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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 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
Sequent and Hypersequent Calculi for Abelian and Łukasiewicz Logics
 ACM Transactions on Computational Logic
, 2005
"... We present two embeddings of infinitevalued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of latticeordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A a ..."
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Cited by 19 (6 self)
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We present two embeddings of infinitevalued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of latticeordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A and ̷L and terminating versions of these calculi; labelled single sequent calculi for A and ̷L of complexity coNP; unlabelled single sequent calculi for A and ̷L. 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
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
A Set Theory Within Fuzzy Logic
"... This paper proposes a possibility of developing an axiomatic set theory, as firstorder theory within the framework of fuzzy logic in the style of [13]. In classical ZFC, we use an analogy of the construction of a Booleanvalued universeover a particular algebra of truth valuesto show the n ..."
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Cited by 2 (1 self)
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This paper proposes a possibility of developing an axiomatic set theory, as firstorder theory within the framework of fuzzy logic in the style of [13]. In classical ZFC, we use an analogy of the construction of a Booleanvalued universeover a particular algebra of truth valuesto show the nontriviality of our theory. We present a list of problems and research tasks. 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.
Tnorm based logics with an independent involutive negation
 Fuzzy Sets and Systems
"... In this paper we investigate the addition of arbitrary independent involutive negations to tnorm based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any tnorm based logic satisfying some basic properties, its extension b ..."
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Cited by 1 (0 self)
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In this paper we investigate the addition of arbitrary independent involutive negations to tnorm based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any tnorm based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We will deal with both propositional and predicate logics. 1
CONTRIBUTIONS to SCIENCE, 2 (1): 922 (2001)
"... Artificial intelligence is a relatively new scientific and technological field which studies the nature of intelligence by using computers to produce intelligent behaviour. Initially, the main goal was a purely scientific one, understanding human intelligence, and this remains the aim of cognitive s ..."
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Artificial intelligence is a relatively new scientific and technological field which studies the nature of intelligence by using computers to produce intelligent behaviour. Initially, the main goal was a purely scientific one, understanding human intelligence, and this remains the aim of cognitive scientists. Unfortunately, such an ambitious and fascinating goal is not only far from being achieved but has yet to be satisfactorily approached. Fortunately, however, artificial intelligence also has an engineering goal: building systems that are useful to people even if the intelligence of such systems has no relation whatsoever with human intelligence, and therefore being able to build them does not necessarily provide any insight into the nature of human intelligence. This engineering goal has become the predominant one among artificial intelligence researchers and has produced impressive results, ranging from knowledgebased systems to autonomous robots, that have been applied to many different domains. Furthermore, artificial intelligence products and services today represent an annual market of tens of billions of dollars worldwide.
Supported by the Austrian Federal Ministry of Education, Science and Culture
"... There is a number of completely integrable gravity theories in two dimensions. We study the metricaffine approach on a 2dimensional spacetime and display a new integrable model. Its properties are described and compared with the known results of Poincaré gauge gravity. PACS: 04.50.+h, 04.20.Fy, 04 ..."
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There is a number of completely integrable gravity theories in two dimensions. We study the metricaffine approach on a 2dimensional spacetime and display a new integrable model. Its properties are described and compared with the known results of Poincaré gauge gravity. PACS: 04.50.+h, 04.20.Fy, 04.20.Jb, 04.60.Kz, 02.30.Ik I.