Spatial Vagueness. 1 Introduction. Brandon Bennett

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1 Spatial Vagueness Brandon Bennett Abstract This chapter explores the phenomenon of vagueness as it relates to spatial information. It will be seen that many semantic subtleties and representational difficulties arise when spatial information is affected by vagueness. Moreover, since vagueness is particularly pervasive in spatial terminology, these problems have a significant bearing on the development of computational systems to provide functionality involving high-level manipulation of spatial data. The paper begins by considering various foundational issues regarding the nature and semantics of vagueness. Overviews are then given of several approaches to spatial vagueness that have been proposed in the literature. Following this, a more detailed presentation is given of the relatively recently developed standpoint theory of vagueness and how it can be applied to spatial concepts and relations. This theory is based on the identification of parameters of variability in the meaning of vague concepts. A standpoint is a choice of threshold values determining the range of variation over which a vague predicate is judged to be applicable. The chapter concludes with an examination a number of particularly significant vague spatial properties and relations and how they can be represented. 1 Introduction Suppose I am asked whether the Leeds City Art Gallery is close to Leeds University. Perhaps am not sure what to say. Suppose I then take an accurate map and determine the distance between the university main entrance and the entrance to the art gallery is 965m. Now, I have a pretty accurate measure of the distance but still I may be undecided as to whether to describe the gallery as close to the university. This is because close to is a vague relation. The example, illustrates a key property of vagueness, which is that it is distinct from uncertainty. Vagueness is not a result of Brandon Bennett School of Computing, University of Leeds, B.Bennett@leeds.ac.uk 1

2 2 Brandon Bennett lack or imprecision of our knowledge about the world, rather it arises from lack of definite criteria for the applicability of certain linguistic terms. Even with complete and accurate knowledge of all relevant objective facts, one may be unsure as to whether a vague description is appropriate to a given situation. In this chapter, I shall examine the phenomenon of vagueness as it relates to spatial information. We shall see that vagueness is particularly pervasive in spatial terminology, and also that, where a description involves both vagueness and spatiality, semantic subtleties are encountered that do not arise when vagueness operates in a non-spatial context. I shall begin by examining some fundamental issues relating to our understanding of vagueness, how it should be represented, and how it interacts with spatial information and representations. Following this, I shall give overviews of some approaches to spatial vagueness that have been proposed in the literature. I shall then present a particular analysis of spatial vagueness, which is based on the identification of parameters of variability in the meaning of vague predicates and on the notion of a standpoint. A standpoint is a choice of threshold values determining the range of variation over which a vague predicate is judged to be applicable. Finally, I shall look in more detail at certain vague spatial predicates that I consider to be of particular significance and examine how these might be modelled in the context of a formal representation system. 2 The Nature of Vagueness 2.1 Distinguishing Vagueness from Generality and Uncertainty Vagueness as I use the term is distinct from both generality and uncertainty. A predicate or proposition may be more or less general according to the range of individuals to which it applies or the range of circumstances under which it holds. However, as long as the boundary between correct and incorrect applications is definite, I shall not consider generality to be a form of vagueness. For example, the sentence Tom s house is within 5 miles of the Tower of London is general but not vague, since its truth can be determined by measuring the distance between the two buildings. By contrast Tom lives near to the Tower of London is vague, since even when I know the exact distance, the truth of this sentence is indeterminate. A proposition is uncertain if we do not know whether it is true or not. In most circumstances we describe a proposition as uncertain when the reason we do not know whether it is true is that we do not possess complete and accurate knowledge about the state of the world. In so far as the term uncertainty is taken to apply only in such cases, vagueness is distinct from uncertainty, since it may occur even where we do have complete and accurate knowledge of the world. However, some have argued (notably Williamson (1992)) that vagueness does indeed arise from lack of knowledge: lack of knowledge about the meanings of terms; and is thus

3 Spatial Vagueness 3 a kind of uncertainty. This position is known as the epistemic view of vagueness. Although it has a devoted following, the epistemic view is not widely accepted. The main objection is that, in the case of a vague linguistic term, there is no fact of the matter as to what is its true precise meaning. Hence, vagueness does not consist in ignorance of any objective fact and is thus not epistemic in nature. Proponents of the epistemic view may counter this objection by claiming that there is a fact of the matter but it is unknowable. 1 My opinion is that vagueness is distinct from uncertainty and that any strong form of the epistemic view is misleading. Nevertheless, vagueness and uncertainty do share significant logical properties and one would expect similar kinds of formal representation to apply to the two phenomena. So, if vagueness is not a kind of generality or uncertainty, what is it? As one might expect, many different theories have been proposed. In fact, even once we have distinguished vagueness from generality and uncertainty, there may well be more than one semantic phenomena operating under the name of vagueness. 2.2 Vagueness in Different Linguistic Categories Whether or not there are multiple kinds of vagueness, it is certainly true that vagueness affects different types of linguistic expression, different syntactic categories, and these must be treated in, at least superficially, and perhaps fundamentally, distinct ways. I need only briefly consider the category of propositions. Obviously, propositions can be and often are vague. However, it is, I believe, uncontroversial that propositions are only vague in so far as they contain constituent parts that are vague. Hence, I shall proceed to examine sub-propositional expressions. But in considering these I shall of course be concerned with the central semantic issue of how they contribute to the truth conditions of propositions in which they occur. Perhaps the most obvious and most studied type of vague expression is adjectives, prime examples being bald, rich and red. This category also includes many spatially oriented adjectives, such as tall, short, large, small, elongated, steep, undulating etc.. For such concepts, their vagueness seems to lie in the lack of clearly defined criteria for their applicability. Thus, when a vague adjective is predicated of an individual, the resulting proposition will be true in some cases and false in others, but for some individuals it will not be clear whether the predication should be considered true or false such individuals are borderline cases. For count noun predicates such as table, mountain, lake, village, the manifestation of vagueness is similar but not exactly the same as in adjectives. As Kamp (1975) observed, the vagueness of count nouns differs from that of adjectives as to 1 Lawry (2008) has proposed a weakened version of the epistemic view, which he calls the epistemic stance. Lawry suggests that while there may be no objective fact of the matter underlying the meaning of vague terms, people tend to use vague language as if there were.

