Abstract
Let t(A) be the result of prefixing the necessity operator ❑ to every proper subformula, save conjunctions and disjunctions, of the formula A of the language of the Heyting propositional calculus H. It is well-known that H can be embedded by t in S4, i.e. A is provable in H iff t(A) is provable in S4. Esakia (1979), and also Blok (1976), have shown that S4Grz (defined below) is the maximal normal extension of S4 in which H can be embedded by t (as a matter of fact, we find in Esakia (1979) not t, but the translation which prefixes ❑ to every subformula; this translation is equivalent to t as far as S4 and its normal extensions are concerned).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
van Benthem, J.F.A.K., and Blok, W.J., 1978, Transitivity follows from Dummett’s axiom, Theoria, 44: 117–118.
Blok, W.J., 1976, “Varieties of Interior Algebras”, dissertation, University of Amsterdam.
Blok, W.J., and Dwinger, Ph., 1975, Equational classes of closure algebras I, Indag. Math., 37: 189–198.
Boolos, G., 1979, “The Unprovability of Consistency: An Essay in Modal Logic”, Cambridge University Press, Cambridge.
Chagrov, A.V., 1985, Varieties of logical matrices (in Russian), Algebra i Logika, 24:426–489 (English translation in: Algebra and Logic, 24:278–325).
Došen, K., 1981, Minimal modal systems in which Heyting and classical logic can be embedded, Publ. Inst. Math. (Beograd) (N.S.), 30 (44): 41–52.
Došen, K., 1985, Models for stronger normal intuitionistic modal logics, Studia Logica, 44: 39–70.
Došsen, K., 1986, Modal translations and intuitionistic double negation, Logique et Anal. (N.S.), 29: 81–94.
Došen, K., 1986a, Higher-level sequent-systems for intuitionistic modal logic, Publ. Inst. Math. (Beograd) (N.S.), 39 (53): 3–12.
Esakia, L.L., 1979, On the variety of Grzegorczyk algebras (in Russian), in: “Issledovaniya po neklassicheskim logikam i teorii mnozhestv”, Nauka, Moscow, 257–287 (Math. Rev. 81j:03097; according to references in this paper, the results presented were announced in 1974 ).
Flagg, R.C., and Friedman, H., 1986, Epistemic and intuitionistic formal systems, Ann. Pure Appl. Logic, 32: 53–60.
Girard, J.-Y., 1987, Linear logic, Theoret. Comput. Sci., 50:1–102.
Lemmon, E.J., and Scott, D.S., 1977, “An Introductio7n to Modal Logic: The ‘Lemmon Notes”, Blackwell, Oxford.
Maksimova, L.L., and Rybakov, V.V., 1974, On the lattice of normal modal logics (in Russian), Algebra i Logika, 13:188–216 (English translation in: Algebra and Logic, 13:105–122).
Ono, H., 1977, On some intuitionistic modal logics, Publ. Res. Inst. Math. Sci. (Kyoto), 13: 687–722.
Rasiowa, H., and Sikorski, R., 1963, “The Mathematics of Metamathematics”, Panstwowe Wydawnictwo Naukowe, Warsaw.
Sotirov, V.H., 1984, Modal theories with intuitionistic logic, in: “Mathematical Logic”, Proceedings of the Conference Dedicated to Markov, Bulgarian Academy of Sciences, Sofia, 139–171.
Veldman, W., 1976, An intuitionistic completeness theorem for intui-tionistic predicate logic, J. Svmboli Logic, 41:159–166.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Plenum Press, New York
About this chapter
Cite this chapter
Došen, K. (1990). Normal Modal Logics in Which the Heyting Propositional Calculus can be Embedded. In: Petkov, P.P. (eds) Mathematical Logic. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0609-2_19
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0609-2_19
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-7890-0
Online ISBN: 978-1-4613-0609-2
eBook Packages: Springer Book Archive