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).
Motivation
Classical accounts of implication validate a range of "paradoxes"—for example, that any truth follows from a contradiction, or that any statement implies a tautology—because material and strict conditionals ignore whether antecedent and consequent are about the same topic. Relevance logic addresses this by requiring a suitable connection between premises and conclusion. A familiar syntactic proxy is variable sharing (or "topic sharing"): no valid inference (and no true conditional <math>A\to B</math>) unless antecedent and consequent share atoms; natural–deduction and sequent systems enforce this by tracking the actual use of premises and by restricting structural rules such as weakening.
History
Early complaints about classical implication predate relevance logic. Hugh MacColl questioned the identification of "if" with truth-functional implication; C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. (For instance, "if this article is an Uncyclopedia article, then two and two is five" is true when translated as a material implication, since this article is a Wikipedia article. But it seems intuitively false if one assumes that a true implication must tie the antecedent and consequent together by some notion of relevance; and whether or not this article is from Uncyclopedia seems in no way relevant to whether two and two is five.) Lewis's strict implication still licensed some irrelevant inferences, however, known as the paradoxes of strict implication.
Relevance logic was proposed in 1928 by Soviet philosopher Ivan E. Orlov (1886 – circa 1936) in his strictly mathematical paper "The Logic of Compatibility of Propositions" published in Matematicheskii Sbornik. The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann, Moh, and Church in the 1950s. Drawing on them, Nuel Belnap and Alan Ross Anderson (with others) wrote the magnum opus of the subject, Entailment: The Logic of Relevance and Necessity in the 1970s (the second volume being published in the nineties). They focused on both systems of entailment and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.
A breakthrough in model theory came in the 1970s with Routley–Meyer ternary-relational semantics, together with the Routley (star) treatment of negation, providing sound/complete frames for many relevance systems and explaining how relevance blocks classical paradoxes. and Kit Fine provided alternative model constructions and algebraic perspectives that further clarified the space of relevant conditionals.
From the late 1970s onward, a family of systems crystallized—ranging from weaker logics such as B (often taken as a minimal relevance base) up through R, E, and their extensions—together with algebraic semantics (e.g., De Morgan monoids) and proof systems (display calculi, natural deduction).