Vague Reference and Approximating Judgments

In what follows, we use lower case roman letters $o,\; o\_1,\; o\_2,\; \backslash dots$ to symbolize objects in reality; $z,\; z\_1,\; z\_2,\; \backslash dots$ to symbolize cells of granular partitions; upper case roman letters from the beginning of the alphabet $A,\; B,\; C,\; \backslash dots$ to symbolize sets of cells; upper case roman letters from the middle of the alphabet $L,\; P,\; \backslash dots$ to symbolize sets of ordered tuples; and upper case Greek letters $\backslash Delta,\; \backslash Delta\_1,\; \backslash dots$ to symbolize sets of objects in reality.

A granular partition $Pt\; =\; ((A,\; \backslash sqsubseteq),\; (\backslash Delta,\; \backslash leq),\; P,\; L)$ is a quadruple such that $(A,\; \backslash sqsubseteq)$ is a system of cells or a cell-structure, $(\backslash Delta,\; \backslash leq)$ is a target domain, $L\; \backslash in\; \backslash text\{Pow\}(\backslash Delta\; \backslash times\; A)$ is a location relation, and $P\; \backslash in\; \backslash text\{Pow\}(A\; \backslash times\; \backslash Delta)$ is a projection relation. The target domain $(\backslash Delta,\; \backslash leq)$ is hereby understood as a mereological structure with $\backslash Delta$ a set of objects and $\backslash leq$ a part-of relation defined on $\backslash Delta$ which satisfies the axioms of general extensional mereology (GEM). A cell structure, $(A,\; \backslash sqsubseteq)$, is a finite set of cells, $z\_0,\; z\_1,\; \backslash dots,\; z\_n$ with a binary subcell relation $\backslash sqsubseteq$. 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:

MA1: The subcell relation $\backslash sqsubseteq$ is reflexive, transitive, and antisymmetric.

MA2: The cell structure of a partition is always such that chains of nested cells are of finite length.

MA3: If two cells have subcells in common, then one is a subcell of the other.

MA4: Each partition contains a unique maximal cell.

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).

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:

(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.)

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:

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):

We require also that projection, considered as a function $p:\; A\; \backslash rightarrow\; \backslash Delta$ between two partially ordered domains $(A\; \backslash text\{\; and\; \}\; \backslash Delta)$, be an order homomorphism:

The root or maximal cell in the cell structure is then the maximal object (the universal or total fusion) in $\backslash Delta$.

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 $Pt\; =\; ((A,\; \backslash subseteq),\; (\backslash Delta,\; \backslash leq),\; P,\; L)$. At the same time we assume a fixed target domain $\backslash Delta$, which the reader can think of as the whole of reality.

What, now, of vagueness? A vague granular partition $Pt^V\; =\; ((A,\; \backslash subseteq),\; (A,\; \backslash leq),\; P^V,\; L^V)$ 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.

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.

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 ‘$\backslash exists!$’ abbreviates: ‘there exists one and only one!’):

We also demand that all $P\_i$ and all $L\_j$ are functional in the sense discussed in the crisp case:

We demand further that cells project onto some object (are non-empty) under every projection:

Again, every particular projection considered as a function from $A$ to $\backslash Delta$ is an order homomorphism:

We now add an axiom that governs the interrelations between projection relations with distinct indexes in the vague partition $\backslash text\{Pt\}^V$. 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:

Given a vague partition as defined above, we can define an equivalence relation between entities in the target domain $\backslash Delta$ as

Clearly, $\backslash approx$ is symmetric and reflexive. To see that $\backslash approx$ is also transitive, assume $o\_1\; \backslash approx\; o\_2$ and $o\_2\; \backslash approx\; o\_3$. 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 $\backslash text\{MB7\}^V$ 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., $o\_1\; \backslash approx\; o\_3$. In the remainder, we write $[o]\_z$ to denote the set $\backslash \{o\; \backslash mid\; \backslash exists\; i\; :\; P\_i(z,\; o)\backslash \}$.

Let $\backslash text\{Pt\}^V\; =\; (A,\; P^V,\; L^V)$ be a vague granular partition. We call all partitions $\backslash text\{Pt\}\; =\; (A,\; P\_b,\; L\_j)$ with $P\_i\; \backslash in\; P^V$ and $L\_j\; \backslash in\; L^V$ which satisfy the axioms MB1–MB6 crispings of the vague partition $\backslash text\{Pt\}^V$. 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 $\backslash approx$. This captures the de dicto view of vagueness.

Given a vague partition $\backslash text\{Pt\}^V\; =\; (A^V,\; \backslash subseteq),\; (A,\; \backslash leq),\; P^V,\; L^V)$, we can for each cell $z\; \backslash in\; A^V$ 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.

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:

Thus the summit of Mount Everest is part of the determinate core of the vague projection of the name 'Everest' and its associated cell.

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:

The dotted region in Figure 2 illustrates the indeterminate zone of the projection of the cell associated with the vague name 'Mount Everest'.

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 $P^V$.

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.

