Substructural Logics and Residuated Lattices — an Introduction

This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebraic structures for substructural logics, and some recent developments of their algebraic study are presented. Based on these facts, we conclude at the end that substructural logics are logics of residuated structures, and in this way explain why sequent systems are suitable for formalizing substructural logics.

[1]  R. P. Dilworth,et al.  Residuated Lattices. , 1938, Proceedings of the National Academy of Sciences of the United States of America.

[2]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[3]  Saul Kripke,et al.  A completeness theorem in modal logic , 1959, Journal of Symbolic Logic.

[4]  Hao Wang,et al.  A survey of mathematical logic , 1963 .

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

[6]  V. N. Grisin.  PREDICATE AND SET-THEORETIC CALCULI BASED ON LOGIC WITHOUT CONTRACTIONS , 1982 .

[7]  Alasdair Urquhart,et al.  The undecidability of entailment and relevant implication , 1984, Journal of Symbolic Logic.

[8]  Hiroakira Ono,et al.  Logics without the contraction rule , 1985, Journal of Symbolic Logic.

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

[10]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[11]  Hiroakira Ono,et al.  Structural Rules and a Logical Hierarchy , 1990 .

[12]  K. I. Rosenthal.  Quantales and their applications , 1990 .

[13]  Eiji Kiriyama,et al.  The contraction rule and decision problems for logics without structural rules , 1991, Stud Logica.

[14]  A. Troelstra.  Lectures on linear logic , 1992 .

[15]  Robert K. Meyer,et al.  The finite model property for BCK and BCIW , 1994, Stud Logica.

[16]  Yves Lafont.  The Finite Model Property for Various Fragments of Linear Logic , 1997, J. Symb. Log..

[17]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[18]  Kazushige Terui,et al.  The finite model property for various fragments of intuitionistic linear logic , 1999, Journal of Symbolic Logic.

[19]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[20]  Giulia Battilotti,et al.  Basic Logic and the Cube of its Extensions , 1999 .

[21]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[22]  Lluis Godo,et al.  Basic Fuzzy Logic is the logic of continuous t-norms and their residua , 2000, Soft Comput..

[23]  Willem J. Blok,et al.  The finite embeddability property for residuated lattices, pocrims and BCK-algebras , 2002, Reports Math. Log..

[24]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[25]  C. Tsinakis,et al.  A Survey of Residuated Lattices , 2002 .

[26]  Franco Montagna,et al.  A Proof of Standard Completeness for Esteva and Godo's Logic MTL , 2002, Stud Logica.

[27]  Hiroakira Ono.  Completions of Algebras and Completeness of Modal and Substructural Logics , 2002, Advances in Modal Logic.

[28]  Hiroakira Ono,et al.  Closure Operators and Complete Embeddings of Residuated Lattices , 2003, Stud Logica.

[29]  Peter Jipsen,et al.  Algebraic Aspects of Cut Elimination , 2004, Stud Logica.