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Tarski, Frege and the Liar Paradox

Published online by Cambridge University Press:  25 February 2009

Sloman Aaron
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A.1. Some philosophers, including Tarski and Russell, have concluded from a study of various versions of the Liar Paradox ‘that there must be a hierarchy of languages, and that the words “true” and “false”, as applied to statements in any given language, are themselves words belonging to a language of higher order’. In his famous essay on truth Tarski claimed that ‘colloquial’ language is inconsistent as a result of its property of ‘universality’: that is, whatever can be said at all can in principle be said in it, with an extended vocabularly if necessary. Thus, in English we can talk about English expressions, what they denote, what they say, whether what they say is true or false, and so on: English contains its own metalanguage. This universality enables us to construct sentences which say of themselves that they are false, and by applying the law of excluded middle to them we easily derive a contradiction. Tarski concludes that ‘these antinomies seem to provide a proof that every language which is universal in the above sense, and for which the normal laws of logic hold, must be inconsistent’ (op. cit., pp. 164—5). He then proposes to avoid such contradictions by the use of a hierarchy of languages such that statements about any one language can be made only in a different language at a higher level.

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Articles
Information
Philosophy , Volume 46 , Issue 176 , April 1971 , pp. 133 - 147
Copyright
Copyright © The Royal Institute of Philosophy 1971

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References

1 Russell, B., An Inquiry into Meaning and Truth, Pelican Books, 1962, p. 17.Google Scholar

2 Tarski, A., ‘The concept of truth in formalized languages’, translated in Logic, Semantics, Metamathematics, Clarendon Press, 1955Google Scholar. All page references are to this article. Readers are advised to acquaint themselves either with the (non-technical) introduction and first section, or else with Tarski's shorter essay ‘The Semantic conception of truth’, in Feigl, H. and Sellars, W. (eds.), Readings in Philosophical Analysis, Appleton-Century-Crofts, 1949.Google Scholar

3 This article was partly stimulated by a paper on ‘Paradoxicality’ by L. Hollings, who also helped by criticizing an earlier version, as did C. J. F. Williams, P. Williams, N. Everitt and Carolyn Stone. Hollings has attempted to carry out the programme mentioned in my concluding paragraph. Unfortunately his paper is not yet published.

4 G. Frege, ‘On sense and reference’ and other papers in Translations from the Philosophical Writings of Gottlob Frege, by P. Geach and M. Black, Basil Blackwell, 1960. Some aspects of his theory are elaborated further in ‘The Thought, A Logical Enquiry’, translated in Mind, 1956 and in Philosophical Logic, edited by P. F. Strawson, O.U.P., 1967. See especially pages 62–5 of Translations.

5 This terminology is suggested by Hollings. See note 3.

6 In ‘Functions and rogators’, in Formal Systems and Recursive Functions, edited by J. N. Crossley and M. A. E. Dummett, North Holland, 1965. The terminology of this paper was unfortunate. The basic aim was to show that the concept of a rule or principle of correlation does for function-signs what Frege's concept of ‘sense’ does for names. In the present article I shall not use ‘function’ to refer to a set of ordered pairs.

7 For example, see pp. 63, 159, 167 of Translations.

8 Dummett, M. A. E., ‘Truth’, in Proc. Aristotelian Soc, 1958–1959Google Scholar, reprinted in Truth, edited by G. Pitcher and in Philosophical Logic, edited by P. F. Strawson. The situation is a bit more complicated than may appear at first, depending on how entailment is defined. For if to say that p entails q is to say that all the truth-conditions of p are included among truth-conditions of q, and if ‘not’ simply exchanges truth-conditions and falsity-conditions, then (a) p can entail q without not-q entailing not-p (though ‘q is not true’ must entail ‘p is not true’), and (b) not-p is not equivalent to ‘p is false’.

9 The outlines of an analysis can be found in section B above, especially B.9–10. Dummett (see note 8) has criticised Frege's theory for failing to explain the asymmetry between truth and falsity, and this criticism would apply equally to my extension of Frege's theory. The reply is (a) that in most of their logical and semantic properties truth and falsity are perfectly symmetrical, and (b) that the lack of symmetry can be completely explained in terms of a pragmatic or communicative convention that, except in special contexts, a complete sentence may be uttered only if that sentence corresponds to the value T. The opposite convention would generate a language grammatically identical, but with every sentence expressing the contradictory of what we understand by it. The whole syntactic and semantic apparatus described in this paper could be embedded in a kind of game in which no assertions are made and in which T and F were perfect duals, provided the above conventions were not involved. It is even arguable that such a game could be learnt by people who had never learnt a language in which true and false assertions could be made. There are further uses of the words ‘true’ and ‘false’ not accounted for in the main text or in these remarks.

10 Since writing this article, I have learnt through Robin Stanton that ideas similar to mine have been developed in connection with computer languages. (See Gorn's, Saul ‘The identification of the computer and information sciences: Their fundamental semiotic concepts and relationships’ in Foundations of Language, Vol. 4, No. 4, Nov. 1968Google Scholar.) Some of the ideas of B.9–10, above, are also closely related to, though developed independently of, Kripke's, Saul ‘Semantical considerations on modal logic’, in Acta Philosophica Fennica, Fasc. XVI, 1963Google Scholar. Kripke's discussion is much more extensional: he is apparently not concerned with identification procedures. I am currently trying to make both discussions more ‘realistic’ by replacing the concept of a possible world with the concept of a possible extension of a part of the actual world.