Vagueness, Incoherence and the Perspective of Rules

Imagine a series of coloured patches having as its first member a patch which is clearly red. As we move along the series, the shade of the patches gradually changes until we reach the last member, which is clearly orange. Suppose also that the change is so gradual that each patch is indistinguishable in shade from its neighbours. Now imagine that a normal English speaker is presented with each of these shades in sequence and asked to respond either "Yes" or "No" to the question "Is this a red patch?".

What might the outcome of such an experiment be? This is, of course, an empirical question but we may safely assume that, at the beginning of the test, a normal subject will confidently answer "Yes", and that at the end he will equally confidently answer "No". We may also assume that, somewhere between these two extremes, there will be patches concerning whose colour status the subject either hesitates before responding, or responds one way but then changes his mind (perhaps repeatedly), or simply looks puzzled and declines to issue a verdict either way. Even in the event that the subject immediately and confidently answers "Yes" or "No" to all of the questions, it is unlikely that he would give precisely the same pattern of responses if he were to run through the series a second time.

Let us call those patches in which no definite, immediate answer is forthcoming borderline cases of redness. I am not suggesting that these cases will form a sharply delineated set since, for example, whether a person has hesitated or not may itself be unclear in some cases. Nor is it being assumed that the responses are definitive since even under identical conditions the responses of the same or different subjects might well result in a different borderline area at some other time. But we may at least assume that, for each subject and for each run of the experiment, a subject's responses will indicate some borderline area. Let us therefore say that, with respect to a given subject's understanding of the expression, ’red’ possesses borderline cases.

Now imagine a similar experiment in which, instead of a sequence of coloured patches, we have a row of men beginning with a man 5’6" in height and ending with one of 6’6". Suppose also that the difference in height between any two adjacent men is exactly an eighth of an inch, and that our subject is presented with each of the men in turn and asked to respond either "Yes" or "No" to the question "Is this man under six feet tall?".

In this case we might expect the results to be similar to those obtained in the coloured patch experiment, i.e. there will probably be cases where a definite response is forthcoming, and cases where the subject hesitates, wavers, or declines to answer. But should we say in this case that, with respect to this subject's understanding of the expression, ’is under six feet tall’ possess borderline cases? The intuitive answer is that we should not, but then what distinguished this expression from ’red’? The answer, surely, is that we can think of no procedure for deciding whether or not ’red’ should apply in cases where application is uncertain. Of course, the subject may complain that, for example, the conditions of illumination are not optimal or that some other factor is standing in the way of his making a correct judgement. But then, given conditions as optimal as the subject pleases, there is still likely to be uncertainty concerning the proper application of the expression in some cases. The point is that, in such cases, this uncertainty will not be dispelled either through the acquisition of further factual information or through a more thorough training in the use of ’red’.

On the other hand, the subject will manifest misunderstanding of ’is under six feet tall’ if he does not know how, at least in principle, to determine the application of the expression in cases of uncertainty. He would need to know that, for example, careful examination with an accurate tape measure would settle the matter, at least in cases where height differences of only an eighth of an inch are in question. I suggest, therefore, that talk of borderline cases is in order only when the uncertainty cannot be removed through the acquisition of further empirical information or a more thorough training in the use of the expression.

One can, of course, always make stipulations. One might, for example, decide to regard a man as bald if he has fewer than a thousand hairs on his head, and such stipulations are indeed frequently made in practical situations. But two things need to be borne in mind. First, such stipulations do not remove any intrinsic uncertainty regarding the correct application of a term which is, after all, in general use, for how could a mere local stipulation do that? Stipulation is really just an attempt to prevent an expression's possession of actual borderline cases from interfering with an activity by removing, for specific practical purposes, potential unclarity or misunderstanding. Secondly, exactly where to draw such boundaries for the purposes at hand is not dictated by the meaning of the expression in question, since the meaning does not determine that sharp boundaries are to be drawn at all. This is what is meant by saying that an expression may possess borderline cases, and is reflected in the fact that its meaning is consistent with the boundary being artificially drawn in any number of places to suit individual or collective requirements.

Let us call an expression which possesses borderline cases, that is, which appears to lack sharply defined boundaries of application, a vague expression. I shall not be especially concerned with the question of whether vague expressions may also be higher-order vague, i.e. whether there may be borderline cases of borderline cases. Mark Sainsbury argues that most of the standard approaches to vagueness founder on the phenomenon of higher-order vagueness since they assume that vagueness can be studied in a precise meta-language, dividing the extension of a vague term into three classes: the positive cases, the negative cases, and the rest. But the boundary between the positive (or negative) cases and the rest is itself vague, so the problem has not been solved but merely shifted to another boundary, and no amount of "upward mobility" is "enough to get on top of all the phenomena. . . . You do not improve a bad idea by iterating it". According to Sainsbury, we must abandon the idea of "providing a system of pigeon-holes, . . . placing a grid over reality" and do away with talk of boundaries altogether. Instead, concepts should be thought of as being

This is a good picture, but it would seem to leave the problem facing the other approaches to vagueness unsolved. For consider the series of coloured patches discussed above. As Dorothy Edgington asks,

So, again, three classes have been specified: those which cluster firmly around the red and orange poles, and the rest. But if these classes exhaust the domain, then

Sainsbury is, however, surely correct to interpret vagueness as boundarylessness, at least if this is read as implying that there are no sharp boundaries to the application of a vague expression. In other words, there are no sharp boundaries to be drawn anywhere in the extension of a vague expression, and so the ’problem’ of higher-order vagueness is not distinct from the problem of ordinary, first-order vagueness: any adequate approach to vagueness must explain both of the phenomena together and not be compelled to give them separate treatment. The root problem of vagueness is well brought out by considering what Sainsbury calls the transition question:

It is, however, precisely this kind of question which any adequate account of vagueness must address. Accordingly, Sorites arguments may usefully be rephrased as transition questions, and the test of any proposed theory of vagueness is now whether it is able to provide adequate answers to them.

In ordinary language, the word ’vague’ is used in a variety of distinct and sometimes conflicting ways. In the remainder of this section we will look at some of the notions with which vagueness is easily confused and attempt to distinguish them from the more philosophically interesting sense just discussed.

Ambiguity and vagueness have frequently not been clearly enough distinguished. Though I have no theory of ambiguity to offer, I want briefly to indicate here that, as intuitively understood, it has little to do with vagueness, at least when this notion is understood in terms of being borderline.

