Grafting Modalities onto Substructural Implication Systems

We investigate the semantics of the logical systems obtained by introducing the modalities □ and ⋄ into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in a previous paper.

[1]  Vito F. Sinisi,et al.  Entailment: The Logic of Relevance and Necessity , 1996 .

[2]  Nuel Belnap,et al.  Linear Logic Displayed , 1989, Notre Dame J. Formal Log..

[3]  James W. Garson.  Modularity and Relevant Logic , 1989, Notre Dame J. Formal Log..

[4]  R. A. Bull,et al.  Basic Modal Logic , 1984 .

[5]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

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

[7]  Kosta Dosen,et al.  Sequent-systems and groupoid models. I , 1988, Stud Logica.

[8]  Vladimir Alexiev,et al.  Applications of Linear Logic to Computation: An Overview , 1994, Log. J. IGPL.

[9]  Marcello D'Agostino,et al.  The Taming of the Cut. Classical Refutations with Analytic Cut , 1994, J. Log. Comput..

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

[11]  Alasdair Urquhart,et al.  Semantics for relevant logics , 1972, Journal of Symbolic Logic.

[12]  Krysia Broda,et al.  Transformation Methods in LDS , 1999 .

[13]  V. Michele Abrusci.  Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic , 1991, J. Symb. Log..

[14]  D. Gabbay.  LDS - Labelled Deductive Systems: Volume 1 - Foundations , 1994 .

[15]  K. Dosen,et al.  Models for normal intuitionistic modal logics , 1984 .

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

[17]  Nuel Belnap,et al.  Display logic , 1982, J. Philos. Log..

[18]  Heinrich Wansing,et al.  The Logic of Information Structures , 1993, Lecture Notes in Computer Science.

[19]  Nuel Belnap,et al.  Gupta's rule of revision theory of truth , 1982, J. Philos. Log..

[20]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[21]  Fiora Pirri,et al.  A uniform tableau method for intuitionistic modal logics I , 1994, Stud Logica.

[22]  Avron Arnon.  The semantics and proof theory of linear logic , 1988 .

[23]  Gisèle Fischer Servi.  On modal logic with an intuitionistic base , 1977 .

[24]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[25]  Wendy MacCaull,et al.  Relational semantics and a relational proof system for full Lambek calculus , 1998, The Journal of Symbolic Logic.

[26]  M. Fitting.  Proof Methods for Modal and Intuitionistic Logics , 1983 .

[27]  Kosta Dosen.  Sequent-systems and groupoid models. II , 1989, Stud Logica.

[28]  Gerard Allwein,et al.  Kripke models for linear logic , 1993, Journal of Symbolic Logic.