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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
Logical Nondeterminism as a Tool for Logical Modularity: An Introduction
 in We Will Show Them: Essays in Honor of Dov Gabbay, Vol
, 2005
"... It is well known that every propositional logic which satisfies certain very ..."
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Cited by 13 (10 self)
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It is well known that every propositional logic which satisfies certain very
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
Finite Schematizable Algebraic Logic
, 1997
"... In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst ..."
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Cited by 9 (1 self)
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In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) nonaxiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasipolyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.
Open questions related to the problem of Birkhoff and Maltsev, Studia Logica 78
 STUDIA LOGICA
, 2004
"... The BirkhoffMaltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are ..."
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Cited by 7 (5 self)
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The BirkhoffMaltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the BirkhoffMaltsev problem are also considered, including ones that stem from the theory of propositional logics.
A sequent calculus for Lukasiewicz's threevalued logic based on Suszko's bivalent semantics
 Bulletin of the Section of Logic
, 1999
"... A sequent calculus S3 for ̷Lukasiewicz’s logic L3 is presented. The completeness theorem is proved relatively to a bivalent semantics equivalent to the non truthfunctional bivalent semantics for L3 proposed by Suszko in 1975. A distinguishing property of the approach proposed here is that we are kee ..."
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Cited by 6 (1 self)
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A sequent calculus S3 for ̷Lukasiewicz’s logic L3 is presented. The completeness theorem is proved relatively to a bivalent semantics equivalent to the non truthfunctional bivalent semantics for L3 proposed by Suszko in 1975. A distinguishing property of the approach proposed here is that we are keeping the format of the classical sequent calculus as much as possible. Mathematics Subject Classification: 03B50, 03F03 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
Behavioral institutions and refinements in generalized hidden logics
 J. Univers. Comput. Sci
, 2006
"... Abstract: We investigate behavioral institutions and refinements in the context of the object oriented paradigm. The novelty of our approach is the application of generalized abstract algebraic logic theory of hidden heterogeneous deductive systems (called hidden klogics) to the algebraic specifica ..."
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Cited by 3 (3 self)
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Abstract: We investigate behavioral institutions and refinements in the context of the object oriented paradigm. The novelty of our approach is the application of generalized abstract algebraic logic theory of hidden heterogeneous deductive systems (called hidden klogics) to the algebraic specification of object oriented programs. This is achieved through the Leibniz congruence relation and its combinatorial properties. We reformulate the notion of hidden klogic as well as the behavioral logic of a hidden klogic as institutions. We define refinements as hidden signature morphisms having the extra property of preserving logical consequence. A stricter class of refinements, the ones that preserve behavioral consequence, is studied. We establish sufficient conditions for an ordinary signature morphism to be a behavioral refinement.
formal and formalized ontologies
 International Journal of HumanComputer Studies
"... 2. Descriptive, formal and formalized ontologies 3. Variants of formalized ontology 4. Some data on formal ontologists 5. A note on Husserl’s conception of formal ontology ..."
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Cited by 3 (1 self)
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2. Descriptive, formal and formalized ontologies 3. Variants of formalized ontology 4. Some data on formal ontologists 5. A note on Husserl’s conception of formal ontology
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.