XEP33FLO Fuzzy Logic

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In the winter semester 2025/26, the course will be held in English or Czech, depending on the students enrolled. The schedule will be decided in September by a Doodle poll so that it would fit as many students as possible.

The curricula will contain a general introduction to fuzzy sets (about 5 weeks), followed by lectures on specific topics chosen by you from several options. Please, read the previous curricula and prepare your comments.

The text on fuzzy propositional operations was updated on March 21, 2025. Comments are welcome.

The lectures can be streamed and recorded on request. However, they are mostly covered by recordings from previous years.

BRUTE

Basic notions. System of cuts of a fuzzy set, theorem on representation of fuzzy sets by cuts, conversion between vertical and horizontal representation.

Fuzzy inclusion. Fuzzy negations. Representation theorem for fuzzy negations.

Fuzzy complements. Fuzzy conjunctions (triangular norms).

Representation theorem for strict conjunctions. Representation theorem for nilpotent conjunctions. Fuzzy disjunctions (triangular conorms)., representation theorems. Fuzzy algebras and their properties. Examples of fuzzy intersections and unions.

Fuzzy disjunctions, representation theorems. Examples of fuzzy intersections and unions. Properties of fuzzy propositional and set operations.

Fuzzy implications and biimplications.

Alternative definition of a sigma-algebra and probability on it.

Comparison to the Kolmogorov axiomatization of probability.

Fuzzifications of sigma-algebras.

Properties of fuzzy sigma-algebras.

Motivation and definitions of probabilities of fuzzy events.

Properties and characterizations of probabilities of fuzzy events.

Syntax of classical logic: formulas, axioms, deduction, theorems.

Deduction theorem in classical logic.

Semantics of classical logic: evaluation, tautologies. Interplay of syntax and semantics of classical logic: soundness, completeness.

Basic logic: axioms, theorems, semantics.

Deduction in basic logic.

Completeness of basic logic.

Other types of fuzzy logics: Gödel logic, product logic (its alternative axiomatization, formulas which are tautologies of product logic but not of Gödel or basic logic), Łukasiewicz logic and its alternative axiomatization.

Rational Pavelka logic.

Compactness of logics.

Testing tautologies in Gödel and Łukasiewicz logic.