Introduction

There is a familiar and ancient Greek paradox called, variously, the Sorites (the Paradox of the Heap), and the Phalakros (the Paradox of the Bald Man). The discovery of the paradox - or, strictly speaking, family of paradoxes - is attributed to the logician Eubulides of Miletus, a contemporary of Aristotle, who was also responsible for various other puzzles, the most famous of which being the Liar - if a man says that he is lying, is he telling the truth?2

The Sorites was originally formulated in terms of a series of questions: Does one grain of sand make a heap? Do two grains of sand make a heap? Do three, ten or ten thousand grains make a heap, and so on. If you admit that one grain does not make a heap, and are unwilling to complain about the addition of any single grain, you are eventually forced to admit that ten thousand grains do not make a heap. The same argument can of course be run in reverse: if you admit that ten thousand grains make a heap, and are unwilling to say that the subtraction of any single grain can make a difference to its being a heap, then you are eventually forced to admit that one or even zero grains make a heap. Taking both of these arguments together, we reach the conclusion that one and ten thousand grains both do, and do not, make a heap - a straightforward contradiction.

The Sorites was fairly widespread in antiquity, and was used particularly by the Sceptics of Plato's Academy in attacking Stoic epistemology. However, for the next thousand years or so, the paradox was discussed little, at least not for sceptical purposes, although it is used by both Leibniz, in his criticism of Locke's theory of natural kinds, and by Hegel, in his ideas about the relationship between the qualitative and the quantitative.

These days - and in fact from quite early on - the Sorites is usually formulated not as a series of questions but as a paradoxical argument. First, there is a premise which everyone will agree to, such as

(1) a single grain of sand does not make a heap;

then a universally quantified conditional premise is introduced:

(2) For any n, if n grains of sand do not make a heap, then n + 1 grains of sand do not make a heap.

Again, this would appear to be uncontroversial, since the addition of a single grain of sand cannot, we intuitively think, make the difference between something's not being a heap and its being one. In Classical two-valued logic, at least, it follows inescapably from (1) and (2) that

(3) For any n, n grains of sand do not make a heap,

whether n be ten or ten million.

It might be thought, at this point, that all that this argument shows is that Classical two-valued semantics is flawed and should be rejected on, for example, Intuitionistic grounds. We might, that is, accept premise (1), but refuse to accept that the second, conditional, premise must be determinately either true or false. But this reply will not work, since the conditional premise can easily be replaced by a finite series of conditionals containing no quantifiers at all: if one grain isn't a heap, then two grains aren't; if two grains aren't, then three aren't, and so on. Since premise (1) is incontrovertibly true, and the conclusion (if n is, say, a million) incontrovertibly false, the challenge now is to say which is the first false (or, at least, not clearly true) conditional in the series.

But what, if anything, do such arguments show? Are they merely interesting puzzles which can be simply dismissed, or do they run deeper? It is notable that only two traditions in the history of philosophy have regarded Sorites paradoxes as a serious threat: one culminated in Stoicism; the other is modern analytic philosophy, from Frege onwards,3 and in the last quarter of a century or so a steady stream of literature on vagueness and the Sorites has appeared.4

Unfortunately, none of the treatments so far proposed have been entirely satisfactory. As Crispin Wright has put it,

In comparison with what is needed, it is not unduly harsh to say that much of the literature on this topic . . . has amounted to little more than tinkering. Writers have been content to devise more or less ad hoc semantics in whose terms the major premises fail of strict truth without doing anything to disclose why their plausibility is specious, still less confront the intuitively powerful arguments on their behalf. It should go without saying that the solution of a paradox requires more than designing wallpaper.5

Any adequate solution must, therefore, take Sorites arguments seriously, and proposing revisions to Classical logic in order to 'handle' vague expressions does not explain the phenomenon of vagueness but merely redescribes it. Vagueness should be accepted as a genuine, important and ineradicable feature of natural language, not as a defect, something to be ignored, eliminated or accommodated ad hoc as part of some wider semantic theory. In particular, it should be borne in mind that

the meaning of vague expressions can be stated only in a language into which those expressions can be translated; it is a mistake to treat the language in which one theorizes about vagueness as though it were precise.6

In addition, any such account should be genuinely explanatory of our mastery of vague expressions, as well as of their meanings.