A word or expression is normally thought to be ambiguous if it has more than one interpretation, meaning or extension. Since nothing much turns on the point, let us for simplicity's sake treat words as graphic (rather than phonic) forms and ignore the fact that some forms are treated by lexicographers and grammarians as homonyms, that is, as functioning as two or more different words rather than as a single ambiguous word. A useful test, if not an actual definition, of ambiguity is the test of contradiction, which states that a word or expression is ambiguous if it may be at once clearly true of an object and clearly false (or, at least, not clearly true) of it. If one thinks of ambiguity in terms of divergent extensions, one may say that a word (graphic form), W, is ambiguous if there can be two W-tokens, one of which denotes something not denoted by the other.

How does vagueness connect with this broad conception of ambiguity? Well, suppose that W has two distinct meanings or divergent extensions, M₁ and M₂, and that there exist borderline instances of M₁. Does this indicate a special connection between vagueness and ambiguity? Not at all, since the fact that M₁ is vague is irrelevant to the fact that it is but one of the meanings of W. It might, for example, have been the case that a word different from W had been used to mean M₁. This would not have affected the vagueness of M₁, although it would have removed the ambiguity of W, since W would have had only the single, unambiguous meaning, M₂.

It might perhaps be more plausible to argue for a connection between vagueness and ambiguity in cases of what Israel Scheffler calls the ambiguity of occurrence, when there is

Scheffler sees ambiguity of occurrence as a special case of vagueness, since there is indecision over whether, in a given context, a term denotes one extension or another.

But does this really show that a type of ambiguity can be a species of vagueness? To answer this, one needs to ask exactly which kind of indecisiveness or uncertainty is involved in such cases. The sort of thing which Scheffler has in mind is presumably exemplified by such sentences as

where there is nothing in the context of utterance of (B) to indicate whether Fred went to the side of a river or to a financial institution, or where the context favours both interpretations equally. One may well be uncertain, on hearing an utterance of (B), about its correct interpretation, but it is not the case that there is, in principle, no way of discovering it, and so this kind of uncertainty is not an example of vagueness but of factual ignorance. The most obvious way to find out what was meant is simply to ask the utterer, so such uncertainty is, in principle, resoluble.

Certain complications may arise, however. It might conceivably be the case that either

or

Case (1) will be seen by those who adopt a realist position towards statements about the past as irrelevant to facts about the utterer's intentions, since he will put any uncertainty about what the utterer meant down to irresoluble ignorance about the facts. An anti-realist about past-tense statements, on the other hand, might see it as implying that both

and

are indeterminate in truth value. However, neither (R) nor (F) is ambiguous, and the indeterminacy alleged by the anti-realist, if it existed, would be due to absence of evidence, not to borderline vagueness. For it always remains possible that evidence should come to light which would either confirm or disconfirm a statement about the past, whereas in the case of a borderline statement there is no such possibility - one cannot imagine what such evidence would look like.

In case (2), both (R) and (F) are unambiguous and false, since the utterer did not mean either ’side of river’ or ’financial institution’. Indeed, it seems that a speaker can mean nothing at all in uttering (B) under such unusual circumstances as described in (2) since, in the absence of any intentions on the part of the speaker, uttering ’bank’ amounts to emitting a meaningless sound. If this is right, then (2) is not even an example of ambiguity of occurrence. I conclude that the uncertainty which arises in cases of ambiguity is not due to borderline vagueness, and that the two notions are therefore distinct.

It is common to find vagueness being identified both with imprecision and generality, the idea being that vagueness may be eliminated either by making our language more precise or more specific. This may be because, as Roy Sorensen points out, ’vague’ is used in two quite different senses, and corresponding to each sense there is a different sense of ’precise’. The first sense of ’vague’, the sense we have been discussing so far, is what Sorensen calls the borderliner sense. In this sense a statement is borderline

The second sense of ’vague’ is ’under-specific’. Here,

The complaint with such statements is not their inaccessibility or lack of truth value, since underspecific statements are often platitudes, and their obvious truth is in stark contrast with borderline statements, which

There is indeed a genuine distinction between vagueness as being borderline and vagueness as generality or lack of specificity. For example, ’living thing’ is more general (less specific) than ’tree’, but is neither more vague nor more precise. Furthermore, an expression, E, must be equally as vague as not-E, but it need not be equally general.

But can vagueness in the sense of being borderline be resolved by introducing greater precision? I think that the answer to this is analogous to the above discussion of underspecificity: borderline statements fail to give us enough determinacy for the purpose at hand. The reason a borderline statement lacks a decidable truth value is that the meaning of at least one of its constituent words fails to determine whether or not the statement is true. Thus a borderline vague word is imprecise only in the sense that its meaning does not specify precise boundaries which sharply divide objects into two groups, one to which the word definitely applies and one to which it definitely does not. Our aim in precisifying such terms is to increase the determinacy of our utterances in particular circumstances by specifying such boundaries in order to communicate our intentions more clearly and to eliminate avoidable confusion on the part of our intended audience. In such cases we are interested in eliminating vagueness only for the purposes at hand, and only insofar as it stands in the way, for example, of successful communication.

Vagueness in the borderliner sense is, I suggest, more interesting, philosophically speaking, than vagueness in the underspecificity sense, since underspecificity can be wholly eliminated, in theory if not always in practice, by adducing more specific factual information in particular contexts. Borderline vagueness, on the other hand, cannot be eliminated in this way, since stipulations which remove uncertainty in particular contexts serve only local interests, and precisely where to draw the line is an arbitrary matter in the sense that it could have been drawn in different places without thereby offending against the meaning of the expression.

In his influential paper ’Verifiability’ Friedrich Waismann wrote:

This passage neatly brings out a distinction which many have failed to draw, namely, that between words which have actual borderline instances and those for which such indeterminacy is only potential. Consequently, they have tended to use the term ’vague’ to cover both actual and possible borderline instances. Thus construed, several writers have held that vagueness is an essential feature of all empirical concepts, such as concepts of colours and natural kinds.

