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that no practical purpose would be served by answering it. At other times, non-philosophers in effect assume without argument a particular treatment of vagueness (not always the same one), without realizing or caring that there are alternatives. The treatment may be good enough for their purposes, or not.

In this case study, our interest in giving a clear and critically reflective answer to a simple, non-technical, non-metalinguistic, nonmetaconceptual question forced us to adjudicate between complex, technical, metalinguistic, and metaconceptual theories. This phenomenon seems to have been overlooked by those who complain about the “arid” technical minuteness of much philosophy in the analytic tradition. A question may be easy to ask but hard to answer. Even if it is posed in dramatic and accessible terms, the reflections needed to select rationally between rival answers may be less dramatic and accessible. Such contrasts are commonplace in other disciplines; it would have been amazing if they had not occurred in philosophy. Impatience with the long haul of technical reflection is a form of shallowness, often thinly disguised by histrionic advocacy of depth. Serious philosophy is always likely to bore those with short attention-spans.¹⁴

      Why should considerations about thought and language play so much more central a role in philosophy than in other disciplines, when the question explicitly under debate is not itself even implicitly about thought or language? The paradigms of philosophical questions are those that seem best addressed by armchair considerations less formal than mathematical proofs. The validity of such informal arguments depends on the structure of the natural language sentences in which they are at least partly formulated, or on the structure of the underlying thoughts. That structure is often hard to discern. We cannot just follow our instincts in reasoning; they are too often wrong (see Chapter 4 for details). In order to reason accurately in informal terms, we must focus on our reasoning as presented in thought or language, to double-check it, and the results are often controversial. Thus questions about the structure of thought and language become central to the debate, even when it is not primarily a debate about thought or language.

The rise of modern logic from Frege onwards has provided philosophers with conceptual instruments of unprecedented power and precision, enabling them to formulate hypotheses with more clarity and determine their consequences with more reliability than ever before. Russell’s theory of descriptions showed vividly how differences between the surface form of a sentence and its underlying semantic structure might mislead us as to its logical relations and thereby create philosophical illusions. The development of formal model-theory and truth-conditional semantics by Tarski and others has provided a rigorous framework for thinking about the validity of our inferences. These theoretical advances have enormous intellectual interest in their own right. They may have made it tempting to suppose that all philosophical problems are problems of language: but they do not really provide serious evidence for that conjecture.

      To deny that all philosophical questions are about thought or language is not to deny the obvious, that many are. We have also seen how in practice the attempt to answer a question which is not about thought or language can largely consist in thinking about thought and language. Some contemporary metaphysicians appear to believe that they can safely ignore formal semantics and the philosophy of language because their interest is in a largely extra-mental reality. They resemble an astronomer who thinks he can safely ignore the physics of telescopes because his interest is in the extra-terrestrial universe. In delicate matters, his attitude makes him all the more likely to project features of his telescope confusedly onto the stars beyond. Similarly, the metaphysicians who most disdain language are the most likely to be its victims. Again, those who neglect logic in order to derive philosophical results from natural science make frequent logical errors in their derivations; their philosophical conclusions do not follow from their scientific premises. For example, some supposed tensions between folk theory and contemporary science depend on fallacies committed in the attempt to draw out the consequences of common sense beliefs.

      Analytic philosophy at its best uses logical rigor and semantic sophistication to achieve a sharpness of philosophical vision unobtainable by other means. To sacrifice those gains would be to choose blurred vision. Fortunately, one can do more with good vision than look at eyes.

Many have been attracted to the idea that all philosophical problems are linguistic or conceptual through the question: if the method of philosophy is a priori reflection, how can it lead to substantive knowledge of the world? Those who find that question compelling may propose that it informs us of relations of ideas rather than matters of fact, or that its truths are analytic rather than synthetic, or that it presents rules of grammar disguised as descriptions, or that its aim is the analysis of thought or language. In short, on this view, philosophical truths are conceptuals truths. We may suspect the presence of empiricist presuppositions in the background – or, as with Ayer, in the foreground. Not starting with such presuppositions, we should be open to the idea that thinking just as much as perceiving is a way of learning how things are. Even if one does not fully understand how thinking can provide new knowledge, the cases of logic and mathematics constitute overwhelming evidence that it does so. The case of the original question, which is philosophical yet queries a theorem of classical logic, shows that we cannot segregate logic from philosophy and claim that armchair thinking illuminates the former but not the latter. In particular, conceptions of logic and mathematics as (unlike philosophy) somehow trivial or non-substantial have not been vindicated by any clear explanation of the relevant sense of “trivial” or “non-substantial.” Whether a given formal system of logic or mathematics is consistent is itself a non-trivial question of logic or mathematics. We know from Gödel’s second incompleteness theorem that the consistency of most standard systems of elementary mathematics cannot be decided in equally elementary mathematics, unless the original system is already inconsistent. The next two chapters investigate in more depth the prospects for conceptual truth and its role in philosophy.

      Notes

      1 ¹ On vagueness in general see, for a start, Graff and Williamson (2002), Keefe (2000), Keefe and Smith (1997), and Williamson (1994a). On vague objects see Williamson (2003b) and references therein.

      2 ² Classical logic is the standard logic of expressions such as “every,” “either … or …” and “not” on the assumption that there is a mutually exclusive, jointly exhaustive dichotomy of sentences into the true and the false.

      3 ³ See also Quine (1970: 11).

      4 ⁴ A recent example of a supervaluationist rejecting such disquotational equivalences for borderline cases is Keefe (2000: 213–20). For further discussion see Williamson (1994a: 162–4) and McGee and McLaughlin (2000).

      5 ⁵ Even if a word retains its linguistic meaning, its reference may shift with the context of utterance (“I,” “now,” “here”). If “dry” undergoes such contextual shifts, T2a and T2b may fail when interpreted as generalizations about utterances of “dry” in contexts other than the theorist’s own. It might be argued that concepts can also undergo contextual shifts in reference: you use the concept I to refer (in thought) to yourself but I use the same concept to refer to myself; at noon we use the concept now to think of noon but at midnight we use the same concept to refer to midnight; at the North Pole we use the concept here to refer to the North Pole but at the South Pole we use the same concept to refer to the South Pole. If so, TC2a and TC2b may also fail when interpreted as generalizations about uses of the concept dry in contexts other than the theorist’s own.

      6 ⁶ For intuitionist logic in general see Dummett (1977). For its application to the problem of vagueness see Graff and Williamson (2002: 473–506) and Chambers (1998).

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