Vagueness is ubiquitous in spatial and geographicalconcepts and tends to persist even where steps are takento give precise definitions. For example, in the guidebook to the Ordnance Survey’s

Land-Line data-set(Ordnance Survey 2000), a ‘road’ is defined as: “Ametalled way for vehicles.” This does tell us somethingabout what is meant by ‘road’ but the definition is stillvague in many respects. We may be unsure about whatsurfaces count as ‘metalled’ – neither the condition ofthe surface nor any restrictions on its spatial extent arespecified. And the term ‘way’ could be understood inmany more or less general senses. ‘Vehicle’ is also avery general term. The OS definition of road wouldseem to apply to bicycle paths, which may not beintended. It also seems to rule out cobbled or pavedstreets which one might expect to be classified as roads.

An understanding of the meanings of vague geo-graphical concepts is relevant to many practicalproblems that involve determining and allocating landtypes. For instance, if we want to answer a questionsuch as ‘How rapidly is the forested area of the earthshrinking?’ the problem of demarcating forest areas iscentral. Similar problems apply to the identification andclassification of ‘deserts’ (and the problem of measuringand monitoring the progress of desertification).Information sources made available by the USGCRP(n.d.) include multiple data sets on topics such as soil,precipitation, vegetation, temperature, land cover, etc.Several of these data sets have ‘desert’ as a specificclass, each with its own method of compilation andconcept of what actually constitutes a desert: absenceof vegetation, annual rainfall below a particular (andvarying) threshold, number of months per yearexceeding a precipitation threshold, type of soil,ecosystem characteristics, etc. The result is a set ofmaps that produces a very different distribution ofdeserts according to which classification you choose touse. Moreover, the concept of desert can itself be vari-

ously classified into sub-types (e.g., ‘desert, mostlybare’, ‘sand desert, partly blowing’, ‘other desert andsemi-desert’, ‘polar desert’, ‘tropical desert’). A furtherexample of the importance in environmental modellingof clarifying vague terms is provided by Alker, Joy,Roberts, and Smith (2000) who consider issues indefining the concept of a ‘Brown-field’ which is oftenused in formulating development policies.

In this paper I shall explore a possible approach tothe logical and semantical analysis of vagueness andapply this to specific problems in the definition ofgeographical concepts. I focus in particular on theexample of the concept ‘forest’, looking at differentways in which the term can be interpreted and how theseeffect the determination of the spatial extensionsassociated with features classified as forests. Beforecolouring the reader’s perception of the issues by sug-gesting a particular theoretical approach, I end theintroduction by enumerating a list of questions, eachof which addresses one of the main aspects of vague-ness associated with the term ‘forest’, and hence hasno clear-cut answer. I shall return to these later andexamine them in the light of a supervaluationist accountof vagueness.

01. Is a forest a natural feature or one determined byconvention and legality?

02. Does ‘forest’ refer to an integral feature or can itbe applied to an arbitrary region of land?

03. What type of vegetation can constitute a forest?(i.e., what species and how big must they be?)

04. How dense must the vegetation be? 05. How large an area must a forest occupy? 06. Are there any constraints on its shape? 07. Must a forest be self connected, or can it consist

of several disjoint parts? 08. Must it be maximal or could it share a border with

another region of forest?

What is a Forest?On the Vagueness of Certain Geographic Concepts Brandon Bennett

Topoi

20: 189–201, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands.

09. Is a clearing a part of or a hole in a forest? 10. Are roads and paths going through a forest parts

of the forest? 11. How should seasonal and other temporal variations

be taken into account? 12. If part of a forest is felled and subsequently re-

grown, does it remain part of the forestthroughout?1

1. The nature of vagueness

To get a purchase on the multiplicity of ways in whichone might bend or stretch the meaning of a concept suchas ‘forest’, we will need a theoretical framework withinwhich properties of the variable meanings of vagueexpressions can be clearly articulated. Unfortunately,the concept of vagueness itself is not completely clear,so I shall first try to be precise about the phenomenonI wish to analyse.

I regard vagueness as a lack of clearly defined criteriafor the applicability of a concept. Thus, it is a propertyof language not of the world itself. Typical examplesof vague propositions in the geographical domain are:‘All mountains are very high’; and ‘Near the marsh isa dense thicket’. The words given in italics are theprincipal sources of vagueness. ‘Mountains’, ‘marsh’and ‘thicket’ are vague feature classifiers: they do nothave precise, universally acknowledged definitions; andconsequently the spatial extent occupied by suchfeatures is potentially controversial. ‘High’ and ‘dense’are adjectives, which give some indication of physicalproperties of a feature but do not specify any definitemeasurable requirement. ‘Very’ accentuates vagueadjectives but does not make them any more definite.

Vagueness should be sharply differentiated fromuncertainty, which is a distinct (though interacting)phenomenon. Uncertainty arises from lack of exactknowledge about an object or situation, and is thus anepistemic state rather than a feature of language.Although modelling of uncertainty is extremelyimportant in the processing and interpretation of spatialinformation, it will not be considered in the currentpaper. I shall assume we are dealing with idealised datawhich is completely certain and accurate. Althoughphilosophers generally have little trouble separating thenotions of vagueness and uncertainty this is not alwaysthe case in certain branches of science.

Vagueness can often lead to uncertainty in that where

a concept such as ‘forest’, ‘desert’ or ‘swamp’ is vaguewe will in many cases be uncertain how to demarcatethe spatial extension of entities to which these conceptsapply. On the other hand, if we are not completelycertain of the exact details of some information we wantto report, we may employ vagueness as a means ofincreasing the certainty of what we say, while at thesame time conveying a sense of imprecision. Forexample, a statement such as ‘The chair is in the cornerof the room’ is vague but can often be said withcertainty, whereas an exact specification of the locationof a chair (or even a range of possible locations) willtypically be uncertain. This example also illustrates thefact that vagueness is not merely a defect of language;it also often facilitates communication without thecumbersome language required to achieve precision.

Vagueness is also sometimes confused with gener-ality. If I say ‘I shall see you again later this month’,this is an example of generality, since the claim can befulfilled in many alternative ways. However, it is notvague, since ‘later this month’ refers to a precise periodof time (I assume all relevant events take place withina single time-zone). But if I say ‘I shall see you in a fewweeks’, this is vague (as well as general) since there isno hard and fast definition of what periods of time canbe described as ‘a few weeks’. Generality per se posesno real problems for the logician, since it is handledperfectly well by classical logic.

Having separated it from some closely related issueswe must ask whether vagueness qua “lack of definitecriteria of application” is a single uniform property oflinguistic expressions or whether it comes in differentvarieties. I argue that it is useful to distinguish betweenat least two kinds, or modes, of vagueness, whichrequire somewhat different logical descriptions. I callthese modes conceptual vagueness and sorites vague-ness.

Conceptual vagueness occurs where there is no singlecompletely adequate definition of a conceptual term.Certain requirements may be clearly identifiable,whereas for other conditions it is arguable whether ornot they are necessary. Certain combinations of theseconditions may capture typical senses of the term butnone is representative of all possible senses. Thus if wetake the intersection of plausible definitions we get aconcept that is much too strict (perhaps even unsatisfi-able), whereas if we take their disjunction we get aconcept that is overly general. This kind of vaguenessis closely related to ambiguity. If a word is ambiguous,

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it has two or more distinct senses that are clearlydistinguishable. However, a conceptually vague termcorresponds to a complex cluster of many overlappingsenses, such that we cannot say exactly what sensesmake up the cluster. Moreover, one can meaningfullyuse a conceptually vague term without being committedto any one of its possible precise interpretations.

Sorites vagueness is the kind of indeterminacy thataffects the thresholds at which we assert properties suchas ‘tall’, ‘large’ or ‘heavy’. Sorites vague predicatesdivide entities with respect to some relevant measurablequantity, without being committed to any specificboundary value. In such cases I suggest that there isnothing vague about the conceptual content of thepredicate – it is just that the content is unspecific aboutthe precise boundaries of application. Indeed theseboundaries could not be made precise without changingthe meaning of the concept.

The two types of vagueness are not mutually exclu-sive. Indeed many natural terms are affected by both. Ina pure case of sorites vagueness it is uncontroversialwhich factors are relevant to the ascription of a term orhow those factors should be measured – it is only thethreshold that is at issue. However, in many cases therelevant factors and suitable thresholds are at issue. Forexample to precisely interpret the concept ‘tall man’ wehave first to decide how we are to measure the heightof a man: must he remove his shoes and hat? what abouthair and prosthetic limbs? what about posture? Once wehave resolved these conceptual issues we then still haveto deal with sorites vagueness in setting the thresholdfor tallness.

Thus, according to my view, the elucidation of vagueterms requires two distinct modes of analysis. First,conceptual vagueness should be tackled by identifyingsubstantial points of controversy among possible defi-nitions. Artificial concepts can then be introduced whichcorrespond to particular choices with regard to theseinterpretational issues. If this analysis were carried outexhaustively (and I don’t want to take a view on whetherthis is always possible) we would end up with a(possibly very large) set of alternative definitions, eachcorresponding to a clarified sense of the original, freefrom conceptual vagueness. However, these artificiallydefined concepts may still be affected by sorites vague-ness. There is little point in attempting to eliminate thismore essential form of vagueness by definition, sinceany such definitions – e.g., stipulating that one senseof ‘tall woman’ is equivalent to ‘woman whose height

exceeds 5

′10″’ – would be more or less arbitrary andwould not serve to explicate the relevant vagueness.However, for certain practical purposes it may be usefulto make such stipulations. More significantly we canplace quite strong constraints on consistent usage ofsorites vague concepts by specifying exactly whichobjectively measurable property the threshold ofapplicability lies on. This will enable us to order sensesof sorites concepts according to where they place thecut-off point.

Both types of vagueness also interact strongly withcontextual phenomena of various kinds. Many conceptsexhibit some form of contextual variability. Forexample ‘large’ in the context of ‘large pond’ has adifferent interpretation than in the context of ‘largelake’. In cases such as this we see that a sorites conceptmay be affected by its context so that location of itsalbeit vague threshold is shifted. The range of possibleinterpretations for a conceptually vague concept canalso be affected, not so much by their immediate syn-tactic context but by their more general context withina particular exchange of information.

Because they are largely independent of vagueness,issues of contextual vagueness will not be addressed inthe current work. I shall assume that context can beeliminated or ignored. For instance we might supposethat some transformation can be carried out that replacescontextually variable concepts with non-contextualconcepts and explicit constraints; or, more simply, wecould just consider composite concepts such as ‘tallman’ and ‘tall child’ as if they were syntactically atomic(although perhaps related by certain meaning postulates,which could be formalised). Despite the fact that inmany cases vagueness seems to be separable fromcontext there may be cases where this distinction isblurred. A close connection between the phenomena isborne out by the fact that formalisations of the logic ofcontext (see e.g. McCarthy, 1993) have much incommon with the supervaluation approach to vagueness.

2. Formalising the logic of vagueness

In modelling vagueness in geographical informationresearchers have tended to adopt techniques which havebeen used within computer science. The most commonapproaches here are based on multi-valued logic(Lukasiewicz and Tarski, 1930) and its generalisationfuzzy logic (Zadeh, 1965; Zadeh, 1975; Goguen, 1969;

WHAT IS A FOREST? 191

Dubois and Prade, 1988). For example, Foody (1992)and Usery (1996) apply a fuzzy representation to geo-graphical features. An extensive discussion of the prosand cons of fuzzy logic can be found in Elkan (1993)and a more philosophically oriented consideration ofthese issues can be found in Williamson (1994). Inaccord with Elkan, I suggest that fuzzy logic may bean appropriate formalism for modelling the relationbetween continuous valued observables and themeanings of vague qualitative predicates; however, itis not suitable as a formalism for carrying out logicalreasoning. This is primarily because propositionaloperators whose values are determined purely in termsof the fuzzy truth values of their arguments cannot takeaccount of either logical or domain-specific constraintsholding among the argument propositions. Moreover, incharacterising the vagueness of geographical concepts,fuzzy logic is a palpably blunt tool. The variety andnuances of possible interpretations of a term such as‘forest’ cannot be adequately characterised in terms ofa probability that a particular piece of land should becounted as a forest.

The approach which I propound in this paper is tomodel vagueness in terms of supervaluation semantics(Fine, 1975), which I believe can provide a much deeperinsight into the nature of the vagueness intrinsic in manygeographical concepts. The fundamental idea uponwhich this theory is based is that a vague language isone which can be made precise in many different andsometimes incompatible ways. A way of making alanguage precise is called a precisification. Eachprecisification p is identified with a precise and con-sistent interpretation, Ip, of the vocabulary of thelanguage. In the simplest case this would be a classicalpropositional or first-order model. A supervaluationmodel then consists simply of a set of precisifications.Given a supervaluation model V, a proposition whichis true under every interpretation Ip ∈ V is called super-true or – in my own terminology – unequivocally true.

Supervaluation semantics by itself does not addanything interesting to logic at the object level. It is easyto see that those formulae that are unequivocally truein every model are just the classically valid formulae.However, the semantics does provide a frameworkwithin which we can define operators that articulatecertain aspects of the logic of vagueness.

One possibility is to take a modal approach andrepresent vagueness in terms of propositional operators(Bennett, 1998). Uφ means that φ is unequivocally true

– i.e., true for all precisifications; Sφ can be read ‘φ isin some sense true’ – i.e., true for some precisification.S is the dual of U and thus can be defined by Sφ ↔¬U¬φ. The inference rules justified by this interpreta-tion are those of the simplest modal logic, S5, where Utakes the place of the usual modal operator

�.We can now qualify assertions according to whether

they hold in some or all precisifications. For examplewe might write S[

Wood(‘Woodsley Clough’)] orU[Wooded(parcel1)]. We can also use these operatorsto specify dependencies between the meanings of vagueconcepts. Thus ∀x[Copse(x) → SWood(x)] meansanything which is a copse (i.e., a small group of trees)is in some sense a wood. Similarly, ∀x[Wood(x) →SForest(x)] captures the intuition that any wood isarguably a (small) forest. If a copse is in some sense awood and a wood is in some sense a forest, this doesnot mean that a copse is in some sense a forest; andindeed according to supervaluation semantics theformula U ¬∃x[Forest(x)

∧ Copse(x)] is consistentwith the previous two formulae. This illustrates theability of the theory to model the blurring of concepts,while still imposing strong constraints upon them.

In many geographical applications we will want toemploy artificial concepts that are designed for empir-ical classification of land types and features. These willin many cases be precisely sharpened versions of naturalterms. The supervaluation operators enable us to relatethese artificial concepts to their vague natural languagecounterparts. For instance, the following formula assertsthat Forest1 is a more precise version of the conceptForest:

∀x[Forest1(x) → (SForest(x))] ∧∀x[(UForest(x)) → Forest1(x)]

By specifying such axioms, ‘soft’ constraints are placedon the meanings of natural concepts. Classifications interms of artificial concepts can be combined withinformation containing natural concepts.

Supervaluation semantics allows one to specify anumber of different entailment relations of varyingstrength. In (Bennett, 1998) I defined seven differententailments each of which has a different force. Theweakest of these is what I call ‘arguable’ entailment,which holds if there is any sense of the concepts in theformulae under which the implication corresponding tothe entailment holds. This gives us entailments that holdunder very flexible (perhaps even inconsistent) inter-pretations of the concepts involved. The strongest is

192 BRANDON BENNETT

‘reliable’ entailment, which holds if: whatever sensesthe premisses are interpreted under, the conclusion holdsin every sense. This can be used to derive entailmentswhich must hold despite the presence of vagueness. Forinstance SDesert(x) → U¬Marsh(x) might hold evenwhere Desert and Marsh are very vague predicates.This ability to derive secure consequences involvingvague concepts is perhaps the main advantage ofsupervaluation semantics over fuzzy logic, where fuzzyconcepts cannot support completely reliable inferences.

Although modal operators allow many logicalproperties of vague concepts to be expressed, they donot provide any way of referring directly to individualprecisifications or to classes of similar precisifications.But in order to carry out a detailed analysis of differentsenses of vague geographical concepts we shall oftenwant to relate natural terms (such as ‘forest’, ‘desert’etc.) to artificially sharpened concepts which give amore precise and objective characterisation of ageographical feature.

In a general reified supervaluation semantics wecould associate arbitrary propositions with precisifica-tion variables and constants. Thus, InPrec(p, φ) wouldassert that φ is true according to precisification p. At theexpense of some elegance we can achieve the sameexpressive power by simply supplementing each pred-icate and function of an ordinary 1st-order language byan additional argument place. For clarity I shall writethis as a prefix to the predicate or function. For instanceSwamp(x) would be replaced by p:Swamp(x) sayingthat, in precisification p, x is a swamp. If we use thisapproach we need not worry about axiomatising thelogical predicate InPrec.

