“Strenge” arithmetics

In Entailment, Anderson and Belnap motivated their modification E of Ackermann’s strenge Implikation Π Π’ as a logic of relevance and necessity. The kindred system R was seen as relevant but not as modal. Our systems of Peano arithmetic R# and omega arithmetic R## were based on R to avoid fallacies of relevance. But problems arose as to which arithmetic sentences were (relevantly) true. Here we base analogous systems on E to solve those problems. Central to motivating E is the rejection of fallacies of modality. Our slogan here for this is, “No diamonds entail any boxes.” Form the strenge Peano arithmetic E# like R#, adding appropriate forms of the Peano axioms to Ackermann’s E..
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