4 4 Brandon Bennett the number of parameters of variation that are usually involved. The applicability of an adjective typically depends on a relatively small number of factors (tallness depends on height, redness on colour). 2 By contrast, the conditions for applicability of a count noun usually involve a large number of factors. For instance, a table should be made of suitably solid material; it should have a flat surface, which is supported by legs; its height and other dimensions should be within certain appropriate ranges; and various other constraints should apply to its shape. The reason these diverse characteristics are gathered together within the meaning of table is that objects with such a combination of properties are useful to people, and hence are frequently encountered and referred to by language users. 3 This correlation of relevant attributes also means that for many count nouns it is difficult to decide which of them are essential and which are merely typical features (c.f. Waismann (1965)). Relational expressions may also be vague, and in the spatial domain the prototypical examples are the relations... is near to... and... is far from.... The vagueness of such relations appears to be similar to that of adjectives. That is, their applicability is dependent on one or two parameters (for instance distance) but the values of the parameter(s) for which the relation is applicable are not precisely determined. As well as affecting predictive expressions, vagueness may also be associated with referential terms, i.e. names and definite descriptions. In the first category we have nominal expressions such as Mount Everest and The Sahara Desert. In the second we have more complex constructs such as the foothills of Mount Everest and the area around the church. There has been considerable debate about whether the vagueness of nominals is parasitic upon that of predicative expressions, or whether it is a separate form of vagueness. One way that vagueness of proper names might be analysed in terms of vagueness of predicates is via the following logical re-writing of a nominal into a definite description, which is then represented using Russell s analysis of definite descriptions: 4 Φ( Mount Everest )!x[mountain(x) Named(x, Mount Everest ) Φ(x)] Under this analysis, the vagueness of Mount Everest is completely explained by the vagueness of mountain. A similar approach could be applied to definite descriptions, although these expressions can take many different forms and may require a variety of different analyses involving constituent vague count nouns, adjectives or relations, or some combination of these. In opposition to this analysis, is the view that the vagueness of nominal expressions cannot or should not be reduced to vagueness of predicative expressions, but has its own mode of operation. Here again two contrasting explanations can be given: one is that vague nominals indeterminately refer to one of many possible 2 There are some adjectives that might be considered exceptions to this. For example, intelligent depends on intelligence, but this quality is manifest in many different attributes. Nevertheless, we often do treat intelligence as if it were a single measurable quantity. 3 The phenomenon of clustering of exemplars, where logically distinct properties tend in the real world to occur in combination, is discussed in (Bennett 2005). 4 Following Russell, the syntax![φ(x)] is used to abbreviate x[φ(x) y[φ(y) y = x]].

5 Spatial Vagueness 5 referents; the other is that the referents of vague nominals are intrinsically vague entities. According to the first view, the semantics for vague nominals would involve an indeterminate denotation function, perhaps modelled by a one-to-many function or by a collection of possible functions (such an account is usually called supervaluationist and will be considered in detail is section 6). According to the second view, the semantics must provide an explicit model of vague entities and incorporate a domain of such entities as the range of the denotation function for nominals. I shall further consider these alternatives in the next section which deals with the issue of whether vagueness is intrinsic or linguistic. Our consideration of vagueness in different categories of linguistic expression may be summarised as follows. Vagueness is present in its most obvious form in adjectives and relations, where conditions of applicability are indeterminate but are typically dependent on a small number of relevant properties. The vagueness of count nouns is similar to that of adjectives, but is typically dependent on more factors and it is often unclear which factors are relevant. The nature of the vagueness of proper nouns and definite descriptions is controversial. In relation to spatial information, the difference between vagueness located in a relation and vagueness located in a nominal expression is illustrated by the following example sentences: 1. The treasure is 20 km from the summit of Mount Everest. 2. The treasure is 20 km from of Mount Everest. 3. The treasure is near to the summit of Mount Everest. 4. The treasure is near to Mount Everest. In sentence 1 both the relation and the reference object are precise. In 2 the relation is precise but the reference object vague, whereas in 3 the relation is vague but the reference object precise. Finally, in 4 both relation and reference object are vague. 2.3 Is Vagueness Intrinsic or Linguistic? A fundamental issue in the ontology of vagueness concerns whether vagueness is an intrinsic property of certain kinds of real-world object (often called de re or ontic vagueness; or whether it is an entirely linguistic phenomenon (de dicto vagueness). For instance Tye (1990) supports the view that there are actual objects in the world that are vague (as well as vague linguistic expressions); whereas Varzi (2001a)) argues that all vagueness is essentially linguistic. 5 It is not clear whether the handling of vagueness within a computational information system requires one to take a philosophically defensible position on ontological status of vagueness. However, the debate does have some bearing on the 5 A much discussed paper by Evans (1978) contains a formal proof that appears to show that some fundamental logical problems may arise from taking an ontic view of vagueness. However, the significance and implications of Evans proof are unclear and controversial.

6 6 Brandon Bennett choice of a suitable representational and semantic framework. If one takes the view that vagueness is purely linguistic, then the phenomenon will be modelled in terms of indeterminate interpretation of predicates and/or naming terms, but the objects to which the predicates and names are applied will be modelled as precisely determined entities. For instance, the denotation of Mount Everest would be considered to be indeterminate, but each possible denotation would correspond to a precisely bounded volume of matter. By contrast, if one adopts an ontic view of vagueness, then the term Mount Everest would denote an entity that is in itself vague. For instance it could be identified with a fuzzy set of points, or with a cluster of possible extensions. Tye (1990) suggests that vagueness can be present both in predicates and also in objects. He argues that the vagueness of objects cannot simply be explained by saying that they are instances of vague predicates. In the case of material objects, Tye proposes that a vague material object is one for which the set of parts of the object is not fully determinate. 6 This condition is of particular relevance to the investigation of spatial vagueness because it specifically identifies ontic vagueness with indeterminacy of spatio-temporal parts. I shall not take a rigid position on whether vagueness is primarily linguistic or ontic. In fact, it seems to me that it may not be possible to make a sharp distinction between ontic and linguistic properties. This is because the entities to which we refer are not given prior to our linguistic conventions; rather the ways that we use language play a significant role in determining the domain of objects. So the entities to which we might ascribe ontic properties are themselves, at least partially, determined by linguistic convention and stipulation. This is especially evident in the case of predicates that are both vague and spatial. As we shall see in the next section, when considering the truth conditions for a predicate that vaguely characterises a spatial entity, one cannot assume that there is a pre-determined and well-defined object to which the predicate is applied. Moreover, from the point of view of establishing formal representations and semantics, the issue of whether vagueness should be modelled in terms of vague entities or in terms of indeterminate linguistic reference may be seen as more a matter of technical convenience than philosophical significance. 3 Vagueness and Spatiality Circumstances where vagueness interacts with spatiality present issues that do not arise where vagueness operates in a non-spatial context. In this section I describe some of the main issues that arise and present some illustrative examples. 6 In fact Tye also suggests the further criterion that there is no determinate fact of the matter about whether there are objects that are neither parts, borderline parts, nor non-parts of o. This condition relates to the issue of second-order vagueness (i.e. the vagueness of the borderline between clear and indeterminate cases), which will not be considered in this chapter.

7 Spatial Vagueness The Sorites Paradox The issue that has dominated much of the debate about vagueness over the last couple of millenia is the logical anomaly known as the sorites paradox. I will not be looking into this in detail here since, although it affects many spatial predicates, the paradox is not essentially spatial in nature. Hence, considering it from a specifically spatial angle would not add much to the a great deal has already been written about this topic elsewhere (see e.g. Williamson (1994), Keefe and Smith (1996), Beall (2003)). The classic form of the sorites paradox involves the predicate... is a heap, 7 and can be stated as follows: 1. This pile of 1,000,000 grains of sand is a heap. 2. If one removes one grain of sand from a heap, the remainder will still be a heap. 3. Even if we removed all the grains from this heap, it would still be a heap. Here, the conclusion, 3, appears to follow from the premisses by a kind of induction, with premiss 1 being the base case and premiss 2 being used as a principle of induction. Similar slippery slope arguments can be formulated using pretty much any vague predicate indeed, susceptibility to sorites arguments is often considered to be a necessary requirement for a predicate to be vague. Many such predicates are spatial in nature, obvious examples being tall, large, near etc.. For example: 1. A man whose height is 2m is tall. 2. If a man of a given height is tall, then a man whose hight is 1mm less is also tall. 3. A man whose height is 1m is tall. A characteristic feature of sorites susceptible predicates is that they are associated with some mode of variation, often a measurable property, that is either continuous or fine grained. In attempting to explain the sorites, it is usually the inductive premiss that comes under most scrutiny. A typical diagnosis is that this premiss must be false, although we are for some reason compelled to believe that it ought to be true. The problem then is explaining why we have a tendency to think it should be true. Accounts of this often refer to the supposition that vague predicates cannot be used to make precise distinctions because their limits of applicability are not welldefined. Hence, if two samples are very similar in all attributes that are relevant to the application of a vague predicate V(x), then it must be that either both of them or neither of them are instances of V. Such reasoning can be used to justify the inductive premiss of a sorites argument. In practice it seems that the sorites paradox has not had a direct impact on existing computational spatial information systems. The types of computer system most in danger of being affected by sorites paradoxes are those concerned with representing 7 Sorites is the Greek word for heap. More accurately, it is an adjective meaning heaped up.

