Relevance logic

From Academic Kids

Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. (It is generally, but not universally, called relevant logic by Australian logicians, and relevance logic by other English-speaking logicians).

Relevance logic aims to capture aspects of implication, or entailment, which 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 am the pope, then 2+2=5" is true. But clearly even if one were the pope, 2+2 would still not be 5 (see counterfactuals). Hence the implication relation ought to be necessary.

Other problems remain even after we eliminate the paradoxes of material implication. Anderson and Belnap (see below) enumerate several "paradoxes of strict implication": for example, a contradiction still implies everything, and everything implies a tautology. The counter-intuition is that implication–as we use that term–requires that there be some kind of connection in subject matter between premises and conclusion.

The fundamental novelty in relevance logic is to change the semantics of implication in such a way that the premises of a valid argument must be "related" to the conclusion. In the propositional calculus, this involves requiring that premises and conclusion share atomic sentences; and certain truth-functional rules, such as addition (the inference from p to p-or-q, for any q at all) are restricted, so that "irrelevant" information cannot be brought in. In predicate calculus, relevance requires sharing of variables and constants between premises and conclusion.

Standard proof theories (such as Fitch-style natural deductions) 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 in the 1950s. Drawing on him, 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.

A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not cause explosion.

References

Bibliography

  • A. R. Anderson and N. D. Belnap, 1975. Entailment:the logic of relevance and necessity, vol. I. Princeton University Press, pp. xxxii+542.
  • A. R. Anderson, N. D. Belnap and J. M. Dunn, 1992. Entailment:the logic of relevance and necessity, vol. II, Princeton University Press, pp. xxvii+749.

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