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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
A Survey on Different Triangular NormBased Fuzzy Logics
, 1999
"... Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizati ..."
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Cited by 13 (1 self)
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Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, manyvalued logic, Frank tnorm 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. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...
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
Generalized MValgebras
 JOURNAL OF ALGEBRA
, 2005
"... We generalize the notion of an MValgebra in the context of residuated lattices to include noncommutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from latticeordered groups via a truncation construction that generalizes the Chang–Mun ..."
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Cited by 9 (5 self)
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We generalize the notion of an MValgebra in the context of residuated lattices to include noncommutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from latticeordered groups via a truncation construction that generalizes the Chang–Mundici Γ functor. This correspondence extends to a categorical equivalence that generalizes the ones established by D. Mundici and A. Dvurečenskij. The decidability of the equational theory of the variety of generalized MValgebras follows from our analysis.
On the Structure of Hoops
 Algebra Universalis
, 1998
"... A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasiv ..."
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Cited by 4 (0 self)
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A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its finite members. Introduction Residuated structures arise in many areas of mathematics, and are particularly common among algebras associated with logical systems. The essential ingredients are a partial order , a binary operation of say multiplication \Delta that respects the partial order, and a binary (left) residuation operation ! characterized by c \Delta a b if and only if c a ! b. In the logical context these represent a partial ordering of an algebra of truth values, (intensional) conjunction and implication, respectively. If the partial order is a semilattice order, and the multiplication the semilattice operation, we obtain the Brouwerian semilattices  the mo...
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
REPRESENTABLE IDEMPOTENT COMMUTATIVE RESIDUATED LATTICES
"... Abstract. It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The ngenerated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebra ..."
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Abstract. It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The ngenerated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of positive relevance logic containing the mingle and GödelDummett axioms has a solvable deducibility problem. 1.
BilatticeBased Squares and Triangles
"... Abstract. In this paper, Ginsberg’s/Fitting’s theory of bilattices is invoked as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and intervalvalued fuzzy sets (IVFSs), serving on one hand to clarify the exact nature of the relationship between these two ..."
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Abstract. In this paper, Ginsberg’s/Fitting’s theory of bilattices is invoked as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and intervalvalued fuzzy sets (IVFSs), serving on one hand to clarify the exact nature of the relationship between these two common extensions of fuzzy sets, and on the other hand providing a general and intuitively attractive framework for the representation of uncertain and potentially conflicting information. 1
Connections between MVn algebras and nvalued LukasiewiczMoisil algebras
 I”; Discrete Mathematics 181
, 1998
"... Abstract: We introduce two chains of unary operations in the MVn algebra of Revaz Grigolia; they will be used in establishing many connections between these algebras and nvalued LukasiewiczMoisil algebras (LMn algebras for short). The study has four parts. It is by and large selfcontained. The ma ..."
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Abstract: We introduce two chains of unary operations in the MVn algebra of Revaz Grigolia; they will be used in establishing many connections between these algebras and nvalued LukasiewiczMoisil algebras (LMn algebras for short). The study has four parts. It is by and large selfcontained. The main result of the rst part is that MV4 algebras coincide with LM4 algebras. The larger class of \relaxed"MVn algebras is also introduced and studied. This class is related to the class of generalized LMn prealgebras. The main results of the second part are that, for n 5, any MVn algebra is an LMn algebra and that the canonical MVn algebra can be identi ed with the canonical LMn algebra. In the third part, the class of good LMn algebras is introduced and studied and it is proved that MVn algebras coincide with good LMn algebras. In the present fourth part, the class ofproper LMn algebras is introduced and studied.proper LMn algebras coincide (can be identi ed) with Cignoli's proper nvalued Lukasiewicz algebras. MVn algebras coincide withproper LMn algebras (n 2). We also give the construction of an LM3 (LM4) algebra from the odd (respectively even)valued LMn algebra (n 5), which proves that LM4 algebras are as much important than LM3 algebras; MVn algebras help to see this point.
Distribution theorems in MValgebras
, 2007
"... Abstract. We prove semantically two classes of distribution theorems for infinitevalued Łukasiewicz logic, A n ∨B n ≡ (A∨B) n and the dual (n·A)∧(n·B) ≡ n·(A∧B) for n ≥ 0. MVAlgebras MValgebras were developed in [2, 3] as an algebraic counterpart to show the completeness of infinitevalued Łukasi ..."
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Abstract. We prove semantically two classes of distribution theorems for infinitevalued Łukasiewicz logic, A n ∨B n ≡ (A∨B) n and the dual (n·A)∧(n·B) ≡ n·(A∧B) for n ≥ 0. MVAlgebras MValgebras were developed in [2, 3] as an algebraic counterpart to show the completeness of infinitevalued Łukasiewicz logic [6, 7]. Definition 1 (MVAlgebra). An MV (for “manyvalued”) algebra is a structure 〈M, ⊕, ¬, 0〉 where 〈M, ⊕, 0 〉 is a commutative (abelian) monoid with the additional conditions ¬¬x = x x ⊕ 1 = 1 x ∨ y = y ∨ x with additional operators defined from 0, ⊕ and ¬ as 1 =de f ¬0 x ∨ y =de f (x ⊃ y) ⊃ y x ⊙ y =de f ¬(¬x ⊕ ¬y) x ∧ y =de f ¬(¬x ∨ ¬y) x ⊃ y =de f ¬x ⊕ y Remark 1. The definitions of the operators ∨, ∧, and ⊃ correspond to the definitions for the same connectives in Łukasiewicz logic. We use the notation from [2], n · x and x n, where n ∈ N, defined as 0 · x =de f 0 x 0 =de f 1 (n + 1) · x =de f (n · x) ⊕ x x n+1 =de f x n ⊙ x ⋆ This work has been partially supported by The EmBounded Project (IST510255), a threeyear FETOpen collaborative project funded by the European Union Framework 6 Programme. Remark 2. An alternative definition of an MValgebra can be given as the structure 〈M, ⊙, ¬, 1〉, with ⊕ and 0 defined by their duals (e.g., see [4]). Definition 2. The ordering relation for MValgebras is defined as x ≤ y if and only if x ∨ y = y. We call an MValgebra linearly ordered (or “linear ” for short) if and only if for all x, y ∈ M, either x ≤ y or y ≤ x. Proposition 1 (Chang Representation Theorem). Every MValgebra is isomorphic to a subdirect product of linear MValgebras.