Fuzzy logic, continuity and effectiveness

Abstract. It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

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

[2]  Giangiacomo Gerla,et al.  Recursively Enumerable L-Sets , 1987, Math. Log. Q..

[3]  Giangiacomo Gerla,et al.  Logics with approximate premises , 1998 .

[4]  Giangiacomo Gerla,et al.  Fuzzy Logic: Mathematical Tools for Approximate Reasoning , 2001 .

[5]  Giangiacomo Gerla,et al.  Decidability, Recursive Enumerability and Kleene Hierarchy For L-Subsets , 1989, Math. Log. Q..

[6]  Jan Pavelka,et al.  On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi , 1979, Math. Log. Q..

[7]  L. A. ZADEH,et al.  The concept of a linguistic variable and its application to approximate reasoning - I , 1975, Inf. Sci..

[8]  P. Hájek.  Fuzzy logic and arithmetical hierarchy , 1995 .

[9]  George Gratzer,et al.  Universal Algebra , 1979 .

[10]  Andrzej W. Jankowski,et al.  On decidable consequence operators , 1986, Stud Logica.

[11]  L. A. Zadeh,et al.  Fuzzy logic and approximate reasoning , 1975, Synthese.

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

[13]  Wolfgang Wechler,et al.  Universal Algebra for Computer Scientists , 1992, EATCS Monographs on Theoretical Computer Science.

[14]  Garrett Birkhoff,et al.  Representations of lattices by sets , 1948 .

[15]  R. Wójcicki.  Theory of Logical Calculi: Basic Theory of Consequence Operations , 1988 .

[16]  Giangiacomo Gerla,et al.  Fuzzy subsets: a constructive approach , 1992 .

[17]  Jan Pavelka,et al.  On Fuzzy Logic III. Semantical completeness of some many-valued propositional calculi , 1979, Math. Log. Q..

[18]  Petr Hájek,et al.  On Logics of Approximate Reasoning , 1992, Logic at Work.

[19]  Giangiacomo Gerla,et al.  Logics with approximate premises , 1998, Int. J. Intell. Syst..

[20]  Giangiacomo Gerla,et al.  Closure Operators in Fuzzy Set Theory , 1999 .

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

[22]  Giangiacomo Gerla,et al.  Decidability and Recursive Enumerability for Fuzzy Subsets , 1988, IPMU.

[23]  Jan Pavelka,et al.  On Fuzzy Logic I Many-valued rules of inference , 1979, Math. Log. Q..

[24]  Lotfi A. Zadeh,et al.  Fuzzy Algorithms , 1968, Inf. Control..

[25]  Giangiacomo Gerla,et al.  An Extension Principle for Closure Operators , 1996 .

[26]  Giangiacomo Gerla,et al.  Generated necessities and possibilities , 1992, Int. J. Intell. Syst..

[27]  Venkat Murali,et al.  Lattice of fuzzy subalgebras and closure systems in I x , 1991 .

[28]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .