Prudence and extensionality in theories of preference and value

Luce's axiom governing probabilities of choice is formulated as a principle governing metalinguistic probabilities. IfX, Y, W are sets of options, and δ(X), δ(Y), δ(W) are sentences asserting that choice is made from these sets, then the axiom is $$\begin{gathered} If \pi \delta (X)] \ne 0 and \pi \delta (X \cap Y)] \ne 0, then \hfill \\ \pi _{\delta (X)} \delta (Y \cap W)] = \pi _{\delta (X \cap Y)} \delta (W)]\pi _{\delta (X)} \delta (Y)] \hfill \\ \end{gathered} $$ where π is a probability on sentences. The axiom is then entailed by extensionality of the probability π in company with a simple condition on probabilities of truth-functions. Conditions are also given under which the probability π is uniquely represented by a probability on the sets of options. What look to be logical constraints on the metalanguage entail a normative or prudential constraint. Debreu's well-known counterinstance to the axiom as a principle governing probability of choice is examined and a novel and consistent interpretation of the axiom is proposed.