SPATIAL COGNTI10N AND COMPUTATION, 3(2&3), 137-156 Copyright © 2003, Lawrence Erlbaum Associates, Inc. Vague Reference and Approximating Judgments Thomas Bittner University of Leipzig Barry Smith University of Leipzig and State University of New York at Buffalo 'Mount Everest' is a vague name. That is (on the account here defended) there are many portions of reality all of which have equal claims to serve as its referent. We propose a new account of such vagueness in terms of a theory of what we shall call granular partitions. We distinguish different kinds of crisp and non-crisp granular partitions and we describe the relations between them, concentrating especially on spatial examples. In addition, we describe the practice whereby subjects use systems of reference grids as a means for tempering the vagueness of their judgments, for example when they say that Libya straddles the Equator or that the meeting will take place between 2 and 3pm. We then demonstrate how the theory of reference partitions can yield a natural account of this practice, which is referred to in the literature as 'approximation'. Keywords: ontology, granular partitions, vagueness, semantic partitions, partition theory, approximation Consider the proper name 'Mount Everest'. This refers to a mereological whole, a certain giant formation of rock:. A mereological whole is the sum of its parts, and Mount Everest certainly contains its summit as part. But it is not so clear which parts along the foothills of Mount Everest are parts of the mountain and which belong to its surroundings. Thus it is not clear which mereological sum of parts of reality actually constitutes Mount Everest. One option is to hold that there are multiple candidates, no one of which can claim exclusive rights to serve as the referent of this name. All of these candidates are involved, in some Correspondence concerning this article should be addressed to Thomas Bittner, Institute for Formal Ontology and Medical Information Science, Univeniity of Leipzig; email thomas.bittner@ifomis.uni-leipzig.de. 138 BITTNER, SMITH sense, when we use the name 'Mount Everest.' We are however not conscious of this multiplicity of candidate referents, effectively because we simply do not care about the question where, precisely, the boundaries around Mount Everest are to be drawn. Each of the many candidates has the summit, with its height of29,028 feet, as part. Each is also a perfectly determinate portion of reality. The candidates differ only regarding which parts along the foothills are included and which are not. V arzi (2001) refers to the above as a de dicto view of vagueness. It treats vagueness not as a property of objects but rather as a semantic property of names and predicates, a property captured formally in terms of a supervaluationistic semantics (Fraassen 1966), (Fine 1975). We shall concentrate our attentions in what follows on the case of singular reference, i.e., reference via names and definite descriptions to concrete portions of reality such as mountains and deserts. We shall also concentrate primarily on spatial examples. As will become clear, however, it is one advantage of the framework here defended that it can be generalized automatically beyond the spatial case. In order to understand vague reference we use the theory of granular partitions we advanced in our earlier papers: (Bittner and Smith 2001a), (Bittner and Smith 2003), (Smith and Brogaard 2002). The fundamental idea is that every use of language to make a judgment about reality brings about a certain granular partition, a grid-like system of cells conceived as projecting onto reality in something like the way in which a bank of flashlights projects onto reality when it carves out cones of light in the darkness. Each judgment, J, can then be conceived as a pair consisting of a sentence, S, and an associated granular partition Pt. We consider reference as a two-step-process. Language tokens are associated with cells in a grid-like structure, and these cells are projected onto reality in the way suggested by our flashlight metaphor. Granular partitions can then be conceived as the cognitive artifacts whereby language gains its foothold in reality. (They thus play a role somewhat similar to that of set-theoretical models in more standard treatments.) In our earlier papers, we showed how this twostep-process allows us to explain the features of selectivity and granularity of reference in judgments. In this paper, we show how the same machinery can help us to understand the phenomena of vagueness and approximation. Crisp Granular Partitions The theory of granular partitions has two parts: (A) a theory of the relations between cells and the structures they form, and (B) a theory of the relations between