Petr Cintula and Libor Behounek

Institute of Computer Science
Academy of Sciences of Czech Republic

Originating as an attempt to provide solid logical foundations for fuzzy set theory, and motivated also by philosophical and computational problems of vagueness and imprecision, Mathematical Fuzzy Logic has become a significant subfield of mathematical logic. Research in this area focuses on many-valued logics with linearly ordered truth values [2] and has yielded elegant and deep mathematical theories and challenging problems, thus continuing to attract an ever increasing number of researchers [5].

Fuzzy logics emerged from Zadeh's fuzzy set theory, which has become extremely popular in computer science and engineering, giving rise to a broad area of research, with countless applications. At the beginning of the 1990s, Petr Hájek started a "tour de force" to provide solid logical foundations for fuzzy logic. In his approach, soon followed by numerous researchers in mathematical logic, fuzzy logics were taken as non-classical many-valued deductive systems with a semantics given by totally ordered algebraic structures
(typically based on t-norms on the real unit interval). Hajek's monograph [1] started the study of t-norm-based fuzzy logics by the methods of algebraic logic, thus giving birth to Mathematical Fuzzy Logic. The last decade has witnessed a significant development of Mathematical Fuzzy Logic, summarized in the new Handbook [3], and a proliferation of various systems of fuzzy logic.

The tutorial will follow Chapter I of the recently published Handbook of Mathematical Fuzzy Logic [4]. The electronic version of the chapter will be made available to the participants of the tutorial.

The tutorial will cover the following topics:

* Propositional logics of continuous t-norms: standard and general semantics, axiomatic systems, completeness theorems.

* Variations of basic propositional fuzzy logics: adding or discarding axioms or connectives.

* Families of fuzzy logics in the logical landscape: fuzzy logics among substructural logics, core fuzzy logics, fuzzy logics as algebraically implicative semilinear logics.

* Metamathematics of propositional fuzzy logics: completeness theorems, functional representation, proof theory, computational complexity.

* Predicate fuzzy logics: syntax, semantics, completeness, notable axiomatic theories.

References

[1] Petr Hajek. Metamathematics of Fuzzy Logic. Volume 4 of Trends in Logic, Kluwer, Dordrecht, 1998.

[2] Libor Behounek, Petr Cintula. Fuzzy logics as the logics of chains. Fuzzy Sets and Systems 157:604-610, 2006.

[3] Petr Cintula, Petr Hajek, Carles Noguera (eds). Handbook of Mathematical Fuzzy Logic. Volumes 37 and 38 of Studies in Logic, Mathematical Logic and Foundations, College Publications, London, 2011.

[4] Libor Behounek, Petr Cintula, Petr Hajek. Introduction to Mathematical Fuzzy Logic. Chapter I of [1] pp. 1-101.

[5] MathFuzzLog, a working group of Mathematical Fuzzy Logic: www.mathfuzzlog.org.

Prerequisites

This is an introductory course and as such will be highly
self-contained. Students will only be assumed to have a basic knowledge of classical propositional and predicate logic and elementary set theory, and some rudiments of universal algebra.