Since a precisification fixes the meanings of all thevague vocabulary of a language, a classification whichmakes precise only part of the vocabulary may becommon to a class of precisifications. In a formalismwith reified precisifications, we can model this byintroducing predicates of precisifications. For example,UNESCOF(p) ↔ Φ(p) might mean that the predicateUNESCOF applies to those precisifications satisfyingsome precise formal specification Φ of the UNESCOforestation classification given in Table I (towards theend of this paper). In this way any refined sense of avague term may be identified with some subset of thespace of all precisifications.

Given that the previous section made much of thedistinction between conceptual and sorites vagueness,I ought to explain how this is manifest in supervalua-

tion semantics. In fact according to my analysis thedifference lies not in the abstract semantics but only inthe types of axioms that constrain the two modes ofvagueness.

Whereas conceptual vagueness is analysed by sharp-ened definitions and axioms relating these definitionswithin the space of precisifications, sorites vagueness isanalysed by specifying the observational propertiesrelative to which a concept has a vague threshold. Astraightforward and general way of specifying therelevant property is via the definition of comparativerelations. For instance the relation taller could bedefined in terms of height by:

∀pxy[p:taller(x, y) ↔ (p:height(x) > p:height(y))]

Clearly, willingness to ascribe the predicate tall to aperson increases the greater their height; so, if any rea-sonable judge calls one person tall, then she must regardall taller people as tall:

∀pxy[(p:tall(x) ∧ p:taller(x, y)) → p:tall(y))]

This encodes a crucial consistency property on precisi-fications regarding their precise interpretation of theadjective ‘tall’.

By comparing the extensions of sorites propertiesaccording to different precisifications we can define anordering on the precisification space. This is similar tothe metrical model for margins of error model of vagueconcepts suggested by Williamson (1994). Furtherdiscussion of this can be found in Bennett (1998).

3. Spatial vagueness

Supervaluation semantics is a very general approach tovagueness but it can only be useful for reasoning abouta specific domain if the peculiar logic of that domainis adequately modelled. In the next section I shall turnto the detailed analysis of the concept of ‘forest’; butprior to that it will be useful to look at the more generalquestion of how vagueness affects the determination ofspatial extensions. The application of supervaluationsemantics to spatial concepts is relatively undeveloped,although it has recently been given some attention(Lewis, 1993; McGee, 1997; Kulik, 2000; Varzi, 2001).

The spatial properties that are easiest to understandsemantically are those that can be defined in terms ofproperties of points – i.e., their extension consists ofall points satisfying some given condition. Examples

WHAT IS A FOREST? 193

of such concepts are ‘the region of the Earth that is morethan 1000 m above sea-level.’ However, in general, a‘region property’ will be associated with a property ofa whole area or volume, which cannot be explicitlyreduced to properties of individual points. For examplea ‘lake’ is not simply made up of the set of points whichare covered by water, it is rather a particular maximalconnected set of water covered points. Indeed, maximalconnectedness is one of the most important factors thatenable us to individuate geographical features fromattributed point data. However, only very basic typesof feature can be regarded simply as maximal connectedsets of points exhibiting a given property. Typically,whether a set of spatial points can be taken as the exten-sion of a feature of a given type is dependent on muchmore complex constraints (consider e.g., how we dif-ferentiate lakes from other hydrological features or howwe might characterise a ‘building’). Some of theserequirements may not even relate to physical proper-ties (e.g., a ‘listed building’).

A further issue that complicates the identification ofregions with sets of points is the status of boundarypoints. For many spatial concepts it is not clear whetherboundary points should be counted as included in theregion to which they refer. This may be regarded as anexample of conceptual vagueness. However, it is also ageneral ontological issue which applies to spatialconcepts that are not in other ways vague.

The ‘Egg-Yolk’ theory (Lehmann and Cohn, 1994;Cohn and Gotts, 1996a; Cohn and Gotts, 1996b) directlymodels the notion of a vague (or uncertain) region interms of its maximal and minimal possible extensions.

The maximal extension is called the ‘egg’ and theminimal is the ‘yolk’, which is required to be a part ofthe egg (see Figure 1a). (The case where the yolk isequal to the egg is allowed, such cases correspondingto ‘crisp’ regions.) This analysis is simple and supportsan account of some significant inferences involvingrelationship between vague regions. However, it cannothandle complex constraints on a region’s possibleextensions between its maxima and minima. Forinstance, although a vague region such as an area ofmarshland might have maxima and minima as illustratedin Figure 1a, the area within the dotted line might notcorrespond to any reasonable precise interpretation of‘marshland’.

Supervaluation semantics is much more general inthat it has the potential to model arbitrary constraintson the distribution of possible extensions, as illustratedin Figure 1b. However, the possible extensions ofnatural vague concepts will not be completely chaoticsince, according to supervaluation theory, they corre-spond to a cluster of precise concepts with similarmeanings. In the case of a purely sorites vague concept,where the vagueness is in the choice of a suitablethreshold for some observable, the possible extensionswill typically (though not necessarily) be contoured asshown in Figure 1c. Each contour corresponds to a moreor less strict sense of a spatial concept. For instance dif-ferent definitions of ‘marshland’ may require more orless water to be present. Where we have mixed vague-ness we will have several sets of contours each corre-sponding to varying the threshold for some conceptuallyunambiguous but still sorites vague concept.

