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A class of decidable intermediate propositional logics

  • C. G. McKay (a1)

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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 (¬PP) and (¬¬PP) 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.

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[1]Kleene, S. C., Logical calculus and realizability, Acta Philosophica Fenilica, vol. 18 (1965), pp. 7180. MR 33 #5485.
[2]Kreisel, G. and Putnam, H., Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül, Archiv für Mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 7478.
[3]McKay, C. G., The decidability of certain intermediate propositional logics, this Journal, vol. 33 (1968), pp. 258264.

A class of decidable intermediate propositional logics

  • C. G. McKay (a1)

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