Normalisation and Language-Games1

Authors

  • Ruy J. G. B. de Queiroz

    1. Departamento de Informática, Pernambuco Federal Univ. at Recife, P.O. Box 7851, Recife, PE 50732-970, Brazil, e-mail: ruy@di.ufpe.br. Visiting Department of computing, Imperial College of Science, Technology and Medicine, University of London, 180 Queen's Gate, London SW7 2BZ, UK.
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  • 1

    An earlier version of this paper was presented at the Logic Colloquium '88, Padova, Italy, August 23–30, 1988, the abstract of which was published in the Journal of Symbolic Logic [de Queiroz, 1990a]. I am indebted to Prof T. Maibaum, Mr M. Sadler and Dr M. Smyth for many useful comments on earlier versions of this paper. Thanks are also due to ‘CNPq’–Conselho National de Desenvolvimento Científico e Tecnológico, Brazilian national council for the scientific and technological development, grant 20.2724/84-CC, and GENESIS, ESPRIT Project 1222 (1041), CEC, for the financial support.

Abstract

The question of finding a suitable formal account of meaning for the (primitive) logical signs has troubled many philosophers and logicians since the early days of formal logic, whenever it is even recognised as a problem. Here I attempt to show how two operational (as opposed to denotational, or truth-based) approaches to the problem can still be shown to be ‘technically’ equivalent, despite having emerged from two different readings of a single philosophical account, and being essentially distinct with respect to the rôle of ‘will’ in the mathematical activity: on the one hand, the ‘semantics of use’, my own reformulation of P. Martin-Löf's Intuitionistic Type Theory canonical-values based semantics by taking the normalisation rules as the key semantical device; and, on the other hand, J. Hintikka's Game-Theoretical Semantics, where the meaning of logical signs is given via semantical games. The philosophical account from which both emerge is precisely Wittgenstein's later account of propositions, where the notion of ‘language-games’ is introduced as a key semantical device. Observing that the normalisation rules seem to be able to formalise the explanation of the (immediate) consequences one can draw from a proposition,3 thus showing the function/purpose/usefulness of its main connective in the calculus of language, it seems reasonable to advocate that such a meta-mathematical device can be a semantically useful notion which would lead to a more reasonable account of the problem of formulating the meaning of logical constants.

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