cells and objects in reality. Consider Figure 1. The left part shows a very simple cell structure, with cells labeled Everest, Lhotse and The Himalayas. The right part shows portions of reality onto which those cells project. Lhotse I I Everest I The Himalayas Figure 1: Left: a partition, with cells Lhotse, Everest and The Himalayas. Right: A part of the Himalayas seen from space, with admissible candidate referents for 'Mount Lhotse' (left) and 'Mount Everest' (right). Language In what follows, we use lower case roman letters o, oi, Oz, .. • to symbolize objects in reality; z, z., 22, to symbolize cells of granular partitions; upper case roman letters from the beginning of the alphabet A. B, C, ... to symbolize sets of cells; upper case roman letters from the middle of the alphabet L, P, ... to symbolize sets of ordered tuples; and upper case Greek letters L1, .e'.11,, ••• to symbolize sets of objects in reality. Theory A A granular partition Pt= ((A, ~), (.e'.1, ~' P, L) is a quadtuple such that (A, ~)is a system of cells or a cell-structure, (L1, ~ is a target domain, LePow(.e'.1 x A) is a location relation, and PePow(A x A) is a projection relation. The target domain (L1, ~. is hereby understood as a mereological structure with A a set of objects and :s; a part-of relation defined on A which satisfies the axioms of general extensional mereology (GEM). A cell structure, (A, ~). is a finite set of cells, z0, zi, ... Zn with a binary subcell relation ~-We say that z1 is a subcell of 22 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 fullows: MAl: The subcell relatioñ is reflerive, 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). 140 BITTNER, SMITH Theory B 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 MB 1 and MB2 hold: MBl L(o, z)-+ P(z, o). MB2 P(z, o)-+ L(o, z). (Here and in what follows initial universal quantifiers are taken as understood. We preserve L and Pas 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: MB3 P(z, o1) and P(z, Oz) -+ 01 = Oz MB4 L(o, z1) and L(o, z2)-+ z1 = Z2 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): MB5 z e A-+ 3o: L(o, z) We require also that projection, considered as a function p: A -+ .6. between two partially ordered domains (A and .6.), be an order homomorphism: MB6: z1 !;;;;; z2 -+ p(z1) ~ p(z2) The root or maximal cell in the cell structure is then the maximal object (the universal or total fusion) in .6.. 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 MBl-6 have been formulated for easy understandability and the system they form is not minimal. (Thus MB2 already follows from MBl, MB3 and MB5.) In order to simplify the notation in what follows, we write Pt= (~ P, L) as an abbreviation for Pt= ((A, ~), (.6., S), P, L). At the same time we assume a fixed target domain .6., which the reader can think of as the whole of reality. VAGUE REFERENCE AND APPROXIMATING 141 Vague Granular Partitions The Theory What. now, of vagueness? A vague granular partition Ptv ==((A. i;;;;;), (A, ~. pv, L v} is a quad.tuple in which the cell structure and target domain are defined as above, and pV and L v are classes of projection and location relations (Bit1ner and Smith 200lb). Again, we wi11 write Ptv =(A, pv, L v) in order to keep the notation simple. Consider Figure 2, which depicts a vague partition Ptv = (A, Pv, L v) of the Himalayas. This has a cell structure A, as shown in the left part of Figure 2, which is in met identical to the corresponding part of Figure l . 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 ooundaries of the actual candidates onto which the cells 'Lhotse' and 'Everest' are projected under the various Pdn pV are continuous ovoids included somewhere within the cloudy regions depicted in the Figure. The projection and location relations in these classes form pairs (Pu ~' which are such that each Pi has a oorresponding unique L; and vice versa, satisfying the following conditions (where the notation '3!i' abbreviates: 'there exists one and only one i'): MBlv Vj: L;(o, z) ~ 31i P;(z, o) MB2v Vi: Pi (z, o} ~ 3!j: L;(o, z} We also demand that all Pi and all L; are functional in the sense discussed in the crisp case: MB3v Pi{z, 01) and Pi(z, 0,2) ~ 01 = 02 MB4v ~o. z1) and L;(o, Zi) ~ z1 = Z2 Lhotse I I Everest I The Himalayas Figure 2: A vague partition of the Himalayas 142 BITTNER, SMITH We demand further that cells project onto some object (are non-empty) under every projection: MB5v: Z(z, A)~ V'j 3o: l._j(o, z) Again, every particular projection considered as a function from A to /1 is an order homomorphism: MB6v: z1 ~ z2 ~ Pi(z1) ~ Pi(z2) We now add an axiom that governs the interrelations between projection relations with distinct indexes in the vague partition Ptv. 