Conditional Probability

Abstract

There are two completely distinct ways in which conditional probability has been thought of. The most common way of thinking of conditional probability is to take as basic the locution, ‘The probability that an A is a C, given that it is a B’. This probability will correspond in a natural way to the measure of C’s among objects belonging to the intersection of A and B. It is the way conditional probability is usually introduced in textbooks. It is what I shall call here conditional measure. In discussions of logical probability, and in some other places, where probability is taken to be a function whose domain is sentences or propositions or facts, conditional probability is introduced by means of a simple generalization of the forgoing definition: we define the conditional probability of s, given t, as the quotient of the probability of both s and t, and the probability of t. It will be claimed in due course that in those cases in which such a definition is plausible, it is because those are cases in which the relation to conditional measure is direct and straightforward. (There are also cases in which it seems just wrong to take the probability of s given t as the probability of the conjunction divided by the probability of t.)