194 BRANDON BENNETT

Fig. 1. Models of vagueness and extension.

4. A supervaluationistic ‘forest’

We are now ready to employ the supervaluation theoryto carry out a detailed analysis of geographical featuredescriptions. Each type of geographical feature hasmany idiosyncrasies in its particular meaning and in theways that its meaning can be stretched or tightened tosuit different purposes. Nevertheless, once an abstractlogical analysis is given, forms of vagueness can bediagnosed that are present in a wide variety of geo-graphical terms. Thus, the supervaluationistic dissectionof ‘forest’ will serve as an example of how one mightanalyse similar concepts such as ‘desert’, ‘marsh’,‘mountain’, or ‘lake’.

One of the most important aspects of the conceptualvagueness of the term ‘forest’ is the ambiguity betweenforests conceived of as natural features and forests asparcels of land upon which is legally or conventionallyconferred the status of being a forest. Although it maybe argued that forests are always originally identifiedwith some natural feature, once forests are named (andthus probably also owned) additional conventional andlegal mechanisms may be employed to individuateforests. Smith (1995, 2001) has investigated theontology of conventional regions of this kind, whichhe calls fiat regions.

In axiomatising the vague term ‘forest’ it is clear thatthe natural and fiat interpretations will obey rather dif-ferent axioms. Hence, any adequate analysis should splitthis concept into two specialisations. The followingaxioms, which employ the reified precisificationnotation, ensure that in any precisification Fiat_Forestand Natural_Forest are sub-concepts of Forest and thatall forests are of one of these two types (they do not ruleout the possibility that something may be both):

∀px[p:Fiat_Forest(x) → p:Forest(x)]

∀px[p:Natural_Forest(x) → p:Forest(x)]

∀px[p:Forest(x) → (p:Fiat_Forest(x) ∨p:Natural_Forest(x))]

Though free from a certain ambiguity, the predicatesFiat_Forest and Natural_Forest are still extremelyvague; each will correspond to a wide range of possiblesenses, and further subdivisions and axioms will berequired to explicate these. In the rest of the analysis Ishall deal only with ‘natural’ forests, since these seemto be vague in a greater variety of ways; however, thesemantics of fiat forests is no doubt also very complex.

To avoid cumbersome terminology, the word ‘forest’shall henceforth be used to mean ‘natural forest’ and theformal predicate Forest shall be used in place ofNatural_Forest.

In clarifying the concept of ‘(natural) forest’ weimmediately encounter a second fundamental ambiguitythat affects this and many similar geographical concepts.When used with an article (‘a forest’ or ‘the forest’)the term typically refers to a particular integral featurewhose boundary (albeit vague) is determined by themeaning of the concept. However, it can also be usedin an adjectival sense to describe an arbitrary regionas ‘forest’. These two uses are not really due tovagueness but rest on a logical distinction that oughtto be explicit in any ontology of geographical descrip-tions.

Though ontologically distinct, features and corre-sponding land-type concepts have strong logical inter-dependence which must be formally specified (seeEschenbach, 2000). Let us use the predicate Forest asa vague feature type and Forested as the correspondingvague land-type classifier and see what axioms onewould expect to link the two concepts. The situation iscomplicated by the fact that Forest refers to a featurewhich is a three-dimensional material object consistingof trees distributed in space, whereas Forested is apredicate of regions of land. To relate these two typesof entity we will need to characterise the mappingbetween a feature and its terrestrial extension, which Iassume to be a two-dimensional region.

We might be inclined to say that a region is ‘forested’just in case it is part of the extension of some forest.However, this definition suffers from a problem ofgranularity, since the extension of a forest may includepockets which are not at all forested or are too small toeven be legitimate candidates for such a description.One might hope to avoid this problem by taking forestedas the more basic property and then defining a forestas a feature whose terrestrial extension is a maximalself-connected forested region. However, since‘forested’ is a land type, the objects that fall under thispredicate are terrestrial regions, not physical objectslike forests. Clearly a type of physical object cannot bedefined from a predicate whose arguments are purelyspatial: in order to carry through a definition we shallhave to add some physical ingredient.

Given any region we can determine, without exces-sive controversy, the vegetative material that is presentin that region. It then seems reasonable to assume that

WHAT IS A FOREST? 195

if this area is the terrestrial extension of a forest (quanatural feature) then all properties of that forestsupervene on properties of that quantity of vegetativematerial.2 Thus the properties of forests (including theiridentity criteria) are determined by the vegetation theycomprise, which in turn is determined by their terres-trial extension.

It must be noted that the terrestrial extension of aforest is not determined directly by its vegetation. Thatis, if we were to simply shrink-wrap the vegetation andproject this volume vertically onto the earth’s surfacewe would define extension at too fine a grain size tocorrespond with our intuitions of the extent of a forest.Rather the extension of a forest is normally understoodas including areas which are surrounded by trees but arenot actually beneath any branch or above any root. Thus,terrestrial extension is a subtle function of vegetationdistribution which will vary according to what preciseinterpretation is placed upon the vague ‘forest’ concept.However, if we can define a predicate ForestExtent,which holds of all those regions that are in someartificially precise sense the terrestrial extension of aforest, we will thereby also determine the physicalconstituents of each forest and an identity criterion forforests.

In terms of the parthood relation P and a predicateCON meaning that a region is self-connected, I definea ForestExtent as a maximal connected forested regionof sufficiently large area:

p:ForestExtent(r) ≡df p:Forested(r) ∧CON(r) ∧ area(r) ≥ p:minfa ∧

¬∃r′[p:Forested(r′) ∧ CON(r′) ∧ P(r, r′)]

The scope of the precisification variable p ensures thatunder any given precisification the meaning ofForestExtent is logically determined by the meaning ofForested under that same precisification. This consis-tency requirement within definitions supports variouspatterns of reliable inference that hold whateverreasonable sense we give the concepts.

