Mathematical Fuzzy Logic

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Vagueness pervades human language, perception, and reasoning since the beginning of their existence, as many natural-language predicates (e.g. "young", "tall", "hot") have no sharp boundaries. A popular approach to this phenomenon (although usually disregarded by philosophers of vagueness) is to stipulate that truth comes in degrees (which can be taken realistically or, at least, as a good "model" of vagueness). Even though many-valued logics were introduced for other purposes already during the first half of the twentieth century by Lukasiewicz [1], a systematic treatment of vagueness by means of the many-valued approach began only after Zadeh`s paper [2] proposed the fuzzy sets paradigm in 1965.

  • Mathematical Fuzzy Logic is a subdiscipline of Mathematical Logic.

This sharply separates it from the usual use of the term fuzzy logic, which is more a fancy label used for anything even very distantly related to some degrees and usually not related to anything which could be called (mathematical) logic. In fact the term fuzzy logic is often used for dealing not with degrees of truth but with degrees of any other quality or modality (possibility, entropy, necessity, or even probability). Thus, for a long time, fuzzy logic has been (and still is) understood as an engineering toolbox. Driven just by applications, it lacked (meta)theoretical foundational grounds and general results; mostly developed by engineers for particular purposes, it suffered from arbitrariness in definitions and often even mathematical imprecision. During the nineties, this situation started to change mainly thanks to the works by Gottwald [3] [4], Hájek [5], Cignoli et al. [6], Novák et al. [7], and by many other mathematical logicians. The efforts of these researchers culminated in establishing Mathematical Fuzzy Logic as a respectable member of the broad family of non-classical logics.

  • Mathematical Fuzzy Logic has its own agenda.

Keeping in mind that Mathematical Fuzzy Logic pertains in the first place to Mathematical Logic we can easily deduce the core of its agenda:

1. Proof systems: Hilbert, Gentzen, natural deduction, tableaux, resolution, computational complexity, etc.

2. Algebraic semantics: residuated lattices, MTL-algebras, BL-algebras, MV-algebras, Abstract Algebraic Logic, functional representations, etc.

3. Game theory: Giles games, Rényi-Ulam games, evaluation games, etc.

4. First-order logics: axiomatizations, arithmetical hierarchy, model theory, etc.

5. Higher-order systems: type theories, Fuzzy Class Theory, and formal fuzzy mathematics.

6. Extended systems: dealing with probability of fuzzy events, adding modalities or truth constants, Dynamic fuzzy logics, evaluated syntax, etc.

Of course we cannot forget the roots of Mathematical Fuzzy Logic and this gives us two final main areas in its agenda (specially important areas in fact, as only good applications can measure a usefulness of a mathematical theory):

7. Philosophical issues: connections with vagueness and uncertainty.

8. Applied logical calculi: foundations of logic programming, logic-based reasoning about similarity, description logics, etc.

  • Mathematical Fuzzy Logic is an open and growing field of study.

One should not be dogmatic as new subfields could emerge any day. Nevertheless, we want to keep two (rather weak) main principles for the scope of our group:

a. being part of Mathematical Logic

b. being related to fuzziness/vagueness/degrees of truth


  1. Jan Lukasiewicz, O logice trojwartosciowej (On Three-Valued Logic), Ruch filozoficzny 5 (1920) 170-171.
  2. Lotfi A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.
  3. Siegfried Gottwald, Fuzzy Sets and Fuzzy Logic: The foundations of application from a Mathematical Point of View, Vieweg, Wiesbaden, 1993.
  4. Siegfried Gottwald, A Treatise on Many-Valued Logics, Studies in Logic and Computation 9, Research Studies Press, Baldock, 2001.
  5. Petr Hájek, Metamathematics of Fuzzy Logic, Trends in Logic 4, Kluwer, Dordrecht, 1998.
  6. Roberto Cignoli, Itala M.L. D'Ottaviano and Daniele Mundici, Algebraic Foundations of Many-Valued Reasoning, Trends in Logic 7, Kluwer, Dordercht, 1999.
  7. Vilém Novák, Irina Perfilieva, and Jiří Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer, Dordrecht, 2000.

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