Waismann himself goes even further than this, however, arguing that open texture is a quite general feature of langauge, since for all terms there is always the possibility that

Examples of such non-empirical terms might include ’real’, ’exist’, ’knowledge’, ’meaningful’ and, indeed, ’vague’ itself. Such terms exhibit "systematic ambiguity", being used on different levels with distinct meanings and criteria of application. The absence of clear and comprehensive rules governing the use of such words means that there are (or could be) borderline cases in which we simply would not know what to say. Though they may not be used on many different levels and do not, therefore, display such systematic ambiguity, other non-empirical concepts, such as ’number’, are also open-textured, according to Waismann: for among the diverse class of entities which we call ’numbers’ we cannot see any common property, and so there is no definition which might settle, for all possible kinds of entity, whether they are to be counted as numbers or not.

Waismann appears to be arguing that whether a term is vague or open-textured is a contingent matter, and depends on what objects actually happen to exist. We can put this point in terms of relativity to domains. A term may be vague for one domain, D, but not for another: if D is the collection of animals in a typical zoo, then the term ’elephant’ will be non-vague relative to D since, for every member of D, either ’is an elephant’ or ’is not an elephant’ is determinately true of it. Likewise, for a domain consisting exclusively of single grains of sand and large collections of grains of sand, ’heap’ will not be vague. It just so happens that the world does actually contain borderline instances of ’heap’, but not (presumably) of ’elephant’, and so the former, though not the latter, is, as a matter of contingent fact, vague in Waismann's sense.

Scheffler goes further, arguing that, generally speaking, "vagueness is relative to term, domain, decision, task, person, and time." In line with his nominalistic and inscriptionalistic approach, Scheffler wants to characterize open texture in terms only of actual tokens of terms, and without reference to possible states of affairs which he regards as obscure and troublesome. To this end he makes use of mention-selection, a kind of captioning device in which one may call a picture, description or other representation of an object by the name of that object itself. For example, one may legitimately point at a portrait of a man and say "There is a man":

Open texture then becomes the thesis that every descriptive term "is uncertain with respect to its mention-selection of some [actual] object".

The problem with this approach, however, is that it does not capture anything like the intuitive notion of open texture which we are concerned to elucidate. If possible objects and states of affairs are to be ruled out, then surely so too are merely possible representations, since representations are objects. But since the number, nature and variety of actual representations is a contingent matter, whether a term is open-textured will also, on this inscriptionalistic account, be dependent on what actually exists. So if the world contains only clear instances of elephants and non-elephants, and representations determinately either elephantish or non-elephantish, then, since we are denied recourse to possibilia of any kind, the term ’elephant’ will be neither vague nor open-textured. But the creation of a single borderline elephantish representation will transform ’elephant’ into an open-textured term. Indeed, it is not clear from what Scheffler says exactly what distinguishes vagueness from open texture, since the device of mention-selection which he exploits allows ’elephant’ to be applied to both elephants and elephantish representations alike. So if there happen to be neither borderline elephants nor borderline elephantish representations in existence, then ’elephant’ is neither vague nor open-textured; but just add either a single borderline elephant or a single borderline elephantish representation to the domain and ’elephant’ becomes both vague and open-textured.

The point about open texture, however, is that, unlike vagueness, it is supposed not to be relative to actual domains in the way Scheffler suggests, for it has to do with the possibility of the existence of borderline objects and so concerns the intension, rather than the extension, of a concept. But nominalists such as Scheffler cannot have recourse to intensions, and so cannot even make sense of the notion of open texture, as characterized by Waismann. So Scheffler has to say that the thesis of universal open texture "is in any case an empirical thesis with great plausibility" because he sees it as being demonstrated by subjects' actual uncertainty over the application of a term when confronted with actual objects or representations. But open texture is not, as intuitively understood, an empirical thesis for it concerns the permanent possibility of the occurrence of unforeseen circumstances, and this is something which empirical evidence can neither confirm nor disconfirm. (Furthermore, the question of whether open texture arises from indeterminacy of meaning or factual ignorance is not itself an empirical one.)

Waismann says that the open texture of concepts is rooted in the incompleteness of our factual knowledge due to the possibility

This is, however, slightly misleading, and should not be taken to imply that, when we are uncertain about the application of a certain term, our uncertainty must in principle be eliminable by acquiring further factual information. This would be to suggest that the meaning of open-textured terms is, contrary to the spirit of Waismann's thesis, determinate, and that the sole source of uncertainty is factual ignorance and not ’incompleteness’ of meaning. The point is, rather, that the meaning of an open-textured term determinately neither rules in nor rules out every possible state of affairs and that, even though all actually existing or imagined cases determinedly either fall or fail to fall under its extension, the future acquisition of factual information may still leave its correct application uncertain in some cases.

Of course, the results of future inquiry may suggest an obvious resolution of our uncertainty in certain cases. But it need not force us to give a clear verdict about the applicability of an expression, even though we may decide confidently to issue one, and even though our verdict strikes us as being the most natural thing to say in the situation. The hypothesis that E is open-textured for S does not imply that S will continue to refuse both to apply and withhold E in cases which S currently regards as borderline. In other words, it does not pretend to be able to predict S's future use of E in all possible circumstances, for what hypothesis could do that? It says only that, whatever S should in fact decide to say about a currently borderline instance of E in the future, his decision need in no way ’follow from’ his previous use or understanding of E. In any case, it should always be possible to think of other cases concerning which the applicability of E would be uncertain (for S).

The discussion so far has rested on the assumption that a genuine distinction may be drawn between cases of uncertainty which involve ignorance of matters of fact and those which involve indeterminacy of meaning. Similar distinctions have, of course, been the subject of influential attacks, and so a few brief remarks are called for in order to try and prevent any attempt to elucidate vagueness, in the sense of indeterminacy of meaning, from being dismissed at the very beginning. Scheffler is alert to Quinean considerations when he writes that, if we abandon the analytic-synthetic distinction,

Instances of indecision

But just because there is no absolutely sharp distinction between questions of fact and questions of meaning, this does not imply that there is no distinction between them at all, or that there really are no such things as questions of meaning, any more than the absence of an absolutely sharp distinction between red and orange implies that there is no distinction at all between them, or that there really are no such things as colours. In the foregoing I have nowhere used the term ’analytic’ and no strict analytic/synthetic distinction has been assumed. Indeed, in the light of Waismann's remarks, we should expect that the predicates ’is due to a deficiency of meaning’ and ’is due to a deficiency of resoluble fact’ are themselves open-textured. It is true that we ought not to prejudge instances of uncertainty as "impervious to further investigation and resolution", but this should not prevent us from passing judgement at all. For example, we have good reason to judge that uncertainty over whether to classify an ostensibly borderline shade as ’red’ or ’orange’ is impervious to resolution, since no-one has the faintest idea what such a resolution might look like. This does not mean, of course, that such judgements are invariably correct. It is, however, a platitude that corrigibility is an ineradicable feature of all genuine judgement.

