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1999 | 46 | 1 | 159-168
Tytuł artykułu

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider the families 𝓛 of propositional superintuitionistic logics (s.i.l.) and NE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between 𝓛 and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig's interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness (HP) etc. have suitable properties of varieties as their images, and many natural algebraic properties are in accordance with natural properties of logics. For example, a s.i.l. L has CIP iff its associated variety V(L) has the amalgamation property (AP); L is Hallden-complete iff V(L) is generated by a subdirectly irreducible Heyting algebra. For any n.m.l. L, the amalgamation property of V(L) is equivalent to a weaker version of the interpolation property for L, and the super-amalgamation property is equivalent to CIP; L is Hallden-complete iff V(L) satisfies a strong version of the joint embedding property. Well-known relational Kripke semantics for the intuitionistic and modal logics seems to be a more natural interpretation than the algebraic one. The categories of Kripke frames may, in a sense, be considered as subcategories of varieties of Heyting or modal algebras. We discuss the question to what extent one may reduce problems on properties of logics to consideration of their semantic models.
Słowa kluczowe
Rocznik
Tom
46
Numer
1
Strony
159-168
Opis fizyczny
Daty
wydano
1999
Twórcy
  • Institute of Mathematics, Siberian Division of Russian Academy of Sciences, 630090, Novosibirsk, Russia
Bibliografia
  • [1] A. Chagrov and M. Zakharyashchev, Undecidability of the disjunction property of propositional logics and other related problems, J. Symbolic Logic 58, 3 (1993), 967-1002.
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  • [8] L. L. Maksimova, Interpolation theorems in modal logics and amalgamated varieties of topoboolean algebras, Algebra i Logika, 18, No. 5, 556-586 (1979).
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  • [12] L. Maksimova, Amalgamation and Interpolation in Normal Modal Logics, Studia Logica, 50, 3/4 (1991), 457-471.
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  • [20] K. Segerberg, An Essay in Classical Modal Logics, Uppsala University, Uppsala, 1971.
  • [21] N.-Y. Suzuki, Intermediate logics characterized by a class of algebraic frames with infinite individual domain, Bull. Section of Logic 18, 2 (1989), 63-71.
  • [22] S. K. Thomason, Categories of frames for modal logic, J. Symbolic Logic, 40, no. 3 (1975), 439-442.
  • [23] J. F. A. K. van Benthem and I. L. Humberstone, Hallden-completeness by gluing of Kripke-frames, Notre Dame J. of Formal Logic, 24 (1983), 426-430.
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Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv46i1p159bwm
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