8 8 Brandon Bennett commonsense knowledge or implementing commonsense reasoning for instance, systems such as CYC (Guha and Lenat 1990). It would be inadvisable for such a system to directly encode any proposition or rule similar to a sorites induction step, since it would be clear that it could quickly lead to problems. Current implemented systems generally avoid the paradox simply by not taking vagueness into account. However, as we shall see later in the chapter, a number of computationally-oriented approaches to representing vagueness have been proposed. In fact, these tend to tackle the phenomenon using modelling techniques that are rather different from the axiomatic analysis that leads to the paradox. Hence, they might be regarded as skirting round the problem rather than solving it. It is not yet clear whether such circumvention will be ultimately satisfactory. As more sophisticated applications and representations are developed, it may turn out that the sorites paradox will need to be confronted more directly. 3.2 The Problem of Individuation That vagueness results in a lack of well-defined criteria for the application of predicates is widely recognised. However, there is a further consequence of vagueness that becomes especially important when considering cases where the vagueness of a predicate affects the determination of the spatial extension of entities satisfying that predicate. The problem is that of individuation i.e. the determination of the class of entities to which the predicate might be applied. The issue of individuation is illustrated in Fig. 1, which shows a section of an extended water body. Suppose we now wish to find instances of the count noun predicates river and lake in relation to this water region. One interpretation is that the region is simply a river section that is rather irregular in width and includes a number of bulges. But another interpretation is that the water body consists of three lakes connected by short river channels. 8 The possibility of these different interpretations arises from the vagueness of the terms river and lake. However, what makes this case especially problematic is that there is no pre-existing division of the water region into segments that we choose to describe as river, lake or whatever. Rather, Fig. 1 Is this just an irregular river, or is it three lakes joined by a river? 8 One might comment that the distinction between lake and river also significantly depends on water flow. This is certainly true but much the same segmentation issue would still arise, and it is much easier to illustrate in terms of shape rather than flow.

9 Spatial Vagueness 9 Fig. 2 Possible forest demarcations for a given tree distribution. Inner contours are based on a high threshold on the tree density. Outer contours are based on lower thresholds. the acceptable ways in which water can be segmented into features is dependent upon the meanings of these feature terms. And since these meanings are affected by vagueness, it is indeterminate what is the most appropriate segmentation. Fig. 2 illustrates a similar case concerning the demarcation of forest regions based on the density of trees. If we vary the minimum tree density threshold required for a region to be classified as a forest, the extension of forest regions will clearly vary. Furthermore, the number of forest regions may alter according to the threshold. 3.3 Consequences of Indeterminate Spatial Extension A well as illustrating the problem of individuation, the examples given in the previous section also show how in association with spatial concepts vagueness not only affects classification but also results in indeterminacy of spatial extension. Consequently vagueness in spatially related terms such as geographic feature types often leads to an indeterminacy in spatial predicates and relations, even where the predicates and relations themselves are apparently precise. Suppose we want to compare the size of a particular expanse of desert at two different time points on the basis of precipitation data collected at two different time points. (Of course the classification of desert may depend on features other than precipitation, but whatever measure of aridity is employed the issue I am about to describe will occur.) In order to demarcate the extension of desert at each time point, we need to choose some threshold for the amount of precipitation below which we will classify a region as desert. But if we do this we may find that whether the demarcated desert region expands or contracts between the two time point depends on the particular threshold value that we choose. This problem is illustrated in Fig. 3. The sub-figures on the left of the diagram show precipitation contour maps for the year In the upper map a threshold of

10 10 Brandon Bennett 30mm per month is taken to bound the shaded desert region, whereas in the lower one a threshold of 20mm per month is used to demarcate the desert. On the right of the diagram we have precipitation contours for the year 2009 and again precipitation thresholds of 30 and 20 mm per month respectively are used to bound the desert region in the upper and lower maps. We see that if we choose the higher threshold (30mm) the demarcated desert region is seen to expand from 1965 to 2009, whereas if we choose the lower threshold (20mm), over the same period the demarcated desert appears to contract. What has happened is that although the driest region has shrunk the region that is dry but somewhat less so has expanded. We could of course avoid this issue by asking about the change in overall precipitation calculated by integrating the precipitation values over the whole map. But this is a different question from that of whether the desert expands or contracts. Moreover, describing the world in terms of average or cumulative measures over a large area can be misleading. For example, if a land region consists of one part that is bitterly cold and another that is intolerably hot, it would be uninformative to classify the whole region as temperate. Fig. 3 Desert demarcation according to different precipitation thresholds.

11 Spatial Vagueness 11 4 A Theory of Crisp and Blurred Regions In this section I give an overview of the theory of vague regions presented by Cohn and Gotts (1996b). This paper is best known for the so-called egg-yolk model of vague spatial regions, which had originally been proposed by Lehmann and Cohn (1994). 9 However, the Cohn and Gotts paper starts by developing a purely axiomatic theory of vague regions, for which the egg-yolk model is given as only one possible interpretation. The style of presentation of this theory suggests a de re ontology of vagueness i.e. vagueness lies in the regions themselves, rather than in the linguistic expressions that refer to these regions. Moreover the egg-yolk semantics can be regarded as a providing a simple model of a spatially vague object. Although presented primarily as a theory of vague spatial regions, the theory is more or less neutral about the type of indeterminacy that it models. Thus, it can equally well be used to represent spatial indeterminacy arising from uncertainty, and indeed the terms vague and uncertain are used more or less interchangeably in the original papers that proposed the theory. The theory is described by means of the following terminology. The term crisp is used to mean that a region is precisely determined, and blurred to mean that it is indeterminate (either vague or uncertain). More generally, a region x may be described as being a crisper version of another region y, or more succinctly, one may say that x is a crisping of y. These propositions both assert that the region x is a more precise version of y. This relation can be understood as meaning that every precise boundary that could be considered as a possible boundary of x could also be considered as a possible boundary of y. 4.1 An Axiomatic Theory of the Crisping Relation A formal theory of crisp and blurred regions is developed in similar fashion to axiomatic theories of mereology (Simons 1987) or the topological Region Connection Calculus (Randell et al. 1992). The theory is based on the primitive relation x y, read as x is a crisping of y. This relation is defined to be a strict partial order: A1) xy[x y (y x)] A2) xyz[(x y y z) x z)] Before stating the further axioms satisfied by, it will be helpful to introduce some definitions: D1) x y def (x y x = y) D2) MA(x,y) def z[z x z y] D3) Crisp(x) def y[y x] 9 Cohn and Gotts (1994, 1996a) are also precursors of Cohn and Gotts (1996b).

