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46
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 52 (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
Probabilistic similarity logic
 in Uncertainty in Artificial Intelligence
"... Many interesting research problems, such as ontology alignment and collective classification, require probabilistic and collective inference over imprecise evidence. Existing approaches are typically adhoc and problemspecific, requiring significant effort to devise and provide poor generalizability ..."
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Cited by 16 (12 self)
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Many interesting research problems, such as ontology alignment and collective classification, require probabilistic and collective inference over imprecise evidence. Existing approaches are typically adhoc and problemspecific, requiring significant effort to devise and provide poor generalizability. In this paper, we introduce probabilistic similarity logic (PSL), a simple, yet powerful language for describing problems which require probabilistic reasoning about similarity where, in addition to reasoning probabilistically, we want to capture both logical constraints and imprecision. We prove that PSL inference is polynomial and outline a wide range of application areas for PSL. 1.
From axioms to analytic rules in nonclassical logics
"... 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 15 (9 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. 1.
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 13 (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
Mathematical fuzzy logic as a tool for the treatment of vague information
 Information Sciences
, 2005
"... The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by ..."
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Cited by 10 (1 self)
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The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1
Triangle algebras: A formal logic approach to intervalvalued residuated lattices, Fuzzy Sets and Systems
"... In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of intervalvalued residuated lattices (IVRLs). Furthermor ..."
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Cited by 7 (6 self)
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In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of intervalvalued residuated lattices (IVRLs). Furthermore, we present Triangle Logic (TL), a system of manyvalued logic capturing the tautologies of IVRLs. Triangle algebras are used to cast the essence of using closed intervals of L as truth values into a set of appropriate logical axioms. Our results constitute a crucial first step towards solving an important research challenge: the axiomatic formalization of residuated tnorm based logics on L I, the lattice of closed intervals of [0,1], in a similar way as was done for formal fuzzy logics on the unit interval. Key words: formal logic, intervalvalued fuzzy set theory, residuated lattices
A characterization of intervalvalued residuated lattices
"... As is wellknown, residuated lattices (RLs) on the unit interval correspond to leftcontinuous tnorms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solv ..."
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Cited by 6 (6 self)
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As is wellknown, residuated lattices (RLs) on the unit interval correspond to leftcontinuous tnorms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solved if it is restricted, making only a few simple and intuitive assumptions, to the class of intervalvalued residuated lattices (IVRLs). More specifically, we derive a full characterization of product and implication in IVRLs in terms of their counterparts on the base RL. To this aim, we use triangle algebras, a recently introduced variety of RLs that serves as an equational representation of IVRLs. Key words: intervalvalued fuzzy set theory, residuated lattices, triangle algebras
Adding Modalities to MTL and its Extensions
"... 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 w ..."
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Cited by 4 (0 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
The pseudolinear semantics of intervalvalued fuzzy logics, Information Sciences 179
, 2009
"... Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called intervalvalued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and co ..."
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Cited by 4 (4 self)
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Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called intervalvalued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the socalled pseudoprelinear triangle algebras are subdirect products of pseudolinear triangle algebras. This can be compared with MTLalgebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. As a consequence, we are able to prove the pseudochain completeness of Pseudolinear Triangle Logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of MTL (Monoidal Tnorm based Logic). This result also provides a better insight in the structure of triangle algebras; it enables us, amongst others, to prove properties of pseudoprelinear triangle algebras more easily. It is known that there is a onetoone correspondence between triangle algebras and couples (L, α), in which L is a residuated lattice and α an element in that residuated lattice. We give a schematic overview of these properties (and a number of others that can be imposed on a triangle algebra), and the corresponding necessary and sufficient conditions on L and α. Key words: intervalvalued fuzzy set theory, residuated lattices, formal logic
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 3 (2 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.