A Gentle Introduction to Mathematical Fuzzy Logic
Petr Cintula
Institute of Computer Science, Academy of Sciences of the Czech Republic
Pod Vodárenskou věží 2, 182 07 Prague, Czech Republic
Carles Noguera
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic
Pod Vodárenskou věží 4, 182 08 Prague, Czech Republic
Abstract
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 (MFL) has become a significant subfield of mathematical logic. Research in this area focuses on many-valued logics with linearly ordered truth theories and challenging problems, thus continuing to attract an ever increasing number of researchers.
The goal of this course is to provide an up-to-date introduction to MFL. Starting with the motivations and historical origins of the area, we present MFL, its main methods, and its core agenda. In particular, we focus on some of its better known logic systems (Łukasiewicz and Gödel–Dummett logics, BL, MTL) and present a general theory of fuzzy logics. Finally, we give an overview of several currently active lines of research in the development and application of fuzzy logics.
Syllabus
1. Motivations and historical origins of mathematical fuzzy logic (slides)
motivation from computer science, engineering, linguistics, and philosophy; introduction to Łukasiewicz and Gödel–Dummett propositional logics: proof system and standard semantics
2. Basic properties of propositional Łukasiewicz and Gödel–Dummett logic (slides)
completeness with respect to general and chain-based semantics; standard completeness; functional representation; finite model property; computational complexity; algebraization, axiomatic extensions
application: fuzzy logic and probability
3. Predicate Łukasiewicz and Gödel–Dummett logic (slides)
syntax, semantics, axiomatizability; undecidability; skolemization; witness semantics; arithmetical hierarchy
application: fuzzy set theory; formal fuzzy mathematics; theory of counterfactual implications
4. Łukasiewicz and Gödel–Dummett logics as logics of continuous t-norms (slides)
Hájek Logic HL; (left-)continuous t-norms and their residua; Mostert–Shield's theorem; Product logic
application: fuzzy logic as logic of resources/costs and logical omniscience paradoxc
5. The growing family of fuzzy logics (slides)
logics in richer propositional languages; Esteva and Godo’s MTL; the quest for basic fuzzy logic; core semilinear logics as a general theory for fuzzy logics
6. Further lines of research and open problems (slides)
recent development in the core theory of MFL; game theory; model theory; proof theory
Exercises
Exercises 1–12 are in the slides for lesson 2
Exercises 13–19 are in the slides for lesson 3
Exercises 20–25 are in the slides for lesson 4
Exercises 26–29 are in the slides for lesson 5
Exercise 30: solve one of the open problems in slides for lesson 6
Prerequisites
This course if rather self-contained: the students are only assumed to have a basic knowledge on classical propositional and predicate logic and elementary knowledge of set theory and lattice theory.
Study material
Libor Běhounek, Petr Cintula, and Petr Hájek: Introduction to Mathematical Fuzzy Logic. In Petr Cintula, Carles Noguera, and Petr Hájek (eds). Handbook of Mathematical Fuzzy Logic. Studies in Logic, Mathematical Logic and Foundations, vol. 37 and 38, London, College Publications, pages 1–101, 2011.
Full course description
Mathematical Fuzzy Logic (MFL) is a subdiscipline of Mathematical Logic that studies a certain family of formal logical systems whose algebraic semantics involve some notion of truth degree. The central role of truth degrees in MFL stems from three distinct historical origins of the discipline:
Philosophical motivations: MFL is motivated by the need to model correct reasoning in the presence of vague predicates (such as 'tall', 'intelligent', 'beautiful', or `simple') when more standard systems, such as classical logic, might be considered inappropriate. Vague predicates correspond to properties without clear boundaries and are omnipresent in natural language and reasoning and, thus, dealing with them is also unavoidable in linguistics. They constitute an important logical problem as clearly seen when confronting sorites paradoxes, where a sufficient number of applications of a legitimate deduction rule (modus ponens) leads from (apparently?) true premises, to a clearly false conclusion: (1) one grain of wheat does not make a heap, (2) a group of grains of wheat does not become a heap just by adding one more grain, therefore: (3) one million grains of wheat does not make a heap. One possible way to tackle this problem is the degree-based approach related to logical systems studied by MFL. In this proposal one assumes that truth comes in degrees which, in the case of the sorites series, vary from the absolute truth of “one grain of wheat does not make a heap” to the absolute falsity of “one million grains of wheat does not make a heap”, through the intermediate decreasing truth degrees of “n grains of wheat do not make a heap”.
