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124
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 64 (4 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
Substructural Logics and Residuated Lattices  An Introduction
, 2003
"... This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebra ..."
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Cited by 25 (2 self)
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This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebraic structures for substructural logics,
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 21 (16 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.
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 16 (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
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 16 (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.
On rational weak nilpotent minimum logics
 J. of Mult.Valued Logic & Soft Computing
"... In this paper we investigate extensions of Gödel and Nilpotent Minimum logics by adding rational truthvalues as truth constants in the language and by adding corresponding bookkeeping axioms for the truthconstants. We also investigate the rational extensions of some parametric families of Weak ..."
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Cited by 15 (12 self)
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In this paper we investigate extensions of Gödel and Nilpotent Minimum logics by adding rational truthvalues as truth constants in the language and by adding corresponding bookkeeping axioms for the truthconstants. We also investigate the rational extensions of some parametric families of Weak Nilpotent Minimum logics, weaker than both Gödel and Nilpotent Minimum logics. Weak and strong standard completeness of these logics are studied in general and in particular when we restrict ourselves to formulas of the kind r → ϕ, where r is a rational in [0, 1] and ϕ is a formula without rational truthconstants.
Advances and challenges in intervalvalued fuzzy logic
, 2006
"... Among the various extensions to the common [0, 1]valued truth degrees of “traditional” fuzzy set theory, closed intervals of [0, 1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncer ..."
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Cited by 13 (5 self)
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Among the various extensions to the common [0, 1]valued truth degrees of “traditional” fuzzy set theory, closed intervals of [0, 1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncertainty without paying too much in terms of computational complexity. From a logical point of view, due to the failure of the omnipresent prelinearity condition, the underlying algebraic structure L I falls outside the mainstream of the research on formal fuzzy logics (including MV, BL and MTLalgebras), and consequently so far has received only marginal attention. This comparative lack of interest for intervalvalued fuzzy logic has been further strengthened, perhaps, by taking for granted that its algebraic operations amount to a twofold application of corresponding operations on the unit interval. Abandoning that simplifying assumption, however, we may find that L I reveals itself as a very rich and noteworthy structure allowing the construction of complex and surprisingly wellbehaved logical systems. Reviewing the main advances on the algebraic characterization of logical operations on L I, and relating these results to the familiar completeness questions (which remain as major challenges) for the associated formal fuzzy logics, this paper paves the way for a systematic study of intervalvalued fuzzy logic in the narrow sense.
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 12 (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 expansions of tnorm based logics with truthconstants, To appear
 in the book Fuzzy Logics and Related Structures
, 2007
"... This paper focuses on completeness results about generic expansions of logics of both continuous tnorms and Weak Nilpotent Minimum (WNM) with truthconstants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truthconstants {r  r ∈ C}, for a suitable countable C ..."
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Cited by 11 (8 self)
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This paper focuses on completeness results about generic expansions of logics of both continuous tnorms and Weak Nilpotent Minimum (WNM) with truthconstants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truthconstants {r  r ∈ C}, for a suitable countable C ⊆ [0, 1], and provide a full description of completeness results when (i) either the tnorm is a finite ordinal sum of Lukasiewicz, Gödel and Product components (and hence continuous) or the tnorm is a Weak Nilpotent Minimum with a finite partition and (ii) the set of truthconstants covers all the unit interval in the sense that each component (in case of continuous tnorm) or each interval of the partition (in the WNM case) contains values of C in its interior. Results on expansions of the logic of a continuous tnorm were already published, while many of the results about WNM are presented here for the first time.
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 9 (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