Abstract
Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose “equality-free fragment” is just the intuitionistic predicate logic.
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References
Bull R. A. and K. Segerberg, 1984, ‘Basic modal logic’, in D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic II, D. Reidel, pp. 1–88.
Dragalin, A. G., 1979,Mathematical intuitionism. Introduction to Proof Theory, (Russian) Nauka, Moskva. English edition; translated by E. Mendelson, Providence, AMS, 1988.
Gabbay, D. M., 1981,Semantical Investigation of Heyting's Intuitionistic Logic, Synthese Library, Studies in Epistemology, Logic, methodology, and Philosophy of Science, vol. 148, D. Reidel Publishing Company, Dordrecht.
Görnemann, S., 1971, ‘A logic stronger than intuitionism’,Journal of Symbolic Logic 36, 249–261.
Hughes, G. E. and M. J. Cresswell, 1968,An Introduction to Modal Logic, Methuen & Co. Ltd.
Jankov, V. A., 1963, ‘The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures’,Soviet Mathematics Doklady 4, 1203–1204.
Jankov, V. A., 1968, ‘Constructing a sequence of strongly independent superintuitionistic propositional calculi’,Soviet Mathematics Doklady 9, 806–807.
Maehara, S., 1954, ‘Eine Darstellung der intuitionistischen Logik in der klassischen’,Nagoya Mathematical Journal 7, 45–64.
McKinsey, J. C. C. and A. Tarski, 1948, ‘Some theorems about the sentential calculi of Lewis and Heyting’,Journal of Symbolic Logic 13, 1–15.
Nagashima, T., 1973, ‘An intermediate predicate logic’,Hitotsubashi Journal of Arts and Sciences 14, 53–58.
Ono, H., 1970, ‘Kripke models and intermediate logics’,Publications of Research Institute for Mathematical Sciences, Kyoto University 6, 461–476.
Ono, H., 1972, ‘A study of intermediate predicate logics’,Publications of Research Institute for Mathematical Sciences, Kyoto University 8, 619–649.
Ono, H., 1987, ‘Some problems in intermediate predicate logics’,Reports on Mathematical Logic 21, 55–67.
Shehtman, V. B. and D. P. Skvortsov, 1990, ‘Semantics of non-classical first-order predicate logics’, inMathematical Logic, edited by P. P. Petkov, Plenum Press, New York, 105–116.
Shimura, T. and N.-Y. Suzuki, 1993, ‘Some super-intuitionistic logics as the logical fragment of equational theories’,Bulletin of the Section of Logic 22, 106–112.
Suzuki, N.-Y., 1993, ‘Some results on the Kripke-sheaf semantics for super-intuitionistic predicate logics’,Studia Logica 52, 73–94.
Troelstra, A. S., 1965, ‘On intermediate propositional logics’,Indagationes Mathematicae 27, 141–152.
Umezawa, T., 1959, ‘On logics intermediate between intuitionistic and classical predicate logic’,Journal of Symbolic Logic 24, 141–153.
Wajsberg, M., 1933–4, ‘Beitrag zur Metamathematik’,Mathematische Annalen 109, 200–229. English translation: ‘A contribution to metamathematics’, inM. Wajsberg, Logical Works, edited by S. J. Surma, Ossolineum 1977, 62–88.
Wroński, A., 1974, ‘The degree of completeness of some fragments of the intuitionistic propositional logic’,Reports on Mathematical Logic 2, 55–62.
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Dedicated to the memory of the late Professor S. Maehara
This research was supported in part by Grant-in Aid for Encouragement of Young Scientists No. 06740140, Ministry of Education, Science and Culture, Japan.
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Suzuki, NY. Constructing a continuum of predicate extensions of each intermediate propositional logic. Stud Logica 54, 173–198 (1995). https://doi.org/10.1007/BF01063151
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DOI: https://doi.org/10.1007/BF01063151