Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity

Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result is established between the category of sober algebras and the category of general Kripke structures. A simple enrichment of the proposed sequent calculi is proved to be complete over standard Kripke structures. The calculi are shown to be analytic in a useful sense.

[1]  Reiner Hähnle,et al.  Tableaux for Many-Valued Logics , 1999 .

[2]  Kosta Dosen,et al.  Sequent-systems for modal logic , 1985, Journal of Symbolic Logic.

[3]  Dov M. Gabbay,et al.  A generalization of analytic deduction via labelled deductive systems. Part I: Basic substructural logics , 1994, Journal of Automated Reasoning.

[4]  J.F.A.K. van Benthem,et al.  Modal logic and classical logic , 1983 .

[5]  Luca Viganò,et al.  Labelled Deduction over Algebras of Truth-Values , 2002, FroCoS.

[6]  Dov M. Gabbay,et al.  Labelled Deductive Systems: Volume 1 , 1996 .

[7]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[8]  Luca Viganò,et al.  Labelled Propositional Modal Logics: Theory and Practice , 1997, J. Log. Comput..

[9]  Luca Viganò,et al.  Labelled non-classical logics , 2000 .

[10]  Krysia Broda,et al.  Labelled Natural Deduction for Substructural Logics , 1999, Log. J. IGPL.

[11]  Patrick Blackburn,et al.  Hybrid Languages and Temporal Logic , 1999, Log. J. IGPL.

[12]  Maarten Marx,et al.  Hybrid logics: characterization, interpolation and complexity , 2001, Journal of Symbolic Logic.

[13]  D. Holdstock.  Past, present--and future? , 2005, Medicine, conflict, and survival.

[14]  D. Gabbay,et al.  Handbook of tableau methods , 1999 .

[15]  Cristina Sernadas,et al.  Modulated Fibring and The Collapsing Problem , 2002, J. Symb. Log..

[16]  Christian G. Fermüller,et al.  Labeled Calculi and Finite-Valued Logics , 1998, Stud Logica.

[17]  Patrick Blackburn,et al.  Internalizing labelled deduction , 2000, J. Log. Comput..

[18]  Dov M. Gabbay,et al.  Chapter 13 – Labelled Deductive Systems , 2003 .

[19]  Cristina Sernadas,et al.  Fibring: completeness preservation , 2001, Journal of Symbolic Logic.

[20]  Patrick Blackburn,et al.  Hybrid Completeness , 1998, Log. J. IGPL.

[21]  Heinrich Wansing,et al.  Sequent Calculi for Normal Modal Proposisional Logics , 1994, J. Log. Comput..

[22]  Walter Alexandre Carnielli,et al.  Systematization of finite many-valued logics through the method of tableaux , 1987, Journal of Symbolic Logic.