Reasoning about mathematical fuzzy logic and its future

This paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author. Our goal is to think of the state of the art in mathematical fuzzy logic (MFL) and to outline some of the tasks on which, in the author's opinion, MFL should focus in the future. In our discussion, we will mention not only the basic theory, but also its extension called fuzzy logic in broader sense (FLb). The paradigm of the latter is to be the logic of natural human reasoning, whose most essential characteristic is the use of natural language. Besides brief description of FLb, we will also mention some of its applications. On the basis of that, we will ponder on other possible directions for research, namely the possibility of using FLn as a metatheory of fuzzy mathematics, as a proper tool for modeling of the main manifestations of the phenomenon of vagueness, and as a reasonable tool for developing models of linguistic semantics.

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

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

[3]  Vilém Novák,et al.  Fuzzy Sets as a Special Mathematical Model of Vagueness Phenomenon , 2006 .

[4]  Vilém Novák,et al.  Submitted/to appear: Fuzzy Sets and Systems Supported by: , 2022 .

[5]  L. Godo,et al.  On the hierarchy of t-norm based residuated fuzzy logics , 2003 .

[6]  Vilém Novák,et al.  Are fuzzy sets a reasonable tool for modeling vague phenomena? , 2005, Fuzzy Sets Syst..

[7]  Vilém Novák,et al.  On the semantics of perception-based fuzzy logic deduction: Research Articles , 2004 .

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

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

[10]  Irina Perfilieva,et al.  Logical foundations of rule-based systems , 2006, Fuzzy Sets Syst..

[11]  Karel Chvalovský.  On the independence of axioms in BL and MTL , 2012, Fuzzy Sets Syst..

[12]  Siegfried Gottwald,et al.  Fuzzy sets and fuzzy logic: the foundations of application - from a mathematical point of view , 1993, Artificial intelligence.

[13]  Irina Perfilieva.  Normal forms in BL-algebra and their contribution to universal approximation of functions , 2004, Fuzzy Sets Syst..

[14]  V. Novák.  Fuzzy sets and their applications , 1989 .

[15]  Vilém Novák,et al.  Towards Formalized Integrated Theory of Fuzzy Logic , 1995 .

[16]  Ingo Glöckner.  Fuzzy Quantifiers: A Computational Theory , 2006 .

[17]  Vilém Novák,et al.  Towards automatic modeling of economic texts , 2007 .

[18]  Vilém Novák,et al.  On fuzzy type theory , 2005, Fuzzy Sets Syst..

[19]  Frank Klawonn,et al.  Fuzzy Control - Fundamentals, Stability and Design of Fuzzy Controllers , 2006, Studies in Fuzziness and Soft Computing.

[20]  Vilém Novák,et al.  On the semantics of perception‐based fuzzy logic deduction , 2004, Int. J. Intell. Syst..

[21]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[22]  Martina Danková.  Representation of logic formulas by normal forms , 2002, Kybernetika.

[23]  Vilém Novák,et al.  Fuzzy logic deduction with crisp observations , 2004, Soft Comput..

[24]  Siegfried Gottwald,et al.  Mathematical Fuzzy Logics , 2008, Bulletin of Symbolic Logic.

[25]  Siegfried Gottwald,et al.  Fuzzy Sets and Fuzzy Logic , 1993 .

[26]  J. A. Goguen,et al.  The logic of inexact concepts , 1969, Synthese.

[27]  Rozsa Peter.  Review: Leon Henkin, Completeness in the Theory of Types , 1951 .

[28]  Dale Jacquette.  A companion to philosophical logic , 2002 .

[29]  Petr Sgall,et al.  The Meaning Of The Sentence In Its Semantic And Pragmatic Aspects , 1986 .

[30]  Vilém Novák,et al.  Logical structure of fuzzy IF-THEN rules , 2006, Fuzzy Sets Syst..

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

[32]  Petr Cintula,et al.  Fuzzy class theory , 2005, Fuzzy Sets Syst..

[33]  A. Mostowski.  On a generalization of quantifiers , 1957 .

[34]  Robert Babuska,et al.  Fuzzy algorithms for control , 1999 .

[35]  Michal Holčapek.  Monadic L-fuzzy Quantifiers of the Type <1n,1> , 2007, EUSFLAT Conf..

[36]  Daniele Mundici.  Ulam Games, Lukasiewicz Logic, and AF C*-Algebras , 1993, Fundam. Informaticae.

