semantic paradoxes a collection of paradoxes involving the semantic notions of truth, predication, and definability. The liar paradox is the oldest and most widely known of these, having been formulated by Eubulides as an objection to Aristotle’s correspondence theory of truth. In its simplest form, the liar paradox arises when we try to assess the truth of a sentence or proposition that asserts its own falsity, e.g.: (A) Sentence (A) is not true. It would seem that sentence (A) cannot be true, since it can be true only if what it says is the case, i.e., if it is not true. Thus sentence (A) is not true. But then, since this is precisely what it claims, it would seem to be true. Several alternative forms of the liar paradox have been given their own names. The postcard paradox, also known as a liar cycle, envisions a postcard with sentence (B) on one side and sentence (C) on the other: (B) The sentence on the other side of this card is true. (C) The sentence on the other side of this card is false. Here, no consistent assignment of truth-values to the pair of sentences is possible. In the preface paradox, it is imagined that a book begins with the claim that at least one sentence in the book is false. This claim is unproblematically true if some later sentence is false, but if the remainder of the book contains only truths, the initial sentence appears to be true if and only if false. The preface paradox is one of many examples of contingent liars, claims that can either have an unproblematic truth-value or be paradoxical, depending on the truth-values of various other claims (in this case, the remaining sentences in the book). Related to the preface paradox is Epimenedes’ paradox: Epimenedes, himself from Crete, is said to have claimed that all Cretans are liars. This claim is paradoxical if interpreted to mean that Cretans always lie, or if interpreted to mean they sometimes lie and if no other claim made by Epimenedes was a lie. On the former interpretation, this is a simple variation of the liar paradox; on the latter, it is a form of contingent liar. Other semantic paradoxes include Berry’s paradox, Richard’s paradox, and Grelling’s paradox. The first two involve the notion of definability of numbers. Berry’s paradox begins by noting that names (or descriptions) of integers consist of finite sequences of syllables. Thus the three-syllable sequence ‘twenty-five’ names 25, and the seven-syllable sequence ‘the sum of three and seven’ names ten. Now consider the collection of all sequences of (English) syllables that are less than nineteen syllables long. Of these, many are nonsensical (‘bababa’) and some make sense but do not name integers (‘artichoke’), but some do (‘the sum of three and seven’). Since there are only finitely many English syllables, there are only finitely many of these sequences, and only finitely many integers named by them. Berry’s paradox arises when we consider the eighteen-syllable sequence ‘the smallest integer not nameable in less than nineteen syllables’. This phrase appears to be a perfectly well-defined description of an integer. But if the phrase names an integer n, then n is nameable in less than nineteen syllables, and hence is not described by the phrase. Richard’s paradox constructs a similarly paradoxical description using what is known as a diagonal construction. Imagine a list of all finite sequences of letters of the alphabet (plus spaces and punctuation), ordered as in a dictionary. Prune this list so that it contains only English definitions of real numbers between 0 and 1. Then consider the definition: ‘Let r be the real number between 0 and 1 whose kth decimal place is ) if the kth decimal place of the number named by the kth member of this list is 1, and 0 otherwise’. This description seems to define a real number that must be different from any number defined on the list. For example, r cannot be defined by the 237th member of the list, because r will differ from that number in at least its 237th decimal place. But if it indeed defines a real number between 0 and 1, then this description should itself be on the list. Yet clearly, it cannot define a number different from the number defined by itself. Apparently, the definition defines a real number between 0 and 1 if and only if it does not appear on the list of such definitions. Grelling’s paradox, also known as the paradox of heterologicality, involves two predicates defined as follows. Say that a predicate is ‘autological’ if it applies to itself. Thus ‘polysyllabic’ and ‘short’ are autological, since ‘polysyllabic’ is polysyllabic, and ‘short’ is short. In contrast, a predicate is ‘heterological’ if and only if it is not autological. The question is whether the predicate ‘heterological’ is heterological. If our answer is yes, then ‘heterological’ applies to itself – and so is autological, not heterological. But if our answer is no, then it does not apply to itself – and so is heterological, once again contradicting our answer. The semantic paradoxes have led to important work in both logic and the philosophy of language, most notably by Russell and Tarski. Russell developed the ramified theory of types as a unified treatment of all the semantic paradoxes. Russell’s theory of types avoids the paradoxes by introducing complex syntactic conditions on formulas and on the definition of new predicates. In the resulting language, definitions like those used in formulating Berry’s and Richard’s paradoxes turn out to be ill-formed, since they quantify over collections of expressions that include themselves, violating what Russell called the vicious circle principle. The theory of types also rules out, on syntactic grounds, predicates that apply to themselves, or to larger expressions containing those very same predicates. In this way, the liar paradox and Grelling’s paradox cannot be constructed within a language conforming to the theory of types. Tarski’s attention to the liar paradox made two fundamental contributions to logic: his development of semantic techniques for defining the truth predicate for formalized languages and his proof of Tarski’s theorem. Tarskian semantics avoids the liar paradox by starting with a formal language, call it L, in which no semantic notions are expressible, and hence in which the liar paradox cannot be formulated. Then using another language, known as the metalanguage, Tarski applies recursive techniques to define the predicate true-in-L, which applies to exactly the true sentences of the original language L. The liar paradox does not arise in the metalanguage, because the sentence (D) Sentence (D) is not true-in-L. is, if expressible in the metalanguage, simply true. (It is true because (D) is not a sentence of L, and so a fortiori not a true sentence of L.) A truth predicate for the metalanguage can then be defined in yet another language, the metametalanguage, and so forth, resulting in a sequence of consistent truth predicates. Tarski’s theorem uses the liar paradox to prove a significant result in logic. The theorem states that the truth predicate for the first-order language of arithmetic is not definable in arithmetic. That is, if we devise a systematic way of representing sentences of arithmetic by numbers, then it is impossible to define an arithmetical predicate that applies to all and only those numbers that represent true sentences of arithmetic. The theorem is proven by showing that if such a predicate were definable, we could construct a sentence of arithmetic that is true if and only if it is not true: an arithmetical version of sentence (A), the liar paradox. Both Russell’s and Tarski’s solutions to the semantic paradoxes have left many philosophers dissatisfied, since the solutions are basically prescriptions for constructing languages in which the paradoxes do not arise. But the fact that paradoxes can be avoided in artificially constructed languages does not itself give a satisfying explanation of what is going wrong when the paradoxes are encountered in natural language, or in an artificial language in which they can be formulated. Most recent work on the liar paradox, following Kripke’s ‘Outline of a Theory of Truth’ (1975), looks at languages in which the paradox can be formulated, and tries to provide a consistent account of truth that preserves as much as possible of the intuitive notion. See also SET-THEORETIC PARADOXES , TRUTH , TYPE THEOR. J.Et.