The Strong Relevance Princeple in Relevant Logic

Abstract

Classical mathematical logic has a well-known problem of "material-implicational paradoxes", and therefore it is not suitable as the logical basis of knowledge engineering. Relevant logics are constructed in order to find a mathematically satisfactory way of grasping the notion of entailment. They are free of material-implicational paradoxes and it seemed that they are suitable as the logical basis of knowledge engineering. However, recently it is pointed out that there still are other types of paradoxes, called "conjunction-implicational paradoxes"and "disjunction-implicational paradoxes"in relevant logics. In order to adopt relevant logics as the fundamental theory of knowledge representation and reasoning, it is necessary to exclude the conjunction-implicational and disjunction-implicational paradoxes from relevant logics. In this paper, we investigate the formal characteristics of conjunction-implicational paradoxes and disjunction-implicational paradoxes, and propose a necessary condition, named the strong relevance principle, for paradox-free relevant logic systems which are free of not only material-implicational paradoxes but also conjunction-implicational and disjunction-implicational paradoxes. We show that relevant logic systems Rc, Ec, and Tc, which are proposed in order to construct a paradox-free relevant logic system, satisfy the strong relevance principle.