[37]  Petr Cintula,et al.  Fuzzy logics as the logics of chains , 2006, Fuzzy Sets Syst..

[38]  M. Ying,et al.  Quantifiers, modifiers and qualifiers in fuzzy logic , 1996, Soft Computing in Intelligent Systems and Information Processing. Proceedings of the 1996 Asian Fuzzy Systems Symposium.

[39]  Stanley Peters,et al.  Quantifiers in language and logic , 2006 .

[40]  Antonín Dvorák,et al.  L-fuzzy quantifiers of type 23291232a determined by fuzzy measures , 2009, Fuzzy Sets Syst..

[41]  Pascual Campoy Cervera,et al.  Computational intelligence in decision and control , 2008, CDC 2008.

[42]  Martine De Cock,et al.  LINGUISTIC MODIFIERS: AN OVERVIEW , 1999 .

[43]  U. Höhle.  Commutative, residuated 1—monoids , 1995 .

[44]  Irina Perfilieva,et al.  Fuzzy Logic Normal Forms for Control Law Representation , 1999 .

[45]  Petr Hájek,et al.  A dialogue on fuzzy logic , 1997, Soft Comput..

[46]  Petr Hájek.  Why Fuzzy Logic , 2007 .

[47]  H. Ono,et al.  Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151 , 2007 .

[48]  P. Sgall,et al.  Topic-focus articulation, tripartite structures, and semantic content , 1998 .

[49]  Petr Cintula,et al.  Relations in Fuzzy Class Theory: : Initial steps , 2008, Fuzzy Sets Syst..

[50]  P. L. Peterson.  Intermediate Quantities , 2020 .

[51]  Vilém Novák,et al.  A General Methodology for Managerial Decision Making Using Intelligent Techniques , 2009 .

[52]  Tomás Kroupa.  Filters in fuzzy class theory , 2008, Fuzzy Sets Syst..

[53]  Petr Hájek,et al.  What is mathematical fuzzy logic , 2006, Fuzzy Sets Syst..

[54]  Da Ruan,et al.  Fuzzy Logic: A Spectrum of Theoretical & Practical Issues , 2007 .

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

[56]  Bernard De Baets,et al.  EQ-algebras , 2009, Fuzzy Sets Syst..

[57]  Lotfi A. Zadeh,et al.  A COMPUTATIONAL APPROACH TO FUZZY QUANTIFIERS IN NATURAL LANGUAGES , 1983 .

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

[59]  Vilém Novák,et al.  ANALYSIS OF PERIODICAL TIME SERIES USING SOFT COMPUTING METHODS , 2008 .

[60]  Vilém Novák,et al.  Linguistic If-Then Rules in Large Scale Application of Fuzzy Control , 2000 .

[61]  Lotfi A. Zadeh,et al.  Quantitative fuzzy semantics , 1971, Inf. Sci..

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

[63]  Lluis Godo,et al.  On Product Logic with Truth-constants , 2006, J. Log. Comput..

[64]  Petr Hájek,et al.  Ten Questions and One Problem on Fuzzy Logic , 1999, Ann. Pure Appl. Log..

[65]  Vilém Novák,et al.  A comprehensive theory of trichotomous evaluative linguistic expressions , 2008, Fuzzy Sets Syst..

[66]  Pavel Materna,et al.  Concepts and objects , 1998 .

[67]  V. Novák,et al.  Mathematical Principles of Fuzzy Logic , 1999 .

[68]  Vilém Novák,et al.  Mathematical Fuzzy Logic in Modeling of Natural LanguageSemantics , 2007 .

[69]  Vilém Novák,et al.  The sorites paradox and fuzzy logic , 2003, Int. J. Gen. Syst..

[70]  Achille C. Varzi.  Supervaluationism and Paraconsistency , 2000 .

[71]  Franco Montagna,et al.  The $L\Pi$ and $L\Pi\frac{1}{2}$ logics: two complete fuzzy systems joining Łukasiewicz and Product Logics , 2001, Arch. Math. Log..

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

[73]  Barbara H. Partee,et al.  Towards an Investigation of the Relation between Topic-Focus Articulation and Tripartite Structures , 1998 .

[74]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[75]  Vilém Novák,et al.  EQ-algebra-based fuzzy type theory and its extensions , 2011, Log. J. IGPL.

[76]  Lluis Godo,et al.  On expansions of WNM t-norm based logics with truth-constants , 2010, Fuzzy Sets Syst..

[77]  Vilém Novák.  On the syntactico-semantical completeness of first-order fuzzy logic. II. Main results , 1990, Kybernetika.

