Graded consequence revisited

In this paper some basic aspects of the theory of graded consequence are reconsidered, based on many results that appeared since the inception of graded consequences in 1986. Relationships of the notion of graded consequence with the notions of consequence in many-valued and fuzzy logics are discussed. Following this, a detailed study is carried out on the necessary conditions that follow from axioms generalizing the rules “ex falso quodlibet” and “reasoning by cases” added to the basic three axioms of graded consequence of reflexivity, monotonicity and cut inspired from Tarski and Gentzen. Towards the converse direction, in the process of finding non-trivial examples satisfying the five graded consequence axioms some sufficient conditions on the algebraic structures and the set of valuations are obtained. The significance of these conditions holding simultaneously is investigated.