On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

[1]  Petr Hájek,et al.  On witnessed models in fuzzy logic , 2007, Math. Log. Q..

[2]  Tadeusz Litak,et al.  Completions of GBL-algebras: negative results , 2008 .

[3]  Xavier Caicedo,et al.  Standard Gödel Modal Logics , 2010, Stud Logica.

[4]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[5]  Frank Wolter,et al.  Handbook of Modal Logic, Volume 3 (Studies in Logic and Practical Reasoning) , 2006 .

[6]  Melvin Fitting,et al.  Many-valued modal logics , 1991, Fundam. Informaticae.

[7]  Melvin Fitting,et al.  Many-valued modal logics II , 1992 .

[8]  Petr Cintula,et al.  From fuzzy logic to fuzzy mathematics: A methodological manifesto , 2006, Fuzzy Sets Syst..

[9]  Franco Montagna,et al.  Axiomatization of Anz Residuated Fuzzy Logic Defined bz a Continuous T-norm , 2003, IFSA.

[10]  Willem J. Blok,et al.  Protoalgebraic logics , 1986, Stud Logica.

[11]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[12]  Xavier Caicedo,et al.  A Godel Modal Logic , 2009, 0903.2767.

[13]  Petr Hájek,et al.  On theories and models in fuzzy predicate logics , 2006, Journal of Symbolic Logic.

[14]  Wlesław Dziobiak,et al.  Concerning axiomatizability of the quasivariety generated by a finite Heyting or topological Boolean algebra , 1982 .

[15]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[16]  Christian G. Fermüller,et al.  Tableaux for Finite-Valued Logics with Arbitrary Distribution Modalities , 1998, TABLEAUX.

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

[18]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[19]  L. Godo,et al.  Exploring a Syntactic Notion of Modal Many-Valued Logics , 2008, SOCO 2008.

[20]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[21]  Bruno Teheux,et al.  Completeness results for many-valued \Lukasiewicz modal systems and relational semantics , 2006 .

[22]  Christian G. Fermüller,et al.  Elimination of Cuts in First-order Finite-valued Logics , 1993, J. Inf. Process. Cybern..

[23]  stanisław j. surma,et al.  an algorithm for axiomatizing every finite logic**The paper was presented at the International Symposium on Multiple-Valued Logic, Morgantown, West Virginia, U.S.A., May, 1974. An abstract of this paper is published in [3]. , 1977 .

[24]  Leon F M Horsten.  Foundations of the Formal Sciences II : Applications of mathematical Logic in Philosophy and Linguistics. Trends in Logic: Studia Logica Library , 2003 .

[25]  H. Ono.  Substructural Logics and Residuated Lattices — an Introduction , 2003 .

[26]  Lluís Godo,et al.  A Fuzzy Modal Logic for Similarity Reasoning , 1999 .

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

[28]  Franco Montagna,et al.  Varieties of BL-algebras , 2005, Soft Comput..

[29]  Ulrich Höhle,et al.  Non-classical logics and their applications to fuzzy subsets : a handbook of the mathematical foundations of fuzzy set theory , 1995 .

[30]  Lluis Godo,et al.  Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results , 2007, Fuzzy Sets Syst..

[31]  Michael M. Richter,et al.  Intelligence and artificial intelligence : an interdisciplinary debate , 1998 .

[32]  Pascal Ostermann,et al.  Many-Valued Modal Propositional Calculi , 1988, Math. Log. Q..

[33]  Umberto Straccia,et al.  Managing uncertainty and vagueness in description logics for the Semantic Web , 2008, J. Web Semant..

[34]  W. Blok,et al.  A finite basis theorem for quasivarieties , 1986 .

[35]  P. Aglianò,et al.  Varieties of BL- algebras I: General properties. , 2003 .

[36]  Petr Hájek.  Logics of Knowing and Believing , 1998 .

[37]  Rineke Verbrugge,et al.  Strong Completeness and Limited Canonicity for PDL , 2008, J. Log. Lang. Inf..

[38]  Petr Hájek,et al.  A many-valued modal logic , 1996 .

[39]  A. U.S..  DUALISABILITY Unary Algebras and Beyond , 1974 .

[40]  Petr Hájek.  On witnessed models in fuzzy logic II , 2007, Math. Log. Q..

[41]  Franco Montagna,et al.  Distinguished algebraic semantics for t-norm based fuzzy logics: Methods and algebraic equivalencies , 2009, Ann. Pure Appl. Log..

[42]  Petr Hájek,et al.  Making fuzzy description logic more general , 2005, Fuzzy Sets Syst..

[43]  Mai Gehrke,et al.  NON-CANONICITY OF MV-ALGEBRAS , 2002 .

[44]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[45]  A. Ciabattoni,et al.  Adding Modalities to MTL and its Extensions , 2022 .

[46]  Petr Hájek,et al.  On Modal Logics for Qualitative Possibility in a Fuzzy Setting , 1994, UAI.

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

[48]  Franco Montagna,et al.  A note on the first-order logic of complete BL-chains , 2008, Math. Log. Q..

[49]  Josep Maria Font,et al.  A Survey of Abstract Algebraic Logic , 2003, Stud Logica.

[50]  Franco Montagna,et al.  Notes on Strong Completeness in Lukasiewicz, Product and BL Logics and in Their First-Order Extensions , 2006, Algebraic and Proof-theoretic Aspects of Non-classical Logics.

[51]  B. Jonnson.  Algebras Whose Congruence Lattices are Distributive. , 1967 .

[52]  Matthias Baaz,et al.  First-order Gödel logics , 2007, Ann. Pure Appl. Log..

[53]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[54]  Nicola Olivetti,et al.  Proof Systems for a Gödel Modal Logic , 2009, TABLEAUX.

[55]  Franco Montagna,et al.  Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras , 2004, Stud Logica.