12 12 Brandon Bennett D1 is just a convenient abbreviation, giving the reflexive counterpart of the crispness ordering. D2 defines the key relation of mutual approximation. MA(x, y) holds just in case there is some third region z that is a crisping of both x and y. Thus x and y could potentially approximate the same region. D3 defines a Crisp region to be one that has no crispings. Further axioms of the theory can now be stated as follows: A3) xy[x y z[z y MA(x,z)]] A4) xy[ma(x,y) z w[w z (w x w y)]] A5) xy[ z[x z y z w[(x w y w) z w]]] A6) xy[ma(x,y) z[x z y z w[(x w y w) z w]]] A7) x y[y x] A8) xy[ z[crisp(z) ((z x) (z y))] (x = y)] A9) x y[y x Crisp(y)] A10) xy[x y z[x z z y]] Axiom A3 says that, if a region can be made more precise in one way, it can also be made precise in another way that is incompatible with the first. Axiom A4 says that, if two regions are mutually approximate, then there is not only a region that is a crisping of both but, more specifically, there is a region that is the least crisp crisping of both. Axiom A5 says that any two regions have a crispest common blurring. (In terms of the egg-yolk model the crispest common blurring of two vague region is the region whose white is the sum of the two whites and yolk is the intersection of the two yolks. This assumes that we may have regions with an empty yolk.) Axiom A6 states that if two regions have a mutual approximation, they must have a maximally blurred mutual approximation. Axiom A7 states that there is a maximally blurred region ( the complete blur ). Cohn and Gotts remark that one may wish to omit this axiom or even assert its negation, which says that for any region one can always find a more blurred region. Axiom A8 specifies a condition for identity. It says that if two regions have the same complete crispings they must be equal. (Cohn and Gotts (1996b) give a slightly more complex but equivalent formula, and they also consider some alternative identity axioms.) Axiom A9 states that every region can be crispened to a completely crisp region. Axiom A10 states that whenever one region is crisper than another, there is a third intermediate region, more blurred than the first but crisper than the second. (This gives makes the crisping ordering dense). Exactly what kinds of relation structure satisfy this axiom set is not known. However, one possible model is considered in the next section.

13 Spatial Vagueness The Egg-Yolk Model The axiomatic theory of crisping can of course be given a semantics based on the general purpose model theory of first-order logic. However, the theory is normally understood in terms of the more specific semantics originally proposed in Lehmann and Cohn (1994). The so-called Egg-Yolk model Lehmann and Cohn (1994) interprets a vague region in terms of a pair of nested crisp regions representing its maximal and minimal possible extensions. The maximal extension is called the egg and the minimal is the yolk, which is required to be a part of the egg (see Fig. 4). (The case where the yolk is equal to the egg is allowed, such cases corresponding to crisp regions.) When an egg-yolk pair is given as the spatial extension of a vague region this means that the region is definitely included in the egg and definitely includes the yolk. Fig. 4 A typical egg-yolk interpretation of a vague region. In terms of the Egg-Yolk model, the crisping relation x y can be understood as holding whenever the egg associated with x is contained within the egg associated with y and the yolk of x contains the yolk of y, and furthermore x and y do not have identical eggs and yolks (if this last condition does not hold, we have x y but not x y). In order to tie the theory of the crisping relation explicitly to the Egg-Yolk model, Cohn and Gotts (1996b) introduce two functions to the vocabulary: egg-of(x) and yolk-of(x), denoting respectively the egg and yolk regions associated with x. The mereological relation P(x, y), meaning x is part of y, is also introduced. The following additional axioms are then specified: A11) x[p(yolk-of(x), egg-of(x))] A12) xy[x y (P(egg-of(x),egg-of(y)) P(yolk-of(y),yolk-of(x)) (P(egg-of(y),egg-of(x)) P(yolk-of(x),yolk-of(y))))] A11 ensures that the yolk of every region is contained within its egg. A12 states that if x is crisper than y then x s egg must be part y s egg and y s yolk must be part of x s yolk and at least one of these relations must be a proper part relation (i.e. the eggs and yolks can t both be equal).

14 14 Brandon Bennett Note that axiom A12 is stated as an implication rather than an equivalence. The explanation given for this is that the egg-yolk condition is proposed as a necessary but not sufficient condition for one vague region being a legitimate crisping of another. The rationale behind this is that there might be a vague region that satisfies the relatively week spatial constraints required to be a crisping of another region and yet would not be considered as a reasonable crisping for other reasons, such as the shape of its egg and yolk. Fig. 5 a) Reasonable, and b) anomalous crispings in the egg-yolk interpretation. Fig. 5 illustrates two potential candidates for the crisping of a given region. The egg and yolk of the initially given region are shown as bounded by solid lines, and the egg and yolk of the candidate crispings are outlined with dashed lines. In case a) the egg and yolk of the candidate crisping are similar in shape to the initial region, so the vague region that they define may be regarded as a legitimate crisping of the original. But in case b) the jagged outline of the egg and yolk of the candidate mean that it is implausible that the original could be crispened in this way. Although, the egg-yolk model has become well-known, the papers of Cohn and Gotts (1994, 1996a,b) suggest that this is just one of a range of possible interpretations that could be given to the axiomatic theory. The egg-yolk model cannot of itself, account for any kind of constraint on a region s plausible extensions between its maxima and minima. However, when dealing with real phenomena, such as vague geographic features, we would expect the range of possible extensions to be structured in accordance with the underlying conditions relative to which a vague region is individuated. For instance, one might expect these extensions to exhibit a contourlike structure, such as we saw in Fig. 2 and Fig. 3 above. Another strand of research on spatial vagueness, somewhat related to the work of Cohn and Gotts and the egg-yolk theory, is work based on rough sets and granular partitions. See for example Bittner and Stell (2002). Consideration of this approach is beyond the scope of this chapter.

15 Spatial Vagueness 15 5 Fuzzy Logic Approaches The most popular approach to modelling vagueness used in AI is that of fuzzy logic and the theory of fuzzy sets. These theories originated with the works of Goguen (1969) and Zadeh (1975) and have given rise to a huge field of research (see (Dubois and Prade 1988, Zimmermann 1996) for surveys of fuzzy logics and their applications). Within the context of the current collected work, I presume that a general introduction to fuzzy logic will not be required. The coverage of fuzzy approaches to vagueness here will be very limited, since these are considered in detail in other chapters of this collection. 5.1 Spatial Interpretation of Fuzzy Sets Fuzzy sets can be given an intuitive spatial interpretation. In the same way that a precise spatial region can be identified by a set of spatial points, a vague spatial region may be associated with a fuzzy set of points. Here, the degree of membership of the point in the fuzzy set corresponds to the degree to which that point is considered as belonging to the vague region. Fuzzy representations have been popular with many geographers as they provide a natural way of representing features with ill-defined boundaries (Wang and Brent Hall 1996). The particular way that fuzzy sets have been employed varies greatly according to the type of feature being analysed, the kinds of data available, and the particular aims of each analysis. Kronenfeld (2003) proposed a fuzzy approach to classification and partitioning of continuously varying land cover types and applies this to the classification of forest types. Arrell et al. (2007) use a fuzzy classification of elevation derivatives to identify natural landforms such as peaks and ridges. Evans and Waters (2007) use fuzzy sets to model the regions referred to by vernacular place names. 5.2 Fuzzy Region Connection Calculus A fuzzy version of the well-known Region Connection Calculus (Randell et al. 1992) of topological relations has been developed by Schockaert et al. (2008, 2009), by treating the primitive connection relation, C, of the RCC theory as a fuzzy relation, and replacing the definitions of other spatial relations by analogous definitions formulated using fuzzy logic operators. Let T (φ,ψ) be the value of a T -norm (fuzzy conjunction) function on the truth values of propositions φ and ψ; and let I T (φ,ψ) be the residual implicator function relative to T, given by I T (φ,ψ) = def sup{λ λ [0,1]andT (φ,λ) ψ}. Then a fuzzy parthood relation P is defined in terms of the connection relation as follows:

