Families of conditionals \(\to\) are characterized in terms of the rules that their members obey. It is shown that for each conditional in a given family, there is a support level \(r\) such that ‘\(C \to B\)’ holds just in case \(P [B|C] \geq r\).
This is a very rich paper and successfully ties together a number of ideas that are ordinarily treated as distinct. The author takes a conditional to be, not a sentential connective, but a relation between sentences. Given his interest, this is perfectly correct: he seeks to model nonmonotonic reasoning – the inferences we commonly make. To refer to the conditionals as nonmonotonic is a little odd (though not so odd if you keep in mind the procedure the author is modelling), and it is quite odd to think of probabilistic conditionalization as nonmonotonic, since the value of \(P (A|B)\) is determined by the axioms of the probability calculus and the input probabilities. As Keynes pointed out long ago, \(P (A|B \& C)\) does not conflict with or provide a new value for \(P (A |B)\) – it is simply a different probability.
Conditional probability, in the sense of Popper, in which no logical properties of \(\&\), \(\neg\) and \(\wedge\) are presupposed, forms the framework for comparing various conditionals. One family of conditionals are the entailment relations, \({\mathbf E} {\mathbf R}\). It is shown that \(B \models A\) if and only if for every relation \(\to\) that is an entailment relation, \(B \to A\). A small set of rules characterizes the set of conditional relations the author calls \({\mathbf O}\); the addition of a further rule yields the system of Adams; this set of conditionals is called \({\mathbf P}\). Adding another rule to \({\mathbf P}\) yields \({\mathbf R}\), the set of conditionals that Lehmann and Magidor call the Rational consequence relations.
After exploring the relations among \({\mathbf O}\), \({\mathbf P}\) and \({\mathbf R}\), the author defines, in terms of \(\to\), a relation \(\ll\) on the sentences of his object language. This relation leads naturally to the discussion of levels of probabilistic support, and the conditional relations that correspond to them. This material is complex, but represents a very interesting and original approach to nonmonotonic logic in terms of inference relations.
Although the list of references is not long, the author does include most of those who have worked on the relation between conditional probability and what has come to be called the logic of conditionals. The one major exception is Charles Morgan, whose article, “Logic, probability theory, and artificial intelligence” [Comput. Intell. 7, 94-109 (1991)], might well have been referred to.