Abstract
Algebraisation is a process of translating the syntax and deductive properties of given logic to an algebraic language. While propositional logics fit various algebraisation frameworks reasonably well, algebraisation of first-order logics has many difficulties. Recently, a class of quasicylindric algebras was introduced and investigated. It was proved that each superintuitionistic predicate logic is strongly complete with respect to a variety of quasicylindric algebras. In this paper, we prove that the semantics of pseudo-boolean models of Rasiowa and Sikorski and the Kripke semantics is subsumed by the semantics of quasicylindric algebras. We also expand results obtained by Larisa Maksimova on algebraisation of non-classical propositional logics to the case of first-order superintuitionistic logics. We consider such deductive properties of first-order superintuitionistic logics as the Beth property and the projective Beth property, the Craig interpolation property, the disjunctive and existential properties. We formulate algebraic equivalents which correspond to these properties in the language of varieties of quasicylindric algebras and establish equivalences of the logical properties and their algebraic counterparts.
This is a preview of subscription content, log in via an institution.
References
Andréka, H., Németi, I. & Sain, I. (1984). Abstract model theoretic approach to algebraic logic, manuscript.
Blok, W. J., & Pigozzi, D. (1989). Algebraizable logics, Memoirs of the American Mathematical Society. Providence: American Mathematical Society.
Czelakowski, J. (1982). Logical matrices and the amalgamation property. Studia Logica, 41(4), 329–341.
Czelakowski, J. (2001). Protoalgebraic logics. Dordrecht: Kluwer Academic Publishers.
Czelakowski, J., & Jansana, R. (2014). Weakly algebraizable logics. Journal of Symbolic Logic, 65, 641–668.
Czelakowski, J., & Pigozzi, D. (2004). Fregean logics. Annals of Pure and Applied Logic, 127, 17–76.
Dragalin, A. G. (1988). Mathematical intuitionism: Introduction to proof theory, Translations of Mathematical Monographs. Providence: American Mathematical Society.
Ershov, Y. L., & Palyutin, E. A. (1986). Mathematical logic. Moscow: Mir Publishers.
Gabbay, D. M., & Maksimova, L. (2005). Interpolation and definability: modal and intuitionistic logics. Oxford: Oxford University Press,Clarendon Press.
Henkin, L., Monk, J. D., & Tarski, A. (1971, 1985). Cylindric algebras, Parts 1, 2. Amsterdam: North Holland.
Hoogland, E. (1997). Algebraic characterizations of two Beth definability properties. Workshop on Abstract Algebraic Logic, Abstracts of talks (pp. 48–52). Bellaterra, Spain: Centre de Recerca Matemática.
Jankov, V. A. (1968). The construction of a sequence of strongly independent superintuitionistic propositional calculi. Soviet Mathematics Doklady, 9, 806–807.
Madarász, J. (1995). The Craig Interpolation theorem in multi-modal logics. Bulletin of the Section of Logic, 24(3), 147–154.
Maksimova, L. (1977). Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras. Algebra and Logic, 16(6), 427–455.
Maksimova, L. (1986). On maximal intermediate logics with disjunction property. Studia Logica, 45(1), 69–75.
Maksimova, L. (1992). Modal logics and varieties of modal algebras: The Beth properties, interpolation, and amalgamation. Algebra and Logic, 31(2), 90–105.
Maksimova, L. (1995). On variable separation in modal and superintuitionistic logics. Studia Logica, 55(1), 99–102.
Maksimova, L. (1998). Explicit and implicit definability in modal and related logics. Bulletin of the Section of Logic, 27(1/2), 36–39.
Maksimova, L. (1999). Interrelations of algebraic, semantical and logical properties for superintuitionistic and modal logics. In Logic, algebra and computer science (Vol. 46, pp. 159–168), Banach center publications. Warszawa: Institute of Mathematics, Polish Academy of Sciences.
Maksimova, L. L. (1979). Interpolation theorems in modal logics and amalgamable varieties of topoboolean algebras. Algebra and Logic, 18(5), 556–586.
Németi, I. (1991). Algebraization of quantifier logics, an introductory overview. Studia Logica, 50(3/4), 485–570.
Ono, H. (1973). A study of intermediate predicate logics. Publications of the Research Institute for Mathematical Sciences, 8, 619–649.
Pinter, C. (1973). A simple algebra of first order logic. Notre Dame Journal of Formal Logic, 14(3), 361–366.
Pinter, C. (1975). Algebraic logic with generalized quantifiers. Notre Dame Journal of Formal Logic, 16, 511–516.
Rasiowa, H. (1974). An algebraic approach to non-classical logics (Vol. 78), Studies in logic and the foundation of mathematics. Amsterdam: North-Holland.
Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics, Monografie matematiczne. Warszawa: Państwowe Wydawnictwo Naukowe.
Rybakov, V. (1997). Admissibility of logical inference rules. Amsterdam: Elsevier.
Tarski, A. (1935). Grundzüge des Systemenkalküls. Erster Teil, Fundamenta Mathematicae, 25, 503–526.
Tishkovsky, D. E. (1999). On algebraic semantics for superintuitionistic predicate logics. Algebra and Logic, 38(1), 36–50.
Tishkovsky, D. E. (2001). Algebraic equivalents of some properties of superintuitionistic predicate logics. Algebra and Logic, 40(2), 218–242.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Tishkovsky, D. (2018). On Algebraisation of Superintuitionistic Predicate Logics. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-69917-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69916-5
Online ISBN: 978-3-319-69917-2
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)