Despite the fact that vagueness and open texture are distinct, though closely related, phenomena, I do not intend to make much of the distinction. This is because I will be primarily concerned with the notion of a semantic rule, and such rules are supposed to determine the application of an expression, not just in actual situations, but in possible or imaginary situations as well. To see this, consider the following statements:

These statements, which might appear as premises in a Sorites argument to demonstrate the vagueness of ’heap’, purport to state some of the rules governing the correct use of the expression. But their truth values are indifferent to whether there are any actual heaps since they are all, in effect, universally quantified conditionals: (H2) is explicitly conditional in form, and (H1) may be re-written as

where the domain of quantification is the set of all possible objects. If one still wishes to maintain that, in the case of a world without borderline heaps, a Sorites argument demonstrates only the intensional vagueness of ’heap’, then so be it. However, nothing much turns on this since the status of statements such as (H1) and (H2), i.e. as rules allegedly governing ’heap’, will not be affected. In what follows I shall therefore use the term ’vague’ to mean ’vague or open-textured’, and so in calling the expression ’cat’ vague, it is not being implied that there are any actual borderline cats.

Much effort in contemporary philosophy of language has been expended in the attempt to construct a theory of meaning for a natural language such as English, that is, an account of what it is that every competent speaker knows when he knows a language. This is usually taken to involve specifying a set of truth conditions or, in anti-realist theories, assertion conditions, for each meaningful sentence of the language.

One idea with which one might begin in connection with such a project goes roughly like this: suppose we had to teach somebody how to play chess without any kind of verbal or written communication. There seems no reason why this could not be done; that is, why an intelligent and receptive subject could not acquire a knowledge of the moves, and so on, just by being immersed in the practice of playing the game. Such a subject might become a reasonably competent chess player without ever learning any of the vocabulary which we use in describing the game. However, once there was unmistakable evidence that the subject had mastered the game, we would not hesitate to attribute to the trainee the knowledge which we express by a statement of the rules and object of the game. We would think of the trainee as possessing this knowledge implicitly, although its content would be the same as the sort of explicit, propositional knowledge which is acquired when someone learns how to play chess in the normal way.

It is tempting to interpret the process of learning a first language along similar lines: language mastery, like mastery of chess, seems to be a rational ability, differing only in that the rules of a natural language are enormously more complex than the rules of chess. As Wright puts it:

In short, speaking a language may be compared with playing a game whose rule book has been lost, and it is the job of the philosopher of language to work out what the rules are, to produce an explicit, propositional description of them, merely by observing the practice of those taking part. The result, if it could be accomplished for a natural language, would be a complete and explicit formulation of the rules which any competent speaker of the language knows, but which is known mostly only implicitly.

There are, however, severe constraints on the nature of the rules implicit knowledge of which may justifiably be ascribed to speakers. These arise from, for example, limitations of human memory, perceptual acuity and cognitive ability. If we find some candidate rule too complex or subtle to be remembered by normal human beings, or which they could not possibly have acquired by direct training, or which implies the ability to discern differences which humans are physically incapable of discerning, then that is good, and often conclusive, reason for rejecting it. Wright mentions five such constraints:

So we have arrived at what Wright calls the Governing View of language. This view is a combination of two claims:

It is at this point, however, that the project of arriving at a coherent theory of meaning along the lines described above begins to run into trouble, since the conditional premises of many Sorites arguments may be supported by one or more of the considerations described in (C1) to (C5). Take, for example, the expression ’red’. It is extremely plausible that if there are semantic rules governing the use of ’red’, then one of them must be

After all, ’red’ is an expression which is taught ostensively and which is typically applied on the basis of casual and unaided observation, so how could it be intelligible to ascribe to a speaker implicit knowledge of a rule which demanded that a distinction be drawn between two items which the speaker cannot tell apart? (Given the limitations of human memory, (R) would also appear to be supported by (C1) and (C5).)

Unfortunately, the supposition that (R) is a rule governing ’red’ leads to paradox. For if a speaker is presented with, say, a series of a thousand coloured patches with clearly red and clearly orange end members, and such that both members of each pair of adjacent patches are indistinguishable in shade to a normal human observer, then if the speaker judges the first patch to be red, he will be inexorably led, by repeated applications of modus ponens, to judge that the final patch is red also. And if the subject then begins from the orange end of the series, and correctly judges the first patch to be not red, then by analogous reasoning he will be forced to judge that the final, clearly red patch is not red either. The conclusion is that if ’red’ is a rule-governed expression, then it is incoherent: it draws no genuine distinctions at all and is therefore vacuous. So if we accept (GV1) and (GV2) we are faced with the problem that Sorites paradoxes

Because of the human limitations described in (C1) to (C5), and since there are real physical differences between things which are too small to notice, then those differences cannot affect the applicability of an expression standardly applied on the basis of causal, unaided observation. Such expressions are, therefore, tolerant to marginal change: their applicability will always survive small degrees of real, physical alteration in relevant respects. The tolerance of ’red’ is reflected in (R), and the applicability of expressions such as ’heap’ and ’bald’, where size and number of hairs are certainly relevant determining features (though not the only ones), will not normally be affected by the removal of a single grain or the extraction of a single hair.

In other words, (C1) to (C5) support quantified conditionals such as:

where x and x’ are adjacent indiscriminable members of a suitable Sorites series, S. Wright puts forward a reductio of the hypothesis that there is a last member of S to which the rules prescribe the application of ’red’. He says:

But the highly plausible assumption that a person's understanding is constrained by (C1) to (C5) is inconsistent with this hypothesis, for

Wright argues that any solution must involve finding a way to avoid having to draw paradoxical conclusions from the above constraints on the use and understanding of the expressions in question. But how can this be possible if, as the Governing View asserts, competence with those expressions consists in internalizing both a set of rules and the limits of their application?

