Conditionals, Comparative Probability, and Triviality: The Conditional of Conditional Probability Cannot Be Represented in the Object Language

In this paper we examine the thesis that the probability of the conditional is the conditional probability. Previous work by a number of authors has shown that in standard numerical probability theories, the addition of the thesis leads to triviality. We introduce very weak, comparative conditional probability structures and discuss some extremely simple constraints. We show that even in such a minimal context, if one adds the thesis that the probability of a conditional is the conditional probability, then one trivializes the theory. Another way of stating the result is that the conditional of conditional probability cannot be represented in the object language on pain of trivializing the theory.