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# Finite and finitely separable intermediate propositional logics

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§0. Introduction and material background. The present paper is devoted to the study of intermediate propositional logics, and it is based on [Be, §§1 and 2].

§2 (§§0 and 1 are introductory) concerns the axiomatization of finite logics. In the literature several effective procedures to axiomatize finite logics are present (cf., for instance, [MK] and [Wr]), but, in each case, the number of propositional variables which are used is redundant. In this direction, Theorem 2.2 provides (a) a criterion to determine, given a finite logic L, the least n such that L is axiomatizable by formulas in n variables, and (b) an effective axiomatization by an n-formula. As a corollary we obtain a negative answer to Problem 7.10 of [Ho/On], showing that there is no connection between the slice to which L belongs and the number of propositional variables necessary to axiomatize L.

The principal results of the paper are in §3. In fact, a great deal of research has been done on the correspondence between conditions on the relation of Kripke-structures from one side, and axioms added to Int from the other. In this section we (a) introduce the concept of finitely separable class of Kripke-frames, and show, by means of several examples, that this concept is “wide”, in the sense that all the most studied classes of frames determined by semantical conditions are finitely separable; (b) show that each finitely separable class is axiomatizable, and that the axioms can be found by means of semantical considerations only; and (c) establish the finite model property for all the finitely separable logics.

## References

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[Be]Bellissima, F., Finitely generated free Heyting algebras, this Journal, vol. 51 (1986), pp. 152165.
[Ga]Gabbay, D. M., Semantical investigations in Heyting's intuitionistic logic, Reidel, Dordrecht, 1981.
[Ga/DJ]Gabbay, D. M. and de Jongh, D. H. J., A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property, this Journal, vol. 39 (1974), pp. 6778.
[Ha]Harrop, R., On the existence of finite models and decision procedures for prepositional calculi, Proceedings of the Cambridge Philosophical Society, vol. 54 (1958), pp. 113.
[Ho]Hosoi, T., On intermediate logics. I, Journal of the Faculty of Science, University of Tokyo, Section I, vol. 14 (1967), pp. 293312.
[Ho/]n]Hosoi, T. and Ono, H., Intermediate prepositional logics (a survey), Journal of Tsuda College, vol. 5 (1973), pp. 6782.
[Ja]Jankov, V. A., The construction of a sequence of strongly independent superintuitionistic propositional calculi, Doklady Akademii Nauk SSSR, vol. 181 (1968), pp. 3334; English translation, Soviet Mathematics Doklady, vol. 9 (1968), pp. 806807.
[Ma]Maksimova, L. L., Pretabular superintuitionistic logics, Algebra i Logika, vol. 11 (1972), pp. 558570; English translation, Algebra and Logic, vol. 11 (1972), pp. 308-314.
[MK]McKay, C. G., On finite logics, Indagationes Mathematicae, vol. 29 (1967), pp. 9596.
[On]Ono, H., Some results on the intermediate logics, Publications of the Research Institute for the Mathematical Sciences, vol. 8 (1972/1973), pp. 619649.
[Se]Segerberg, K., Prepositional logics related to Heyting's and Johansson's, Theoria, vol. 34 (1968), pp. 2661.
[Sm]Smoryński, C., Investigations on intuitionistic formal systems by means of Kripke models, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1973.
[Wr]Wronski, A., An algorithm for finding finite axiomatizations of finite intermediate logics, Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic, vol. 2 (1972), pp. 3844.

# Finite and finitely separable intermediate propositional logics

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