FIRST-ORDER ROUGH LOGIC I: APPROXIMATE REASONING VIA ROUGH SETS

Abstract

Earlier the authors have shown that rough sets can be characterized by six topological properties. In this paper, a new formal logic system based on such axioms is proposed. It will be called First-Order Logic for Rough Approximation or simply Rough Logic. The axiom schemas of rough logic turn out to be the same as those of the modal logic S₅. In other words, topological and modal logic considerations led to the same conclusion. So rough logic must have captured the intrinsic meaning of approximate reasoning However, their interpretations are different. To reflect the differences in semantics, possible worlds are renamed as observable worlds. Each observable world represents a different rough observation of the actual world. Rough logic also provides a frame work for approximation. It integrates imperfect observations (observable worlds) into an approximation of actual world. Any good approximation theory should have a convergence theorem-its details are deferred to next paper. A sample theorem is as follows: If there is a convergent sequence of rough observations (namely, equivalence relations), then the corresponding rough models converge to the Tarskian model of first-order classical logic.