Abstract

What do well-known theories look like if formulated with a relevant rather than a standard classical or intuitionist logic? Do familiar reconstructions of these theories go through, or do we change the reconstruction when we change the logic? I show in this paper that a new class of fallacies arises when we take the familiar Peano postulates as the foundation for a relevant theory of the natural numbers N. For these postulates fail in the relevant context to establish the relevant cancellation theorem. Put otherwise, there are fallacies of division in arithmetic formulated relevantly!