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A Note on 3-valued Rough Logic Accepting Decision Rules

Rough sets carry, intuitively, a 3-valued logical structure related to the three regions into which any rough set x divides the universe., viz., the lower definable set i(x), the upper definable set c(x), and the boundary region c(x)\setminus i(x) witnessing the vagueness of associated knowledge. In spite of this intuition, the currently known way of relating rough sets and 3-valued logics is only via 3-valued Lukasiewicz algebras (Pagliani) that endow spaces of disjoint representations of rough sets with its structure. Here, we point to a 3-valued rough logic RL of unary predicates in which values of logical formulas are given as intensions over possible worlds that are definable sets in a model of rough set theory (RZF). This logic is closely related to the Lukasiewicz 3-valued logic, i.e., its theorems are theorems of the Lukasiewicz 3-valued logic and theorems of the Lukasiewicz 3-valued logic are in one-to-one correspondence with acceptable formulas of rough logic. The formulas of rough logic have denotations and are evaluated in any universe U in which a structure of RZF has been established. RZF is introduced in this note as a variant of set theory in which elementship is defined via containment, i.e., it acquires a mereological character (for this, see the cited exposition of Lesniewski's ideas). As an application of rough logic RL, decision rules and dependencies in information systems are characterized as acceptable formulas of this logic whereas functional dependencies turn out to be theorems of rough logic RL.