Relevance logic and entailment

‘Relevance logic’ came into being in the late 1950s, inspired by Wilhelm Ackermann, who rejected certain formulas of the form A→B on the grounds that ‘the truth of A has nothing to do with the question whether there is a logical connection between B and A’.

The central idea of relevance logic is to give an account of logical consequence, or entailment, for which a connection of relevance between premises and conclusion is a necessary condition. In both classical and intuitionistic logic, this condition is missing, as is highlighted by the validity in those logics of the ‘spread law’, A &∼A→B; a contradiction ‘spreads’ to every proposition, and simple inconsistency is equivalent to absolute inconsistency. In relevance logic the spread law fails, and the simple inconsistency of a theory (that a set of formulas entails a contradiction) is distinguished from absolute inconsistency (or triviality: that a set of formulas entails every proposition). The programme of relevance logic is to characterize a logic, or a range of logics, satisfying the relevance condition, and to study theories based on such logics, such as relevant arithmetic and relevant set theory.