Jabarkan Asumsi Cox dari Téoréma Cox dalam bahasa sunda

Cox hayang sistimna bisa nedunan desiderata di handap ieu

Divisibilitas jeung komparabilitas - The plausibility of a statement is a réal number and is dependent on information we have related to the statement.

Common sense - Plausibilities should vary sensibly with the assessment of plausibilities in the modél.

Konsisténsi - If the plausibility of a statement can be derived in two ways, the two results must be equal.

Postulat nu dicutat di handap ieu disalin tina Arnborg & Sjödin (1999).

"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 (1999a, 1999b).  However it appéars 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 decréases as the other incréases.  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 [A & B] 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 multiplication of probabilities is associative, and then that it may as well be ordinary multiplication of réal numbers.)  Because of the associative nature of the "and" opération 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 [0, 1].  This function therefore may be taken to be multiplication.

Suppose [A & B] is equivalent to [C & D].  If we take acquire new information A and then acquire further new information B, and update all probabilities éach 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 réal numbers, this becomes a functional equation

where f is as above.

Cox's théorem implies that any plausibility modél that meets the

postulates is equivalent to the subjective probability modél, i.e.,

can be converted to the probability modél by rescaling.