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 z1 and z2 under different projections Pi and Pj then the targeting cells must be identical: MB7v Pi(zi, o) and Pj(Zi, o) ~ z1 = z2 Equivalence of Candidate Referents Given a vague partition as defined above, we can define an equivalence relation between entities in the target domain /1 as ~ 01 ~ Oz = 3z, i, j : Pi(z, 01) and Pj(Z, Oz). Clearly,~ is symmetric and reflexive. To see that~ is also transitive, assume o1 ~ Oz and Oz ~ O]. This means that there exist Zi. :q, i, j, k, m such that Pi(Zi. 01), P~(zi, Oz), Pk(z2, Oz) and Pm(z2, 03). From Pj(Zi, Oz) and Pk(z2, Oz) it follows by MB7 that z1 = z2, and hence for some cells z and some projections P1 and Pm it holds that P;(z, o1) and Pm(z, 03), i.e., o1 ~ 03. In the remainder, we write [o]z to denote the set {o !Ji: P;(z, o) }. Let Pt v = (~ P , L v) be a vague granular partition. We call all partitions Pt= (A, P1, l._j) with Pi e Pv and l._j E Lv which satisfy the axioms MB1-MB6 crispings of the vague partition Ptv. Consider a partition with cells labeled with vague proper names. Intuitively, each crisping (A, Pit l._j) then recognizes exactly one candidate precisified referent for each such cell. The precise candidates carved out by the separate (A, Pit Lj) 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 ~. This captures the de dicto view of vagueness. Semantic partition Given a vague partition Ptv=((A v, ~). (11, ~. Pv, L v), we can for each cell z E Av 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. VAGUE REFERENCE AND APPROXIMATING 143 We say that xis part of the determinate core of the vague projection Pv of the cell z if and only if, under all projections pi(z) in Pv, xis a part of the targeted candidate referent: determinatev(x, z) =Vi: x ~ Pi(z) 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 xis a part of the indeterminate zone of the vague projection pv of the cell z if and only ifthere are some projections Pi in pv under which xis part of the targeted candidate referent and other projections pj(z) in pv under which this is not the case: indeterminatev(x, z) = 3i: x ::$; Pi(z) and 3j: -,(x ::$; pj(z)) 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 pV of the cell z if and only if xis not reached by any projection Pi(z) in Pv. exteriorv(x, z) = Vi: -,(x :s; Pi(z)) 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 ofreality into determinate core, indeterminate zone, and exterior zone. Let axw(x) denote the mereological sum of all x satisfying w(x) and let z be the cell in the partition Pv 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: detv(z) = ox determinatev(x, z) indetv(z) =ox indeterminatev(x, z) extv(z) = ox exteriorv(x, 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 detv(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. 144 BITTNER, SMITH Approximating Judgments Approximation in Egg-yolk Partitions 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 ofMount 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. Egg-yolk Partitions vs. Semantic partition 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 VAGUE REFERENCE AND APPROXIMATING 145 exterior boundary where-tho-boundaiy-candidates are Figure 3: Egg-yolk like reference partitions 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. Approximation in Complex Partitions 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 ofldaho. In the sorts of contexts which we humans normally inhabit, it is impossible to refer to any crisp bowidary when making judgments about the location of a region ofbad 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 [BJ. 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 ofthis 146 BITTNER, SMITH 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 finergrained 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. Approximation and Judgments 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 Figure 4: States of the United States with Hurricane Walter VAGUE REFERENCE AND APPROXIMATING 147 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 JA, if uttered successfully, imposes two partitions onto reality: a vague partition Ptv and a reference partition I>f, along the lines above, whereby the latter serves to delimit the vagueness of the former. An approximating judgment JA is thus a triple (S, Pt v, I>f ), consisting of a sentence, S, together with two granular partitions, Ptv and I>f. In the approximating judgment JA = ([B], Pt v, I>f), expressed by the sentence: 'Hurricane Walter extends over parts of Wyoming, Montana, Utah, and Idaho', the corresponding vague partition Ptv 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 I>f. The latter constrains the admissible projections of the cell labeled 'Hurricane Walter' in Ptv in such a way that each candidate referent that is