Universal Logic

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 [5], including axiomatization, completeness and incompleteness results, results on complexity etc. The more general logic MTL was introduced in [1]. The results show that these logics have very good properties. Some of many new results on them will be presented.

Lecture notes (see A and B )

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



Mathematical fuzzy logic Petr Hájek and Petr Cintula Institute of Computer science Academy of Sciences of the Czech republic - Prague Czech Republic
	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 [5], including axiomatization, completeness and incompleteness results, results on complexity etc. The more general logic MTL was introduced in [1]. The results show that these logics have very good properties. Some of many new results on them will be presented. Lecture notes (see A and B ) 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
		References 1. F. Esteva and L. Godo: Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124/3, (2001), pp.271-288. 2. S.Gottwald, A Treatise on Many-Valued Logics, Research Studies Press, Baldock, 2000. 3. K. Gödel, Zum intuitionistischen Aussagenkalkül, Anzieger Akademie der Wissenschaften Wien, Math. - naturwissensch. Klasse, 69 (1932), pp.65-66. 4. P. Hájek, Fuzzy logic and arithmetical hierarchy III, Studia Logica, 68 (2001), pp.129-142. 5. P.Hájek, Metamathematics of Fuzzy Logics, Kluwer, Dodrecht, 1998. 6. P. Hájek, Why Fuzzy Logic?, in A Companion to Philosophical Logic, J. Dale (ed), Blackwell Publishers, Massachusetts, 2002, pp.595-605. 7. P.Hájek, J. Paris, and J. Shepherdson, The Liar Paradox and Fuzzy Logics, Journal of Symbolic Logic, 65/1 (2000), pp.339-346. 8. Novák, I. Perfilieva, and J. Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer, Norwell, 1999. 9. J. Lukasiewicz, A. Tarski, Untersuchungen über den Aussagenkalkül. Comptes Rendus de la Siciete des Sciences et des Letters de Varsovie, cl. iii 23 (1930), 1-21. Lecture Notes will be available here by January 31th 2005.