An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)–(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.

Cattaneo, G., Ciucci, D., Dubois, D. (2011). Algebraic models of deviant modal operators based on de Morgan and Kleene lattices. INFORMATION SCIENCES, 181(19), 4075-4100 [10.1016/j.ins.2011.05.008].

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

CATTANEO, GIANPIERO;CIUCCI, DAVIDE ELIO;
2011

Abstract

An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)–(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.
Articolo in rivista - Articolo scientifico
Modal logic, rough sets, Brouwer Zadeh lattice, Kleene lattice, Heyting–Wajsberg algebra, Fuzzy sets
English
2011
181
19
4075
4100
open
Cattaneo, G., Ciucci, D., Dubois, D. (2011). Algebraic models of deviant modal operators based on de Morgan and Kleene lattices. INFORMATION SCIENCES, 181(19), 4075-4100 [10.1016/j.ins.2011.05.008].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/23656
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