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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 80 (5 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
A Survey of Residuated Lattices
"... Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered ..."
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Cited by 62 (6 self)
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Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered groups, ideal lattices of rings, linear logic and multivalued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability. We end with a list of open problems that we hope will stimulate further research.
Abstract algebraic logic and the deduction theorem
 Bull. of Symbolic Logic
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Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL
, 2006
"... Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem ..."
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Cited by 19 (5 self)
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Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
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 15 (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
On the Minimum ManyValued Modal Logic over a Finite Residuated Lattice
, 2009
"... This article deals with manyvalued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum mod ..."
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Cited by 11 (4 self)
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This article deals with manyvalued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truthconstants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truthconstants in the language.
Cancellative Residuated Lattices
 ALGEBRA UNIVERSALIS
, 2003
"... Cancellative residuated lattices are natural generalizations of latticeordered groups (ℓgroups). Although cancellative monoids are defined by quasiequations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that c ..."
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Cited by 10 (6 self)
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Cancellative residuated lattices are natural generalizations of latticeordered groups (ℓgroups). Although cancellative monoids are defined by quasiequations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to ℓgroups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an orderpreserving injection of the lattice of all lattice varieties into the subvariety lattice of CanRL. We define generalized MValgebras and generalized BLalgebras and prove that the cancellative integral members of these varieties are precisely the negative cones of ℓgroups, hence the latter form a variety, denoted by LG −. Furthermore we prove that the map that sends a subvariety of ℓgroups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of LG to the lattice of subvarieties of LG −. Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie’s characterization of categorically equivalent varieties.
Tutorial: Complexity of ManyValued Logics
 In Proc. 31st International Symposium on MultipleValued Logics, IEEE CS Press, Los Alamitos
, 2001
"... this article selfcontained. ..."
The logic of equilibrium and abelian lattice ordered groups
 Transactions of the AMS
, 2002
"... We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. “Truth values ” are interpreted as deviations From a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statem ..."
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Cited by 5 (1 self)
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We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. “Truth values ” are interpreted as deviations From a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and its equivalent algebraic semantics BAL is definitionally ∗ Funding for the first and third author has been provided by FOMEC. † Funding for the second author has been provided by FONDECYT 1020621, Facultad de Ciencias Exactas, U.N. de La Plata, and FOMEC. 1 equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the coNPcompleteness of the tautology problem of Bal. 1