Relevance logic, also called relevant logic, is a kind of non-classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 requiring the antecedent

Antecedent (logic)

An antecedent is the first half of a hypothetical proposition.Examples:* If P, then Q.This is a nonlogical formulation of a hypothetical proposition...

 and consequent

Consequent

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then".Examples:* If P, then Q.Q is the consequent of this hypothetical proposition....

 of implications

Entailment

In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...

 be relevantly related. They may be viewed as a family of substructural

Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules , such as weakening, contraction or associativity...

 or modal

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

 logics. (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 that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication

Paradoxes of material implication

The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment....

 such as the principle that a falsehood implies any proposition

Vacuous truth

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...

. Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance. And whether or not I'm a donkey seems in no way relevant to whether two and two is four.

How does relevance logic formally capture a notion of relevance? In terms of a syntactical constraint for a propositional calculus

Propositional calculus

In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

, it is necessary, but not sufficient, that premises and conclusion share atomic formula

Atomic formula

In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...

e (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

Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...

 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

Sequent calculus

In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...

 can be modified by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequent

Sequent

In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.-...

s.

A notable feature of relevance logics is that they are paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

s: the existence of a contradiction will not cause "explosion

Principle of explosion

The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...

". 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).

History

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.
The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann

Wilhelm Ackermann

Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....

,
Moh,
and Church

Alonzo Church

Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

in the 1950s. Drawing on them, Nuel Belnap

Nuel Belnap

Nuel D. Belnap, Jr. is an American logician and philosopher who has made many important contributions to the philosophy of logic, temporal logic, and structural proof theory. He has taught at the University of Pittsburgh since 1961; before that he was at Yale University. His best known work is...

 and Alan Ross Anderson

Alan Ross Anderson

Alan Ross Anderson was an American logician and professor of philosophy at Yale University and the University of Pittsburgh....

 (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

Entailment

In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...

 and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.

Semantics

Relevance logic is, in syntactical terms, a substructural logic because it is obtained from classical logic by removing some of its structural rules (e.g. explicitly of some sequent calculus or implicitly by "tagging" inferences of a natural deduction system). It is sometimes referred to as a modal logic because it can be characterized as a class of formulas valid over a class of Kripke (relational) frames

Kripke semantics

Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...

. In Kripke semantics for relevant logic, the implication operator is a binary modal operator, and negation is usually taken to be a unary modal operator. As such, the accessibility relation governing the operator is ternary rather than the usual binary ones that govern unary modal operators often read as "necessarily".

A Kripke frame F for a propositional relevance language is a triple (W,R,*) where W is a set of indices (or points or worlds), R is a ternary accessibility relation between indices, and * is a unary function taking indices to indices. A model M for the language is an ordered pair (F,V) where F is a frame and V is a valuation function mapping sets of worlds (propositions) to propositional letters. Let M be a model and a,b,c indices from M. An implication is defined

• .

Negation is defined

• .

One obtains various relevance logics by placing appropriate restrictions on R and on *. Details need to be filled in.

External links

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