The precisification-relative area constant minfa givesthe minimal size of the extension of a forest. Typicallya forest is taken to be a rather large feature coveringmany square kilometers of land. Where the area issmaller the feature is likely to be called a ‘wood’. If itis smaller still, say less than a hectare, one wouldperhaps qualify it as a ‘small wood’ or use another termsuch as ‘spinney’. We could define ‘wood’ for exampleby using a ‘minimal wood area’ constant minwa and

perhaps also a maximal wood area constant. The con-straint

∀p[p:minfa ≥ p:minwa]

means that in every precisification forests must be atleast as large as woods. Nevertheless, there need be nosuch consistency between different precisifications:what is a wood in one particular precisification couldbe counted as a forest in another. One might also wantto place constraints on the shape of legitimate exten-sions of woods and forests, since a predominantly lineardistribution of trees is not normally considered a foresthowever large an area it covers. Finding reasonableshape constraints is surprisingly difficult and is beyondthe scope of the present work.

Having defined ‘forest’ in terms of ‘forested’ weneed to consider how observable measurements of thephysical world determine which regions fall under theconcept ‘forested’; or rather, given our supervaluationmethodology, we need to elucidate how these observ-ables relate to different precise interpretations of‘forested’. It will be useful at this stage to introduce avague definition which seems to take into account allof the most salient requirements of the concept. I shallsay that a region is forested if it is densely covered bytrees. Although this definition does not make theconcept any more precise, it does focus our attention onthe key sources of vagueness in any characterisation offorested: what is a tree? how should we measure thedensity of trees? and, when can a terrain type which isintrinsically vague and granular be said to ‘cover’ agiven area?

Apart from the case of metaphorical usage of‘forested’, which shall not be considered here (thoughit could be argued that metaphor sheds considerablelight on the meanings of vague terms), it is uncontro-versial that the distribution of trees is the primary factorin determining whether a region is forested. Other landproperties and vegetation may be relevant to propertiesof a forest but are not essentially relevant to its exten-sion. However, the term ‘tree’ is itself to a certain extentvague.

My dictionary (the Concise Oxford, 1999) defines‘tree’ as “A woody perennial plant, typically with asingle stem or trunk, growing to a considerable heightand bearing lateral branches.” This definition exhibitsboth conceptual vagueness, as to which plant forms orspecies count as trees, and also sorites vagueness, in that‘considerable height’ prescribes a vague threshold

196 BRANDON BENNETT

relative to an objective physical property. Artificiallyprecise definitions of ‘tree’ may be given either genet-ically, in terms of a set of tree species, or by stipulatingconditions of physical stature. In the literature onvegetation mapping these modes of classification arereferred to respectively as floristic and physiognomic(see e.g., USGS (1994b) for a discussion of theseclassifications). Perhaps the most intuitively reasonableclassifications can be obtained by some combination offloristic and physiognomic constraints. The super-valuation approach is well suited to formalising logicalrelationships among natural and artificially defined treeconcepts. For instance the formula

∀px[p:Tree57(x) → p:Tree(x)]

asserts that Tree57 is a sharpened version of the natural,unrefined concept of tree.

Once we know what a tree is we can try to formu-late possible precise versions of the notion of a densecoverage of trees. This turns out to be a surprisinglydifficult problem. There are various possible ways onecan quantify trees within an area. Practical forestmensuration techniques employ at least the following:the number of individual tree specimens, the totalvolume of vegetation, the cross-sectional area occupiedby tree trunks (at some stipulated height from theground). In each case the measure may be applied tojust the dominant tree species, to all vegetation, or tosome restricted sub-class. Another consideration is thatthese measures only work well for regions of a suffi-cient size. A very small region will, most likely, lieoutside every tree or within or on the edge of anindividual tree; and in each case density cannot sensiblybe measured. In fact it seems to be generally true thatto determine whether a point belongs to the region of aparticular land-type we often have to look not only atwhat is present at that point but at what is present insome significantly extended region including the point.

Given that one can specify measurement schemeswhich quantify the trees in a region (provided the regionis sufficiently large) it might at first sight seem a simplematter to divide this quantity by the area of the regionto arrive at a measure of tree density. Indeed we coulddefine a family of possible measures and articulate theinterdependencies between them in a supervaluation-based logic. However, if we calculate, by whatevermeans, the tree density of an arbitrary region we willget a value that is an average over the whole area. Butthe region might include one or more parts which are

very densely forested while other parts might be com-pletely treeless. In order to identify forests we must havesome way of separating regions of high and low treedensity. But since our method of calculating tree densityrequires one to start with a predefined region we comeup against a chicken and egg problem.

I have identified several ways of tackling thisproblem each of which has a rather different flavour.One is to impose on the terrain a tessellation ofappropriate granularity. Density is then only computeddirectly for the cells of this grid – larger regions ofdense tree structure are constructed as sums of theseunits. Another approach is to partition the land byreference to a more easily measurable secondary indi-cator. Both these methods are pragmatic solutions whichare widely adopted in actual geographical field work. Iwill comment further on them in the next section.

