Williamson
Vagueness
The difference between artificial and natural languages stands out sharply in The Sound of Music, when Julie Andrews tries to teach her pupils solfeggio (do, re, mi...) But it doesn't mean anything, one of them protests. That's why we put words into it, says Andrews, and bursts into song: Do, a deer, a female deer, Re, a drop of golden sun, Mi, a name I call myself, Fa, a long, long way to run... In Vagueness, Timothy Williamson shows the kind of significance problems created when we put words into it, i.e., when we apply the rules of perfect languages, such as formal logic, to the vague concepts of everyday life. Using the classic Sorites Paradoxes – 'The Bald Man' and 'The Heap' – he reviews the countless attempts that have been made to resolve the conflict between logic and natural language. 'The Heap', for example: if we assume that one grain of wheat is not a heap, and that the addition of one more makes hardly any difference, then we are ultimately forced to admit that ten thousand do not form a heap either. If one doesn't form a heap, then two don't either, and if two don't, then three don't, etc. ad infinitum. The problem here would seem not be the rules of derivation, but the vagueness of the word 'heap'. There is no number n of grains which can be considered to definitely constitute a heap; n +1 grains still definitely do not form a heap. For various quantities n, the question: do n grains form a heap? cannot be answered yes or no. Williamson's book is a defence of the stoic view that such vagueness is an epistemological phenomenon: there is indeed a clear boundary between what is and is not a heap of wheat, but users of the language cannot determine exactly where it is.
The Sorites paradoxes sank into relative obscurity after ancient times. Only at the end of the nineteenth century did the concept of 'vagueness' acquire a technical philosophical meaning. It was confined to cases of 'blurred boundaries': the vague boundaries between 'little' and 'much', 'bald' and 'hairy', between a heap or no heap. Especially for analytical philosophers, with their interest in formal logic, the correctness of the proof in the Sorites paradoxes creates a problem: if the premises are acceptable, the bizarre conclusion cannot be avoided. Frege believed that vagueness, like madness, could only be designated in order to subsequently exclude the phenomenon from the perfect language, but Pierce and Russell attached more theoretical value to the concept of vagueness: no language is precise. For Russell, vagueness was a characteristic of representation. Therefore, the concept should not be projected onto reality. Idealism is disposed of in a few words: It is thought that there must be some kind of identity between the knower and the known, and hence the knower infers that the known also is muddle-headed. The systematic description of vagueness really only got rolling and took shape in the Black-Hempel debate. Black had defined vagueness as a consistency profile, a function of the use of words in everyday language. Hempel countered that exclusive attention to the use of a term abstracts from the syntax and semantics that play a determining role in language. According to Hempel, Black's use of the concept of vagueness had no relevance for logic. Black later conceded this methodological point: When logical puzzles exploit vagueness, they are to be solved not by the choice of some particular non-classical system of logic appropriate to vagueness, but by a better understanding of the very direct relation between logical systems and linguistic practice. For Williamson, this insight was of crucial importance for research into vagueness and he harshly concludes: If that is the specific moral of the Black-Hempel debate, it has not been widely learned.
The philosophy of vagueness took shape soon after in the development of a 'surplus value logic'. The principle of 'two-valuedness' teaches that a pronouncement is either true or false; vague concepts seem to demand a classification system with more than two categories: three-valued logic, continuum-valued logic, fuzzy logic. The problem with such research is that in practice, it continues to take classic meta-logic as its point of departure. And where a complete and sound system of axioms and derivation rules for a continuum-valued logic is sought, it turns out that completeness and soundness can only be obtained for fragments of such a logical system. Classical logic would seem simpler and more attractive in all respects.
In a separate chapter, Williamson describes the method of supervaluations: a method that combines various 'precisely made' semantic descriptions (interpretations) of a language into a semantic description of the originally vague language. A pronouncement is called 'supertrue' when it is true in all accepted interpretations and 'super untrue' when untrue in all accepted interpretations. A new operator – 'definitely' – can be added to classical predicate logic to express supertruth. 'Definitely a' then is only true when a is supertrue: Definitely can be given a formal semantics very like the possible world semantics for the modal operator necessarily. Williamson disposes of the supervaluation method in the same rigorous fashion as all other attempts to use a precise meta-language to provide vague languages with a formal semantic description. All such attempts are doomed to failure, because we cannot express in a precise meta-language what is said by pronouncements in the vague object-language, as in order to do so, we would have to speak vaguely. These limitations to expression make such a meta-language incapable of conveying the meaning of pronouncements in object language; thus, it can hardly be considered suitable for a real semantic description of an object language. The dream of a perfect, precise meta-language is a false one; we must accept that our assignment of truth and untruth contains an element of vagueness. After hundreds of pages of finely-honed argument against more and less sophisticated attacks on the principle of two-valuedness, the conclusion is blindingly simple: It is incoherent to suppose that vague utterances in borderline cases both say something and fail to be either true or false. It is coherent to suppose them to be neither true nor false only at the cost of treating them as though they said nothing. Formal semantics pays the cost by affecting to use a precise meta-language in which one cannot say what utterances in the vague object language mean. Since we are inescapably committed to the practice of using vague language, we cannot permanently afford that price.' Vagueness (here Williamson has returned to the Stoa) is a distinct species of the genus inexact knowledge.
The theories that are refuted in Vagueness are all a great deal more exciting than the theory that remains in the end, especially because the latter is so convincing. Balanced against the disadvantage of seeming unreasonableness, Williamson's theory has the advantage of according an important role to everyday language use. When we are confronted with the accusation that the meaning of a pronouncement has not been established clearly when we put words into it, Williamson's Epistemicism can come to our aid in the same way as Humpty Dumpty's Nominalism: The question is, said Alice, whether you can make words to mean so many different things. The question is, said Humpty Dumpty which is to be master – that's all.
