Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak's rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f -- to the negative region, and the undefined value u -- to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a "common denominator" for Kleene and Lukasiewicz 3-valued logics, which represent its two different "determinizations". In turn, the weak semantics -- where both t and u are treated as designated -represents such a "common denominator" for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same pro...

• Type: Journal
• Year: 2008
• Where: SLOGICA
• Authors: Arnon Avron, Beata Konikowska
• Keywords: Logics, Lukasiewicz 3-valued Logic, Rough Set, SLOGICA 2008