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A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property1

  • D. M. Gabbay (a1) and D. H. J. De Jongh (a2)
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The intuitionistic propositional logic I has the following (disjunction) property

.

We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψα))→ ((∼ϕψ) ∨ (∼ϕα)) and Scott's system I + ((∼ ∼ϕϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.

In this note we shall construct a sequence of intermediate logics Dn with the following properties:

These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).

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1

Earlier version appeared in Technical Report, Jerusalem, 1969.

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References
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[1]Kreisel, G. and Putnam, H., Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül, Archiv für mathematische Logic und Grundlagenforschung, vol. 3 (1957), pp. 7478.
[2]Kripke, S., Semantic analysis for intuitionistic logic, Formal system and recursive functions (Crossley, J. and Dummett, M., Editors), North-Holland, Amsterdam, 1965.
[3]Gabbay, D. M., Decidability results in nonclassical logics. I, Annals of Mathematical Logic (to appear).
[3a]Gabbay, D. M., On decidable, finitely axiomatizable, modal and tense systems without the finite model properly, Israel Journal of Mathematics, vol. 10 (1971), pp. 478503.
[3b]Gabbay, D. M., Decidability of the Kreisel-Putnam system, this Journal, vol. 35 (1970), pp. 431437.
[3c]Gabbay, D. M., Model theory for intuitionistic logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 18 (1971), pp. 4954.
[4]Rabin, M. O., Decidability of second order theories and automata on trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.
[5]Troelstra, A. S., Intermediate logics, Indagationes Mathematicae, vol. 27 (1965), pp. 141152.
[6]Segerberg, K., Propositional logics related to Heyting's and Johansson's, Theoria, vol. 34 (1968), pp. 2661.
[7]Gabbay, D. M., Semantical methods in non-classical logics, North-Holland, Amsterdam (to appear).
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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