Page 285 - Encyclopedia of Philosophy of Language

Rescher N 1969 Many-Valued Logic. McGraw Hill, New York
Routley R, Meyer R K, Plumwood V, Brady R T 1982 Relevant Logics and Their Rivals. Ridgeview, Atacasdero, CA
Schock R 1968 Logics Without Existence Assumptions. Almqvist and Wiksell, Stockholm
Synthese 1975 Vol. 30 (special issue devoted to logic and semantics of vagueness)
Woods J H 1974 The Logic of Fiction. Mouton, The Hague
1
The phrase '/> entails q was introduced in Moore
1922 to describe the internal relation between two propositions when the one, q, follows from the other, p. Different theories of this relation have been proposed.
1. The Classical View
The orthodox view, sometimes called the 'classical view,' is that p entails q when it is impossible for p to be true and q false. This view was codified by C. I. Lewis in his theory of 'strict implication.' It is thought to have some strange and unintuitive consequences, often called the 'paradoxes of strict implication,' that inconsistent propositions entail any proposition (for if they are inconsistent, they cannot be true, and so it is impossible they are true and any other proposition false) and that necessary propositions are entailed by any proposition (for a similar reason—being necess- ary they cannot be false). These paradoxes are either explained and justified, or lead to abandoning this theory for another (see Sects. 3 and 5). One attempt at justification was given by Lewis, in an argument originally developed in the twelfth-century school of Petit-Font in Paris, to show that a contradiction entails any proposition. The argument proceeds by intuitively sound steps from/? and not-/?, top, top or q, and also to not-/?, and so to q.
2. MaterialImplication
A view which was attributed by Lewis (1912) to Russell, the 'material implication' view, is that p entails q when either p is false or q is true. This view is thought to have similar paradoxical consequences, called the 'paradoxes of material implication,' that a false proposition entails any proposition and that a true one is entailed by any proposition.
3. TheLogicandRelevanceandNecessity
Following earlier work by E. Nelson, Wilhelm Acker- mann presented in 1956 a codification of an account
of entailment in which a 'logical connection' between p and q is demanded. His work was extended by Alan Anderson and Nuel Belnap Jr. and given the pre-
sumptive title, 'the logic of entailment,' drawing from Lewis a requirement of necessity and from Acker- mann a requirement of relevance.
4. The Rejection of Transitivity of Entailment
One answer to Lewis's argument, from/? and not-/? to q, is to admit that every step in the argument is sound, but to deny the transitivity of entailment, that the succession of sound steps builds into a single sound step. One explication of this idea, due to von Wright and Geach, restricts sound entailments '/>entails q" to substitution-instances of classically valid entailments A -+ B in which A is not logically false and B is not logically true.
5. Connexive Logics
Another account of entailment draws inspiration from some remarks of Aristotle's, when he appears to say that no proposition entails both some other propo- sition and its contradictory. McCall has developed logics containing such theses as (p-»q)-*~(p->
~#)» ~(p-+~P), and ~ ((/?-> q).(p -» ~ q)). How- ever, as Routley and Montgomery (1968) show, any consistent system of logic incorporating such theses as those above must lack very many principles which one would think highly desirable. Their main objec- tion is that the failure of these principles makes it hard to conceive of a suitable semantic interpretation of the logic.
See also: Deviant Logics; Logic: Historical Survey; Relevant Logic.
Bibliography
Anderson A R, Belnap N D Jr 1975,1992 Entailment, 2 vols. Princeton University Press, Princeton, NJ
Entailment S. Read
Entailment
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