It is notable that here, unlike in the previous passages quoted, Wright is concerned not with ’red’ but with ’looks red’. He seems to have conceded that it is not necessary that every distinction in real colour be detectable by casual observation, but he thinks that, in any case, at least ’looks red’ is Sorites-amenable and that this will, after all, affect ’red’ itself since "Looking red suffices for being red when other things are equal". And if only ’looks red’ is now in question, paradox follows straightforwardly. For it is a truism that the role of phenomenal predicates such as ’looks red’ is purely to record public appearances, thus on the Governing View the rules for such predicates can relate only to appearance and can prescribe application only on the basis of appearance. But then "they must prescribe application of such a predicate to both members, if either, of any pair of items whose appearances are the same." Wright's conclusion is that if there are governing semantic rules for phenomenal predicates at all, then the truism is bound to be directly involved in any specification of their content.

Wright sees no definite prospect that his arguments concerning tolerance will generalize to other classes of expression. In Chapter 4, however, we will see reasons for thinking that observational expressions do not, in respect of their Sorites-susceptibility, constitute a special class, and that the Sorites will appear to threaten any expression whose rules fail in certain cases to prescribe whether or not it should be applied. Furthermore, Wright consistently under-estimates the subtlety and richness of our use of observational terms such as ’red’, and thus also their affinities with non-observational expressions. This makes it easier to restrict the problematic class in the way that he does, for, if we are to take seriously the idea that ascriptions of implicit knowledge of rules to speakers should take into account their cognitive and physical limitations, then just about any concept can be reduced to incoherence in a similar way. If we consider the rules governing the use of the expression ’table’, for example, it would appear that if there are such rules, then they will have to respect the same human limitations as will the rules for ’red’. But no graspable and followable rule could require that a speaker competent with the use of ’table’ be capable of making distinctions at, say, the atomic level. And even if we were capable of making such fine distinctions, the meaning of ’table’ certainly does not depend on them.

According to Wright, there are only three possible responses to the above arguments against the Governing View: (i) accept both (GV1) and (GV2) and conclude that many of our expressions are incoherent; (ii) accept (GV1) but reject (GV2) and conclude that we do internalize semantic rules but that (C1) to (C5) do not constitute constraints on how they are to be determined; or (iii) reject (GV1) and conclude that some alternative, non-rule-based conception of speakers' mastery of observational expressions is called for. The only other alternative, as Wright sees it, is to uphold (some version of) the Governing View, and

In Chapters 2 and 3 we will see that (iii), the response Wright favours - that we should adopt instead a "more purely behavioristic" conception of what competence with such expressions involves - is untenable, and that (ii) is not viable either.

A number of writers have adopted response (i). Michael Dummett, for example, concludes that the paradox must be accepted and that our colour words are indeed incoherent because they are governed by inconsistent rules. But, as Wright points out, this view does not explain our obvious ability to use colour words successfully and coherently, and to communicate effectively by means of them, and it therefore leaves it unclear why we should continue to think of such expressions as being governed by rules at all, if those rules must be inconsistent. It also fails to explain why

Our conviction is, indeed, that nobody ought to be disturbed by the paradox, for otherwise we would have to think ourselves so irrational that "we cannot recognize our confusion even when it is completely explicit".

Another intriguing approach to the Sorites is epistemicism, an extreme form of semantic realism according to which there exist sharp, but unknowable, cut-off points to the application of vague expressions. Thus in the Sorites series for ’red’ there really would exist two adjacent patches, one of which was red and the other not, i.e. there would be a definite first non-red patch, but we would have no means, even in principle, of discovering which patch it was. Similarly with ’heap’ and other vague expressions: if we begin removing grains of sand from a heap there will be a definite first grain, G, such that removing G will make the difference between there being a heap and not being one, even though we can never discover G's identity. Although the epistemic theory does take vagueness seriously - it does not, for example, attempt to eliminate it or play down its significance - it does not see it as arising from any indeterminacy in the meanings of vague expressions; vagueness is, rather, a matter of irremediable ignorance or semantic uncertainty.

Although at first sight wildly implausible, epistemicism has been compellingly argued for. I shall not, however, be discussing it further here since my primary concern is with semantic rules, and the theory may be interpreted as in effect denying that the rules for the use of vague expressions wholly capture their meaning, i.e. as denying (GV1) and (GV2). But then, even if epistemicism were tenable, it would be of little help since the rules which we actually follow would, in the light of Wright's criticisms of the Governing View, still appear to be inconsistent. But it is this apparent incoherence which needs to be dispelled, and which will be the main subject of this thesis.

In the remainder of this section I shall discuss three responses to the Sorites: (i) the theory that the paradox may be resolved by adopting a non-Classical, many-valued logic; (ii) the idea that vagueness may be eliminated by adducing more contextual information; and (iii) an attempt by Mark Platts to show that rules sufficient for grasping vague expressions need not in fact imply the truth of Wright's tolerance rules.

There are senses in which we speak, in ordinary language, of "degrees of truth". We do say such things as:

An interesting feature of such sentences is that they tend to produce in their audience the feeling that they are, as they stand, incomplete and not wholly informative, and the expectation that some further explanation will be forthcoming. This is because ’degrees of truth’ is not a simple, unambiguous notion, but stands in need of qualification and explanation, the character of which will vary from context to context.

Thus with (T1) one might go on to clarify one's assertion by pointing out that ’won’ is ambiguous, that it may or may not apply depending, for example, on the particular aims of military intervention, and in what respects they were, or were not, fulfilled. In such a context (T1) might be readily paraphrasable as

where an explanation of that assertion will then presumably be forthcoming. The utterer of (T2) might be intended to convey come such thought as

and the person making the accusation in (T3) might have meant something like

The above are instances of the notion of ’partial truth’ and are all being used to express reports of omission, ambiguity and errors of detail. (There are doubtless other types of circumstance in which it is appropriate to use such language.) But note that in none of the above is it being suggested or implied that the truth predicate itself admits of degrees. All of the examples may be paraphrased, without loss of information, so that it is only truth simpliciter that is in question. Talk of degrees of truth in such cases is just a façon de parler, and it remains to be seen whether there is any interesting philosophical sense, and if so, any utility, in the idea that truth can genuinely admit of degrees when ordinary language does not seem to require it. Indeed, as we have seen, it invites ambiguity when used in everyday situations, though this may be dispelled when the context of a given utterance is clearly appreciated by its intended audience.

