Logical systems based on the principle that logical consequence, or entailment, only holds between propositions which are relevant to each other.
They were developed, notably by Alan Ross Anderson and Nuel D Belnap (1920- ), as a reaction to the claim of Clarence Irving Lewis and Cooper H. Langford (in Symbolic Logic (1932), chapter 8) that logical entailment is the same as strict implication, where this is defined so that (where P and Q are propositions) P strictly implies Q if and only if it is logically impossible (that is to say a contradiction) for P to be true and Q false.
A contradiction therefore strictly implies any proposition, and any proposition strictly implies a logical truth. These so-called paradoxes of strict implication seem counterintuitive, and relevance logics restrict entailment to apply more narrowly than strict implication. The need to do this, however, is disputed.
J Bennett, 'Entailment', Philosophical Review (1968); general survey, questioning need for relevance logics