Rough Truth, Consequence, Consistency and Belief Revision

The article aims at re-visiting the notion of rough truth proposed by Pawlak in 1987 [11] and investigating some of its ‘logical’ consequences. We focus on the formal deductive apparatus $\mathcal{L}_\mathcal{R}$ [12], that is sound and complete with respect to a semantics based on rough truth (extended to rough validity). The notion of rough consequence [4] is used in a modified form to formulate $\mathcal{L}_\mathcal{R}$. The system has some desirable features of ‘rough’ reasoning – e.g. roughly true propositions can be derived from roughly true premisses in an information system. Further, rough consistency [4] is used to prove completeness. These properties of $\mathcal{L}_\mathcal{R}$ motivate us to use it for a proposal of rough belief change. During change, the operative constraints on a system of beliefs are that of rough consistency preservation and deductive closure with respect to $\mathcal{L}_\mathcal{R}$. Following the AGM [1] line, postulates for defining revision and contraction functions are presented. Interrelationships of these functions are also proved.