A major motivation, among recent writers, for the thesis that truth is a matter of degree is that it has been seen as a promising approach to the Sorites paradox. One may distinguish two kinds of degree-theories, the finite-valued and infinite-valued approaches, though many degree-theorists seem to remain neutral on the question of which approach to adopt, presumably because it is not relevant to their particular projects.

The idea of finite-valued semantics, or multivalence is to jettison Bivalence in favour of a greater number of discrete values. A natural position is to adopt the Principle of Trivalence which uses, say, the values ’true’, ’false’ and ’borderline’, but the theory, being committed only to the Principle of Valence, is compatible with the adoption of an definite number of discrete values. The finite-valued approach has been well criticized, and I will not rehearse the criticisms at length here. Briefly, the problem, is that adopting, for example, a tri-partite division does nothing to block a Sorites paradox which moves from ’true’ to ’borderline’, for it is as implausible to maintain that there is a sharp cut-off point between a statement’s being true and its being borderline as it is to maintain that there is a sharp cut-off point between its being true and its being false. It can be shown, by analogue arguments, that the same problem, besets any attempt to create an n-partite division along a range of cases. Indeed, one may construct the following Sorites whose major premise seems highly plausible:

The problem with finite-valued theories, apart from their lack of success in blocking the Sorites, is that they seem to impose an artificial discreteness on phenomena which strike us as continuous. The idea that continua in nature ought to be reflected by continua in language is certainly an appealing one, but it is fraught with difficulties. In what follows I shall outline objections to continuum-valued theories raised by Wright which show convincingly that no such approach is a viable option.

One of the great attractions of degree-theoretic accounts is that it purports to show how the major premises of Sorites arguments are plausible by assigning to them a degree of truth sufficiently high enough to make it rational to accept them. The ’leakage’ of truth from each of the conditionals is very small, too small, in fact, to be discerned, but over a sufficiently large number of iterations of modus ponens, the leakage builds up until, imperceptibly, we move from (nearly) true premises to a false conclusion. Wright's first objection is that, contrary to appearances, the degree-theoretic account really cannot explain the plausibility of the major premises. He notes that any conditional which constitutes the major premise of a Sorites argument can be rewritten as a conjunction of the form

(x) - [F(x) & - F(x’)],

where x’ is a value which differs by a suitably small degree from x. The problem however is that, in general, on any plausible and non-arbitrary assignment of truth values to conjunctions, P & -Q "cannot be almost false if P has a middling sort of degree of truth and Q . . . a marginally smaller degree."

Such penumbral conjunctions could never have a degree of truth high enough to be rationally accepted, which would run counter to the intuitive assumption that each of the major premises has a very high degree of truth. One common method of determining the truth of a conjunction is to assign it the minimum of the value of its conjunction. On this approach the idea would be that if P had a value of, say, 0.5, and -Q a value of 0.49, then the conjunction would assume a value of 0.49, a value which, it is true, is far too low to give us a great deal of confidence. An approach which multiplied the values of the conjunctions would produce an even worse result (0.5 x 0.49 = 0.245).

A second problem with degree-theoretic approaches is that the notion of difference of degree which they demand is not fully intelligible. For example, Peacocke explains the notions of identity and difference of degree in terms of the two conditions:

But, as Wright points out, one can often discriminate between two shades in isolation without being able to say which is redder than the other "unless one was assured that the series was moving uniformly away from red". But this, because of the effects on colour due to saturation, intensity and hue, need not be so. It would seem that "we make no use of the comparative except in cases involving differences much greater than just-discriminability." In any case, (D1) and (D2) are circular, for they "involve quantification over instances of the concept [whose identity criteria] we are trying to explain".

Another serious problem is that

If a degree-theoretic account is to stand any chance at all, it will have to respect the intuition that predicates can apply definitely and absolutely, that, for example, "x is red" can be true(1). For if not, then we will encounter counter-intuitive results such as that a man with 100,000 hairs on his head is balder than a man with 100,001 hairs. But both men are clear cases of non-baldness, and denying this is tantamount to denying that any man can be clearly not bald, no matter how man hairs he may have.

The crucial question is: if it is true(1) that a man with 100,000 hairs is not bald, and true(1) that a man with 100,001 hairs is not bald, then how many hairs must a man have for the value to dip to true(0.9999. . .)? The Sorites problem is introduced all over again. The case of ’red’ illustrates the point more vividly, for both scarlet and vermilion are definitely, and paradigmatically, red and it is not plausible to think that one is any redder than the other, i.e. there is no total ordering of shades of red which degree-theoretic approaches require. (Nor, in fact, is there a total-ordering of instances of concepts such as ’bald’, for whether someone is bald is not dependent merely on the number of hairs on his head, but also on other factors, such as the length, thickness and distribution of the hairs. Holding all other features constant except the number of hairs will give a total ordering, but then this will not give us a total ordering for the concept which we actually use.) But at which point does a patch fall below true(1)? It would seem that the degree-theoretic approach is committed to the existence of a first patch which is not definitely red. But then this undercuts the whole motivation for using degrees of truth in the first place.

Wright's final objection to degree-theoretic approaches is that they do not apply to Sorties paradoxes involving phenomenal predicates. The thesis that any pair of indistinguishable items must be red if either is implies that there is no difference between looking red and being red. But this thesis is false, since there is an appearance/reality distinction for colour predicates. That this is so can be seen by the fact that differently coloured objects can be indistinguishable when bathed, for example, in red light of suitable intensity and hue. So it might be the case that "x is red" might possess different degrees of truth when applied to two indistinguishable items. But if ’red’ is not observational in this sense, the same cannot be said for phenomenal predicates such as ’looks red’. As Wright remarks, it is utterly unclear how ’looks red’ might be true of indistinguishable objects to differing degrees, and so a degree-theoretic approach, by itself, will do nothing to block the paradox. For rules for the use of phenomenal predicates would have to "assign exactly similar appearances exactly similar worth".

According to infinite-valued theories, any difference, no mater how small, will make a real difference to the degree to which a given predicate applies. The above considerations should merely serve to reinforce the feeling that it is absurd to think that our predicates are capable of making infinitely fine discriminations, for is it really coherent to suppose that we can somehow get the meanings of our words, which derive from beings who are sensitive only to finite and discrete changes, to ’shadow’ the continua of nature? Discrete distinctions have to be drawn, and we draw them to suit our purposes, but such distinctions are rarely stable. There is still a sense, of course, in which expressions such as ’red’ and ’bald’ admit of degrees, though one cannot hope to quantify those degrees with any precision. The notion of a degree must inevitably remain rough and ready. We should not expect it to be precise since its imprecision reflects the nature of the uncertainty characteristic of our use of vague expressions in borderline cases.

