Probabilistic entailment of conditionals by conditionals

Symbolic logics that embody different theories of natural-language conditionals have been developed. One such logic is that of Ernest Adams. An Adams conditional α>β expresses the idea that the conditional probability Pr(β|α) 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 α is a β, provided that such a statement is taken to mean that the conditional probability that a randomly selected object is a β-given that it is an α-is close to one. In Adams´ logic, a finite set of premises {φ₁>ψ₁ ,...,φ_n>ψ_n} is said to probabilistically entail a finite set of alternative conclusions {η ₁>μ₁,...,η_m>μ_m } iff, roughly speaking, whenever the conditional probabilities Pr(ψ₁|φ₁),...,Pr(ψ_n|φ _n) are all close to one, at least one of the conditional probabilities Pr(μ₁|η₁),..., Pr(μm|η _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