Topics in the logic of relevance : towards a theory of entailment

I investigate the claim of those who believe that considerations of relevance are essential to an analysis of entailment. A hallmark of this claim is that A&-A does not entail any random B, and hence there must be an error in Lewis' proof that it does. Their attack on the "Official View" of entailment has generally suffered from their neglect to give a formal analysis of relevance. Without such an analysis, the arguments of the proponents of relevance cannot compete with the well established classical and intuitionistic theories of deduction. One exception may be the work in relevance logic begun by Anderson and Belnap. Thus I devote a section to their system R and its important neighbors. A systematic and critical appraisal of the system shows that without an independent account of relevance the choice of axioms for the system must be ad hoc. This is so despite the development of formal semantics, and a primitive move towards a theory of relevance embodied in Belnap's claim that "a necessary condition for A's being relevant to B is that they share a propositional variable." Hence it is essential to give a formal characterization of relevance before we may ascertain the part it plays in the analysis of entailment. I give two formal criteria of relevance, truth functional relevance and occurrence relevance, and investigate their formal properties The investigation of systems which embody these formal relevance criteria enables me to develop a theory of the part relevance plays in logical study. My position is that relevance logic is not, and should not be, the primary tool for logical studies. In most cases we are primarily interested in the kind of logical connection exemplified by the classical concept of deducibility. However, there are logical inquiries in which relevance logic plays a crucial role, and our intuitions in these areas appear to be compatible with the formal relevance logics that I develop. For when we study the strength of assumptions, alternative foundations, or the structure of proofs or systems, we are very much concerned with relevance. In particular, relevance appears to be involved with the concept of mathematical elegance. Thus, relevance is perhaps not essential to deduction, but is a valid area of concern for the logician