PETR CINTULA, LIBOR BĚHOUNEK: MATHEMATICAL FUZZY LOGIC

Tutorial (6-8 hours), Autumn School of Logic, Pec p. Sn. Oct 20-25, 2004.

Mathematical fuzzy logic or fuzzy logic in the narrow sense is symbolic many-valued logic with a comparative notion of truth. Best understood systems are t-norm based, i.e. using continuous t-norms on the interval [0,1] as standard semantics of conjunction and their residua as standard semantics of implication. General algebras of truth functions are so-called BL-algebras (MTL algebras). The corresponding logic BL – basic fuzzy logic, both propositional and predicate calculus – is elaborated in Hajek’s monograph, including axiomatization, completeness and incompleteness results, results on complexity etc. The results show that these logics have very good properties. Some of many new results on them will be presented.

Description of the contents of the tutorial:

  • Vagueness, fuzziness vs. probability, comparative degrees of truth, standard [0,1]-valued semantics, t-norms, residua
  • Fuzzy propositional calculi: axiomatic systems MTL, BL, and their schematic extensions (incl. Lukasiewicz and Gödel-Dummett logics)
  • Algebraic semantics: Hajek’s BL-algebras, Chang’s MV-algebras, etc.
  • Theorems: completeness, subdirect decomposition, deduction, compactness, complexity issues
  • Adding truth constants: Pavelka style extensions
  • Fuzzy predicate calculi: Tarski semantics, completeness and incompleteness results, arithmetical hierarchy
  • Proof theory: hypersequent calculi
  • Alternative semantics: Kripke and game-thoretic semantics
  • Functional representation: McNaughton functions, Pierce-Birkhoff conjecture
  • Extending language: logics with globalization, involutive negation, additional conjunction
  • Higher-order fuzzy logic, fuzzy mathematics