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Canonical extensions of double quasioperator algebras: an algebraic perspective on duality for certain algebras with binary operations
, 2006
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On uniform word problems involving bridging operators on distributive lattices
 Proceedings of TABLEAUX 2002. LNAI 2381
, 2002
"... Abstract. In this paper we analyze some fragments of the universal theory of distributive lattices with many sorted bridging operators. Our interest in such algebras is motivated by the fact that, in description logics, numerical features are often expressed by using maps that associate numerical va ..."
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Abstract. In this paper we analyze some fragments of the universal theory of distributive lattices with many sorted bridging operators. Our interest in such algebras is motivated by the fact that, in description logics, numerical features are often expressed by using maps that associate numerical values to sets (more generally, to lattice elements). We first establish a link between satisfiability of universal sentences with respect to algebraic models and satisfiability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences, and provide some decidability results based on the use of resolution or hyperresolution. Links between hyperresolution and tableau methods are also discussed, and a tableau procedure for checking satisfiability of formulae of type t1 ≤ t2 is obtained by using a hyperresolution calculus. 1
Autoreferential semantics for manyvalued modal logics
"... ABSTRACT. In this paper we consider the class of truthfunctional modal manyvalued logics with the complete lattice of truthvalues. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarc ..."
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ABSTRACT. In this paper we consider the class of truthfunctional modal manyvalued logics with the complete lattice of truthvalues. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The nonconstructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative pseudocomplements as in intuitionistic logic. We show that the complete lattices are intrinsically modal, with banal identity modal operator. We define the autoreferential setbased representation for the class of modal algebras, and show that the autoreferential Kripkestyle semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truthvalues. The philosophical assumption is based on the consideration that each possible world represents a level of credibility, so that only propositions with the right logic value (i.e., level of credibility) can be accepted by this world, then we connect it with paraconsistent properties and LFI logics. The bottom truth value in this complete lattice corresponds to the trivial world in which each formula is satisfied, that is, to the world with explosive inconsistency. The
New Representation Theorem for Manyvalued Modal Logics
"... We propose a new definition of Representation theorem for manyvalued modal logics, based on a complete latice of algebraic truth values, and define the stronger relationship between algebraic models of a given logic L and relational structures used to define the Kripke possibleworld semantics for ..."
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We propose a new definition of Representation theorem for manyvalued modal logics, based on a complete latice of algebraic truth values, and define the stronger relationship between algebraic models of a given logic L and relational structures used to define the Kripke possibleworld semantics for L. Such a new framework offers clear semantics for the satisfaction algebraic relation, based on manyvalued models of a logic L, avoiding the necessity to define a designated subset of logic values for a satisfaction relation, often difficult to determine for manyvalued logic, especially for bilattice based logic. We define the subclass of manyvalued modal logics based on distributive lattices which have compact autoreferential cannonical representation. The significant member of this subclass is the paraconsistent fuzzy logic extended by new logic values in order to deal with incomplete and inconsistent information also. The Kripkestyle semantics for this subclass of modal logics have as set of possible worlds the jointirriducible subset of the carrier set of manyvalued algebras. Such a new theory is applied for the case of autoepistemic intuitionistic manyvalued logic based on Belnap’s 4valued bilattice as minimal extension of classic logic used to manage incomplete and inconsistent information also. 1
A New Representation Theorem for Manyvalued Modal Logics
"... We propose a new definition of the representation theorem for manyvalued logics, with modal operators as well, and define the stronger relationship between algebraic models of a given logic and relational structures used to define the Kripke possibleworld semantics for it. Such a new framework of ..."
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We propose a new definition of the representation theorem for manyvalued logics, with modal operators as well, and define the stronger relationship between algebraic models of a given logic and relational structures used to define the Kripke possibleworld semantics for it. Such a new framework offers a new semantics for manyvalued logics based on the truthinvariance entailment. Consequently, it is substantially different from current definitions based on a matrix with a designated subset of logic values, used for the satisfaction relation, often difficult to fix. In the case when the manyvalued modal logics are based on the set of truthvalues that are complete distributive lattices we obtain a compact autoreferential Kripkestyle canonical representation. The Kripkestyle semantics for this subclass of modal logics have the jointirreducible subset of the carrier set of manyvalued algebras as set of possible worlds. A significant member of this subclass is the paraconsistent fuzzy logic extended by new logic values in order to also deal with incomplete and inconsistent information. This new theory is applied for the case of autoepistemic intuitionistic manyvalued logic, based on Belnap’s 4valued bilattice, as a minimal extension of classical logic used to manage incomplete and inconsistent information as well. 1