Label-free Modular Systems for Classical and Intuitionistic Modal Logics

In this paper we show for each of the modal axioms d, t, b, 4, and 5 an equivalent set of inference rules in a nested sequent system, such that, when added to the basic system for the modal logic K, the resulting system admits cut elimination. Then we show the same result also for intuitionistic modal logic. We achieve this by combining structural and logical rules.

[1]  Lutz Straßburger,et al.  Modular Sequent Systems for Modal Logic , 2009, TABLEAUX.

[2]  Rajeev Goré,et al.  On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics , 2011, Log. Methods Comput. Sci..

[3]  B. Nebel.  Introduction to Modal Logic Introduction , 2009 .

[4]  Lutz Straßburger,et al.  Cut Elimination in Nested Sequents for Intuitionistic Modal Logics , 2013, FoSSaCS.

[5]  Kazushige Terui,et al.  From Axioms to Analytic Rules in Nonclassical Logics , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[6]  Valeria de Paiva,et al.  On an Intuitionistic Modal Logic , 2000, Stud Logica.

[7]  Gordon Plotkin,et al.  A framework for intuitionistic modal logics: extended abstract , 1986 .

[8]  Francesca Poggiolesi,et al.  The Method of Tree-Hypersequents for Modal Propositional Logic , 2009, Towards Mathematical Philosophy.

[9]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[10]  Revantha Ramanayake,et al.  Structural Extensions of Display Calculi: A General Recipe , 2013, WoLLIC.

[11]  Melvin Fitting,et al.  Prefixed tableaus and nested sequents , 2012, Ann. Pure Appl. Log..

[12]  Ryo Kashima,et al.  Cut-free sequent calculi for some tense logics , 1994, Stud Logica.

[13]  Gordon D. Plotkin,et al.  A Framework for Intuitionistic Modal Logics , 1988, TARK.

[14]  Kai Br.  Deep Sequent Systems for Modal Logic , 2006 .

[15]  Yakoub Salhi,et al.  Label-free natural deduction systems for intuitionistic and classical modal logics , 2010, J. Appl. Non Class. Logics.

[16]  Kai Brünnler,et al.  Deep sequent systems for modal logic , 2009, Arch. Math. Log..

[17]  Frank Pfenning,et al.  A judgmental reconstruction of modal logic , 2001, Mathematical Structures in Computer Science.

[18]  Alex K. Simpson,et al.  The proof theory and semantics of intuitionistic modal logic , 1994 .