The Cretan philosopher Epimenides said that Cretans always lie. Assuming, for the sake of argument, the mendacity of all other statements by Cretans, we get a paradox: if what Epimenides said was true, it must have been a lie, whereas if what he said was a lie, it would have made his statement true. The citizens of Crete have long since forgiven the insult, but semantics has never recovered.
Alfred Tarski perceived the consequences of Epimenides’ paradox with particular clarity. Our common-sense intuitions about truth follow the paradigm: ‘Snow is white’ is true if and only if snow is white. As Tarski rigorously shows, if the language we are describing (the object language) is the same as the language in which we are formulating our theory (the metalanguage), this paradigm will be inconsistent with the rudimentary laws of syntax. The conclusion Tarski drew was that, if we are to develop a satisfactory theory of truth, our metalanguage must be essentially richer in expressive power than the object language. Since there is no human language essentially richer than English (or any other natural language), there can be no satisfactory theory of truth for English.
One earnestly hopes that this is not the end of the matter. Tarski’s analysis leaves open the prospect that we can develop a fully satisfactory theory of truth for a substantial fragment of English; also the prospect that we can develop a theory of truth for English as a whole which, while not fully satisfying our intuitions, is none the less useful and illuminating.
Both prospects have been substantially advanced by Saul Kripke’s ‘Outline of a Theory of Truth’, which exploits the idea that there are truth-value gaps, statements which are neither true nor false, and that Epimenides’ insult was one of them.
Invocation of truth-value gaps does not resolve the paradox in any straightforward way. If we let the phrase ‘the simple liar sentence’ refer to the sentence ‘The simple liar sentence is false’, we see that we can readily account for the paradoxical features of the sentence by declaring the sentence neither true nor false; but if we let the strengthened liar sentence be ‘The strengthened liar sentence is not true’, we get a sentence we cannot dispose of so tidily. If the strengthened liar is neither true nor false, then it is not true; but that it is not true is precisely what the sentence says.
Truth-value gaps have not vanquished the liar paradox. Nor have any of the alternatives, the most prominent of which are a contextualist account, which sees the English word ‘true’ as radically ambiguous, and so-called ‘revision theory’, which investigates the cyclic reasoning that occurs when we try to evaluate the simple liar sentence: if the sentence is true, then it must be false; but if, then, it is false, it must be true; and so on. While these approaches have not eliminated the paradox, they have opened new approaches that have greatly improved our prospects for finding a comfortable way to live with it.
A Tale of Excluding the Middle
