Definition
Relevance logic is a non-classical logic designed to fix the paradoxes of the material conditional by demanding that the antecedent and consequent of a conditional be genuinely relevant to each other. A conditional is acceptable only if the antecedent actually bears on the consequent.
This directly rejects the classical results that "if the moon is cheese, then 2 plus 2 is 5" and similar irrelevant conditionals are true. In relevance logic, a true antecedent and true consequent are not enough — there must be a real connection of content or meaning between them.
Why it matters
How it works
Relevance logicians redesign the semantics of the conditional so that an inference counts as valid only when the premises are actually used in deriving the conclusion. Adding an irrelevant premise no longer licenses arbitrary conclusions, and a false antecedent no longer makes a conditional automatically true.
Priest presents relevance logic as a leading example of his larger theme: logic is not finished. Classical logic is one theory of validity among several, and where it conflicts with our reasoning intuitions — as it does with conditionals — rival logics step in. Choosing between them is a substantive philosophical decision rather than a matter of mere convention.