Lossy inference rules and their bounds

a brief review

David Makinson

pp. 385-407

in: Arnold Koslow, Arthur Buchsbaum (eds), The road to universal logic I, Basel, Birkhäuser, 2015

Abstract

This paper reviews results that have been obtained about bounds on the loss of probability occasioned by applying classically sound, but probabilistically unsound, Horn rules for inference relations. It uses only elementary finite probability theory without appealing to linear algebra, and also provides some new results, in the same spirit, on non-Horn rules. More specifically, it does the following: (i) recalls Adams' well-known sum bound for the rule Right∧+ and shows how it is inherited by the rules CM, CT and Left∨+; (ii) draws attention to lesser known but tighter bounds for CM, CT and Left∨+ due respectively to Bourne & Parsons, Adams, and Gilio, and provides elementary verifications for those that were originally obtained using linear algebra; (iii) shows that the sum bound for Right∧+ and the improved bounds for CM, CT and Left∨+ are all in a natural sense optimal; (iv) distinguishes two kinds of loss for almost-Horn rules, "distributed" and "pointed"; and (v) finds bounds for both distributed and pointed loss, optimal in the distributed case, for the specific almost-Horn rules of disjunctive rationality (DR) and rational monotony (RM).