Page 318 - Encyclopedia of Philosophy of Language

Language and Logic
view to convincing others, and which is only rarely about crocodiles!
See also: Logic: Historical Survey. Bibliography
Blair J A, Johnson R H 1980 Informal Logic; The First International Symposium. Edgepress, Pt Reyes, CA
Eemeren F H van, Grootendorst R, Kruiger T 1987 Hand- book of Argumentation Theory. Foris, Dordrecht
Eemeren F H van, Grootendorst R 1984 Speech Acts in Argumentative Discussion. Foris, Dordrecht
Freeman J B 1988 Thinking Logically. Prentice Hall, Engle- wood Cliffs, NJ
Freeman J B 1991 Dialectics and the Macrostructure of Argu- ment: A Theory of Argument Structure. Foris Publications, Berlin/New York
Frege G 1879 Begriffsscrift. Louis Nebert, Halle
Govier T 1985 A Practical Study of Argument. Wadsworth,
Belmont, CA
Johnson R H, Blair J A 1977 Logical Self-Defense. McGraw-
Hill, New York
Johnson-Laird P N 1983 Mental Models. Cambridge Uni-
versity Press, Cambridge
Johnson-Laird P N 1991 Deduction. Lawrence Erlbaum,
Hove
Kneale W, Kneale M 1962 The Development of Logic.Oxford
University Press, Oxford
Perelman C, Olbrechts-Tyteca 1969 The New Rhetoric: A
Treatise on Argumentation. University of Notre Dame
Press, Notre Dame, IN
Prior A N 1962 Formal Logic. Oxford University Press,
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Scriven M 1976 Reasoning. McGraw-Hill, New York
S.
An orthodox account of logical consequence requires that one proposition is a consequence of others if the latter cannot be true and at the same time the former false. This view would seem, however, to ignore any connection of relevance between the propositions involved. An alternative account, designed to expli- cate this notion of logical relevance of one proposition to another, was developed by Anderson and Belnap out of earlier work by Ackermann. It is called 'relevant,' or 'relevance,' logic.
1. Relevance
Anderson and Belnap focused on two aspects of relevance:
1.1 Meaning-connection
If one proposition entails another, there must, they said, be some connection of meaning between them. How can this be explicated in the context of modern formal logic? Anderson and Belnap concentrated initially on propositional logic, and within that on so- called first-degree entailments, entailments of the form A-»B, where A and B contain no occurrences of the entailment connective '-»,' but only truth-functional connectives '&' (and),' v ' (or), and' ~ ' (not). Belnap proposed a test of meaning-connection relevance in this context, that of variable-sharing. A necessary con- dition for A to entail B is that A and B (combinations of propositional variables) should share a variable.
1.2 Derivational Utility
V ariable-sharing can be criticized as over-technical and parochial, and difficult to generalize beyond prop-
Read
ositional logic. An alternative criterion which And- erson and Belnap developed is based on the idea of the use of an assumption in a chain of derivations. Only what has actually been used in some essential way is relevant. However,what is essential is difficult to pin down. It cannot mean 'necessary,' given that they accept the validity of (p-*q)-*((q-*p)-*(p-*q))\ and it must be restricted to occurrences or tokens, sincetheyreject p-*(p-*p).
2. EntailmentandRelevantImplication
Anderson and Belnap's first interest was in the logic of entailment, E, which respects considerations of both relevance and necessity. Before long, however, a the- ory of a non-modal conditional connective was developed, called the calculus of relevant implication, R. In the early 1970s, it was discovered (by Max- imova) that extending relevant implication by an S4- type modal connective, yielding the system R°, did not give a theory identical to that of entailment, and
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Relevant Logic
since then interest in the calculus E of entailment has D
wanedinfavorofRandR .
3. Semantics
3.1 Relational Semantics
At the end of the 1960s, relevant logic was well-worked out as a formal syntactic theory, but it had no real semantics. Following Kripke's lead in providing a set- theoretic semantical analysis of modal logic, several authors independently hit upon the way to provide a similar semantics for relevant logic. The preferred version has settled on a relational semantics, in which