LT9UAS 3.3.2 Logical Entropy
      - 5.3.3 #F9M3T6 3.3.3 The Relationship to Shannon Entropy
    - 5.4 #H59FZG 3.4 The Dual Creation Stories: Ex Nihilo and Big Bang
    - 5.5 #XQMUBJ 3.5 Classical Metaphysics
    - 5.6 #MUCGHX 3.6 Quantum Mechanics Math as the Hilbert Space Version of Partition Math
      - 5.6.1 #M7ZGQQ 3.6.1 Introduction: A Logical Basis for Superposition
      - 5.6.2 #U886EP 3.6.2 Quantum States
      - 5.6.3 #AZQEU9 3.6.3 Quantum Observables
      - 5.6.4 #2QCJ8W 3.6.4 Quantum Measurement
      - 5.6.5 #LVRMM9 3.6.5 The Objective Indefiniteness Interpretation of QM
      - 5.6.6 #THR8RK 3.6.6 Commuting, Non-commuting, and Conjugate Operators
      - 5.6.7 #J4NVLA 3.6.7 Group Representation Theory
    - 5.7 #UXVXWD 3.7 Selectionist and Generative Mechanisms in the Life Sciences
      - 5.7.1 #KZZHDH 3.7.1 Introduction: The Basic Ideas
      - 5.7.2 #R53RQ8 3.7.2 Partitions and Codes
      - 5.7.3 #3H794M 3.7.3 The genetic code
      - 5.7.4 #B6N2ZZ 3.7.4 The Principles &amp; Parameters Mechanism for Language Acquisition
      - 5.7.5 #Y84EKA 3.7.5 Embryonic stem cell development
      - 5.7.6 #MBBW95 3.7.6 Selectionist and Generative Mechanisms Redux
  - 6 #L62FZR 4 Discussion and Conclusions
  - 7 #FZCQ66 5 Declarations
  - 8 #ANBS9T References

### 11. Assistant

A useful starting point is that **RDF Surfaces and partition logic operate at different logical levels**:

- **RDF Surfaces extends RDF toward classical first-order logic.** A surface groups triples into a conjunction; a negative surface asserts the negation of that whole conjunction. Nesting supplies implication, disjunction, and quantifier patterns.
- **Ellerman’s partition logic is not propositional or first-order syntax over facts.** Its values are partitions of a fixed universe \(U\), ordered by refinement. A partition represents which pairs are distinguished and which remain indistinguishable (#3AJ2XX).

So an RDF negative surface is **not** Ellerman’s partition negation. Indeed, Ellerman’s absolute negation is nearly degenerate:

$$
\neg \sigma=\sigma\Rightarrow\mathbf 0=
\begin{cases}
\mathbf 1,&\sigma=\mathbf 0,\\
\mathbf 0,&\sigma\neq\mathbf 0.
\end{cases}
$$

That is why relative negation or implication, \(\sigma\Rightarrow\pi\), does the real work (#2TNJYT).

## The strongest bridge: RDF descriptions induce partitions

Let \(U\) be a set of RDF resources and let an RDF graph determine an observational signature

$$
f_G:U\longrightarrow S,
$$

where \(f_G(x)\) could record selected properties of \(x\). Then \(f_G\) induces the kernel partition

$$
x\sim_G y\quad\Longleftrightarrow\quad f_G(x)=f_G(y).
$$

Resources belong to the same block when the graph does not distinguish them under the chosen vocabulary. Adding informative triples can refine this partition by introducing new distinctions. This is Ellerman’s function-to-partition construction: inverse-image fibres of a function form a partition (#BLF82B).

For example, with terrain cells:

```n3
:cell1 :drainsTo :outletA .
:cell2 :drainsTo :outletA .
:cell3 :drainsTo :outletB .
```

the `:drainsTo` map induces

$$
\pi_{\mathrm{drain}}
=
\{\{\texttt{cell1},\texttt{cell2}\},
  \{\texttt{cell3}\}\}.
$$

These are catchment blocks. This is an exact partition when every cell drains to exactly one outlet, as already noted in #TUCFMG.

## What RDF Surfaces contributes

Ordinary RDF can state the classifications. RDF Surfaces can state the **constraints under which they constitute a partition**, such as:

- no cell drains to two distinct outlets;
- every cell has an outlet;
- two named outlets are distinct;
- one classification entails or conflicts with another.

Thus the division of labour could be:

> **RDF Surfaces expresses and checks the theory of a classification; partition logic describes the informational structure induced by that classification.**

If graph \(G_2\) distinguishes every pair distinguished by \(G_1\), then

$$
\operatorname{dit}(\pi_{G_1})
\subseteq
\operatorname{dit}(\pi_{G_2}),
$$

so

$$
\pi_{G_1}\preceq\pi_{G_2}.
$$

This makes refinement a precise account of “the newer RDF description is more discriminating.” Ellerman defines refinement exactly through ditset inclusion (#3AJ2XX).

## Two different notions of implication

The comparison becomes especially interesting around implication.

An RDF Surfaces implication pattern expresses a classical statement such as

$$
\forall x\,[P(x)\rightarrow Q(x)].
$$

Ellerman’s implication instead compares two partitions:

$$
\sigma\Rightarrow\pi=\mathbf 1
\quad\Longleftrightarrow\quad
\sigma\preceq\pi.
$$

It asks whether the distinctions of \(\sigma\) are preserved by the finer classification \(\pi\). Operationally, it discretizes precisely those blocks of \(\pi\) already contained within blocks of \(\sigma\) (#YUEKZJ). It is therefore better read as an **indicator of refinement** than as material implication between propositions.

The promising synthesis is consequently not:

> RDF surface negation = partition negation.

It is:

> RDF Surfaces supplies classical statements about resources and relations; those relations induce quotient structures whose refinement, combination, and information content can be analysed using partition logic.

One caveat is crucial: not every RDF relation determines a partition. The relevant relation must be reflexive, symmetric, and transitive—or arise as the kernel of a function—and its blocks must be mutually exclusive and jointly exhaustive (#ZBQE96). Overlapping tags, spatial centres, and multiple classifications are generally covers or families of subsets, not single partitions.