A third approach is to identify a region of uniformlyhigh tree coverage by ensuring that all significantlylarge sub-regions maintain the required density. This isnot easy to specify, but the following definition seemsto capture the idea reasonably well:

p:Forested(r) ≡df ∀d[(Disc(d) ∧ area(d) = p:fgran∧ area(d ∩ r) ≥ p:fbgran) →

tree-density(d ∩ r) ≥ p:fdthresh]

The specification is in terms of discs of a certain size(i.e., having an area of p:fgran), which is deemed appro-priate (according to precisification p) to the granularitywith which forest density can sensibly be measured. Forall such discs overlapping the region by more than acertain amount (given by the ‘forest border granularity’parameter, p:fbgran), the density in the area of overlapmust be higher than a certain threshold, p:fdthresh. Irestrict attention to discs since we will probably wantto exclude sub-regions which have a high tree densitybut are fragmented or very linear in form. The formulais complicated by the requirements of modelling theedges of a forested region to avoid unwanted shrinkageof legitimate extensions. This requires special attentionand makes the definition far less intuitive than onewould like. Despite being rather artificial, the definitiondoes succeed in providing a conceptually unambiguouscharacterisation of a forested region. The variabilityof reasonable values for the density and granularityparameters models purely sorites vagueness.

A further way that one could address the issue ofdense coverage is to identify high density areas in termsof the local distribution of vegetation. That is, we con-

WHAT IS A FOREST? 197

struct forested regions by a consideration of thelocations of individual trees and their spatial relation-ship to other neighbouring trees. Finding sensible waysof grouping trees is far from straightforward but maylead to a fruitful analysis. There are certainly a largenumber of incompatible ways in which this could bedone, and the lack of any reason to choose a particularapproach is perhaps the main reason why this kind ofclassification has not been widely studied. However,using the framework of supervaluation theory one canexplore many possible definitions without committingto any one.

A simple example is the following definition, inwhich a forested region is defined as one such that allof its points are within a certain distance of a tree:

p:Forested(x) ≡df

∀(π ∈ x)∃t[p:Tree(t) ∧ dist(π, t) ≤ p:d]

This gives a family of possible precise conceptsdetermined by the parameter d which is a function ofthe precisification p. There is a strong logical constrainton this family in that the extension of this concept witha small d is always a subset of the extension determinedby any larger d. Hence we would get a contoured dis-tribution of possible extensions similar to that shownin Figure 2. An obvious problem with this definition isthat each isolated tree constitutes a small forest. Oneway of countering this would be to require that eachpoint of a forested region is close to some moderatelylarge number of trees. Another similar approach would

be to construct the region as the sum of the convex-hullsof sets of ‘sufficiently close’ trees.

The ad hoc nature of my characterisations of‘forested’ may indicate that the concept is in need ofdeeper ontological analysis. On the other hand it mightbe that there is an essential conceptual vagueness in theconcept, so that no complete finite specification of itspotential interpretations is possible. Nevertheless theartificial concepts do seem to capture much of what isintended by the naive description of ‘forested’ and togive a reasonable account of the parameters of itsintrinsic sorites vagueness. Moreover, I suggest thatintuitive classifications of areas into forested and non-forested will not greatly deviate from classificationsobtainable by plugging reasonable parameters into theartificial concepts.

A further facet of the interpretation of the word‘forest’ which is worth a brief mention is its connota-tion relative to other terms for similar geographicalfeatures. I am thinking here of the contrast between,for example, ‘forest’ and ‘jungle’. Jungle almost alwaysrefers to a tropical or sub-tropical vegetation cover,whereas forest is more general but with a suggestion ofa temperate climate. The logic of connotations is likelyto be extremely tangled; but in so far as any clear dif-ferences in sense can be identified they can be straight-forwardly encoded within the supervaluationframework.

5. Comparison with the geographer’s forest

Let us round off our examination of forests by consid-ering how the supervaluationist approach can be relatedto a particular definition of forest that has been widelyused in geographical applications. Table I shows aphysiognomic classification of levels of forestation thatwas proposed in UNESCO (1973) and later adopted inUSGS (1994). The range of different terms employedin the table illustrates the way that a precisification (orclass of precisifications) is not merely associated witha collection of senses of individual terms but withcomplex system of logical constrains concerning themeanings of multiple interrelated concepts.

This classification carries with it a lot of implicitconceptual baggage which may not be compatible withother ways of defining forests. For instance, any pre-cisification satisfying it must enforce the constraint thatwoodland and shrub-land are necessarily disjoint. There

198 BRANDON BENNETT

Fig. 2. Possible forest demarcations for a given tree distribution.

is also some lack of specificity in the classification. Itis not clear whether a population of fairly widely spacedtall trees growing among a dense cover of small shrubsshould be counted as ‘sparse woodland’ or ‘shrub-land’.So some precisifications satisfying UNESCOF mightrequire height to take precedence over density whileothers could require the converse.

One facet of the problem of characterising foreststhat is not tackled at all by the UNESCO classificationis the question of how forest boundaries should bedemarcated. The scheme seems to assume that a candi-date area has already been identified which can then bemeasured in terms of average vegetation height (pre-sumably the height of a species which is in some sensedominant in the area) and the percentage canopycoverage over that area.

USGS (1994) surveys a number of methods whichfield workers use to elicit stand boundaries from otherrelevant and more directly measurable factors, such asclimate and topography. Similarly, in relation to soil-type boundaries, Mark and Csillag (1989) note thatboundaries are often introduced on the basis of surfacefeatures that are correlated with, but not essential, tosoil classification. Influences of inessential features onforest stand demarcation from aerial photographs arehinted at by the results of Johnston and Lowell (2000).Although indirect methods may be effective for manypurposes they do not elucidate how to partition vegeta-tion-types in terms of properties of the vegetation itselfand hence, from an ontological point of view, they aresuspect because they define something in terms offactors that are only contingently related to the phe-nomenon in question.

