Sequent Calculi for Multi-modal Logic with Interaction

This paper studies Gentzen-style sequent calculi for multi-modal logics with interaction between the modalities. We prove cut elimination and some of its usual corollaries for two such logics: Standard Deontic Logic with the Ought-implies-Can principle, and a non-normal deontic logic where obligation, permissions and abilities interact in a complex way. The key insight of these results is to make rules sensitive to the shape of the formulas on either sides of the sequents. This way one can devise rules in a much more modular fashion. This feature of Hilbert-style systems is notoriously lost when one moves to sequent calculi. By partly restoring modularity the method proposed here can potentially provide a unified approach to the proof theory of multi-modal systems.

[1]  Johan van Benthem,et al.  Merging Frameworks for Interaction , 2009, J. Philos. Log..

[2]  Xin Sun,et al.  Conditional Ought, a Game Theoretical Perspective , 2011, LORI.

[3]  Albert J. J. Anglberger,et al.  The Logic of Best Actions from a Deontic Perspective , 2014, Johan van Benthem on Logic and Information Dynamics.

[4]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[5]  Guido Governatori,et al.  Labelled Modal Sequents , 2000 .

[6]  Thierry Lucas,et al.  Sequent Calculi and Decision Procedures for Weak Modal Systems , 2000, Stud Logica.

[7]  Paul McNamara,et al.  Deontic logic , 2006, Logic and the Modalities in the Twentieth Century.

[8]  Yde Venema,et al.  Dynamic Logic by David Harel, Dexter Kozen and Jerzy Tiuryn. The MIT Press, Cambridge, Massachusetts. Hardback: ISBN 0–262–08289–6, $50, xv + 459 pages , 2002, Theory and Practice of Logic Programming.

[9]  H. Wansing.  Displaying Modal Logic , 1998 .

[10]  Francesca Poggiolesi,et al.  Gentzen Calculi for Modal Propositional Logic , 2010 .

[11]  Francesca Poggiolesi.  From Single Agent to Multi-Agent via Hypersequents , 2013, Logica Universalis.

[12]  Francesca Poggiolesi,et al.  A Contraction-free and Cut-free Sequent Calculus for Propositional Dynamic Logic , 2010, Stud Logica.

[13]  Riccardo Pucella.  Review of Dynamic Logic (Foundations of Computing): authors of book: D. Harel, D. Kozen and J. Tiuryn , 2001, SIGA.

[14]  Francesca Poggiolesi,et al.  A CUT-FREE SIMPLE SEQUENT CALCULUS FOR MODAL LOGIC S5 , 2008, The Review of Symbolic Logic.

[15]  A. Avron.  The method of hypersequents in the proof theory of propositional non-classical logics , 1996 .

[16]  Albert J. J. Anglberger,et al.  The Logic of Obligation as Weakest Permission - (Short Version) , 2012, DEON.

[17]  Alexandru Baltag,et al.  Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision , 2006, WoLLIC.

[18]  Arnon Avron,et al.  Hypersequents, logical consequence and intermediate logics for concurrency , 1991, Annals of Mathematics and Artificial Intelligence.

[19]  Martin Lange,et al.  Cut-free sequent systems for temporal logic , 2008, J. Log. Algebraic Methods Program..

[20]  Johan van Benthem,et al.  Deontics = Betterness + Priority , 2010, DEON.

[21]  Greg Restall,et al.  Proofnets for S5: sequents and circuits for modal logic , 2007 .

[22]  Tomohiro Hoshi,et al.  Taking Mistakes Seriously: Equivalence Notions for Game Scenarios with off Equilibrium Play , 2011, LORI.