Page 319 - Encyclopedia of Philosophy of Language

the conditional '->' receives its truth-condition rela- tive to a ternary relation of accessibility between worlds or indices: 'A-»B' is true at x if B is true at z whenever A is true at y and Rxyz—where Rxyz can be read as 'x and y are compossible at z'.
3.2 OtherSemanticAnalyses
The ternary relation Rxyz obeys the same principles as a basic relation of spherical geometry: 'z lies within the minor arc of the great circle through x and y.' The worlds semantics also has an algebraic counterpart— the Lindenbaum algebra is a distributive lattice with a semigroup operation, called a De Morgan monoid, whose prime filters correspond to the worlds. By focusing on the semigroup operation ° and all the filters one can also develop an elegant operational semantics, whose truth clause for '-»•' reads: 'A->B' is true at x if and only if wheneverA is true at y, Bis true at x°y.
4. Decidability
It was thought for some time that R (and E) were decidable, that is, that there was an effective method for testing for validity in them. No proof could be found, however, and eventually in the early 1980s Urquhart established their undecidability.
Perhaps the most famous thesis valid in classical logic but invalid in R and E is the inference from A v B and ~ A to B, Disjunctive Syllogism. The cor- responding rule form, that if A v B and ~ A are in some theory T then so is B, is known as the Gamma conjecture for T. (y was Ackermann's name for this rule in his forerunner of E.) E and R are Gamma-
that the extension to first order and quantifiers was relatively straightforward. In point of fact, there are two tricky problems here. The first concerns the cor- rect analysis of All As are B. If it is represented as (Vx)(Ax-»Bx), with the '-»' of relevant logic, while Some As are B remains as (9x)(Ax & Bx), then Not all As are B and Some As are not B are no longer equi- valent. This is counter-intuitive. Second, straight- forward extensions of the proof theory and semantics in fact result in incompleteness, as shown in Fine 1988. This was rectified in Fine 1989 by revising the semantics. Quantified relevant logic is still a fairly unstable theory.
6. SubstructuralLogics
Research into relevant logics since 1990 has been sub- sumed within the broader genus of sub-structural logics, a neologism coined to denote logics with restricted structural rules—rules such as Contraction, Weakening, Permutation etc.; and the rise of linear logic in theoretical computer science, in which the tracking of resources is a central concern, has both drawn on earlier results in relevant logic and has extended them, as a closely related substructural logic.
See also: Deviant Logics; Formal Semantics; Modal Logic.
Bibliography
Anderson A R, Belnap N D 1975, 1992 Entailment, 2 vols. Princeton University Press, Princeton, NJ
Avron A 1992 Whither relevance logic? Journal of Philo- sophical Logic 21: 243-81
Dunn J M 1986 Relevance logic and entailment. In: Gabbay D M, Guenthner F (eds.) Handbook of Philosophical Logic, vol. III. Reidel, Dordrecht
Fine K 1988 Semantics for quantified relevance logic. Journal of Philosophical Logic 17:27-59
Fine K 1989 Incompleteness for quantified relevance logics. In: Norman J, Sylvan R (eds.) Directionsin Relevant Logic. Kluwer, Dordrecht
Read S 1988 Relevant Logic. Blackwell, Oxford
Routley R, Plumwood V, Meyer M K, Brady R T 1982 Relevant Logics and Their Rivals. Ridgeview, Atascadero,
CA
Schroeder-Heister P, Dosen K (eds.) 1993 Substructural
Logics. Clarendon Press, Oxford
theories. Much work in the 1970s and 1980s focused #
on R , effectively Peano arithmetic based on R. To much surprise, this was found in 1987 not to be a Gamma-theory.
5. Quantified Relevant Logic
With the exception of the study of R*, much of the work on relevant logic has concentrated on the prep- ositional fragment (indeed, in its first phase, the first- degree part—theses A-»B where A and B contain no arrows—dominated discussion), perhaps in the belief
Many propositions depend for their truth values on facts about particular individuals, but only singular propositions are directly about individuals. The prop-
osition that the President of the USA is not tall, depends for its truth value on the height of a particular person, for example Bill Clinton, but it is not directly
Singular/General Proposition M. Crimmins
SingularIGeneral Proposition
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