Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications 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 Australian logicians, and relevance logic by other English-speaking logicians.)

Relevance logic was proposed in 1928 by Soviet (Russian) philosopher Ivan E. Orlov (1886–circa 1936) in his strictly mathematical paper "The Logic of Compatibility of Propositions" published in Matematicheskii Sbornik.

Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic holds, for example, that a falsehood implies any proposition. Hence "if I'm a donkey, then two and two is four" is true. Yet that it is false that I am a donkey is irrelevant for inferring that two and two is four. And so such implications ought to be false.

It is also thought that another necessary condition of true implications is that they are necessary, that is, the antecedent necessary entails the conclusion. Clearly this fails for material implications. However other problems remain even after we eliminate the paradoxes of material implication. For example, Anderson and Belnap (see below) enumerate several "paradoxes of strict implication": e.g. a contradiction implies everything, and everything implies a tautology. The counter-intuition is that implication—as we use that term in ordinary language—requires that there be some kind of connection in subject matter between premises and conclusion. The challenge is spelling out this connection in purely formal terms.

In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae. 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 a derivation indicating the "relevant" premises. Gentzen-style 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.

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. They focused on both systems of entailment and systems of relevance, where the former is supposed to be both relevant and necessary.

A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not cause explosion. This follows from the fact that a conditional with a contradictory antecedent that shares no propositional or predicate letters with the consequent cannot be true.

References

• Alan Ross Anderson and Nuel Belnap, 1975. Entailment:the logic of relevance and necessity, vol. I. Princeton University Press.
• ------- and J. M. Dunn, 1992. Entailment: the logic of relevance and necessity, vol. II, Princeton University Press.
• Mares, Edwin, and Meyer, R. K., 2001, "Relevant Logics," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.

External links

• Stanford Encyclopaedia of Philosophy: "Relevance logic" -- by Edwin Mares.