What follows may be read as an extended response to Wright's 1987 article 'Further Reflections on the Sorites Paradox', which contains the most compelling argument yet put forward against the supposition that our use of vague observational expressions is governed by substantial and consistent rules for their correct use. Wright's approach thus involves construing the premises of Sorites arguments (such as (1) and (2), above) as formulations of semantic rules governing the use of such expressions, and it is this approach which I shall adopt here.7 The real significance of Wright's work on vagueness has not, I think, been widely enough appreciated and if, as I suspect, his arguments against the Governing View8 will generalize to non-observational expressions, then we are in urgent need of an adequate resolution of the Sorites which is also consistent with the supposition that the expressions of our language are governed by substantial rules which competent speakers genuinely and consistently follow. One of the main aims of this thesis is to indicate how such a resolution might be possible.

My strategy is as follows. In Chapter 1 I clarify the philosophical notion of vagueness and distinguish it from other, related notions with which it is easily confused; I also explain why Wright rejects the Governing View and why he thinks that the supposition that observational expressions such as colour words are governed by substantial rules for their correct use leads to versions of the Sorites paradox; some responses to the paradox are also discussed and criticized. In Chapter 2 I present a detailed critique of Wright's own preferred approach to the problem, which involves dropping the assumption that observational terms are rule-governed and replacing it with a more behaviouristic or statistical conception of language mastery. Such attempts, I argue, must fail, and in Chapter 3 I examine the nature of rules in general, and semantic rules in particular, and try to show that we cannot afford to dispense with the notion of a semantic rule in any adequate account of human linguistic ability. Chapter 4 criticizes a recent, ingenious attempt to solve the Sorites put forward by Linda Burns, and introduces further paradoxical arguments, this time relating to the notions of judgement and justification, which also, it is argued, pose problems for non-observational expressions. So we arrive at an impasse, which I attempt to overcome in Chapter 5 by suggesting that, if we heed what the later Wittgenstein had to say about rules, rule-following and justification, the supposition that language use is rule-governed is not, contra Wright, after all paradoxical, even for the 'problematic' class of observational expressions. Accepting this conclusion does, however, involve abandoning a certain dominant, rationalistic conception of the foundations of language, and of the nature of language mastery. Finally, in Chapter 6, I offer an interpretation of Wittgenstein's use of the term 'criterion', a species of rule which plays a pivotal role in his later philosophy of language, and show how a number of rival interpretations misrepresent Wittgenstein's distinctive philosophical methodology. I also show how criteria are connected with, and shed considerable light on, his general approach to language and rules. A comparison of his use of 'criterion' with ordinary or non-philosophical uses of the term is given as an Appendix.

1 Wittgenstein [1953], §98.

2 For a fascinating survey of the early history of Sorites paradoxes, see Williamson [1994], Chapter 1.

3 See ibid., p. 36.

4 Williamson [1994] contains an exhaustive bibliography on the subject.

5 Wright [1987], p. 282.

6 Williamson [1994], p. xi.

7 Since my concern, like Wright's, is thus primarily with language, I shall not specifically deal with the question of whether the world itself might be vague, or whether there can be vague objects. Since the publication of a note by Evans [1978] there has been a deluge of literature on this subject (see, e.g., Salmon [1982], pp. 243-6, Noonan [1982], [1984] and [1990], Broome [1984], Wiggins [1986], Rasmussen [1986], Cook [1986], Garrett [1988] and [1991], Lewis [1988], Burgess [1989], Johnsen [1989], Over [1989], Tye [1990], Zemach [1991] and Williamson [1994], Chapter 9). I am, however, sympathetic to Mark Sainsbury's [1995] contention that objects may be said to be vague only in virtue of the fact that the expressions used to refer to them are, i.e. there is no further, substantive thesis of vague objects.

8 See below, §1.2.