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25
From axioms to analytic rules in nonclassical logics
 23RD ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 2008
"... We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of propositional nonclassical logics including intermediate, fuzzy and ..."
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Cited by 29 (14 self)
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We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of propositional nonclassical logics including intermediate, fuzzy and substructural logics. Our work encompasses many existing results, allows for the definition of new calculi and contains a uniform semantic proof of cutelimination for hypersequent calculi.
Uniform rules and dialogue games for fuzzy logics
 LPAR 2004, Springer LNCS
"... Abstract. We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental tnorm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequentsofrelations. Such ..."
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Cited by 20 (10 self)
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Abstract. We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental tnorm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequentsofrelations. Such a framework can be interpreted via a particular class of dialogue games combined with bets, where the rules reflect possible moves in the game. The problem of determining the validity of atomic relational hypersequents is shown to be polynomial for each logic, allowing us to develop CoNP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics. 1
Expanding the realm of systematic proof theory
"... Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language ..."
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Cited by 7 (3 self)
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Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language of classical linear logic without exponentials. We then give a systematic procedure to transform axioms up to the level P ′ 3 of the hierarchy into inference rules in multipleconclusion (hyper)sequent calculi, which enjoy cutelimination under a certain condition. This allows a systematic treatment of logics which could not be dealt with in the previous approach. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P ′ 3. The case study of Abelian and ̷Lukasiewicz logic is outlined. 1
Adding modalities to MTL and its extensions
 Proceedings of the Linz Symposium
, 2005
"... Abstract. Monoidal tnorm logic MTL and related fuzzy logics are extended with various modalities distinguished by the axiom (A ∨ B) → (A ∨ B). Such modalities include Linear logiclike exponentials, the globalization (or Delta) operator, and truth stressers like “very true”. Extensions of MTL with ..."
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Cited by 7 (1 self)
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Abstract. Monoidal tnorm logic MTL and related fuzzy logics are extended with various modalities distinguished by the axiom (A ∨ B) → (A ∨ B). Such modalities include Linear logiclike exponentials, the globalization (or Delta) operator, and truth stressers like “very true”. Extensions of MTL with modalities are presented here via axiomatizations, hypersequent calculi, and algebraic semantics, and related to standard algebras based on tnorms. Embeddings of logics, decidability, and the finite embedding property are also investigated. 1
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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Cited by 6 (4 self)
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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
Combining supervaluation and degree based reasoning under vagueness
"... Abstract. Two popular approaches to formalize adequate reasoning with vague propositions are usually deemed incompatible: On the one hand, there is supervaluation with respect to precisification spaces, which consist in collections of classical interpretations that represent admissible ways of makin ..."
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Cited by 4 (2 self)
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Abstract. Two popular approaches to formalize adequate reasoning with vague propositions are usually deemed incompatible: On the one hand, there is supervaluation with respect to precisification spaces, which consist in collections of classical interpretations that represent admissible ways of making vague atomic statements precise. On the other hand, tnorm based fuzzy logics model truth functional reasoning, where reals in the unit interval [0,1] are interpreted as degrees of truth. We show that both types of reasoning can be combined within a single logic SŁ, that extends both: Łukasiewicz logic Ł and (classical) S5, where the modality corresponds to ‘... is true in all complete precisifications’. Our main result consists in a game theoretic interpretation of SŁ, building on ideas already introduced by Robin Giles in the 1970s to obtain a characterization of Ł in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. In our case the experiments are replaced by random evaluations with respect to a given probability distribution over permissible precisifications. 1
Giles’s Game and the Proof Theory of ̷Lukasiewicz Logic
"... Abstract. In the 1970s, Robin Giles introduced a game combining Lorenzenstyle dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in ̷Lukasiewicz logic. In this pa ..."
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Cited by 3 (2 self)
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Abstract. In the 1970s, Robin Giles introduced a game combining Lorenzenstyle dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in ̷Lukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work on the proof theory of ̷Lukasiewicz logic.
Proof theory for Casari’s comparative logics
 J. Log. Comput
"... Abstract. Comparative logics were introduced by Casari in the 1980s to treat aspects of comparative reasoning occurring in natural language. In this paper Gentzen systems are defined for these logics by means of a special mix rule that combines calculi for various substructural logics with a hyperse ..."
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Abstract. Comparative logics were introduced by Casari in the 1980s to treat aspects of comparative reasoning occurring in natural language. In this paper Gentzen systems are defined for these logics by means of a special mix rule that combines calculi for various substructural logics with a hypersequent calculus for Meyer and Slaney’s Abelian logic. Cutelimination is established for all these systems, and as a consequence, a positive answer is given to an open problem on the decidability of the basic comparative logic. 1
Neutrosophic logics on NonArchimedean Structures
 Critical Review, Creighton University, USA
"... We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ ax ..."
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We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ axiom. These logics are built as different extensions of standard manyvalued logics (namely, Lukasiewicz’s, Gödel’s, Product, and Post’s logics). The informal sense of Archimedes ’ axiom is that anything can be measured by a ruler. Also logical multiplevalidity without Archimedes ’ axiom consists in that the set of truth values is infinite and it is not wellfounded and wellordered. We consider two cases of nonArchimedean multivalued logics: the first with manyvalidity in the interval [0, 1] of hypernumbers and the second with manyvalidity in the ring Zp of padic integers. On the base of nonArchimedean valued logics, we construct nonArchimedean valued interval neutrosophic logics by which we can describe neutrality phenomena.