Representations of the lower and upper approximations of a set in the context of an approximation space as modal operators in the first order language of modal logics, are quite natural and widely familiar now to the rough-set community. According to the perception of an observer, objects of a universe (of discourse) are clustered. These are the basic information granules (or quanta). With respect to the information available, objects belonging to the same cluster are indistinguishable. It may not always be the case that the clusters are mutually disjoint.

Rough logics with possible applications to approximate reasoning

A notion of rough consequence is investigated in detail. Two algebraic structures emerge from the modal system S5. Properties of these structures, called rough algebras, have been studied. A link between rough consequence and one of the rough algebras is established.

Rough Consequence and Rough Algebra

This paper considers a number of alternative rough consequence logics which come in a natural way from the logics Lr and LR previously studied by Chakraborty and Banerjee. The systems have been compared to variants of S5, and the logic Triv of Hughes and Cresswell. A comparison has also been made with Lr and LR, and therefore with Jaskowski's discussive logic J, as J is equivalent to LR.

Some rough consequence logics and their interrelations

Pawlak had proposed the notion of rough truth in 1987 [16]. The article takes a fresh look at this "soft" truth, and presents a formal system L$_R$, that is shown to be sound and complete with respect to a semantics determined by this notion. L$_R$ is based on the modal logic S5. Notable is the rough consequence relation defining L$_R$ (a first version introduced in [9]), and rough consistency (also introduced in [9]), used to prove the completeness result. The former is defined in order to be able to derive roughly true propositions from roughly true premisses in an information system. The motivation for the latter stems from the observation that a proposition and its negation may well be roughly true together. A characterization of L$_R$-consequence shows that the paraconsistent discussive logic J of Jaskowski is equivalent to L$_R$. So, L$_R$, developed from a totally independent angle, viz. that of rough set theory, gives an alternative formulation to this well-studied logic. It is further observed that pre-rough logic [3] and 3-valued Lukasiewicz logic are embeddable into L$_R$.

Rough logics with possible applications to approximate reasoning

References

Rough Consequence and Rough Algebra

Some rough consequence logics and their interrelations

Logic for Rough Truth

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