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 <i>x</i> divides the universe., viz., the lower definable set <i>i(x)</i>, the upper definable set <i>c(x)</i>, and the boundary region <i>c(x)</i>\<i>i(x)</i> 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 Łukasiewicz 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 Łukasiewicz 3-valued logic and theorems of the Łukasiewicz 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 <i>U</i> 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 <i>RL</i>, 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 <i>RL</i>.