16 16 Brandon Bennett Fig. 6 a, b and c are three fuzzy regions. Under Schockaert s definition of fuzzy parthood, b will be evaluated as being part of a to a higher degree than is c. P(a,b) def ( x U)[C(x,a) C(x,b)] = inf x U {I T (C(x,a),C(x,b))} So the universal quantification in the classical definition is replaced by the infimum of the truth values for all regions in the domain, and implication is replaced by the residual implicator. Similar mappings from first-order logic to fuzzy relations can be given for all the topological relations of the original RCC theory. This fuzzification method is certainly well-principled, and various strong results can be proved regarding the structure and reasoning capabilities of the resulting fuzzy region connection calculus. However, some properties of the relations defined in this way appear to be somewhat unexpected. Once source of counter-intuitive aspects of the theory is the use of the infimum (and supremum) as corresponding to universal (and existential) quantification. For example, consider the case illustrated in Fig. 6. Here we have one region b which seems to be very much part of region a, except from a thin protruding spike, whereas c lies on the edge of a but has no part that is strongly within a. In fuzzy RCC one may find that in such a situation c will be evaluated as being part of a to a higher degree than is b. This is because a region near the tip of b may have a high degree of connection with b but a very low degree of connection with a. Hence, even though regions connected to most parts of b will be highly connected to a, the infimum in the fuzzy P definition means that the evaluation of the overall degree of truth of the P relation is determined by that part of b that is least connected to a. By contrast, the whole of c is on the periphery of a but no part of c is very far from a. Since the furthest part of c is nearer to a than is the furthest part of b, P(c,a) is likely to be evaluated as having a higher degree of truth than P(b,a). Of course, this depends on the particular details of the situation and the distribution of the fuzzy C relation. However, it is evident that very natural measures of the degree of connection (e.g. based on distance) can give rise to measures of parthood that may seem un-natural. The reason for this is that, when we hold the view that a topological relation between spatial regions is almost true, we often mean that it would hold if we disregard some comparatively insignificant part of one or both of the regions. But the use of the infimum in the fuzzy part definition means that we cannot disregard any part of either of the regions.

17 Spatial Vagueness 17 6 Supervaluationist Approaches 6.1 Origins and Motivations of Supervaluation Semantics The fundamental idea of the supervaluationist account of vagueness, is that a language containing vague predicates can be interpreted in many different ways, each of which can be modelled in terms of a more precise version of the language, which is known as a precisification. In specifying a formal supervaluation semantics each precisification is associated with a valuation of the symbols (i.e. predicates and constants) of the language. In some accounts each precisification is simply a classical assignment, corresponding to a completely precise interpretation; but it is common to allow precisifications that are not completely precise, which are associated these with partial assignments. The interpretation of the vague language itself is determined by a supervaluation, which is the collection of assignments at all precisifications. The view that vagueness can be analysed in terms of multiple senses was proposed by Mehlberg (1958), and a formal semantics based on a multiplicity of classical interpretations was used by van Fraassen (1969) to explain the logic of presupposition. It was subsequently applied to the analysis of vagueness by Fine (1975) and independently by Kamp (1975). Thereafter, it has been one of the more popular approaches to the semantics of vagueness adopted by philosophers and linguists, and to a lesser extent by logicians. Only a brief overview of supervaluation can be given here. A more thorough account and extensive discussion can be found in works such as Williamson (1994) and Smith (2008). A major strength of the supervaluation approach is that it enables the expressive and inferential power of classical logic to be retained (albeit within the context somewhat more elaborate semantics) despite the presence of vagueness. In particular, necessary logical relationships among vague concepts can be specified using classical axioms and definitions. These analytic interdependencies will be preserved, even though the criteria of correspondence between concepts and the world are ill-defined and fluid. For example, in the sentence Tom is tall and Simon is short, both tall and short are vague. However, their meaning is coordinated in that, when applied in the same context relative to a given comparison class, it must be that in any admissible precisification the minimal height at which an object is ascribed the property tall is higher than the maximal height at which an object is ascribed the property short. Thus if it were to turn out that Tom s height is less than that of Simon, the claim would be false in any admissible precisification. Investigation of supervaluation semantics in the philosophical literature tends, as one might expect, to be drawn towards subtle foundational questions (such as those concerning the sorites paradox and second-order vagueness). Consequently, there has been relatively little development towards practical use of supervaluation semantics in formal representations designed for computational information processing applications. In so far as statistically oriented formalisms (such as fuzzy logic) replace classical logical operators by statistical functions, supervaluationism

18 18 Brandon Bennett is often regarded as fundamentally opposed to any kind of statistical approach to language interpretation. This may turn out to be a simplistic view, since it is certainly possible to add some form of probability distribution over the set of precisifications. However, this possibility will not be investigated here. 6.2 Admissible Precisifications and Supertruth In standard supervaluation semantics the concept of an admissible precisification plays a key role. It is used to elaborate the simple classical concept of truth by defining the concept of supertruth, which is applicable to propositions that include vague terminology. In a semantics where all precisifications are associated with complete classical models, the following definition is may be given: φ is supertrue just in case φ is true according to every admissible precisification. In many accounts (e.g. Fine (1975)) the notion of admissible precisification is taken as primitive. It is normally assumed that, in addition to satisfying conditions of logical consistency (in virtue of being associated with classical models), an admissible precisification also satisfies an appropriate theory specifying analytic properties and relationships between vocabulary terms (e.g. nothing can be an instance of both the predicates tall and short here we are assuming the predicates are applied in the same context). Stipulating such semantic conditions may be complex in general, but poses no particular theoretical problems. But if the only restrictions on admissible precisifications are that they must satisfy logical and analytical axioms, then the only propositions that will count as supertrue will be those that are analytically true. This is usually regarded as too strong a requirement. What we would like is a notion of supertruth such that a proposition is true if it come out as true on any reasonable interpretation of its terms not on every possible interpretation. Consequently, the set of admissible precisifications is normally taken to be those that are in some sense reasonable. For instance if a man is 6 6 in hight, one would expect him to be tall according to all admissible precisifications. However, this begs the question of what precisifications should be counted as admissible. The problem of determining the set of admissible precisifications is often sidestepped in presentations of supervaluation semantics. However, it causes difficulties both from a theoretical and practical point of view. From a theoretical perspective, any attempt to stipulate a set of admissible precisifications brings to the fore the problem of second-order vagueness. To explain briefly: the admissible precisifications are intended to correspond to possible ways of deciding the truth of predications applied to borderline cases. That is, every complete precisification corresponds to a set of decisions as to whether each borderline case is an instance of a given vague predicate. By contrast, a non-admissible precisification would one one that assigns a predicate as true of an object to which it is clearly not applicable (or as not

