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A class of decidable intermediate propositional logics

Published online by Cambridge University Press:  12 March 2014

C. G. McKay
C. G. McKay*
Affiliation:
Lakehead University, Ontario, Canada
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Extract

For the terminology and motivation of this note, my earlier paper [3] should be consulted. Here I make a slight change by requiring the Intuitionist Propositional Logic H to be given in terms of axiom schemata rather than axioms.

Definition. An axiom schema F is essentially negative iff each schematic letter appearing in F is negated.

Thus the schemata (¬P ∨ ¬¬P), (¬¬(¬P ∨ ¬Q)→(¬P ∨ ¬Q)) are essentially negative, whereas (¬P ∨ P) and (¬¬P→P) are not.

Lemma. Let F be an essentially negative axiom schema. Then F yields at most finitely many intuitionistically nonequivalent axioms whose atoms are chosen from a fixed finite set.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 127 - 128
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Kleene, S. C., Logical calculus and realizability, Acta Philosophica Fenilica, vol. 18 (1965), pp. 71–80. MR 33 #5485.Google Scholar
[2]Kreisel, G. and Putnam, H., Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül, Archiv für Mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 74–78.CrossRefGoogle Scholar
[3]McKay, C. G., The decidability of certain intermediate propositional logics, this Journal, vol. 33 (1968), pp. 258–264.Google Scholar