The Semantics of First Degree Entailment

Abstract

We argue that the semantics of the first degree paradox-free implication system FD supports the claim it is superior to strict implication as an analysis of entailment at the first degree level. The semantics also reveals that Disjunctive Syllogism, A & (,.A v B) -AB, far from being a paradigmatic entailment, is invalid, and allows the illegitimate suppression of tautologies. We consider systems which take as primitive connectives the truthfunctional connectives '&' and ' (with 'v' defined: A v B DI(~-~Q-A & -B)) and the symbol 'a' read 'entails'. A first degree well formed formula (abbreviated f.d. wff) is a wff which contains no arrows (i.e. no instances of symbol '--') nested within an arrow.1 An f.d. wff is a truth-function of first degree entailments, where an f.d. entailment is a wff of the form (B -* C) with B and C truthfunctional, that is with B and C of zero degree or, in other words, containing no arrows. At the first degree level, then, the issue of the implicational behaviour of implications is not considered. We concentrate on first degree semantics because this sharpens and simplifies the choice among rival systems. For different systems may have a common first degree theory. In particular a large class of strict implication systems (including all Lewis systems) have the same first degree; and the system FD coincides with the f.d. theory of the system E of Anderson and Belnap (presented in [1] and [2]) and also of a number of rival entailment systems both included in, including, and only intersecting E (see [11]). Thus the divergence of entailment from strict implication, and from other implications such as connexive implication, is already clear at the first degree. Most of the traditional and current disputes come up and can be settled at the first degree level, for example, disputes about the paradoxes and their effects, and as to the adequacy of principles such as Disjunctive Syllogism. Thus