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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.
Satisfiability in Fuzzy Logics
 Neural Network World
, 2000
"... The notion of validation set of a formula in a fuzzy logic was introduced by Butnariu, Klement and Zafrany. It is the set of all evaluations of the formula for all possible evaluations of its atomic symbols. We generalize this notion to sets of formulas. This enables us to formulate and prove ge ..."
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Cited by 13 (1 self)
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The notion of validation set of a formula in a fuzzy logic was introduced by Butnariu, Klement and Zafrany. It is the set of all evaluations of the formula for all possible evaluations of its atomic symbols. We generalize this notion to sets of formulas. This enables us to formulate and prove generalized theorems on satisfiability and compactness of various fuzzy logics. We also propose and study new types of satisfiability and consistency degree of a set of formulas. Key words: Fuzzy logic, manyvalued logic, triangular norm, compactness of a logic. 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. There are different ways of interpretation of the implication; this, together with the choice of the triangular norm, leads to a large collection of fuzzy logics with different semantics. Here we wor...
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
New foundations for imperative logic I: Logical connectives, consistency, and quantifiers
 Noûs
, 2008
"... Abstract. Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: “kiss me and hug me ” is the conjunction of “kiss me ” with “hug me”. This example may suggest that declarative and imperative logic are isomorphic: just as th ..."
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Cited by 4 (4 self)
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Abstract. Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: “kiss me and hug me ” is the conjunction of “kiss me ” with “hug me”. This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied⎯what more is there to say? Much more, I argue. “If you love me, kiss me”, a conditional imperative, mixes a declarative antecedent (“you love me”) with an imperative consequent (“kiss me”); it is satisfied if you love and kiss me, violated if you love but don’t kiss me, and avoided if you don’t love me. So we need a logic of threevalued imperatives which mixes declaratives with imperatives. I develop such a logic. 1.
Beyond Two: Theory and applications of multiplevalued logic
"... Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the wellknown LindenbaumTarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainl ..."
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Cited by 3 (0 self)
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Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the wellknown LindenbaumTarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the socalled protoalgebraic logics. I will show how the idea (already explored by WÓJCKICI and NOWAK) ofdeÞning logics using a scheme of “preservation of degrees of truth ” (as opposed to the more usual one of “preservation of truth”) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MValgebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a “strong version ” deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truthpreserving scheme, that is, the corresponding Þnitevalued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for nonprotoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnitevalued logic of Łukasiewicz. 1 1
A formal study of linearity axioms for fuzzy orderings, Fuzzy Sets and Systems
, 2004
"... This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and onetoone ..."
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Cited by 2 (2 self)
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This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and onetoone correspondence between linearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case. If we do not require a onetoone correspondence between linearity and maximality, however, we obtain that an implicationbased definition appears to constitute a sound compromise, in particular, if Łukasiewicztype logics are considered. Key words: completeness, fuzzy ordering, fuzzy preference modeling, fuzzy relation,
On approximate reasoning with graded rules
 Fuzzy Sets and Systems
"... This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IFTHEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are ..."
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Cited by 2 (1 self)
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This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IFTHEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.
Monadic Fragments of Gödel Logics: Decidability and Undecidability Results
"... Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of ..."
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Cited by 2 (1 self)
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Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important subcase, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G↑, like all other infinitevalued logics, is shown to be undecidable if the projection operator △ is added, while all finitevalued monadic Gödel logics remain decidable with △. 1
Maximal and premaximal paraconsistency in the framework of threevalued semantics. Studia Logica, 2011
"... Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the c ..."
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Cited by 2 (2 self)
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Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or nondeterministic threevalued matrices. We show that all reasonable paraconsistent logics based on threevalued deterministic matrices are maximal in our strong sense. This applies to practically all threevalued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on threevalued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these nondeterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) threevalued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core ” of maximal paraconsistency of all possible paraconsistent determinizations of a nondeterministic matrix, thus representing what is really essential for their maximal paraconsistency. 1