Probabilistic entailment of conditionals by conditionals

Abstract

Symbolic logics that embody different theories of natural-language conditionals have been developed. One such logic is that of Ernest Adams. An Adams conditional /spl alpha/>/spl beta/ expresses the idea that the conditional probability Pr(/spl beta/|/spl alpha/) is close to one; his logic may be used to reason about such ideas. In particular, Adam's logic may be used to reason about imperfect generalizations such as nearly every /spl alpha/ is a /spl beta/, provided that such a statement is taken to mean that the conditional probability that a randomly selected object is a /spl beta/-given that it is an /spl alpha/-is close to one. In Adams' logic, a finite set of premises {/spl phisub 1/>/spl psisub 1/,...,/spl phisub n/>/spl psisub n/} is said to probabilistically entail a finite set of alternative conclusions {/spl etasub 1/>/spl musub 1/,...,/spl etasub m/>/spl musub m/} iff, roughly speaking, whenever the conditional probabilities Pr(/spl psisub 1/|/spl phisub 1/),...,Pr(/spl psisub n/|/spl phisub n/) are all close to one, at least one of the conditional probabilities Pr(/spl musub 1/|/spl etasub 1/),..., Pr(/spl mu/m|/spl etasub m/) will also be close to one. Adams has developed a test for ascertaining whether a set of premises probabilistically entails a set of alternative conclusions. However, his test is computationally intensive. A new, more efficient test is presented in this paper. It also proves that the new test is valid. >