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25
Hypersequent calculi for Gödel logics: a survey
 Journal of Logic and Computation
, 2003
"... Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for ..."
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Cited by 19 (4 self)
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Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinitevalued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to firstorder as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1
Herbrand’s theorem for prenex Gödel logic and its consequences for theorem proving
 IN LOGIC FOR PROGRAMMING AND AUTOMATED REASONING LPAR’2001, 201–216. LNAI 2250
, 2001
"... Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calc ..."
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Cited by 16 (11 self)
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Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.
Terminological difficulties in fuzzy set theory  the case of “intuitionistic fuzzy sets”, Fuzzy Sets and Systems 156 (3
, 2005
"... Abstract: This note points out a terminological clash between Atanassov's “intuitionistic fuzzy sets ” and what is currently understood as intuitionistic logic. They differ both by their motivations and their underlying mathematical structure. Furthermore, Atanassov's construct is isomorph ..."
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Cited by 15 (1 self)
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Abstract: This note points out a terminological clash between Atanassov's “intuitionistic fuzzy sets ” and what is currently understood as intuitionistic logic. They differ both by their motivations and their underlying mathematical structure. Furthermore, Atanassov's construct is isomorphic to intervalvalued fuzzy sets and other similar notions, even if their interpretive settings and motivation are quite different, the latter capturing the idea of illknown membership grade, while the former starts from the idea of evaluating degrees of membership and nonmembership independently. This paper is a plea for a clarification of terminology, based on mathematical resemblances and the comparison of motivations between “intuitionistic fuzzy sets ” and other theories. 1.
Firstorder Gödel logics
, 2006
"... Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It i ..."
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Cited by 14 (5 self)
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Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negationfree, and existential fragments of all firstorder Gödel logics are also characterized.
A SchütteTait style cutelimination proof for firstorder Gödel logic
 In Automated Reasoning with Tableaux and Related Methods (Tableaux’02), volume 2381 of LNAI
, 2002
"... Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity o ..."
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Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity of cutformulas in it. We compare this SchütteTait style cutelimination proof to a Gentzen style proof. 1
Density Elimination
, 2008
"... Density elimination, a close relative of cut elimination, consists of removing applications of the TakeutiTitani density rule from derivations in Gentzenstyle (hypersequent) calculi. Its most important use is as a crucial step in establishing standard completeness for syntactic presentations of fu ..."
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Cited by 6 (3 self)
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Density elimination, a close relative of cut elimination, consists of removing applications of the TakeutiTitani density rule from derivations in Gentzenstyle (hypersequent) calculi. Its most important use is as a crucial step in establishing standard completeness for syntactic presentations of fuzzy logics; that is, completeness with respect to algebras based on the real unit interval [0,1]. This paper introduces the method of density elimination by substitutions. For general classes of (firstorder) hypersequent calculi, it is shown that density elimination by substitutions is guaranteed by known sufficient conditions for cut elimination. These results provide the basis for uniform characterizations of calculi complete with respect to densely and linearly ordered algebras. Standard completeness follows for many firstorder fuzzy logics using a DedekindMacNeillestyle completion and embedding.
Characterization of the axiomatizable prenex fragments of firstorder Gödel logics
 IN 33RD INTERNATIONAL SYMPOSIUM ON MULTIPLEVALUED LOGIC. MAY 2003
, 2003
"... The prenex fragments of firstorder infinitevalued Gödel logics are classified. It is shown that the prenex Gödel logics characterized by finite and by uncountable subsets of [0,1] are axiomatizable, and that the prenex fragments of all countably infinite Gödel logics are not axiomatizable. ..."
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Cited by 6 (1 self)
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The prenex fragments of firstorder infinitevalued Gödel logics are classified. It is shown that the prenex Gödel logics characterized by finite and by uncountable subsets of [0,1] are axiomatizable, and that the prenex fragments of all countably infinite Gödel logics are not axiomatizable.
Herbrand’s theorem, skolemization, and proof systems for firstorder ̷Lukasiewicz logic
 Journal of Logic and Computation
"... Abstract. An approximate Herbrand theorem is established for firstorder infinitevalued Łukasiewicz Logic and used to obtain a prooftheoretic proof of Skolemization. These results are then used to define proof systems in the framework of hypersequents. In particular, a calculus lacking cutelimin ..."
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Abstract. An approximate Herbrand theorem is established for firstorder infinitevalued Łukasiewicz Logic and used to obtain a prooftheoretic proof of Skolemization. These results are then used to define proof systems in the framework of hypersequents. In particular, a calculus lacking cutelimination is defined for the firstorder logic characterized by linearly ordered MValgebras, a cutfree calculus with an infinitary rule for the full firstorder Łukasiewicz Logic, and a cutfree calculus with finitary rules for its onevariable fragment. 1
Cut elimination for first order Gödel logic by hyperclause resolution
"... Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one ..."
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Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one of the main formalizations of fuzzy logic.
Standard completeness for extensions of MTL: an automated approach
, 2012
"... We provide general conditions on hypersequent calculi that guarantee standard completeness for the formalized logics. These conditions are implemented in the PROLOG system AxiomCalc that takes as input any suitable axiomatic extension of Monoidal Tnorm Logic MTL and outputs a hypersequent calculus ..."
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Cited by 3 (3 self)
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We provide general conditions on hypersequent calculi that guarantee standard completeness for the formalized logics. These conditions are implemented in the PROLOG system AxiomCalc that takes as input any suitable axiomatic extension of Monoidal Tnorm Logic MTL and outputs a hypersequent calculus for the logic and the result of the check. Our approach subsumes many existing results and allows for the computerized discovery of new fuzzy logics.