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Formal fuzzy logic
Libor Behounek, Tuesday 16:40, room 137, Celetna 20, Prague
The course will present formal systems of fuzzy logic from the point of view of substructural
logics and give an introduction to formal fuzzy mathematics in the framework of higher-order
fuzzy logic. The contents will center around the newest results in the field of study.
Syllabus:
Strike-through items were omitted due to too different backgrounds of the audience. Items
with grey background were addressed at seminars of applied logic at ICS AS CR (not part of
the course, but recommended for students)

A motivation for fuzzy logic as a logic of comparative truth: the paradox of the heap, the
philosophy of vagueness, mathematical and logical modeling, historical remarks, topicality.

Prerequisities in logic and the foundations of mathematics: multi-sorted logic with identity,
Hilbert-style and Gentzen-style calculi, substructural logics, Tarski structural consequence, matrix
semantics, Lindenbaum algebra, semantical completeness, varieties and quasivarieties, residuated
lattices, local and global consequence relations, the deduction theorem, second and higher-order
logic, Henkin-style semantics, simple type theory, the foundationalistic architecture of
mathematics.

Propositional fuzzy logics among substructural logics: weakly implicative logics,
internalization of the local consequence relation, the role and equivalents of prelinearity,
propositional connectives, Hilbert-style calculi for fuzzy logics, the meaning of special axioms,
particular systems of fuzzy logics and their properties, related logics, algebraic completeness
theorems, the deduction theorem, hypersequent calculi, standard completeness theorems, t-norms
and uninorms, functional representability.

First and higher-order fuzzy logic: semantics and axiomatics of first-order fuzzy logics,
completeness, formal theories over fuzzy logic, properties of non-classical and non-contractive
theories, non-contractive solutions to Russell's paradox, formal fuzzy set theories, higher-order
fuzzy logic.

Formal fuzzy mathematics: the theory of fuzzy classes, the theory of fuzzy relations, inner truth
values and characteristic functions, quantifiers, fuzzy modalities, intensional semantics.

Interpretations and applications of formal fuzzy mathematics: a solution to the paradox of the
heap, higher-order vagueness, the role of non-contractivity, interpretations of truth-functionality, a
comparison with probability theory and other theories of vagueness, traditional fuzzy sets,
engineers' fuzzy mathematics, attitudes of the philosophy of vagueness, the limits of applicability
of formal fuzzy mathematics.
Literature:
Paoli F.: Substructural Logics: A Primer. Kluwer 2002.
Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht 1998.
Gottwald S.: Treatise on Many-Valued Logic. Research Studies Press, Baldock 2001.
Metcalfe G.: Proof theory for propositional fuzzy logics. PhD thesis, King’s College, London
2004. (Available online at http://www.dcs.kcl.ac.uk/pg/metcalfe)
Cintula P.: Weakly implicative (fuzzy) logics I: Basic properties, Archive for Mathematical
Logic 45 (2006): 673–704.
Cintula P., Hájek P.: Triangular norm based predicate fuzzy logics. To appear in the
proceedings of Linz Seminar 2005.
Běhounek L., Cintula P.: Fuzzy class theory. Fuzzy Sets and Systems 154 (2005): 34–55.