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Fragments of the propositional calculus

Published online by Cambridge University Press:  12 March 2014

Leon Henkin
Leon Henkin*
Affiliation:
Princeton University
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Extract

Of the several methods for proving the completeness of sets of axioms for the prepositional calculus perhaps the simplest is due to Kalmár, although it does not appear to be widely known. In this paper we generalize Kalmár's method to indicate how to obtain a complete axiomatization of any fragment of the propositional calculus which includes material implication. We shall carry through the description and proofs for the case where, in addition to a symbol for implication, there is just one other primitive truth-function symbol. For systems in which there are no other function symbols, or more than one other such symbol, notational changes but not conceptual changes will be required.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 14 , Issue 1 , 16 May 1949 , pp. 42 - 48
Copyright
Copyright © Association for Symbolic Logic 1949

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References

2 László, Kalmár, Über die Axiomatisierbarkeit des Aussagenkalküls, Acta scientiarum mathematicarum, vol. 7(1935), pp. 222–243.Google Scholar The author wishes to thank Prof. Church for bringing Kalmár's paper to his attention after an earlier version of the present paper had been submitted.

3 Prof. Church points out that the use of an arbitrary wff С in this way may be considered as arising from Kalmár's proof by first expressing negation in terms of implication and falsehood, inserting double negations in defining А* for the case where А′ is Т, and then replacing the symbol used to denote falsehood by a prepositional variable. The use of propositional variables to express falsehood was first undertaken by Wajsberg, in his Metalogische Beiträge, Wiadomości matematyczne, vol. 53 (1936), pp. 131–168.Google Scholar An alternative interpretation is to consider that we have replaced

in Kalmár'a proof (

being either the variable x i or its negation, ∼x i, with А* in a similar relation to А) by

which is seen to be valid for arbitrary С. This line may now be expressed without the symbols ∼ and v by using Β ⊃ С for ∼Β v С and Β ⊃ А for Β v С.

4 By essentially different methods this lemma can be generalized to the case of systems in which there is a non-denumerable set of variables. David Gale has obtained one such proof based on the theorem that an arbitrary product of compact topological spaces is compact. For another method of proof see my forthcoming paper The completeness of the first-order functional calculus. The method used here was employed by Gödel, K.; cf. Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37(1930), pp. 349–360.CrossRefGoogle Scholar