Page 185 - Encyclopedia of Philosophy of Language

to 'define' a real number which is distinct from every 'definable' real number. Berry's paradox (see Russell 1908) used the definition: 'the least integer not name- able in fewer than nineteen syllables,' a definition which itself contains only eighteen syllables. Kurt Grelling created the paradox of heterologicality: he defined a 'heterologicaF predicate as a one-place predicate which is not true of itself (e.g., the predicate 'is long' is heterological, since 'is long' is not long). Grelling then raised the paradoxical problem: is 'is heterological' a heterological predicate? Finally, in 1932, Tarski modified Godel's 1931 proof of the incompleteness theorem (which was itself inspired by Richard's paradox) to create a formal version of Eub- ulides' Liar Paradox, which Tarski used to prove the undefinability of'truth'. Tarski introduced a schema, called Convention T, which constitutes a necessary condition of adequacy for any purported definition of truth: s is true if and only if s (e.g., an instance of this schema would be: 'snow is white' is true if and only if snow is white). Tarski demonstrated that no language can contain a predicate 'true' which makes every instance of convention T true. He did this by con- structing a Liar sentence L which says, in effect, that L is false. Convention T then implies that L is true if and only if L is false.
2. TheSemanticParadoxesasDiagonalArguments
In 1891, Cantor developed an argument known as the diagonal argument. The argument includes a method for constructing, given an infinite list of infinite sequences, an infinite sequence which does not belong to the list. The two-dimensional array which results from arranging the infinite sequences one after the other is examined below. Then the diagonal of this array is looked at, and a sequence is constructed which differs from the diagonal at every step. The resulting sequence will be different from every sequence on the original list, since it will differ with each such sequence at at least one point: the point at which the diagonal crosses the sequence. The relevance of this con- struction to the semantic paradoxes can be illustrated by means of Grelling's heterological paradox. First, all of the one-place predicates expressible in English are placed in some fixed order. The predicates are then arranged both along the top and the left-hand side of a two-dimensional array. At each point in the array, a T is placed if the predicate of the row is true of the predicate of the column, otherwise an F is inserted. The predicate 'is heterological' can now be defined by reference to the diagonal of the array: if the nth row has a T in the nth column, then the nth predicate is not heterological, so the predicate 'is heterological' gets an F in the nth column. Similarly, if the nth row has an F in the nth column, then 'is heterological' gets a T in the nth column. By Cantor's argument, 'is heterological' cannot appear in the list of predicates, that is, it is not a predicate expressible in English. Yet,
this cannot be, since we have in fact so expressed it (see Simmons 1990).
3. AvoidingtheParadoxesinFormalLanguages: Type Theory
Both Bertrand Russell and Tarski recommended that mathematicians work in a rigidly typed formal language which avoids the semantical paradoxes. Rus- sell called his language the language of 'ramified type theory'. Tarski proposed the distinction between 'object language' and 'meta language'. Alonzo Church (1979) demonstrated that Tarski's distinction is implied by Russell's type theory. According to Tarski, the semantic theory for a language L (the object language) must be carried out in a distinct language L' (the metalanguage). Thus, the predicates 'is a true sentence of V or 'is a heterological predicate of U cannot be expressed in L itself but only in a distinct metalanguage for L. If one wishes to develop a sem- antic theory for the metalanguage L', one must do this in yet another language L>, the meta- metalanguage. This series of increasingly powerful languages, each with the capacity for expressing the semantic theories of its predecessors, is known as 'the Tarskian hierarchy'.
4. SemanticParadoxinNaturalLanguage: Soluble or Insoluble?
Tarski characterized natural languages as incon- sistent, since they plainly violate his principle of the distinctness of object language and metalanguage. According to Tarski, natural languages purport to be universal, to be able to express anything which can be expressed. Tarski believed that the semantic para- doxes demonstrate that no language can in fact be universal. Consequently, formal semantics cannot be carried out in a natural language like English, for if it could, then the semantics for any language (including English itself) could be carried out in English, causing us to run afoul of the semantic paradoxes. Further- more, no fully satisfactory semantics for English as a whole can be given (in any language), since the sem- antic rules for words like 'true' or 'definable' implicit in ordinary practice are logically inconsistent. Thus, Tarski held that the semantic paradoxes constitute an insoluble problem for the semantics of natural language. Defenders of this view in the 1980s and 1990s, such as Anil Gupta and Stephen Yablo, study the semantic paradoxes in order to describe math- ematically the incoherences and instabilities of natural language, and to diagnose exactly how natural language goes wrong (see Gupta 1982; Yablo 1985).
Beginning in the 1970s, a number of proposals have been made to solve the semantic paradoxes in natural languages. According to these proposals, it is possible to consistently assign semantic values of some kind to the sentences of natural language, including instances of paradoxes like the Liar. It is claimed that properly
Paradoxes, Semantic
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