targeted by a pr~ection Pi of Ptv must extend over parts of reality targeted by the cells ofJ>t1<labeled 'Wyoming', 'Utah', 'Montana', and 'Idaho' respectively. Partition Theory and Approximation 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 148 BITTNER, SMITH cells of this reference partition. The constants Jo, 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 ~o, z, ID) the rough location of the portion ofreality o with respect to the cell z and the relation pRcz, o, ID) 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, ID 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 Jo, po, or no. Consider the left part of Figure 4. There the relation po holds between all admissible candidate referents HW; and Montana, i.e., Vi: L R(HWh Montana, po). The relation no holds between all the HW; and Oregon, i.e., Vi: L R(HW;, Oregon, no). We can characterize the relationships between exact and rough location and exact and rough projection in reference partitions as follows: L ~o, z,fo) = 3x (L(x, z) and x ~ o) pR(z, o,fo) = 3x (P(z, x) and x ~ o) L ~o, z,po) = 3x( L(x, z)and 3y (y~ x and y~ o) and 3y (y ~ x and-.(y ~ o))) pR(z, o,po) = 3x (P(z, x) and 3y (y ~ x and y ~ o) and 3y (y ~ x and-.(y ~ o))) L ~o, z, no)= 3x (L(x, z)and -.3y: y ~ x and y ~ o) pR(z, o, no)= 3x (P(z, z) and --,3y: y ~ x and y ~ o) 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 pf with rough location relation LR, as follows: D01 0:2 = Vz, ID: L ~o., z, ID) tt L ~0:2, z, ID). Thus two objects are equivalent with respect to the granular partition pf if and only if they have an identical rough location with respect to all cells of this partition. The relation can thus be interpreted as meaning: indiscernibilitywith respect to the frame ofreference provided by pf, In an approximating judgment JA = (S, Ptv, pf), the reference partition pf will be chosen in such a way that the candidate referents targeted by a single cell in Pt v are equivalent with respect to -. 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 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 VAGUE REFERENCE AND APPROXIMATING 149 familiarity. We will discuss different choices of reference partitions in later sections. We define a reference partition as a quintuple, pf= ((A, ~), (A, ~). pR, LR, n) where (A, ~) and (A, ~) are a cell structure and target domain as specified above, pR and LR are rough projection and location relations, and n is the set of values (fo, po, no) indicating degrees of overlap (coarser and finer distinctions are possible, as discussed in (Bittner and Stell 2003)). We then can prove that the following counterparts ofMBl-3 hold for reference partitions: TRI L ~o, z, co) ~ pR(z, o, co) TR2 pR(z, o, co)~ L ~o, z, co) TR3 (Vz, co: PR(z, 01, ro) ttpR(z, 02, co)) ~ 01 02 TRI follows from MBl. To see this assume LR(o, z, ro) and let oo= fo. We have 3x (L(x, z) and x ~ o). By MBl we have 3x (P(z, x) and x ~ o), hence pR(z, o, fo) and similarly for co= po and co= no. TR2 follows from MB2 in a similar manner. To see TR3 assume Vz,oo ~z, oi, oo) ttpR{z, 02, co)). By TRI and TR2 LR and pR are logically equivalent and can be substituted for each other. Therefore we have LR(o1, z, oo) ttLR(02, z, oo), i.e., 01 02. Corresponding to MB4 we now demand that if all objects have the same relation oo E {fo,po, no} with respect to the cells z1 and z2 then these two cells are identical: Rl (Vo, oo: L R(o, Zt. ro) ++L ~o, Zi, ro)) ~ z1 = z2. Constraining Approximation Well-Formed Approximations If an approximating judgment like ([A], Pt v, pf) 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 pf. Thus, in the hiker case, each value of p vi(' Everest') must be such that its boundary can be crossed in one hour from the time when the judgment is made (Fi~e 3). Let (S, Pt v, pf) be an approximating judgment and let Pt be a vague partition with a cell for each vague name in the sentence S. We then demand that in such an ~oximating judgment the reference partition pf and the vague partition Pt 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~) have the same rough approximation in the underlying reference partition (i.e., are also equivalent in the sense of-): EP Ot ~ 02 ~ 01 02. 150 BITTNER, SMITH We call BP the equivalence principle. BP rules out many reference partitions Pt1l which cannot be used for constraining the vagueness of the vague partition Ptv. 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 lm2 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~ does not imply equivalence in the sense of-. 