Reference via a vague name 'N' creates a partition of reality into determinate core, indeterminate zone, and exterior zone. Let $\backslash sigma\; x\; \backslash psi(x)$ denote the mereological sum of all $x$ satisfying $\backslash psi(x)$ 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 $z$:

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 $\backslash text\{det\}\_V(z)$ is a partial function since there does not necessarily exist a portion of reality which is a part of all projections of the cell $z$.

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?

Suppose you are an expert mountain guide hiking through the Himalayas with your friends and you assert:

[A]: We will cross the boundary of Mount Everest within the next hour.

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.

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.

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.

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.

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:

[B] Hurricane Walter extends over parts of Wyoming, parts of Montana, parts of Utah, and parts of Idaho.

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].

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).

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.

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.

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.

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 $Pt^R$.

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.

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.

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.

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.

We call the relation $L^R(o,\; z,\; \backslash omega)$ the rough location of the portion of reality o with respect to the cell z and the relation $P^R(z,\; o,\; \backslash omega)$ 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, $\backslash omega$ 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., $\backslash forall\; i:\; L^R(HW\_i,\; Montana,\; po)$. The relation no holds between all the $HW\_i$ and Oregon, i.e., $\backslash forall\; i:\; L^R(HW\_i,\; Oregon,\; no)$.

We can characterize the relationships between exact and rough location and exact and rough projection in reference partitions as follows:

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:

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 $\backslash sim$ 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 $\backslash sim$. 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 $\backslash sim$ 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.

We define a reference partition as a quintuple, $Pt^R\; =\; ((A,\; \backslash subseteq),\; (\backslash Delta,\; \backslash leq),\; P^R,\; L^R,\; \backslash Omega)$ where $(A,\; \backslash subseteq)$ and $(\backslash Delta,\; \backslash leq)$ are a cell structure and target domain as specified above, $P^R$ and $L^R$ are rough projection and location relations, and $\backslash Omega$ is the set of values $\backslash \{fo,\; po,\; no\backslash \}$ 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:

TR1 follows from MB1. To see this assume $L^R(o,\; z,\; \backslash omega)$ and let $\backslash omega\; =\; fo$. We have $\backslash exists\; x\; (L(x,\; z)\; \backslash text\{\; and\; \}\; x\; \backslash leq\; o)$. By MB1 we have $\backslash exists\; x\; (P(z,\; x)\; \backslash text\{\; and\; \}\; x\; \backslash leq\; o)$, hence $P^R(z,\; o,\; fo)$ and similarly for $\backslash omega\; =\; po$ and $\backslash omega\; =\; no$. TR2 follows from MB2 in a similar manner. To see TR3 assume $\backslash forall\; z,\; \backslash omega\; (P^R(z,\; o\_1,\; \backslash omega)\; \backslash leftrightarrow\; P^R(z,\; o\_2,\; \backslash omega))$. By TR1 and TR2 $L^R$ and $P^R$ are logically equivalent and can be substituted for each other. Therefore we have $L^R(o\_1,\; z,\; \backslash omega)\; \backslash leftrightarrow\; L^R(o\_2,\; z,\; \backslash omega)$, i.e., $o\_1\; \backslash sim\; o\_2$.

Corresponding to MB4 we now demand that if all objects have the same relation $\backslash omega\; \backslash in\; \backslash \{fo,\; po,\; no\backslash \}$ with respect to the cells $z\_1$ and $z\_2$ then these two cells are identical:

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).

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 $\backslash approx$) have the same rough approximation in the underlying reference partition (i.e., are also equivalent in the sense of $\backslash sim$):

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 $\backslash approx$ does not imply equivalence in the sense of $\backslash sim$.

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 ($x\; \backslash sim\; y$), 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'.

In this section, we discuss the relationship between semantic partition and the kinds of reference partitions previously discussed.

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 $Pt^R$.

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 $\backslash sim$. 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.

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 $o$:

where $\backslash text\{co\}(x)$ is defined as above.

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 $o$:

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'.

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.

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.

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'.

In formal terms, we describe this as follows. Let $J\; =\; (N,\; Pt^V,\; Pt^R)$ be an approximating judgment with $Pt^V\; =\; ((A^V,\; \backslash subseteq),\; (\backslash Delta,\; \backslash leq),\; P^V,\; L^V)$ and $Pt^R\; =\; ((A^R,\; \backslash subseteq),\; (\backslash Delta,\; \backslash leq),\; P^R,\; L^R,\; \backslash Omega)$. We then call $J^A$ precise if and only if

Here $z\; \backslash in\; A^V$ 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.

We now discuss constraining approximations, defined as those approximations that do not have the property of being precise but still satisfy the equivalence principle.

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 $o\; \backslash in\; [o]\_z$ with respect to the approximating partition $Pt^R$ is called constraining if and only if the following holds:

either:

Lower(o) is defined and $\backslash text\{Lower\}(o)\; \backslash leq\; \backslash text\{det\}\_V(z)\; \backslash leq\; (\backslash text\{det\}\_V(z)\; +\; \backslash text\{indet\}\_V(z))\; \backslash leq\; \backslash text\{Upper\}(o)$
or:

Consider Figure 4 and assume that the determinate zone of the vague reference of the name ‘Hurricane Walter’, $\backslash text\{det\}\_V(\backslash text\{HW\})$, 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.

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.