[78]  Lotfi A. Zadeh,et al.  Precisiated Natural Language (PNL) , 2004, AI Mag..

[79]  Zeungnam Bien,et al.  Fuzzy Logic and its Applications to Engineering, Information Sciences, and Intelligent Systems , 1995, Theory and Decision Library.

[80]  John McCarthy,et al.  Programs with common sense , 1960 .

[81]  Lotfi A. Zadeh,et al.  Precisiated Natural Language , 2007, Aspects of Automatic Text Analysis.

[82]  Vilém Novák.  Perception-Based Logical Deduction , 2004, Fuzzy Days.

[83]  R. Mesiar,et al.  Logical, algebraic, analytic, and probabilistic aspects of triangular norms , 2005 .

[84]  Vilém Novák,et al.  Formal theories and linguistic descriptions , 2004, Fuzzy Sets Syst..

[85]  Bernd Reusch.  Computational Intelligence, Theory and Applications , 1997 .

[86]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[87]  Antonín Dvorák,et al.  Basic Properties of L-fuzzy Quantifiers of the Type <1> Determined by Fuzzy Measures , 2009, IFSA/EUSFLAT Conf..

[88]  Franco Montagna,et al.  MV-algebras with internal states and probabilistic fuzzy logics , 2009, Int. J. Approx. Reason..

[89]  G. Klir,et al.  Fuzzy logic in geology , 2004 .

[90]  B. Russell.  Mathematical Logic as Based on the Theory of Types , 1908 .

[91]  Vilém Novák,et al.  A formal theory of intermediate quantifiers , 2008, Fuzzy Sets Syst..

[92]  Diderik Batens.  Frontiers of Paraconsistent Logic , 2000 .

[93]  R. Keefe.  Theories of vagueness , 2000 .

[94]  Michal Holčapek,et al.  Monadic L-fuzzy Quantifiers of the Type . , 2007 .

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

[96]  Dag Westerstaåhl,et al.  Quantifiers in Formal and Natural Languages , 1989 .

[97]  Irina Perfilieva.  Logical approximation , 2002, Soft Comput..

[98]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[99]  Petr Cintula,et al.  The L[Pi] and L[Pi1/2]propositional and predicate logics , 2001, Fuzzy Sets Syst..

[100]  Petr Cintula,et al.  Weakly Implicative (Fuzzy) Logics I: Basic Properties , 2006, Arch. Math. Log..

[101]  Vilém Novák,et al.  On good EQ-algebras , 2011, Fuzzy Sets Syst..

[102]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[103]  Vilém Novák,et al.  Analysis of seasonal time series using fuzzy approach , 2010, Int. J. Gen. Syst..

[104]  Vilém Novák,et al.  Learning rule base in linguistic expert systems , 2002, Soft Comput..

[105]  Etienne E. Kerre,et al.  Fuzzy If-Then Rules in Computational Intelligence: Theory and Applications , 2012 .

[106]  Petr Hájek,et al.  Formal systems of fuzzy logic and their fragments , 2007, Ann. Pure Appl. Log..

[107]  Vilém Novák.  Fuzzy logic with countable evaluated syntax revisited , 2007, Fuzzy Sets Syst..

[108]  Martina Danková,et al.  Relational compositions in Fuzzy Class Theory , 2009, Fuzzy Sets Syst..

[109]  E. Trillas,et al.  in Fuzzy Logic , 2002 .

[110]  Petr Hájek,et al.  Fss 5598 Modc+(sc) Triangular Norm Based Predicate Fuzzy Logics , 2022 .

[111]  Leora Morgenstern,et al.  Introduction: Progress in formal commonsense reasoning , 2004, Artif. Intell..

[112]  Roberto Cignoli,et al.  Ulam’s game , 2000 .

[113]  Bernadette Bouchon-Meunier,et al.  Fuzzy Logic And Soft Computing , 1995 .

[114]  Moataz Saleh El-Zekey,et al.  Representable good EQ-algebras , 2010, Soft Comput..

[115]  Johan van Benthem,et al.  Handbook of Logic and Language , 1996 .

[116]  S. Gottwald,et al.  Triangular norm-based mathematical fuzzy logics , 2005 .

[117]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[118]  Rostislav Horcík,et al.  Solution of a system of linear equations with fuzzy numbers , 2008, Fuzzy Sets Syst..

[119]  Irina Perfilieva.  Normal forms in BL and ŁΠ algebras of functions , 2004, Soft Comput..