Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but not universally, called relevant logic by British and, especially, Australian logicians, and relevance logic by American logicians.

In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae (formulae that do not contain any logical connectives). In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusion. This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system. In particular, a Fitch-style natural deduction can be adapted to accommodate relevance by introducing tags at the end of each line of an application of an inference indicating the premises relevant to the conclusion of the inference. Gentzen-style sequent calculi can be modified by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequents.

A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not necessarily cause an "explosion." This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable).