Paradoxes solved by simple relevance criteria. (English) Zbl 0622.03007

The authors recall two well-known criteria that have been suggested in the literature, for the relevance of premises and conclusion of an argument scheme: the ”Aristotelian” criterion that every propositional and predicate variable occurring in the conclusion must already occur in one of the premises, and the ”Körner criterion” that no single occurrence of a propositional or predicate variable in the scheme may be replaced by its negation without modification of validity. Taking these two criteria jointly, they contend that they ”solve” a number of ”paradoxes” of deontic and epistemic logic and in the theory of the nature of explanation, confirmation, and dispositions. By this they mean that a number of intuitively controversial theses of systems of deontic and epistemic logic may be rejected, since their corresponding argument schemes fail at least one of the relevance criteria, and that a number of apparent counterexamples to well-known formal explications of methodological concepts (such as Hempel’s classic definition of a covering-law explanation) cease to be counterexamples because they too fail at least one of the two relevance criteria.
The authors appear quite unconcerned that their two criteria, between them, reject such standard argument forms as \(A\wedge B\vdash A\), \(A\vdash A\vee B\), and \(A\vee (A\wedge B)\vdash A\), as well as more exotic ones. Application of the two criteria thus rules out a lot more than the targeted ”paradoxes”. At the same time, the authors appear not to consider the question of whether variant formulations of some of their ”paradoxes” escape the criteria. For example, although the so-called paradox of Ross in deontic logic, OA\(\vdash O(A\vee B)\) where O is the unary operator for obligatoriness, is rejected by both criteria, the variant OA\(\vee OB\vdash O(A\vee B)\) survives both. Nor do the authors consider the significance of the fact that the force of their criteria depends on the particular choice of primitive truth-functional operators. For example, \(A\equiv B\vdash A\supset B\) survives when \(\equiv\) is primitive, but fails the Körner criterion when \(A\equiv B\) is defined as (A\(\supset B)\wedge (B\supset A)\). Similarly for \(A+B\), \(\neg A\vdash \neg B\) according as exclusive disjunction \(+\) is taken as primitive or not.
Reviewer: D.Makinson


03A05 Philosophical and critical aspects of logic and foundations
03B05 Classical propositional logic
03B45 Modal logic (including the logic of norms)