XEP33FLO Fuzzy Logic

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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.