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

Abstract

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 nondistributive ones, in the scope of knowledge representation and approximation spaces is discussed.

Orthopairs and granular computing

“…-Clarify the algebraic picture and define orthopairs and related operations on more abstract settings. Some results in this sense are already known, we have algebraic approaches to orthopairs (Cattaneo et al 2011) and also of pair of nested/ortho pairs defined on Boolean or Heyting algebras (Monteiro 1980;Vakarelov 1977;Walker 1994). For more details see Ciucci (2011) and [Pagliani (2008), Frame 10.11].…”

“…We notice that we are not dealing here with an algebraic approach to these operations (such as Cattaneo et al 2011;Moraschini 2014), nor to an abstraction of orthopairs to Boolean, or even more general, algebras. For some hint on this topic see Ciucci (2011) and [Pagliani (2008), Frame 10.11].…”

Tense operators in fuzzy logic

“…It is worth noticing that the operators G and H can be considered as certain kind of modal operators which were already studied for intuitionistic calculus by Wijesekera [21], further by Dellunde, Godo and Marchioni [10] and by Mukherjee and Dasgupta [18], in the de Morgan framework by Cattaneo, Ciucci and Dubois [4], for tense symmetric Heyting algebras by Figallo, Pelaitay and Sanza [14] and in a general setting by Ewald [13]. For the logic of quantum mechanics (see e.g.…”

“…for all x ∈ A and for all s ∈ T , s(H (x)) = L {t (x); tR G,H s},(4) for all x ∈ A and for all s ∈ T , s(F (x)) = L {t (x); sR G,H t} then the map i T A is an order reflecting morphism of tense fuzzy dynamic algebras into the complete fuzzy tense algebra (L T ; G, P , H , F ). The fuzzy tense algebra (A; G, P , H, F ) is then said to be representable in L with respect to T .…”

Covering-based rough sets and modal logics. Part I

“…In some cases, only a fragment of logical studies connected with various rough set models is indicated in the literature [28,2,37,4,7,20,11,5,30,15,14,38,24,40,16,17,22] mostly from algebraic perspective. We shall work in the domain of covering-based rough sets.…”