At first blush, classical logic seems to be incompatible with certain intuitively valid principles. These principles include the T-schema for truth (for liar and curry paradox reasons) and tolerance principles for vague predicates (for sorites paradox reasons). When these principles are added to full classical logic, so it seems, the result is overstrong: unacceptable consequences follow. Sometimes this is taken as evidence that these intuitively valid principles are in fact invalid; sometimes it is taken as evidence that classical logic itself needs to be weakened so that the principles can be preserved.

This talk will argue  that neither of these options need be taken. There is a third way: we can preserve full classical logic, the T-schema, and tolerance, all at the same time. This is possible if---and only if---we do not impose the requirement of transitivity on our resulting logical systems. The resulting systems validate every classically valid inference over the full vocabulary---including the truth predicate, for example. But because these inferences cannot unrestrictedly be chained together, the systems are in fact conservative extensions of classical logic.

An inferentialist (and in particular a bilateralist) understanding of these systems is sketched. This provides one natural way to understand how inference can fail to be fully transitive: there are some sentences---paradoxical ones---that fall either into the underlap or the overlap of appropriate assertion and denial.