This paper introduces the notion of the Borel probabilistic rough truth degree of a formula in a special kind of rough logic, by employing Borel probability measures on the valuation set endowed with the usual product topology. It facilitates a special form of rough logic with integration to quantitative logic. The axiomatic definition of probabilistic rough truth degree is given and its representation theorem is also presented. The proposed notion of Borel probabilistic rough truth degree can be regarded as the quantitative analysis of rough logic, as well as the advancing research of the existing notion of truth degree from rough set perspective. Based upon the fundamental notion of rough truth degree, some graded versions of the existing notions, including the roughness degree, accuracy degree and the rough similarity degree, are also presented. Subsequently, three different kinds of approximate reasoning models are established. The obtained results achieve a combination of rough logic and quantitative logic and provide a possible framework for rough truth based approximate reasoning.