19 Spatial Vagueness 19 true of something to which it is clearly applicable). The problem is that any sharp distinction between admissible and non-admissible precisifications assumes a correspondingly sharp distinction between borderline instances and clear instances (or clear non-instances). But given that we are considering vague predicates, it seems untenable that there should be a precise boundary between borderline and clear-cut cases. The problem with admissibility from a practical point of view, is that if we actually wanted to reason with or implement a formal system in which the set of admissible precisifications plays a key role, we would have to explicitly specify all the conditions required of an admissible precisification. This would not only be complex but would also seem to require one to make stipulations without any obvious means of justification. 6.3 Computational Applications of Supervaluationism The uptake of supervaluation-style approaches in computational applications has been relatively limited. One reason for this is probably the difficulty in specifying the set of admissible precisifications, as mentioned above. Nevertheless it is worth mentioning some works which have taken preliminary steps in the application of supervaluation semantics. Bennett (1998) proposed a two-dimensional model theory and a corresponding modal logic, in which the interpretations of propositions are indexed both by precisifications and possible worlds. Relative to this semantics a spectrum of entailment relations were defined corresponding to more or less strict requirements on how the vague senses of premisses and conclusion are allowed to vary. In Bennett (2006) the semantics of vague adjectives was characterised in terms of their dependence on relevant objective observables (e.g. tall is dependent on height ). This may be seen as a precursor of the standpoint semantics, which will be presented in the next section. An example of the use of a supervaluationist approach in an implemented computer system for processing geographic information can be found in Bennett et al. (2008). Applications of supervaluation or similar theories to geographic information have been proposed by Smith and Mark (2003) Varzi (2001a) and Kulik (2003). Halpern (2004) analyses vagueness in terms of the subjective reports of multiple agents, but these play a similar role in his semantics to precisifications in the semantics proposed in this paper.

20 20 Brandon Bennett 7 Standpoint Semantics Standpoint Semantics is both a refinement and an extension of supervaluation semantics whose purpose is to make more explicit the modes of variability of vague concepts and to support a definition of truth that is relative to a particular attitude to the meanings of terms in a vague language. Whereas supervaluation semantics provides a very general framework within which vagueness can be analysed formally, standpoints semantics is more geared towards detailed modelling of specific vague concepts within some particular application domain. 7.1 What is a Standpoint? In making an assertion or a coherent series of assertions, one is taking a standpoint regarding the applicability of linguistic expressions to describing the world. Such a standpoint depends partly on one s beliefs about the world and partly on one s linguistic judgements about the criteria of applicability of words to a particular situation. This is especially so when some of the words involved are vague. For instance, one might take the standpoint that a certain body of water should be described as a lake, whereas another smaller water-body should be described as a pond. It is not suggested that each person/agent has fixed standpoint, which they stick to in all situations. Rather an agent adopts a given standpoint at a particular time as a basis for describing certain features of the world. In a different situation the agent might find that adopting different standpoint is more convenient for describing salient features of the world. This is somewhat misleading since even a person thinking privately may be aware that an attribution is not clear cut. Hence a person may change their standpoint. Moreover this is not necessarily because they think they were mistaken. It can just be that they come to the view that a different standpoint might be more useful for communication purposes. Different standpoints may be appropriate in different circumstances. The core of standpoint semantics does not explain why a person may hold a particular standpoint or the reasons for differences or changes of standpoint, although a more elaborate theory dealing with these issues could be built upon the basic formalism. In taking a standpoint, one is making somewhat arbitrary choices relating to the limits of applicability of natural language terminology. But a key feature of the theory is that all assertions made in the context of a given standpoint must be mutually consistent in their use of terminology. Hence, if I take a standpoint in which I consider Tom to be tall, then if Jim is greater in height than Tom then (under the assumption that height is the only attribute relevant to tallness) I must also agree with the claim that Jim is tall.

21 Spatial Vagueness Parameterised Precisification Spaces By itself, supervaluation semantics simply models vagueness in terms of an abstract set of possible interpretations, but gives no analysis of the particular modes of semantic variability that occur in the meanings of natural language vocabulary. A key idea of standpoint semantics is that the range of possible precisifications of a vague language can be described by a (finite) number of relevant parameters relating to objectively observable properties; and the limitations on applicability of vocabulary according to a particular standpoint can be modelled by a set of threshold values, that are assigned to these parameters. To take a simple example, if the language contains a predicate Tall (as applicable to humans), then a relevant observable is height. And to determine a precisification of Tall we would have to assign a particular threshold value to a parameter, which could be called tall human min height. One issue that complicates this analysis is that vague adjectives tend to be context sensitive in that an appropriate threshold value depends on the category of things to which the adjective is applied. This is an important aspect of the semantics of vague terminology but is a side issue in relation to our main concerns in this chapter. Here we shall assume that vague properties are applied uniformly over the set of things to which they can be applied. To make this explicit we could always use separate properties such as Tall-Human and Tall-Giraffe, although we won t actually need to do this for present purposes. A formal treatment of category dependent vague adjectives is given in Bennett (2006). In general a predicate can be dependent on threshold valuations of several different parameters (e.g. Lake might depend on both its area and some parameter constraining its shape.) Thus, rather than trying to identify a single measure by which the applicability of a predicate may be judged, we allow multiple vague criteria to be considered independently. In the initial development of the standpoint approach Santos et al. (2005a), Mallenby and Bennett (2007), Third et al. (2007), Bennett et al. (2008)), it was assumed that standpoints can be given a model theoretic semantics by associating each standpoint with a unique threshold valuation characterising a complete precisification. In so far as standpoints may be identified with an aspect of a cognitive state, this idea is perhaps simplistic. It is implausible that an agent would ever be committed to any completely precise value for a threshold determining the range of applicability of a vague predicate. Cognitive standpoints are more plausibly associated with constraints on a range of possible threshold values rather than exact valuations of thresholds. For instance, if I call someone tall, then my claim implies an upper bound on what I consider to be a suitable threshold for tallness the threshold cannot be higher than the height of that person. This elaboration of the status of standpoints in relation to thresholds is being developed in ongoing research. But, in the context of implementing cartographic displays showing the spatial extensions of instances of vague terms, modelling a standpoint as a fully determinate parameterised precisification has been found to be useful and informative. It has the advantage that the regions displayed in accordance with a standpoint always corresponds to some precise definition. This is desirable if one wants to compare different instances of

