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# The decidability of the Kreisel-Putnam system1

## Extract

The intuitionistic propositional logic I has the following disjunction property

This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiom

has the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom:

In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.

## Footnotes

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1

This research was sponsored under contract no. N00014–69-G-0192 U.S. Office of Naval Research, Information System Branch.

## References

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[0]Diego, A., Sur les algebras de Hilbert, Gauthiers-Villars, Paris, 1966.
[1]Gabbay, D., Decidability results in nonclassical logics. I, to appear; Part II, in preparation.
[2]Gabbay, D., Model theory for intuitionistic logic. I, to appear.
[3]Gabbay, D. and de Jongh, D., Sequences of decidable, finitely axiomatizable intermediate logics with the disjunction property, to appear.
[4]Harrop, R., On the existence of finite models and decision procedures, Proceedings of the Cambridge Philosophical Society, vol. 54 (1958), pp. 116.
[5]Kreisel, G. and Putnam, H., Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül, Archiv für Mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 7478.
[6]Kripke, S., Semantic analysis for intuitionistic logic, Formal systems and recursive functions, Amsterdam, J. Crossley-M. Dummett, Editors, 1965.
[7]McKay, C., Decidability of certain intermediate logics, this Journal, vol. 33 (1968), pp. 258265.
[8]Medvedev, T., Interpretations of logical formulae by means of finite problems, Doklady, vol. 7 (1966), pp. 857860.
[9]Rabin, M. O., Decidability of second order theories and automata on trees, Transactions of the American Mathematical Society, vol. 121 (1969), pp. 135.
[10]Segerberg, K., Propositional logics related to Heyting and Johansson, Theoria, vol. 34 (1968), pp. 2661.

# The decidability of the Kreisel-Putnam system1

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