Relevance logic

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.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.

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 such as the principle that a falsehood implies any proposition. 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, 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 an application of an inference indicating the premises relevant to the conclusion of the inference. 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.

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 (or derivable).


Relevance logic is syntactically speaking 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. 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 a quadruple (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
* M,amodelsphi opsiiffforall b,c((Rabcland M,bmodelsphi)Rightarrow M,cmodelspsi).

Negation is defined
* M,amodelslnotphiiff M,a^* otmodelsphi.One obtains various relevance logics by placing appropriate restrictions on R and on *. "Details need to be filled in."


*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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • relevance logic — relevance logics …   Philosophy dictionary

  • Relevance (disambiguation) — Relevance is a measure of how pertinent, connected, or applicable something is.Relevance may also refer to:*Relevance (information retrieval), a measure of a document s applicability to a given subject or search query *Relevance (law), regarding… …   Wikipedia

  • Logic — For other uses, see Logic (disambiguation). Philosophy …   Wikipedia

  • logic, history of — Introduction       the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic       There was a medieval tradition according to which the Greek philosopher …   Universalium

  • Relevance — is a term used to describe how pertinent, connected, or applicable something is to a given matter. A thing is relevant if it serves as a mean to a given purpose. Imagine a patient suffering a well defined disease such as scurvy caused by lack of… …   Wikipedia

  • logic — logicless, adj. /loj ik/, n. 1. the science that investigates the principles governing correct or reliable inference. 2. a particular method of reasoning or argumentation: We were unable to follow his logic. 3. the system or principles of… …   Universalium

  • logic — noun Etymology: Middle English logik, from Anglo French, from Latin logica, from Greek logikē, from feminine of logikos of reason, from logos reason more at legend Date: 12th century 1. a. (1) a science that deals with the principles and criteria …   New Collegiate Dictionary

  • 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 (or… …   Wikipedia

  • Outline of logic — The following outline is provided as an overview of and topical guide to logic: Logic – formal science of using reason, considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and… …   Wikipedia

  • History of logic — Philosophy ( …   Wikipedia