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On witnessed models in fuzzy logic III – witnessed Gödel logics
Author(s) -
Häjek Petr
Publication year - 2010
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200810047
Subject(s) - truth value , mathematics , unit interval , t norm fuzzy logics , countable set , monoidal t norm logic , truth function , axiom , discrete mathematics , pairwise comparison , tautology (logic) , fuzzy logic , semantics (computer science) , propositional calculus , fuzzy set , algebra over a field , propositional variable , fuzzy number , intermediate logic , pure mathematics , description logic , computer science , theoretical computer science , artificial intelligence , programming language , statistics , geometry
Abstract Gödel (fuzzy) logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise different sets of (standard) tautologies, all these sets being non‐arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is Π 2 ‐complete. Further similar results are obtained (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)