Cox's Theorem - Cox's Assumptions

Cox's Assumptions

Cox wanted his system to satisfy the following conditions:

1. Divisibility and comparability – The plausibility of a statement is a real number and is dependent on information we have related to the statement.
2. Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model.
3. Consistency – If the plausibility of a statement can be derived in many ways, all the results must be equal.

The postulates as stated here are taken from Arnborg and Sjödin. "Common sense" includes consistency with Aristotelian logic when statements are completely plausible or implausible.

The postulates as originally stated by Cox were not mathematically rigorous (although better than the informal description above), e.g., as noted by Halpern. However it appears to be possible to augment them with various mathematical assumptions made either implicitly or explicitly by Cox to produce a valid proof.

Cox's axioms and functional equations are:

• The plausibility of a proposition determines the plausibility of the proposition's negation; either decreases as the other increases. Because "a double negative is an affirmative", this becomes a functional equation

saying that the function f that maps the probability of a proposition to the probability of the proposition's negation is an involution, i.e., it is its own inverse.

• The plausibility of the conjunction of two propositions A, B, depends only on the plausibility of B and that of A given that B is true. (From this Cox eventually infers that conjunction of plausibilities is associative, and then that it may as well be ordinary multiplication of real numbers.) Because of the associative nature of the "and" operation in propositional logic, this becomes a functional equation saying that the function g such that

is an associative binary operation. All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in the interval . This function therefore may be taken to be multiplication.

• Suppose is equivalent to . If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation

where f is as above.

Cox's theorem implies that any plausibility model that meets the postulates is equivalent to the subjective probability model, i.e., can be converted to the probability model by rescaling.