Another way in which geographical informationsystems and other land surveys often avoid the diffi-culty of boundary identification is to employ some form

of atomic area (usually the cells of a grid) as a minimalunit for which a land-type is determined. Using thisontology the difficulty of assigning a boundary to anintricate natural object is largely avoided. Instead,measurements are applied to whole cells (or randomsamples from cells) and a land-type inherits its boundaryfrom the already given boundaries of a group of cells.This is fine as long as we always take a coarse view ofthe world, where we can treat the cells as atomic units.However, if we are in the business of accounting for thedifferent senses of a term like ‘forest’ we will also wantto account for perspectives that go right down to thelevel of individual trees. For example we might havedata that tells us that a garden is within a dense forestbut we cannot infer that the garden contains trees unlessthe atomic units that have been classified as forest aresmaller than the garden.

Given the slipperiness of the concept ‘forest’ onemight assume that the problem of boundary demarca-tion would have been tackled exhaustively within thesubject of forestry. However, this does not seem to bea major concern in the literature of that field (thestandard textbook Husch et al. (1963) considers onlytechnical problems of surveying a boundary, and doesnot mention conceptual problems in defining bound-aries). In fact, this is not surprising when we considerthe nature of forestry and the kinds of information itrequires. For most purposes a forester can assume thathis forest consists of a collection of stands whoseboundaries are well-defined. The properties of eachstand can then be determined by random sampling tech-niques; and from these measurements, economicallyimportant quantities such as ‘forest volume’ can thenbe derived by simple computations or by the use ofempirically verified tables. The problem of determiningboundaries is not of great importance because the

WHAT IS A FOREST? 199

TABLE IA physiognomic classification of vegetation types (UNESCO 1973)

Plant-form/Height Percent canopy cover of vascular vegetation

100%–60% 60%–25% 25%–10% 10%–1%(interlocking) (touching) (spaced)

Trees > 5 m Forest Woodland Sparse woodland

Shrubs/Trees 0.5 m–5 m Shrub-land Sparse shrub-land Sparsely

Shrubs < 0.5 m Dwarf shrub-land Sparse Dwarf shrub-land vegetated

Herbs Herbaceous

statistical approach to measurements works with anyreasonable bounding of the forest area and, in all butexceptional cases, mitigates the effect of any uncertaintyin this boundary.

Thus, while the identification of boundaries is amajor concern in the ontology of geographical features,for certain purposes they can be taken for granted.However, in situations where we need to identify land-types for some high-level evaluation or planningproblem, the problem of demarcation is crucial, andontological analysis serves a useful function in pro-viding a basis upon which consistent reasoning can becarried out.

In the analysis of the last section, I avoided thoseaspects of the vagueness of ‘forest’ which are associ-ated with persistence through time. In what we mightcall the ordinary usage of ‘forest’, it is applied to anarea that is densely covered by trees at the time thedescription is made. However, in the context of forestmanagement and ecological classification it is quitecommon to regard a land-type in terms of a cyclical orprogressive process that takes place in some area of landover a considerable period of time. From this perspec-tive it is perfectly natural to consider an area as‘forested’ even when all its trees have been felled andcarted off to the log mill. In this sense ‘forested’ wouldinclude areas of land at all stages in the arboriculturalprocess. Thus for many geographical applications onewill need to carefully differentiate between senses offorest which employ conflicting notions of its temporalstatus.

6. Conclusion

The geographical sciences face increasing pressures toassimilate huge quantities of information and exploitthem consistently for a host of diverse applications inindustrial, environmental, and social management. Theneed to provide a firm foundation for the interpretationand manipulation of this data has led to recent interestof geographers in ontological questions (e.g., Frank,1997). But, while there is a high level of awareness ofissues of uncertain and imprecise information, the dif-ficulty of taking account of the intrinsic vagueness ofnatural concepts does not seem to have been fully appre-ciated. As I have shown, an adequate analysis of a singlebasic geographical feature type may involve enormouscomplexity.

Vagueness is often regarded as a phenomenon whichdefies detailed theoretical explication; but, in this paper,I have attempted to show that a concerted analysis canreveal logical constraints underlying the apparentlynebulous meanings of vague concepts. I have suggestedsupervaluation semantics as a powerful theoretical toolby which one may make inroads into the tangledsemantics of ill-defined concepts by articulating whatis fixed and what is variable among a space of possibleprecise senses. A key aspect of my analysis is thedivision of the phenomenon of vagueness into concep-tual and ‘sorites’ modes, which allows one to separatethe concerns of analytical ontology from problems offormalising the logic of pure sorites vagueness.

I must confess that, in presenting various formal def-initions purporting to characterise senses of the word‘forest’, I have often met with some scepticism. Criticshave argued that, since the term is so obviously vaguein such a numerous variety of ways, any attempt to pindown its meaning formally is completely hopeless. Itis for the reader to decide whether the analysis givenin this paper cuts anywhere near the heart of themeaning of ‘forest’ or merely wanders among its manybranches. It is true that vagueness, especially in its puresorites form, is as yet very poorly understood, and thereis no consensus on how inference should be conductedin the presence of vague concepts. However, to equatewhat is not understood with what is unintelligible is todeny the value of philosophy. Many geographicalconcepts may be extremely vague but I do not believethey are completely unprincipled.3

Notes

1 Accompanying its ‘Land Usage of the World’ data the web sitewww.ecoworld.com gives the following definition of forest: “Forest:Land under natural forests or planted stands of trees. Also includeslogged areas to be replanted in the near future, after logging.”2 Here I am ignoring the temporal aspects of forests but I do notsee any obvious impediment to reconstructing this analysis, albeit ina rather more complex form, taking into account temporal persistenceand evolution.3 This work was supported by the EPSRC under grant GR/M56807.

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