Logic of approximation reasoning

Abstract

An algebraic and set-theoretical approach to approximation reasoning as proposed in [10] and [5] leads to a formulation of a class of first order logics. They are certain intermediate logics equipped with approximation operators d_t for t ε T — where (T, ≤) is a poset establishing a type of logic under consideration — and with modal connectives C_t of possibility and I_t of necessity, t ε T and possibly with C_T and I_T. Their semantics is based on the idea that a set of objects to be recognized in a process of an approximation reasoning is approximated by means of a family of sets covering this set and by their intersection. Approximating sets with equivalence classes of equivalence relations, as connected with Pawlak's rough sets methods (see [8], [7], [12], [13]) is an additional tool. The main task of this paper is to formulate and prove the completeness theorem for the logics under consideration. For that purpose a theory of plain semi-Post algebras as introduced and developed in [3] has been applied. These algebras replace more complicated semi-Post algebras occurring in [10].