Reduced models for relevant logics without ${\rm WI}$.
John Slaney · Notre Dame Journal of Formal Logic · 1987
A major motivating concern of the relevant logics has been to devise a decent theory of the logical behavior of the operation of implication, expressed in the object language by '->' and corresponding to the relation of implication which is naturally expressed in the metalanguage.That being so, it should be expected that P-+Q be true if and only if p implies q> and more generally that the conditional be true in model m iff p implies q according to m.A conditional is true in m iff it holds at the base world (the real world) in m, and its antecedent implies its consequent according to m iff the consequent holds in m at every world at which the antecedent holds.The motivationally natural thought just sketched emerges in the Routley-Meyer semantics as the requirement that the base world O should satisfy, for all a and b, a B) & A -> B, supported as insightful by such as Brady [2], Priest [7], Routley [8,10], and the author.The object of the present paper is to prove a reduced modeling theorem