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 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 Q.eft) 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'. Figure 5 Left: Yellowstone National Park (YNP) and Zion. National Park (ZNP). Right: Hurricane Walter. Precise Approximation In this section, we discuss the relationship between semantic partition and the kinds of reference partitions previously discussed. Consider the approximating judgment J = ([A], PtV, Pf!l). The reference partition Pt1l shown in the left part of Figure 3 imposes two fiat boundaries onto reality: an interior boundary of the approximation and an exterior boundary of the approximation. AB 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 to'W&I'ds the summit of Mount Everest results in the reference partition Ptt. Consider now the semantic partition imposed by the cell labeled 'Everest' in the vague partition Ptv. We can see that the relationship betW\leD. Ptv and pf VAGUE REFERENCE AND APPROXIMATING 151 satisfies the equi:wlenoe principle EP, wbiab &:mands that all candidate TI!finnts oftho'VllgUOname 'Eva:ost' are equivalent1Ulder-. Consiclar nowthe locatian of two pairs af boundaries: (a) the interior and emrior boundaries ~ed by 1he judaina subject 11.!1 a ftame of refb'ence fur her spprmimation; md (b) the bounclariea im.IJOBCd by 1he seu:nmtie partitfon into de!aminatc: ZQile, inclebmninate .zañ and exl«i.or zono via tho 4.1ella of Ptv. Wo say 1hat tho approximating judgment is precise if and ooly if (I) the interior boundary aftho appmximation coincides with fhe boundary separating the ~ zone fi:om the 8'11TOlll!dillg pan. af the semantic partition; and (2) the ezterior boundary af tho approximation coincides with the boundary separating tho sterior zone fi:om the indeterminato zone af tho sanantic partitiDll. This mean& that the semantic partitiOll and the re~ partition coincide. In order to tab more compla xdi::ren1:e panition1 into -.nt., we now define tippet and tawer approximatiam of an object o with telptJCt to such pertitioos. (Again, we use the 11otioo1 oflowe:r 11114 upper approximatiOll in order to rmphaaize the euu:apoocJmi:e 1o r:ough sci theory of Pawlak (1982).) The law« app:roxh:wttion af an cbjec:t o with IeSp«t to a reference partition J.>{is the mereological sum af all those pol1i011S of reality which are mgeted by cells af Pili-an.t which are co11taitted in o: Lower(o) = ao'(3z(o' = p(z) & ?'(z, o,/o))), where ox+(x) is defitted 111 above. The upper epprOllimation is lhe men»l\laieal swn of all those pcx1ions of rcality whim an: targeted by ciells aflhe xefc:mwe partition and whim overlap o: Upper(o) = CICl(3'4,<I = p(z) &: (P8(z, o,/o) or P1'(z, o,po)))) ('.onsidtt 11ow the re&renoe partitiO!lj shown in the left part or Yiaute 6, which is a refined vc:nion oflhe egg-yolk reference partition in the right part of Figure 6. The core cell of the la1ta' ii subdivided into easla'n and westmn Sllhcellt {~ imcl we for eutem pert of the core region and~ part of the care tegion, ~vely). MorecM:l' the region whc:re 1he bo!mdmii:a me is A B Ftgwe 6; .Approximation in eompkx panitions. Left:: a relined egg-~ partition. Right: a raster partition. D 152 BITTNER, SMITH 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 'we'. Its upper approximation has as parts in addition portions ofreality targeted by the cells 'nb' and 'sb'. For another ex.ample of lower and upper approximations, consider the right part ofFigure 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, 0, 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 pR(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, Ptv, l'f) be an approximating judgment with Ptv =((Av,~). (A,~). Pv, L v) and l'f= ((AR,~), (A, ~), pR, LR, Q). We then call JA precise if and only if Voe [o]z: Lower(o) = detv(z) and Upper(o) = (detv(z) + indetv(z)). Here z e Av 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. Constraining Approximation 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 Ptv 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 Pv. The approximation of candidate referents o e [o]z with respect to the approximating partition l'f is called constraining if and only ifthe following holds: Vo E [o]z: VAGUE REFERENCE AND APPROXIMATING 153 either: Lower(o) is defined and Lower(o) ~ detv(z) ~ (detv(z) + indetv(z)) ~ Upper(o) or: detv(z) ~ (detv(z) + indetv(z)) ~ Upper(o) Consider Figure 4 and assume that the determinate zone of the vague reference of the name 'Hurricane Walter', detv(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 ofldaho, Montana, Wyoming, and Utah. Hence the resulting approximation is constraining. Consider the class of constraining approximations. As already Aristotle repeatedly emphasized (at 1094bl 1 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 2001 b ), 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. Properties of Reference Partitions Reference partitions are of central importance for approximating judgments. Examples of