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Meaning approached via proofs. (English) Zbl 1103.03055
Summary: According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book [The logical basis of metaphysics. London: Duckworth (1991)]. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence.

MSC:
03F03 Proof theory in general (including proof-theoretic semantics)
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[3] Dummett, M.: 1973, The Justification of Deduction, The British Academy, London (republished in M. Dummett, Truth and Other Enigmas, Duckworth, London 1978).
[4] Dummett, M.: 1976, ’What is a Theory of Meaning II’, in G. Evans et al. (eds.), Truth and Meaning, Oxford, pp. 67–137 (republished in M. Dummett, The Seas of Language, Clarendon Press, Oxford 1993).
[5] Dummett, M.: 1983, ’Language and Truth’, in R. Harris (ed.), Approaches to Language, Oxford (republished in M. Dummett, The Seas of Language, Clarendon Press, Oxford 1993). · Zbl 0875.03033
[7] Gentzen, G.: 1934, ’Untersuchungen Über das Logische Schließen’, Mathematische Zeitschrift 39, 176–210 and 405–431. · JFM 60.0020.02
[9] Martin-Löf, P.: 1985, ’On the Meanings of the Logical Constants and the Justification of the Logical Laws’, in Atti degli Incontri di Logica Matematica, Siena, Vol. 2, pp. 203–281 (reprinted in Nordic Journal of Philosophical Logic 1, 11–60). · Zbl 0593.03005
[17] Prior, A. N.: 1960, ’The Runabout Inference-Ticket’, Analysis 24.
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