22 22 Brandon Bennett vague predicates. Moreover, it is relatively easy to design an interface such that a user can easily change their standpoint by altering the thresholds assigned to one or more of the parameters that define the standpoint. To summarise, the key ideas of standpoint semantics are: a) to identify precisifications with threshold valuations i.e. assignments of threshold values to a set of parameters that model the variability in meaning of the vague concepts of a language; and b) to always evaluate information relative to a standpoint, which in the simplest case corresponds to a single precisification, but could correspond to a set of precisifications compatible with an agent s current attitudes to language use. A threshold valuation appropriate for specifying a standpoint in relation to the domain of hydrographic geography might be represented by something like the following: V = [ pond vs lake area threshold = 200 (m 2 ), desert max precipitation = 20 (mm per month), elongated region min elongation ratio = 2.2,...] Here, the first parameter determines a cut-off between ponds and lakes in terms of their surface area and the second sets the maximum precipitation at which a region could be considered to be a desert. The last parameter might be used to specify conditions under which a region is considered to be elongated (how this property might be defined will be discussed in section 9.2 below). 7.3 Defining and Interpreting Vague Concepts using Parameters As well as providing a formal structure that defines the semantic choices associated with a particular precisification, the parameters of semantic variation and the threshold assignments to these parameters play further key roles in standpoint semantics. As well as their role in the semantics, the parameters may also be referred to explicitly in the formal object language in which we both axiomatise or define vague concepts and in which we also represent information expressed in terms of these concepts. We here assume that the object language is first-order logic or rather it is firstorder logic with a small syntactic innovation (in fact a similar extension is likely to be possible for other formal languages, but we will not examine this possibility here). The innovation is that, for each vague predicate, we allow additional arguments to be attached to it corresponding to semantic variation parameters, relating to the variability in the meaning of that concept. Specifically, where a vague n-ary predicate V depends on m parameters we write it in the form: V[p 1,...,p m ](x 1,...,x n )

23 Spatial Vagueness 23 The following examples illustrate the use of this language augmented with precisification parameters to define some vague spatial concepts: 1. Tall[tall thresh](x) def height(x) > tall thresh 2. Forested[ forest max tree dist ](r) def p[in(p, r) t[tree(t) (dist(p, t) < forest max tree dist)]] 3. Forest[ forest max tree dist, forest min area ](r) def Forested[forest max tree dist](r) (area(r) > forest min area) r [Forested[forest max tree dist](r ) PP(r,r )] Example 1 is a simple definition of tall as a predicate that applies to anything whose height is greater than a particular threshold. Definition 2 specifies that a region is forested just in case every point in that region is less than a certain threshold distance from a tree. Finally, example 3 defines a forest as being a forested region whose area is greater than a given minimum and that is not contained within some large forested region (here PP(x,y) means that x is a proper part of y). Actually, the additional parameter syntax [p 1,...,p m ] is not really essential since we could either just treat the variability parameters as ordinary additional arguments or we could simply omit them from the predicate arguments altogether and just have them as constants embedded within the definitions. However, the extended syntax seems to be both convenient and informative as it ensures that the parameters of variability of each vague predicate are clearly indicated and highlights the conceptual difference between the objects to which a predicate is applied and the parameters used to precisify the predicate s meaning. We can now understand how a threshold valuation (associated with a standpoint) is used to interpret each vague predicate in a precise way. All we need do is to substitute the values given by the threshold valuation in place of the corresponding threshold parameters given in the definition. If we then remove the [p 1,...,p m ] argument lists we end up with ordinary first-order formulae, defining precise versions of the vague predicates, in accordance the given threshold valuation. 8 Comparison Between Approaches It may be useful to compare how the variable extension of a vague spatial region is modelled according to the different approaches we have considered. We assume that we wish to model the spatial extension of an instance r of some vague spatial predicate V(x). Fig. 7 illustrates the different models that arise. In Fig. 7.a we see the egg-yolk model with its inner yolk corresponding to the region that is definitely part of the extension of r, and its outer egg boundary within which the extension of r must lie. The dotted lines indicate the boundaries of crisp regions that are possible crispings of r. We note that by itself, the egg-yolk model does not place any further restrictions on the shape of these possible crispings.

24 24 Brandon Bennett Fig. 7 Comparison of extensions of a region instantiating a vague predicate, as modelled by: a) the Egg-Yolk model, b) a fuzzy set of spatial points, c) the precisifications of a supervaluation semantics, and d) the standpoints of a standpoint semantics. Fig. 7.b depicts region r represented as a fuzzy set of spatial points. The shading represents the degree to which each point is considered to lie within r the darker the shade the higher the degree of membership. Fig. 7.c is supposed to indicate possible extensions of a feature under a general supervaluation semantics. This is rather misleading as it would depend very much on what semantic conditions were specified in the theory determining the set of admissible precisifications. Most likely the set of extensions according to different precisifications would be much more structured. What this diagram is intended to indicate is just that supervaluation semantics by itself does not impose particular conditions on the range of possible extensions of a region characterised by a vague predicate. Fig. 7d shows possible extensions according to a standpoint semantics. Here we assume that a single parameter has been determined that is relevant to application of a predicate that is used individuate the region. According to different choices of a threshold of applicability, we get different extensions. These will typically be

25 Spatial Vagueness 25 structured like a contour map. If a strict threshold is set (which may be high or low depending on whether the parameter is a positive or negative indicator of the feature under consideration) then a small region is identified, and with less strict thresholds monotonically larger regions are demarcated. We note that models a, b and d have similar structure. The Egg-Yolk boundaries of a can be regarded as representing two distinguished contours chosen from the continuum of contours resulting from threshold choices made in relation to the standpoint model d. Similarly, the fuzzy membership function depicted in b could be chosen so that its α-cuts also correspond to contours in d. Moreover, although the supervaluation model c shows extensions corresponding to arbitrary precisifications, suitable conditions on the set of admissible precisifications could make this too correspond with the contours of d. A further connection is that the egg and yolk of a (or indeed the contours of d) could be associated with limitations on the extensions of r that are possible in admissible precisifications of a supervaluation semantics. c would then look more like a (or d). Nevertheless, despite the similarity between the resulting models. There are deep conceptual differences. In the fuzzy model, b, the meaning of the vague predicate V is considered to be static and the points are associated with V to a more or less strong degree. But with the standpoint semantics it is the meaning of V that varies, and a point may be part of the extension of an instance of V for some but not others of these meanings. The differences between these approaches would become more evident if we considered examples involving a combination of inter-related classifications for instance the extensions of two or more regions that are instances of different but related vague predicates. In fuzzy logic the conjunction of multiple vague predications is modelled by a function of the truth degrees associated with each extensions. This mode of combination makes it difficult to capture the significance of dependencies between vague concepts. But in the standpoint approach, interdependent concepts will share parameters of variability, so that a choice of threshold value may affect the meaning of several different vague concepts. Consequently, for any given standpoint the extensions of related features are coordinated. For example, suppose that the terms forest and heath-land are defined so that areas with a tree density above a chosen threshold are classified as forest but those with tree density less than this threshold (but above some lower threshold) are classified as heath-land. According to the standpoint semantics the border between forest and heath-land would move according to the adopted standpoint; and for any particular standpoint no region would be both forest and heath-land. By contrast, in the borderline between forest and heath-land we would have an area that is forest to some degree and is also heath-land to some degree. The ways in which separate items of vague information can be combined and collectively interpreted within each of the various formalisms is a very significant and interesting topic. However, further consideration of this aspect of vagueness is beyond the scope of this chapter.