reference partitions include: any political subdivision, rastershaped partitions adjusted to latitude and longitude, the block structure in American cities, the subdivision of Vienna into Bezirke and of France into Departements, etc. Other important groups of reference partitions are partitions imposed by quantity-scales of all kinds (Johansson 1989, chapter 4), including temporal partitions like calendars (Bittner 2002). Consider again the judgment [B]: 'Hurricane Walter extends over farts of W~oming, Montana, Utah, and Idaho' and the corre~onding structure J = ([B], Pt , i>f ). The skeleton of the reference partition l¥" is the partition Pt8, which recognizes the United States (Figure 4) and thereby establishes the frame of reference for the approximation. Consider Figure 3. Here the skeleton Pt8 of the reference partitions is an egg-yolk structure containing the cells labeled 'core', 'exterior', and 'where the candidate boundaries are'. Every reference partition i>f =((A.~). (A,~). pR, LR, Q) has a crisp partition Pt8 =((A,~). (A,~. F8, L8) called the skeleton ofi>f. Both, Pt8 and I>f share the cell structure (A, ~) and the target domain (A, ~). In order to ensure that the intuitions sketched in the previous paragraph (and implicitly assumed in our 154 BITTNER, SMITH definitions of rough location LR and rough projection pR) are satisfied we demand that the skeleton has following properties: 1. If Ps(z, o) holds in Pt8 then so does pR(z, o,fo) in ft1l. For all other cells z1 in the shared cell structure A we have pR(zi. o, no). That is: P8(z, o) ~ {eR(z, o,fo) and (Vz1 EA: z1 '# z ~ pR(zi. o, no))). 11. If Ls(o, z) holds in Pt8 then so does L ~o, z,fo) in ft1l. For all other cells z1 in A we have LR( o, zi, no). That is: L8(o, z) ~ (L R(o, z,fo) and (Vz1 e A: z1 *z ~ L R(o, Zi. no))). iii. Skeletons satisfy MBl--6. Often skeletons are also full, exhaustive, and complete in the sense of (Bittner and Smith 2003), which means in effect that they create subdivisions of the targeted domain into jointly exhaustive and pairwise disjoint portions. The skeletons of reference partitions which serve as frames of reference are often spatial or temporal in nature. They are relatively stable, i.e., they do not change over time. This implies in turn: (a) that the pertinent cell structure is fixed and (b) that the objects onto which the skeleton projects do not change (they are, for example, spatial regions tied to the surface of the Earth). Consider again the examples in Figure 4. The granular partition projecting onto the United States has existed for more than one hundred years without significant changes. (Hurricane Walter, in contrast, changes continuously throughout the course of its (brief) existence.) In fact, Figure 4 itself needs to be considered as a snapshot of reality at some determinate point in time (Smith and Brogaard, 2002, Bittner and Smith 2003a, Grenon 2003). It provides us with useful information when we are told that Hurricane Walter was located in parts of Montana, Idaho, Wyoming, and Utah at such and such a time. Every American child learns the corresponding reference partition in school, and uses it for all sorts of purposes thereafter (Stevens and Coupe 1978). Reference partitions are characteristically built out of boundaries with which human beings can become easily familiar, objects which facilitate easy learning. Conclusions We have proposed an application of the theory of granular partitions to the phenomenon of vagueness, a phenomenon which is itself seen in de dicto terms, i.e. as a semantic property of names and predicates. We defended a supervaluationistic theory of the underlying semantics and expressed it in terms of the theory of granular partitions. We showed that the use of frames of reference in making approximating judgments can be formulated very naturally in partitiontheoretic terms, and that the framework of granular partitions then helps us to understand the relationships between vagueness and approximation. While the bulk of our examples were derived from the spatial domain, the generality of the theory of granular partitions allows an easy generalization to other sorts of cases. VAGUE REFERENCE AND APPROXIMATING 155 Acknowledgments The authors thank Wolfgang Heydrich and the reviewers for helpful comments. Part of this work was carried out under the auspices of the Wolfgang Paul Program of the Alexander von Humboldt Foundation. It was also supported by DARPA under the Command Post of the Future program, and by the National Science Foundation under its Research on Learning and Education program and also under NSF Research Grant BCS-9975557: "Geographic Categories: An Ontological Investigation". References Bittner, T. 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