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Simple Consequence Relations
 Information and Computation
, 1991
"... We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (incl ..."
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Cited by 99 (18 self)
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We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and nonmonotonic logics) and for a general, semanticsindependent classification of standard connectives via equations on consequence relations (these include Girard's "multiplicatives" and "additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as good representations in each case (especially from the implementation point of view) are explained. We end by briefly outlining (with examples) some methods for developing nonuniform, but still efficient, representations of consequence relations.
The Method of Hypersequents in the Proof Theory of Propositional NonClassical Logics
 IN LOGIC: FROM FOUNDATIONS TO APPLICATIONS, EUROPEAN LOGIC COLLOQUIUM
, 1994
"... ..."
A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 53 (3 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
Residuated fuzzy logics with an Involutive Negation
"... Residuated fuzzy logic calculi are related to continuous tnorms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant 0, namely :' is ' ! 0. However, thi ..."
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Cited by 26 (7 self)
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Residuated fuzzy logic calculi are related to continuous tnorms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant 0, namely :' is ' ! 0. However, this negation behaves quite differently depending on the tnorm. For a nilpotent tnorm (a tnorm which is isomorphic to / Lukasiewicz tnorm), it turns out that : is an involutive negation. However, for tnorms without nontrivial zero divisors, : is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous tnorms without nontrivial zero divisors and extended with an involutive negation. 1 1 Introduction Residuated fuzzy (manyvalued) logic calculi are related to continuous tnorms which are used as truth functions for the conjunction connective, and their residua as truth functions for the implication. Main examples are / Lukasiewicz (/L), Godel...
Hypersequents and the proof theory of intuitionistic fuzzy logic
 Computer Science Logic CSL’2000. Proceedings, LNCS 1862
, 2000
"... Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and ..."
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Cited by 20 (10 self)
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Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, and allows cutelimination. A question by Takano regarding the eliminability of the TakeutiTitani density rule is answered affirmatively. 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 15 (12 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.
Quantified propositional Gödel logics
 In Proceedings of LPAR’2000, LNAI 1955
, 2000
"... Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also ..."
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Cited by 14 (7 self)
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Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also shown that the compact infinitevalued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation. 1
A Deterministic Terminating Sequent Calculus for GödelDummett logic
, 1999
"... We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4 ..."
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Cited by 13 (0 self)
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We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4LC are invertible, thus allowing a deterministic proofsearch procedure.
A Survey on Different Triangular NormBased Fuzzy Logics
, 1999
"... Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizati ..."
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Cited by 13 (1 self)
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Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, manyvalued logic, Frank tnorm 1 Introduction A manyvalued propositional logic with a continuum of truth values modelled by the unit interval [0; 1] is quite often called a fuzzy logic. In such a logic, the conjunction is usually interpreted by a triangular norm. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...
A Tableaux System for GödelDummett Logic Based on a Hypersequential Calculus
 In Automated Reasoning with Tableaux and Related Methods (Tableaux’2000), volume 1847 of Lectures Notes in Artificial Intelligence
, 2000
"... We present a terminating contractionfree calculus GLC for the propositional fragment of GodelDummett Logic LC. GLC uses hypersequents, and unlike other Gentzentype calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a ..."
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Cited by 11 (0 self)
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We present a terminating contractionfree calculus GLC for the propositional fragment of GodelDummett Logic LC. GLC uses hypersequents, and unlike other Gentzentype calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a deterministic tableaux system for LC. This tableaux system is presented in the last section. I A Review of LC and GLC In [Go33] Godel introduced a sequence fG n g of nvalued logics, as well as an infinitevalued matrix G ! in which all the G n s can be embedded. He used these matrices to show some important properties of intuitionistic logic. The logic of G ! was later axiomatized by Dummett in [Du59] and is known since then as Dummett's LC. It probably is the most important intermediate logic, one that turns up in several places, like the provability logic of Heyting's Arithmetics ([Vi82]) and relevance logic ([DM71]) and recently fuzzy logic([Ha98]). semantically LC corresponds to lin...