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Inferential equivalence and natural deduction

Published online by Cambridge University Press:  12 March 2014

H. Hiż*
Affiliation:
Pennsylvania State University

Extract

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists of

Another one, closely connected with A1–A3, is comprised of

B1–B3 jointly assert the inferential equivalence of a sentence of the form (pq) ⊃ r and two sentences of the forms qr and ∼pr respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.

The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1957

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References

1 Łukasiewicz, Jan, Ein Vollständigkeitsbeweis des zweiwertigen Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, vol. 24 (1931), Classe iii, pp. 153183.Google Scholar

2 The phrase ‘is a theorem’ shall be tacit in the subsequent metatheorems.

3 Łukasiewicz, Jan, Elementy logiki matematycznej, Warsaw, 1929Google Scholar. Also see Church, Alonzo, Introduction to mathematical logic, vol. 1, (1956), p. 160Google Scholar.