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Canonical rules

  • Emil Jeřábek (a1)
Abstract

We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

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[1] Baader, Franz and Narendran, Paliath, Unification of concept terms in description logics, Journal of Symbolic Computation, vol. 31 (2001), pp. 277305.
[2] Blackburn, Patrick, van Benthem, Johan, and Wolter, Frank (editors), Handbook of modal logic, Studies in Logic and Practical Reasoning, vol. 3, Elsevier, Amsterdam, 2007.
[3] Blok, Willem J., Varieties of interior algebras, Ph.D. thesis, University of Amsterdam, 1976.
[4] Blok, Willem J., On the degree of incompleteness in modal logics and the covering relation in the lattice of modal logics, Technical Report 78-07, Department of Mathematics, University of Amsterdam, 1978.
[5] Chagrov, Alexander V., A decidable modal logic with the undecidable admissibility problem for inference rules, Algebra and Logic, vol. 31 (1992), no. 1, pp. 5355.
[6] Chagrov, Alexander V. and Zakharyaschev, Michael, Modal logic, Oxford Logic Guides, vol. 35, Oxford University Press, 1997.
[7] W. Wistar Comfort and Stylianos Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, vol. 211, Springer, Berlin, 1974.
[8] Esakia, Leo L., To the theory of modal and superintuitionistic systems, Logical inference. Proceedings of the USSR symposium on the theory of logical inference (Smirnov, V. A., editor), Nauka, Moscow, 1979, pp. 147172 (Russian).
[9] Fine, Kit, An ascending chain of SA logics, Theoria, vol. 40 (1974), no. 2, pp. 110116.
[10] Fine, Kit, Logics containing K4, Part 11, this Journal, vol. 50 (1985), no. 3, pp. 619651.
[11] Friedman, Harvey M., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), no. 2, pp. 113129.
[12] Ghilardi, Silvio, Unification in intuitionistic logic, this Journal, vol. 64 (1999), no. 2, pp. 859880.
[13] Ghilardi, Silvio, Best solving modal equations, Annals of Pure and Applied Logic, vol. 102 (2000), no. 3, pp. 183198.
[14] Iemhoff, Rosalie, On the admissible rules of intuitionistic propositional logic, this Journal, vol. 66 (2001), no. 1, pp. 281294.
[15] Iemhoff, Rosalie, Intermediate logics and Visser's rules, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 1, pp. 6581.
[16] Iemhoff, Rosalie, On the rules of intermediate logics, Archive for Mathematical Logic, vol. 45 (2006), no. 5, pp. 581599.
[17] Jankov, V. A., The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures, Mathematics of the USSR, Doklady, vol. 4 (1963), no. 4, pp. 12031204.
[18] Jeřábek, Emil, Admissible rules of modal logics, Journal of Logic and Computation, vol. 15 (2005), no. 4, pp. 411431.
[19] Jeřábek, Emil, Frege systems for extensible modal logics, Annals of Pure and Applied Logic, vol. 142 (2006), pp. 366379.
[20] Jeřábek, Emil, Complexity of admissible rules, Archive for Mathematical Logic, vol. 46 (2007), no. 2, pp. 7392.
[21] Jeřábek, Emil, independent bases of admissible rules, Logic Journal of the IGPL, vol. 16 (2008), no. 3, pp. 249267.
[22] Katětov, Miroslav, A theorem on mappings, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), no. 3, pp. 432433.
[23] Kracht, Marcus, An almost general splitting theorem for modal logic, Stadia Logica, vol. 49 (1990), no. 4, pp. 455470.
[24] Kracht, Marcus, Review of [29], >Notre Dame Journal of Formal Logic, vol. 40 (1999), no. 4, pp. 578587.
[25] Kracht, Marcus, Modal consequence relations, In Blackburn et al. [2], pp. 491545.
[26] Lorenzen, Paul, Einführung in die operative Logik und Mathematik, Grundlehren der mathematischen Wissenschaften, vol. 78, Springer, 1955 (German).
[27] Maksimova, Larisa L. and Rybakov, Vladimir V., Lattices of modal logics, Algebra and Logic, vol. 13 (1974), pp. 105122.
[28] Rautenberg, Wolfgang, Splitting lattices of logics, Archiv für mathematische Logik und Grundlagenforschung, vol. 20 (1980), no. 3-4, pp. 155159.
[29] Rybakov, Vladimir V., Admissibility of logical inference rules, Studies in Logic and the Foundations of Mathematics, vol. 136, Elsevier, 1997.
[30] Rybakov, Vladimir V., Logical consecutions in discrete linear temporal logic, this Journal, vol. 70 (2005), no. 4, pp. 11371149.
[31] Rybakov, Vladimir V., Logical consecutions in intransitive temporal linear logic of finite intervals, Journal of Logic and Computation, vol. 15 (2005), no. 5, pp. 663678.
[32] Rybakov, Vladimir V., Linear temporal logic with Until and Before on integer numbers, deciding algorithms, Computer science-theory and applications (Grigoriev, Dima, Harrison, John, and Hirsch, Edward A., editors), Lecture Notes in Computer Science, vol. 3967, Springer, 2006, pp. 322333.
[33] Wolter, Frank, Tense logic without tense operators, Mathematical Logic Quarterly, vol. 42 (1996), no. 1, pp. 145171.
[34] Wolter, Frank and Zakharyaschev, Michael, Modal decision problems, In Blackburn et al. [2], pp. 427489.
[35] Wolter, Frank and Zakharyaschev, Michael, Undecidability of the unification and admissibility problems for modal and description logics, ACM Transactions on Computational Logic, vol. 9 (2008), no. 4.
[36] Zakharyaschev, Michael, Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems, Mathematics of the USSR, Sbornik, vol. 68 (1991), no. 1, pp. 277289.
[37] Zakharyaschev, Michael, Canonical formulas for K4. Part I: Basic results, this Journal, vol. 57 (1992), no. 4, pp. 13771402.
[38] Zakharyaschev, Michael, Canonical formulas for K4. Part IT. Cofinalsubframe logics, this Journal, vol. 61 (1996), no. 2, pp. 421449.
[39] Zakharyaschev, Michael, Canonical formulas for K4. Part III: the finite model property, this Journal, vol. 62 (1997), no. 3, pp. 950975.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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