1 Chapter 1 Two Versions of the Liar Paradox The Liar paradox is the most widely known of all philosophical conundrums. Its reputation stems from the simplicity with which it can be presented. In its most accessible form, the Liar appears as a kind of parlor game. One imagines a multiple choice questionnaire, with the following curious entry: * This sentence is false. The starred sentence is a) True b) False c) All of the above d) None of the above. There follows a bit of informal reasoning. If a) is the right answer, then the sentence is true. Since it says it is false, if it is true it must really be false. Contradiction. Ergo, a) is not the right answer. If b) is the right answer, then the sentence is false. But it says it is false, so then it would be true. Contradiction. Ergo, b) is not the right answer. If both a) and b) are wrong, surely c) is. That leaves d). The starred sentence is neither true nor false. The conclusion of this little argument is somewhat surprising, if one has taken it for granted that every grammatical declarative sentence is either true or false. But the reasoning looks solid, and the sentence is a bit peculiar anyhow, so the best advice would seem to be to accept the conclusion: some sentences are neither true nor false. This conclusion will have consequences when one tries to formulate an explicit theory of truth, or an explicit 2 semantics for a language. But it is not so hard after all to cook up a semantics with truth-value gaps, or with more than the two classical truth values, in which sentences like the starred sentence fail to be either true or false. From this point of view, there is nothing deeper in the puzzle than a pathological sentence for which provision must be made. In more sophisticated parlors, the questionnaire is a bit different: * This sentence is not true. The starred sentence is a) True b) False c) All of the above d) None of the above. By a similar piece of reasoning we are left with d) again, but our conscience is uneasy. If the answer is "none of the above", then a) is not right, but if a) is not right, then the sentence is not true, but that is just what the starred sentence says, so the starred sentence is true and a) is right after all. But if a) is right, then the sentence is true, but it says it isn't true, so.... Problems become yet more acute with a third questionnaire: * This sentence is not true. The starred sentence is a) True b) Not True c) All of the above d) None of the above. Both the first and second answers lead straight to contradictions, and so can be ruled out by reductio. This leaves d) again, but the intrinsic tenability of d) is suspect. Maintaining that a sentence is neither true nor false appears to be a 3 coherent option: one might, for example, assert that nonsense sentences, or ungrammatical strings, or commands, are neither. Falsity is, as it were, a positive characteristic, to which only appropriately constructed sentences can aspire (prefixing "It is not the case that.." to a false sentence should yield a true sentence). But "Not True" takes in even ungrammatical strings: it is merely the complement class of "True". Surely every sentence, ungrammatical or not, either falls within the extension of "True" or outside it, so how could d) even be an option? The only available alternative is that the extension of "True" is somehow not well defined, that "True" is a vague predicate with an indeterminate boundary, and that the starred sentence falls in the region of contention. But this does not explain how there could be apparently flawless arguments demonstrating that the sentence cannot consistently be held to be true or not true. For other vague predicates, the opposite obtains: a borderline bald person can consistently be regarded either as bald or not. What conclusion is one to draw from these three puzzles? Perhaps nothing more than that these sentences are peculiar, and that, if we trust the first argument, we should make room in our semantics for sentences which are neither true nor false. Let the final semantics determine their ultimate fate: since we have no coherent strong intuition about what to do with them, it is a clear case of spoils to the victor. In many semantic schemes, the Liar sentence, in both its forms, falls into a truth value gap. Perhaps there is no more to be said of it. This rather benign assessment of the Liar is belied as soon as one turns to the locus classicus of serious semantics. At the end of the first section of "The Concept of Truth in Formalized Languages", Alfred Tarski arrives at a stark and entirely negative assessment of the problem of defining truth in a language which contains its own truth predicate. The allegedly insuperable 4 difficulties are motivated by the antinomy of the Liar, leading Tarski to this conclusion: If we analyse this antinomy in the above formulation we reach the conviction that no consistent language can exist for which the usual laws of logic hold and which at the same time satisfies the following conditions: (I) for any sentence which occurs in the language a definite name of this sentence also belongs to the language; (II) every sentence formed from (2) [i.e. "x is a true sentence if and only if p"] by replacing the symbol 'p' by any sentence of the language and the symbol 'x' by a name of this sentence is to be regarded as a true sentence of this language; (III) in the language in question an empirically established premise having the same meaning as (a) [i.e. the sentence which asserts that the denoting term which occurs in the Liar sentence refers to the sentence itself] can be formulated and accepted as a true sentence. Tarski 1956, p. 165 Tarski assumed that all natural ("colloquial") languages satisfy conditions (I) and (III), and also that any acceptable account of a truth predicate must entail (II), at least in the metalanguage. Consequently, no natural language which serves as its own metalanguage can consistently formulate its own account of truth. If one adds to this the claim that every user of a natural language must regard the T-sentences formulable in that language as true (whether there exists any explicit "theory of truth" or not), one arrives at the striking conclusion that every user of a natural language who accepts all the classical logical inferences must perforce be inconsistent. 5 This last conclusion has not met with universal approbation. Robert Martin, for example, remarks: In at least three places Tarski argues essentially as follows: Every language meeting certain conditions (he speaks earlier of universality, later of semantic closure), and in which the normal laws of logic hold, is inconsistent. Although Tarski acknowledges difficulties in making precise sense in saying so, it is fairly clear that he thinks that colloquial English meets the conditions in question, and is in fact inconsistent. This argument has made many philosophers, including, perhaps, Tarski himself, quite uncomfortable. It is not clear even what it means to say that a natural language is inconsistent; nor is one quite comfortable arguing in the natural language to such a conclusion about natural language. Martin 1984, p. 4 Martin goes on to suggest that Tarski ought to have drawn a different conclusion, viz. that the concept of truth is not expressible in any natural language (Ibid. p. 5), and hence, presumably, that speakers of only natural languages either have no concept of truth, or else are unable to express it. This is surely as odd a conclusion as Tarski's. If one has no such concept available before constructing a formalized language, what constraints are to guide the construction? It is rather like saying that although there is no concept expressed by "Blag" in English, one can set about constructing a theory of Blag in a formalized language, which theory can somehow be judged intuitively as either right or wrong. On the other hand, if speakers of natural language do have a concept of truth, why can't they express it simply by introducing a term for it (e.g. "truth")? Expressing a concept does not require offering an analysis of it in other terms. 6 Pace Martin, Tarski's claim can be made quite precise, and anyone attempting to construct a theory of truth for natural languages must be prepared to answer it. There is, indeed, a swarm of theories of truth designed to apply to languages which contain their own truth predicate, including, for example, Bas van Fraassen's supervaluation approach [1968, 1970], Saul Kripke's fixed-point theory [1975], and Anil Gupta's revision theory [1982]. At first glance, it even seems obvious how these theories will block Tarski's conclusion. But on second glance, matters become rather more complicated, and it is not so clear after all how Tarski's objections are to be met. Fully responding to Tarski's concerns will lead us to a slightly different theory of truth than the aforementioned, one more adequate to our actual practice of reasoning about truth. Our first task, then, is to make Tarski's argument explicit, and take the first and second glances. The Liar in Language L Consider the simplest possible language in which the Liar paradox can be framed, which language we will call L.