Fuzzy Set Theory: In 1965 Lotfi Zadeh proposed fuzzy sets as a new mathematical paradigm for dealing with imprecision and gradual change in engineering applications [Zad]. Their conceptual simplicity (a fuzzy set is nothing more than a classical set endowed with a [0,1]-valued function which represents the degree to which an element belongs to the fuzzy set) provided the basis for a substantial new research area and applications such as a very popular engineering toolbox used successfully in many technological applications, in particular, in so-called fuzzy control. This field is referred to as fuzzy logic, although its mathematical machinery and the concepts investigated are largely unrelated to those typically used and studied in (Mathematical) Logic. Nevertheless, there have been some attempts to present fuzzy logic in the sense of Zadeh as a useful tool for dealing with vagueness paradoxes (see e.g. [Gog]).
Many-valued logics: The 20th century witnessed a proliferation of logical systems whose intended algebraic semantics, in contrast to classical logic, have more than two truth values. Some systems even have infinitely-many truth values, like Łukasiewicz logic [LukTar] or Gödel–Dummett logic [Dum]. Many-valued systems were inspired by a variety of motivations, only occasionally related to the aforementioned vagueness problems. More recently, Algebraic Logic has developed a paradigm in which most systems of non-classical logics can be seen as many-valued logics, because they are given a semantics in terms of algebras with more than two truth values. From this point of view, many-valued logics encompass wide well-studied families of logical systems such as relevance logics, intuitionistic and superintuitionistic logics and substructural logics in general (see e.g. [GJKO]).
MFL was born at the crossroads of these three areas. At the beginning of the nineties of last century, a small group of researchers (including among others Esteva, Godo, Gottwald, Hájek, Höhle, and Novák), persuaded that fuzzy set theory could be a useful paradigm for dealing with logical problems related to vagueness, began investigations dedicated to providing solid logical foundations for such a discipline. In other words, they started developing logical systems in the tradition of Mathematical Logic that would have the [0,1]-valued operations used in fuzzy set theory as their intended semantics. In the course of this development, they realized that some of these logical systems were already known such as Łukasiewicz and Gödel–Dummett infinitely valued logics. Both systems turned out to be strongly related to fuzzy sets because they are [0,1]-valued and the truth functions interpreting their logical connectives are, in fact, of the same kind (t-norms, t-conorms, negations) as those used to compute the combination (resp. intersection, union, complement) of fuzzy sets. These pioneering efforts produced a number of important papers and even some monographs (especially [Haj], but also [Got] or [NPM]).
As a result of this work, fuzzy logics have become a respectable family in the broad landscape of non-classical logics studied by Mathematical Logic. It has been clearly shown that fuzzy logics can be seen as a particular kind of many-valued systems (or substructural logics) whose intended semantics is typically based on algebras of linearly ordered truth values. In order to distinguish it from the works on fuzzy set theory misleadingly labeled as fuzzy logic, the study of these systems has been called Mathematical Fuzzy Logic. In the last years we have seen the blossoming of MFL, with a plethora of works going far beyond the developments of Hájek's landmark monograph [Haj] and resulting into an extensive corpus of results (partly) collected in the new reference handbook [CHN].
The aim of this course is to present an up-to-date introduction to MFL. Starting with the motivations and historical origins of the area, we will present MFL as a subdiscipline of Mathematical Logic which, as such, has acquired the typical core agenda of this field. We will study in details some of its better known logic systems (Łukasiewicz and Gödel–Dummett logics, BL, MTL) and present a general theory of fuzzy logics. We will finish with an overview of several currently active lines of research in the development and application of fuzzy logics.
References
[CHN] Petr Cintula, Petr Hájek, Carles Noguera (eds). Handbook of Mathematical Fuzzy Logic. Studies in Logic, Mathematical Logic and Foundations, vol. 37 and 38, London, College Publications, 2011.
[Dum] Michael Dummett. A propositional calculus with denumerable matrix. Journal of Symbolic Logic, 24:97–106, 1959.
[GJKO] Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics, vol. 151, Amsterdam, Elsevier, 2007.
[Gog] Joseph Amadee Goguen. The logic of inexact concepts. Synthese, 19:325–373, 1969.
[Got] Siegfried Gottwald. A Treatise on Many-Valued Logics, volume 9 of Studies in Logic and Computation. Research Studies Press, Baldock, 2001.
[Haj] Petr Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998.
[LukTar] Jan Łukasiewicz and Alfred Tarski. Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, 23:30–50, 1930.
[NPM] Vilém Novák, Irina Perfilieva, and Jiří Močkoř. Mathematical Principles of Fuzzy Logic. Kluwer, Dordrecht, 2000.
[Zad] Lotfi A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965.