The relevance of context in helping to determine the correct application of vague expressions can hardly be denied. Apart from the obvious cases of indexicals and ambiguous terms its role is to be seen most vividly in connection with syncategorematic terms such as ’tall’ and ’quick’, where an awareness of the intended class of comparison, for example, the class of Swedish males or the class of modes of public transport in the nineteenth century, is essential if one is to make informed judgements concerning the correct applications of such terms. Context can also be highly relevant to many other kinds of expression, such as colour terms. For example, someone going into a clothes shop intending to buy a white summer dress, as opposed to, say, a red one, may be prepared in such circumstances to count as white some slightly ’off-white’ shades; on the other hand, for the artist or colour scientist concerned with extremely subtle nuances of shade such a loose classification might be wholly inadequate, and a shade classified as ’white’ in the former situation might not be so classifiable in the latter. In such cases, a knowledge of the context and purpose of the judgement in question is certainly essential in order to evaluate its appropriateness.

It might be thought, therefore, that vagueness arises because, in borderline statements, the context has not been specified in sufficient detail. If such contextual details were added, one might argue, the statements would cease to be borderline and would assume determinate truth-values. This view has the virtue that borderline vagueness as defined above, namely, as ignorance not resoluble by any increase in ascertainable factual knowledge, does not really exist at all, since contextual information, such as the location of a given speaker, is ordinary factual information which can, in principle, always be obtained. Thus when all the contextual facts are in, there could not, on this view, be any residual ignorance due to borderline vagueness, as previously defined. Thus the need for further investigation into the nature of borderline vagueness would be obviated, and the resulting account would have the great virtue of simplicity.

Unfortunately, this view is not tenable. It has some plausibility in the case of syncategorematic terms such as ’tall’ and ’heavy’, where reference to some standard, implicit or explicit, is required in order to determine whether or not the term applies. For example, a height of five feet eight inches is tall for a jockey or a pygmy but not for a British adult male; and a weight of twenty kilos is heavy for a domestic cat but not for a lion. So the truth-value of sentences such as "A height of 5’8" is tall" and "A weight of twenty kilos is heavy" is indeterminate so long as information about the comparison class is unavailable. So given that the correct application of syncategorematic terms is context-relative, can this relativity account in all cases for their possession of borderline instances?

An argument for an affirmative answer to this question might run as follows: the list of classes of objects to which a word such as ’tall’ might legitimately be applied is open-ended. ’Tall’ is currently applied to people, buildings, and to a host of other vertically inclined physical objects, and it would be impossible and, indeed, absurd, to limit the classes of objects to which ’tall’ may legitimately be applied in the future, while keeping faith with the normal meaning of ’tall’. To make ’tall’ non-vague would require specifying, for each of an indefinite number of classes of objects, a precise height over which a member of that class is to be considered tall: one for British males, another for skyscrapers, and so on indefinitely. (With respect to specifying precise cut-off points, ’tall’ is a simple example of a syncategorematic tern, since only one dimension of difference, namely height, is in question. But whether someone is, for example, fat is not just a matter of weight, or of girth, or of any other single measurable quantity, but rather of ratios of a number of quantities. In such a case, a precise set of ratios, rather than single points, will need to be specified for each class of objects.) And even if one managed to get as far as all currently existing classes of objects, there would still remain those kinds of objects which do not currently exist. The task is pointless and, indeed, impossible. Therefore, the vagueness of syncategorematic terms like ’tall’ is unavoidable since it is impossible to foresee all the possible contexts in which it might be applied.

This argument certainly supports the view that terms such as ’tall’ are vague and that their vagueness is not readily eradicable due to the open-endedness of the list of classes of objects to which such terms might be applied. Indeed, I cannot think of an example of a syncategorematic term that is not vague. It might be thought that words which has an ’absolute’ use, such as ’flat’, are examples of words that are syncategorematic but not vague, since an object whose surface has the tiniest bump or scratch ceases to be (absolutely) flat. Absolute flatness may be given a precise mathematical definition, and is never actually encountered in the natural world. When used in this non-vague way, however, ’flat’ also ceases to be syncategorematic, since no comparison class needs to be contextually specified in order for a hearer to interpret sentences containing the word. But in ordinary usage ’flat’ is syncategorematic, since tiny deviations from perfect flatness do not turn ordinarily flat surfaces into ones that are not flat, and the standards of flatness required, for example, by amateur cricketers for a cricket pitch are far lower than those required by astronomers for the mirrors of large telescopes, where even microscopic scratches are unacceptable.

So it would seem that syncategorematicity and context-relativity tend to lead to vagueness, and for good reason. However, it cannot be the root cause of vagueness, since syncategorematic terms may remain vague even after their comparison classes have been fully specified. For example, the predicate ’tall for a British male’ is neither syncategorematic nor otherwise context-relative, and so anyone who hears the sentence "Bill is tall for a British male" will know all that he needs to know in order to judge that it is true, provided that he knows that Bill is a British male who is over six feet in height. But ’tall for a British male’ is still vague, since it has borderline instances. To this, it might be objected that some contextual information is still missing. But what kind of contextual information could settle the question of the tallness of such a borderline case one way or the other? It looks as though the only thing which would eliminate the residual vagueness of ’tall for a British male’ would be the existence, somewhere, of a precise point, P, such that any British male taller than P counts as tall. But given that the meaning of ’Bill is tall for a British male’ does not imply the existence of such a point (or rather, pace epistemicists, no discoverable point), must the utterer of the sentence have had some precise height in mind? And if not, which features of the context are supposed to provide it? ’Tall for a British male’ is an example of a predicate which is neither syncategorematic nor otherwise context-relative. In fact, many of the examples which are typically given as paradigms of vague predicates, such as ’red’, ’bald’ and ’heap’, are not obviously context-relative either. This suggests that the context-relativity of a syncategorematic term, although it may contribute to the possession of borderline instances due to the kind of open-endedness described above, is not the main source of the term's vagueness.

