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Canonical formulas for K4. Part II: Cofinal subframe logics

  • Michael Zakharyaschev (a1)

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This paper is a continuation of Zakharyaschev [25], where the following basic results on modal logics with transitive frames were obtained:

• With every finite rooted transitive frame and every set of antichains (which were called closed domains) in two formulas α (, , ⊥) and α(, ) were associated. We called them the canonical and negation free canonical formulas, respectively, and proved the Refutability Criterion characterizing the constitution of their refutation general frames in terms of subreduction (alias partial p-morphism), the cofinality condition and the closed domain condition.

• We proved also the Completeness Theorem for the canonical formulas providing us with an algorithm which, given a modal formula φ, returns canonical formulas α(i, i), ⊥), for i = 1,…, n, such that

if φ is negation free then the algorithm instead of α(i, i, ⊥) can use the negation free canonical formulas α(i, i). Thus, every normal modal logic containing K4 can be axiomatized by a set of canonical formulas.

In this Part we apply the apparatus of the canonical formulas for establishing a number of results on the decidability, finite model property, elementarity and some other properties of modal logics within the field of K4.

Our attention will be focused on the class of logics which can be axiomatized by canonical formulas without closed domains, i.e., on the logics of the form

Adapting the terminology of Fine [11], we call them the cofinal subframe logics and denote this class by . As was shown in Part I, almost all standard modal logics are in .

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[1]Bull, R. A., That all normal extensions of S4.3 have the finite model property, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 341344.
[2]Chagrov, A. V., On the polynomial finite model property of modal and intermediate logics, Mathematical logic, mathematical linguistics and algorithm theory, Kalinin State University, Kalinin, 1983, Russian, pp. 7583.
[3]Chagrov, A. V. and Zakharyaschev, M. V., On the independent axiomatizability of modal and intermediate logics, Journal of Logic and Computation, vol. 5 (1995), pp. 287302.
[4]Chagrov, A. V. and Zakharyaschev, M.V., The undecidability of the disjunction property of propositional logics and other related problems, this Journal, vol. 58 (1993), pp. 9671002.
[5]Chagrova, L. A., On first-order definability of intuitionistic formulas with restrictions on occurrences of connectives, Logical methods for constructing effective algorithms, Kalinin State University, Kalinin, 1986, Russian, pp. 135136.
[6]Chagrova, L. A., On the preservation of first-order properties under the embedding of intermediate logics in modal logics, Proceedings of the Xth USSR conference for mathematical logic, Alma-Ata, 1990, Russian, p. 163.
[7]Chagrova, L. A., An undecidable problem in correspondence theory, this Journal, vol. 56 (1991), pp. 12611272.
[8]Feys, R., Modal logic, Louvain: E. Nauwelaerts, Paris, 1965.
[9]Fine, K., The logics containing S4.3, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 371376.
[10]Fine, K., Some connections between elementary and modal logic, Proceedings of the third scandinavian logic symposium (Kanger, S., editor), North-Holland, Amsterdam., 1975, pp. 1531.
[11]Fine, K., Logics containing K4, Part 11, this Journal, vol. 50 (1985), pp. 619651.
[12]Kracht, M., Internal definability and completeness in modal logic, Ph.D. thesis, Freie Universität, Berlin, 1990.
[13]Logic notebook, Institute of Mathematics, 1986.
[14]Maksimova, L. L., Pretabular extensions ofLewis S4, Algebra and Logic, vol. 14 (1975), pp. 1633.
[15]Maksimova, L. L. and Rybakov, V.V., On the lattice of normal modal logics, Algebra and Logic, vol. 13 (1974), pp. 188216, Russian.
[16]McKay, C. G., The decidability of certain intermediate logics, this Journal, vol. 33 (1968), pp. 258264.
[17]Ono, H. and Nakamura, A., On the size of refutation Kripke models for some linear modal and tense logics, Studia Logica, vol. 39 (1980), pp. 325333.
[18]Rodenburg, Rh., Intuitionistic correspondence theory, Ph.D. thesis, University of Amsterdam, 1986.
[19]Segerberg, K., An essay in classical modal logic, Philosophical Studies, vol. 13 (1971), University of Uppsala.
[20]Shimura, T., Kripke completeness of some intermediate predicate logics with the axiom of constant domain and a variant of canonical formulas, Studia Logica, vol. 52 (1993), pp. 2340.
[21]Zakharyaschev, M. V., On intermediate logics, Soviet Mathematics Doklady, vol. 27 (1983), pp. 274277.
[22]Zakharyaschev, M. V., Syntax and semantics of superintuitionistic and modal logics, Ph.D. thesis, Moscow, 1984, Russian.
[23]Zakharyaschev, M. V., Modal companions of intermediate logics: syntax, semantics and preservation theorems, Mathematical Sbornik, vol. 180 (1989), pp. 14151427, English translation: Mathematics of the USSR Sbornik, vol.68 (1991), pp.277–289.
[24]Zakharyaschev, M. V., Syntax and semantics of intermediate logics, Algebra and Logic, vol. 28 (1989), pp. 262282.
[25]Zakharyaschev, M. V., Canonical formulas for K4. part 1: Basic results, this Journal, vol. 57 (1992), pp. 13771402.
[26]Zakharyaschev, M. V., Intermediate logics with disjunction free axioms are canonical, IGPL Newsletter, vol. 1 (1992), no. 4, pp. 78.
[27]Zakharyaschev, M. V., Canonical formulas for modal and superintuitionistic logics: A short outline, Modal logic audits neighbours (de Rijke, M., editor), Dordrecht, Kluwer Academic Publishers, 1995, in print.
[28]Zakharyaschev, M. V. and Popov, S.V., On the complexity of countermodels for intuitionistic calculus, Institute of Applied Mathematics, the USSR Academy of Sciences no. 45, 1980, preprint, Russian.

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