relevance logic any of a range of logics and philosophies of logic united by their insistence that the premises of a valid inference must be relevant to the conclusion. Standard, or classical, logic contains inferences that break this requirement, e.g., the spread law, that from a contradiction any proposition whatsoever follows. Relevance logic had its genesis in a system of strenge Implikation published by Wilhelm Ackermann in 1956. Ackermann’s idea for rejecting irrelevance was taken up and developed by Alan Anderson and Nuel Belnap in a series of papers between 1959 and Anderson’s death in 1974. The first main summaries of these researches appeared under their names, and those of many collaborators, in Entailment: The Logic of Relevance and Necessity (vol. 1, 1975; vol. 2, 1992).
By the time of Anderson’s death, a substantial research effort into relevance logic was under way, and it has continued. Besides the rather vague unity of the idea of relevance between premises and conclusion, there is a technical criterion often used to mark out relevance logic, introduced by Belnap in 1960, and applicable really only to propositional logics (the main focus of concern to date): a necessary condition of relevance is that premises and conclusion should share a (propositional) variable.
Early attention was focused on systems E of entailment and T of ticket entailment. Both are subsystems of C. I. Lewis’s system S4 of strict implication and of classical truth-functional logic (i.e., consequences in E and T in ‘P’ are consequences in S4 in ‘ ‘ and in classical logic in ‘/’). Besides rejection of the spread law, probably the most notorious inference that is rejected is disjunctive syllogism (DS) for extensional disjunction (which is equivalent to detachment for material implication): A 7 B,ÝA , B. The reason is immediate, given acceptance of Simplification and Addition: Simplification takes us from A & ÝA to each conjunct, and Addition turns the first conjunct into A 7 B. Unless DS were rejected, the spread law would follow. Since the late 1960s, attention has shifted to the system R of relevant implication, which adds permutation to E, to mingle systems which extend E and R by the mingle law A P (A P A), and to contraction-free logics, which additionally reject contraction, in one form reading (A P (A P B)) P (A P B). R minus contraction (RW) differs from linear logic, much studied recently in computer science, only by accepting the distribution of ‘&’ over ‘7’, which the latter rejects. Like linear logic, relevance logic contains both truth-functional and non-truth-functional connectives. Unlike linear logic, however, R, E, and T are undecidable (unusual among propositional logics). This result was obtained only in 1984. In the early 1970s, relevance logics were given possible-worlds semantics by several authors working independently. They also have axiomatic, natural deduction, and sequent (or consecution) formulations. One technical result that has attracted attention has been the demonstration that, although relevance logics reject DS, they all accept Ackermann’s rule Gamma: that if A 7 B and ÝA are theses, so is B. A recent result occasioning much surprise was that relevant arithmetic (consisting of Peano’s postulates on the base of quantified R) does not admit Gamma. See also IMPLICATION, MODAL LOGIC. S.L.R.