In any case, the claim that adducing more contextual information is always bound to reduce the degree of vagueness of an utterance is not correct. Suppose, for example, that we are eavesdropping on a private conversation and someone says of a 6’2" man "He's tall". In many contexts this utterance will be taken to be clearly true. And suppose that, as we continue to eavesdrop, it becomes clear that the utterance was made in the context of a discussion about Swedish men. Still, probably, the utterance was true, though perhaps less clearly so. But then suppose that, after a little longer, we finally learn that the whole conversation concerned (say) Swedish firemen. At this point, however, in possession of as much contextual information as we could hope for, the original utterance begins to look a little borderline (I have not carried out an extensive study of Swedish firemen, but I imagine that 6’2" is not especially tall). At each stage, as more of the context gets filled in, the degree of vagueness of the original statement is increased rather than reduced.

The contextualist argument in fact appears to come to much the same as the identification of vagueness with generality, and the absence of vagueness with specificity, a view which was criticized in §1.1(ii): a specific statement may convey more information than a more general one, but it need not be less vague. Similarly, providing further contextual information may help to make clear exactly what is being asserted or denied - which statement is being made - but it does not necessarily reduce the (borderline) vagueness of the statement; indeed, as we have seen, it may even increase it.

Mark Platts has argued that rules sufficient for attaining a grasp of vague predicates need not imply the truth of the conditional premises of Sorites arguments, including Wright's tolerance rules, and that we may therefore deny such premises without having to deny that the predicates are rule-governed. Consider the following Sorites argument:

Therefore:

Platts rightly points out that we can quite easily master the concept ’picture of a face’ without having to master any dot-vocabulary - we might not even know that the picture is made up of arrangements of dots - and that therefore premise (D2) cannot be the correct formulation of a semantic rule governing the use of the concept ’picture of a face’.

Platts then goes on to argue, rather less plausibly, that similar considerations also apply to the Sorites paradox for ’heap’. For the conditional premise of this paradox is

but in order to master the concept ’heap of sand’ we do not need also to have mastered the concept ’single grain of sand’, or be able to perceive exact quantities of sand, and so the truth of (H2), which makes reference to those concepts, is not implied by the rules governing ’heap of sand’. (H2) may therefore be rejected as irrelevant to the meaning of ’heap of sand’ and paradox is avoided. At most, all that is required for a grasp of the concept is the ability to detect imprecise quantities of grains and a grasp of the vague concepts ’many’, ’few’ and so on. Platts writes that

We may concede that (D2), (H2), and many other typical conditional premises should not be attributed to speakers in order to explain their competence in the use of Sorites predicates, and that such premises will not find their way into any final account of the rules governing such predicates. However, Platts' argument from conceptual independence still leaves us with a big problem. For, although (D2) may not itself constitute an acceptable formulation of a rule a grasp of which is necessary for mastery of ’picture of a face’, it does nevertheless seem to be true, and would certainly be regarded as highly plausible by most competent speakers of English. That is, any person possessing the concept ’picture of a face’, and who also has some knowledge of the ways in which arrangements of dots can constitute pictures of faces, as in newspaper photographs, would most likely take (D2) to be true, for there will be no one point in the dot removal process at which one can say with any justification or confidence that a picture of a face clearly no longer exists.

Another, stark example of this is Peter Unger's Sorites paradoxes involving removals of single atoms from everyday physical objects. Platts would have it that the rules for the use of ’table’, for example, would not imply the truth of the conditional premises of these arguments, since one can master the concept of ’table’ without knowing anything about atomic structure. The premises still remain highly compelling, however, and Platts does not explain why this should be so. The question is: what is the connection between the (perceived) truth of these premises and the particular semantic rules for ’picture of a face’ or ’table’ which, we are assuming, make no essential reference to dots or atoms?

Let us concentrate on observational expressions. Abstracting from particular observational expressions, the following seems to be a plausible candidate for a rule governing their use in general:

This amounts to the idea that there are differences too small to matter to the application of any given observational expression and is supposed, I take it, to be part of what it is to be an observational expression. However, I do not need to know anything about dots, grains of sand, atoms, or whatever, in order to be able to grasp (O1).

It follows from (O1) that

where (o + D) is the object resulting from applying D to o, for example, the object obtained by the addition or removal of a single dot from a picture of a face. Furthermore, anyone who knows about black dots such as those to be found in newspaper photographs knows that

Add to this the uncontroversial

and given that ’picture of a face’ is an observational expression, (D2) comes out true. (’Picture of a face’ is in fact not a particularly good example of an observational expression since factors other that how the thing looks may be relevant to its application, for example, whether a picture in a newspaper actually is a picture of a face, rather than of a face-like cloud formation. But this does not really matter for present purposes.)

So even though the rules specifically governing the expression ’picture of a face’ do not on their own imply the truth of (D2), nevertheless those rules, together with other highly plausible facts and rules concerning the use of observational expressions in general, of which competent speakers ought to be aware, do imply its truth. If, like Platts, one wishes to deny (D2), then one will also have to deny at least one of these further premises. But none of them is any less plausible than (D2) itself.

Similar considerations apply to the paradox of the heap. For again, Platts' argument would force us to admit that

cannot be a rule governing ’heap of sand’ since it makes reference to an exact quantity of grains, and one does not need to grasp the idea of such exact quantities in order to have mastery of ’heap of sand’. Nevertheless, (H1) is true, and we are even more certain of its truth than we are of the truth of (H2). Also, as with (H2), if we were to deny (H1) we would have to deny at least one obvious fact about grains of sand or heaps. I conclude that the fact that the premises of Sorites arguments need not express rules for the use of an expression, E, since they contain references to expressions mastery of which is not required for mastery of E, does not undermine the force of those arguments. As long as there exist unit differences of some sort or other which are too small to matter to the application of E, in the sense that they can never make the difference between E's applying and failing to apply, then the arguments go through (pending, of course, a refutation of them more convincing than Platts' attempt). It just so happens that, for the expression ’heap of sand’, a single grain of sand constitutes such a unit difference (if it didn't, we could choose a smaller unit difference). That fact need not enter into the rules governing ’heap of sand’, but it is nevertheless true, and known by competent speakers to be true.

All that the argument for ’picture of a face’ requires to go through is the truth of (D2) given, of course, the truth of (D1). If the paradoxes are to be solved, and the picture of language as a coherent, rule-governed activity is to be retained, then a conception of rules is required on which the unrestricted truth of the conditional premises of Sorites arguments is not implied.