Semantic Analyses of Rough Truth for Axioms in Modal Logic

Rough truth which lies between truth and falsity is one of the five logic values in Pawlak rough logic. Through considering relations between any two approximate spaces among all of them on domain U\+n, a lattice is constructed, which is a kind of algebraic structure, and just using the lattice, a special Kripke model is developed. Within this model, semantic analyses are discussed for axioms of the formal reasoning system in modal logic. Instead of discussing only two values of truth and falsity, the discussions mainly focus on the analyses of rough truth. The conclusions show that the axioms of the formal reasoning system in modal logic are almost rough truth validity within the special Kripke model. Thus soundness would be gained when using some of the axioms to make formal reasoning.