26 26 Brandon Bennett 9 Some Significant Vague Spatial Predicates In this section we shall examine certain vague spatial predicates in more detail. Several significant kinds of spatial concept and relation will be covered. But, since there are a large number of ways in which vagueness affects spatial predication, this is not an exhaustive analysis. 9.1 Vague Distance Relations: Near and Far Arguably the most fundamental spatial relations are those relating to distance. Indeed Tarski (1959) showed that all precise geometric predicates can be defined starting from the basic relation of the equidistance of two pairs of points. In natural language we describe distances in a variety of ways. Sometimes we use numerical distance measurements, typically with a very high degree of approximation. For instance, if I say that Leeds is 200 miles from London, I do not imply an exact measurement if the distance were 10 or even 20 miles more or less, this would still be regarded as a reasonable claim (the semantics of numerical approximations has been studied by a number of authors e.g. (Corver et al. 2007, Krifka 2007)). Although these measurement approximations are closely related to vagueness they are essentially numerical rather than spatial in character and will not be considered further here. Another equally, if not more, common way of describing distances is by means of terms referring to vague distance relationships i.e. words such as near, far, close, distant. Examination of the informal concept of nearness in geography goes back at least to the work of Lundberg and Eckman (1973). More recent studies of how people use the words near and far include (Fisher and Orf 1991), (Frank 1992), (Gahegan 1995) and (Worboys 2001) Application of fuzzy logic techniques to representing spatial relations such as near are investigated by Robinson (1990, 2000), who describe a system that uses question answering by human subjects to learn a fuzzy representation of the concept near by constructing a fuzzy set of the places near the reference place. Fisher and Orf (1991) in their survey of subjects ascription of near (in a university campus setting) found that, rather than judgements of nearness being clustered within a single distance range, several clusters were found (three in fact), which seemed to indicate that different semantic interpretations of near were being employed by different subjects. What is clear (as indicated by the experimental results of Gahegan (1995)) is that many contextual factors, apart from the actual distances involved, have a strong influence on how subjects apply the description near. These include the relative sizes of objects or regions involved, connection paths between places, scale, and the perceived significance of objects.

27 Spatial Vagueness Elongation vs. Expansiveness A qualitative distinction that appears to be of general importance in our description of the world, and has particular significance in relation to many geographic features, is that between elongated and expansive regions. A typical example is the distinction between a river, which is elongated, and a lake, which is expansive. The term elongated is used here to refer to a region that is long an thin, but not necessarily straight. Thus, a river may curve and wind but is still, in this sense, elongated. Given the infinite possible shapes that a region may take, it is not obvious how to measure the degree to which an arbitrary region is elongated. One relatively simple idea is to consider the ratio of the radius of a region s minimal bounding circle to the radius of its maximal inscribed circle. For the case of a 2-dimensional region, this measure is illustrated in Fig. 8. A similar measure can be defined for 3-dimensional regions, using spheres instead of circles. Such a measure can easily be utilised within a standpoint semantics formalism. For example, one might define the vague property Elongated, using the standpoint semantics notation, as follows: Elongated[elong ratio](x) def (min bound rad(x)/max insc rad(x)) elong ratio This formulation is fine as long as we have already demarcated the boundaries of all regions that we wish to classify. In this case we can directly evaluate the truth of the Elongated predicate, relative to a given value of the threshold τ(elong ratio), for any given region. However, in many domains certainly in geography and biology we often encounter cases where we are looking for an elongated part of a larger system. For example we may want to individuate a river that is part of a complex hydrological system. A method for identifying elongated parts of a larger region (and hence partitioning it into elongated and expansive segments) was proposed by Santos et al. (2005b) and further developed in (Mallenby and Bennett 2007). Initial geometrical process- Fig. 8 Calculation of elongation ratio. The elongation ratio is obtained by dividing the radius of the minimum bounding circle by the radius of the maximal inscribed circle i.e. L = R/r.

28 28 Brandon Bennett Fig. 9 Medial axis skeleton of the Humber estuary and its tributaries. ing is carried out to find the medial axis skeleton of the region under analysis. 10 This is the locus of all points that are equidistant from two or more boundary points of the region. Fig. 9 shows the medial axis skeleton computed for the Humber estuary in the UK. It can be seen that the skeleton includes line segments of two somewhat different kinds. Some of the segments run along what we might naturally think of as the middle of the channels of the water system, whereas others run from the middle to the edge of the water region. The latter type of segment arise from relatively small indentations in the river boundary and are not particularly relevant to the shape of the water system as a whole. In order to remove these unwanted segments, the skeleton is then pruned by removing all segments whose rate of approach towards the boundary is greater than a certain threshold. Once this pruning has been carried out, the remaining, more globally significant, part of the medial axis skeleton is used to identify elongated parts of the region. The idea is that these are associated with segments of the medial axis along which the width remains approximately constant. As shown in Fig. 10, the width variance is calculated for each point along a (pruned) medial axis section. The calculation is based on the highest and lowest widths evaluated along a sample segment of the medial axis, extending a certain distance either side of the point under consideration. In order to make the measure of width variation scale invariant, the length of medial axis over which it is calculated is taken to be equal to the width of the region at that point. The width variation is then computed as the ratio of the highest to lowest width (i.e. distance from a point on the medial segment to the nearest boundary 10 This was carried out using the software VRONI (Held 2001).

29 Spatial Vagueness 29 point) along the sample segment. 11 Fig. 10 illustrates the width variation measure at two points p and q along the medial axis of a region. At p the width variation is low since all widths at medial points within the maximal inscribed circle centred at p are similar, whereas at q the width variation is high. Once this variation measure has been computed for each point on the (pruned) medial axis, the elongated parts of the region are determined as those parts of region such that every point on the segment of medial axis running through that region has a width variation below a given threshold value. Points with width variation above the threshold are considered to lie in expansive sections of the region (as are points that lie on branching points of the medial axis, for which the measure is not well defined). The application of this method to the Humber estuary region is shown below in Fig. 11. Here we see two possible demarcations of the elongated sections of the estuary and the river Hull from which it opens. The upper map shows the demarcation obtained by using a strict threshold of 1.09 on the maximum width variance, whereas the lower map was produced using the much more tolerant threshold of This provides a good example of how a standpoint semantics approach can be used to visualise a geographic feature according to different interpretations of a vague concept. 9.3 Geographic Feature Types and Terminology Geographers, and more especially surveyors and cartographers, have long been aware of the difficulties of giving precise definitions of spatial features (see for example (Maling 1989, chapters 5 and 12)). The prevalence of cartographic maps as the prime medium for geographic information may have hidden the true extent of indeterminism in geographic features and their boundaries. Constructing a map Fig. 10 Determination of elongated segments of a larger region. 11 A number of variants of this simple calculation also give reasonable results it is as yet unclear which is the most appropriate.

30 30 Brandon Bennett Max width variance = 1.09 Max width variance = 1.32 Fig. 11 River stretches identified according to different standpoints. involves the use of complex procedures and conventions for converting observations measurements into cartographic regions and entities. Moreover, many stages of these procedures require a certain amount of subjective judgement in order to transform the multifarious characteristics of the world into precise cartographic objects. Thus the resulting map representation gives an impression that the world is far more neatly organised and compartmentalised than is really the case. As we have seen from many of the illustrative examples given above, many of the representations proposed by researchers in AI, formal logic and ontology have been developed with geographic applications in mind. Relevant works include that of Varzi (2001a) and the collection of papers in Varzi (2001b). Certain geographic feature types have received particular attention. I now briefly summarise some contributions in this area. The nature of forests and how they should be defined and identified has been examined by Bennett (2001) and Lund (2002). The case of mountains has been considered by Smith and Mark (2003). As mentioned above a number of papers have tackled the definition and individuation of hydrographic features As well as being interested in specific types of geographic feature, geographers are also concerned with general aspects of the way humans describe the world. Such descriptions are of course greatly affected by the vagueness of our natural language terminology. The question of how vernacular terms related to geographic regions