Relevance and conformity (Q1078164)

This is a discussion of some of the features of the author's systems of relevant logic, which he calls ''conforming'' relevant logics. They obey the principle of conformity to Boolean ordering: theorems are entailed by their negations. An example of a conforming relevant logic already in the literature is J. M. Dunn's R-Mingle. The author gives semantics for first-degree conforming implication that diverge somewhat from Dunn's for first-degree entailment. For example, the non-entailment \(A\vee B\to B\vee \sim B\) is validated. The author's semantics for his full systems in effect uses semilattices for possible worlds but no inconsistent worlds. This semantics is easily generalized to quantification, unlike the semantics of R. Sylvan, R. K. Meyer, and K. Fine for the nonconforming relevant logic R. It also extends nicely the Kantian content containment claimed by Anderson and Belnap for their variable- sharing property. In conforming relevant logic, contraposition is invalid and transitivity holds only in the weak imported from (A\(\to B)\&(B\to C)\to (A\to C)\).