Probabilities For Two Properties

Let R(X, B) denote the class of probability functions that are defined on algebra X and that represent rationally permissible degrees of certainty for a person whose total relevant background evidence is B. This paper is concerned with characterizing R(X, B) for the case in whichX is an algebra of propositions involving two properties and B is empty. It proposes necessary conditions for a probability function to be in R(X, B), some of which involve the notion of statistical dependence. The class of probability functions that satisfy these conditions, here denoted P_I, includes a class that Carnap once proposed for the same situation. Probability functions in P_I violate Carnap's axiom of analogy but, it is argued, that axiom should be rejected. A derivation of Carnap's model by Hesse has limitations that are not present in the derivation of P_I given here. Various alternative probability models are considered and rejected.