A Tableau Method for Checking Rule Admissibility in S4

Rules that are admissible can be used in any derivations in any axiomatic system of a logic. In this paper we introduce a method for checking the admissibility of rules in the modal logic S4. Our method is based on a standard semantic ground tableau approach. In particular, we reduce rule admissibility in S4 to satisfiability of a formula in a logic that extends S4. The extended logic is characterised by a class of models that satisfy a variant of the co-cover property. The class of models can be formalised by a well-defined first-order specification. Using a recently introduced framework for synthesising tableau decision procedures this can be turned into a sound, complete and terminating tableau calculus for the extended logic, and gives a tableau-based method for determining the admissibility of rules.

[1]  Frank Wolter,et al.  Undecidability of the unification and admissibility problems for modal and description logics , 2006, TOCL.

[2]  Silvio Ghilardi,et al.  Best Solving Modal Equations , 2000, Ann. Pure Appl. Log..

[3]  Renate A. Schmidt,et al.  A General Tableau Method for Deciding Description Logics, Modal Logics and Related First-Order Fragments , 2008, IJCAR.

[4]  Emil Jerábek,et al.  Complexity of admissible rules , 2007, Arch. Math. Log..

[5]  Vladimir V. Rybakov,et al.  A criterion for admissibility of rules in the model system S4 and the intuitionistic logic , 1984 .

[6]  S. V. Babenyshev.  The decidability of admissibility problems for modal logics S4.2 and S4.2Grz and superintuitionistic logic KC , 1992 .

[7]  Silvio Ghilardi,et al.  Unification in intuitionistic logic , 1999, Journal of Symbolic Logic.

[8]  Vladimir V. Rybakov,et al.  Rules of inference with parameters for intuitionistic logic , 1992, Journal of Symbolic Logic.

[9]  Harvey M. Friedman,et al.  One hundred and two problems in mathematical logic , 1975, Journal of Symbolic Logic.

[10]  Rosalie Iemho.  On the Admissible Rules of Intuitionistic Propositional Logic , 2008 .

[11]  Emil Jerábek,et al.  Admissible Rules of Modal Logics , 2005, J. Log. Comput..

[12]  Vladimir V. Rybakov.  Semantic admissibility criteria for deduction rules in S4 and Int , 1991 .

[13]  Ronald Harrop,et al.  Concerning formulas of the types A→B ν C,A →(Ex)B(x) in intuitionistic formal systems , 1960, Journal of Symbolic Logic.

[14]  Vladimir V. Rybakov,et al.  On Finite Model Property for Admissible Rules , 1999, Math. Log. Q..

[15]  Rosalie Iemhoff,et al.  On the admissible rules of intuitionistic propositional logic , 2001, Journal of Symbolic Logic.

[16]  Silvio Ghilardi,et al.  A Resolution/Tableaux Algorithm for Projective Approximations in IPC , 2002, Log. J. IGPL.

[17]  Renate A. Schmidt,et al.  Using Tableau to Decide Expressive Description Logics with Role Negation , 2007, ISWC/ASWC.

[18]  Paul Rozière,et al.  Admissible and derivable rules in intuitionistic logic , 1993, Mathematical Structures in Computer Science.

[19]  P. Lorenzen.  Einführung in die operative Logik und Mathematik , 1955 .

[20]  Vladimir V. Rybakov,et al.  Admissibility of Logical Inference Rules , 2011 .

[21]  Silvio Ghilardi,et al.  Filtering unification and most general unifiers in modal logic , 2004, Journal of Symbolic Logic.

[22]  Renate A. Schmidt,et al.  Automated Synthesis of Tableau Calculi , 2011, Log. Methods Comput. Sci..

[23]  Vladimir V. Rybakov,et al.  Logics with the universal modality and admissible consecutions , 2007, J. Appl. Non Class. Logics.

[24]  G. Mints,et al.  Derivability of admissible rules , 1976 .