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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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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
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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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
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 4 (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
Paraconsistency in Chang's Logic with Positive and Negative Truth Values
"... In [6], C. C. Chang introduced a natural generalization of Lukasiewicz infinite valued propositional logic L. In this logic the truth values are extended from the interval [0,1] to the interval [1,1]. We will call L # the logic whose designated values are those greater or equal than 0. (Chang calls ..."
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Cited by 2 (2 self)
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In [6], C. C. Chang introduced a natural generalization of Lukasiewicz infinite valued propositional logic L. In this logic the truth values are extended from the interval [0,1] to the interval [1,1]. We will call L # the logic whose designated values are those greater or equal than 0. (Chang calls this logic p # [0].) In this semantics, for a truth assignment v the value of the negation is v(#) = v(#) . This implies that there are sentences for which v(#) = v(#) = 0 , that is, both sentences are tautologies. Moreover, the sentence # #) is not a tautology so # L # is paraconsistent. Two are the main results of this paper. First we axiomatize the system # L # 0 , the logic whose only designated truth value is 0, that is, the paraconsistent sentences of L # . Then, we prove that the categories # , whose objects are MV algebras and MV #  # Funding for the first author has been provided by FONDECYT grant 1990433 and FOMEC. 1 algebras respectively, with their corresponding morphisms, are equivalent. These categories are associated with Lukasiewicz' infinite valued calculus and with Chang's logic L # , respectively. 1
On the infinitevalued ̷Lukasiewicz logic that preserves degrees of truth Josep Maria Font
, 2005
"... ̷Lukasiewicz’s infinitevalued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the ̷Lukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be seman ..."
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̷Lukasiewicz’s infinitevalued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the ̷Lukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from ̷Lukasiewicz algebra by using a “truthpreserving” scheme. This deductive system is algebraizable, nonselfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, nonprotoalgebraic, and satisfies a “graded ” deduction theorem. Moreover, the Gentzen system is algebraizable. The first mentioned deductive system turns out to be the extension of the second by the rule of Modus Ponens.
On Lukasiewicz's fourvalued modal logic
, 2000
"... . # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behav ..."
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. # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counterintuitive aspects of this logic are discussed under the light of the presented results, # Lukasiewicz's own texts, and related literature. 1 Introduction The Polish philosopher and logician Jan # Lukasiewicz (Lwow, 1878  Dublin, 1956) is one of the fathers of modern manyvalued logic, and some of the systems he introduced are presently a topic of deep investigation. In particular his infinitelyvalued logic belongs to the core systems of mathematical fuzzy logic as a logic of comparative truth, see [5, 15, 14, 16]. However, it must be stressed here that # Lukasiewicz's logical work bears also a special relationship to modal logic. Actually, modal notions were part of #...
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.
Partial algebras for ̷lukasiewicz logic and its extensions
, 2004
"... It is a wellknown fact that MValgebras, the algebraic counterpart of ̷Lukasiewicz logic, correspond to a certain type of partial algebras: latticeordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct ..."
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It is a wellknown fact that MValgebras, the algebraic counterpart of ̷Lukasiewicz logic, correspond to a certain type of partial algebras: latticeordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓgroups in a straightforward manner. In this paper, we consider several logics differing from ̷Lukasiewicz logics in that they contain further connectives: the P̷L, P̷L ′, P̷L ′ △, and ̷LΠlogics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing frings. All in all, we get threefold correspondences: the total algebras the partial algebras the representing rings. 1