 So far, we have a very useful consequence of the classical notion of validity. Now Adams extends this consequence to arguments involving conditionals. Take a language with “and”, “or”, “not” and “if” — but with “if” occurring only as the main connective in a sentence. (We put aside compounds of conditionals.) Take any argument formulated in this language. Consider any probability function over the sentences of this argument which assigns non-zero probability to the antecedents of all conditionals — that is, any assignment of numbers to the non-conditional sentences which conforms to the Partition Principle, and to the conditional sentences which conforms to Supp’s thesis: \(\mathbf{p}(B\) if \(A) = \mathbf{p}_A (B) = \mathbf{p}(A \amp B)/\mathbf{p}(A)\). Let the improbability of the conditional “If \(A, B\)” be \(1 - \mathbf{p}_A (B)\). Define a valid argument as one such that there is no probability function in which the improbability of the conclusion exceeds the sum of the improbabilities of the premises. And a nice logic emerges, which is now well known. It is the same as Stalnaker’s logic over this domain (see §4.1). There are rules of proof, a decision procedure, consistency and completeness can be proved. See Adams (1998 and 1975).