Implications of Cox's Postulates
The laws of probability derivable from these postulates are the following. Here w(A|B) is the "plausibility" of the proposition A given B, and m is some positive number.
- Certainty is represented by w(A|B) = 1.
- wm(A|B) + wm(AC|B) = 1
- w(A, B|C) = w(A|C) w(B|A, C) = w(B|C) w(A|B, C).
It is important to note that the postulates imply only these general properties. These are equivalent to the usual laws of probability assuming some conventions, namely that the scale of measurement is from zero to one, and the plausibility function, conventionally denoted P or Pr, is equal to wm. (We could have equivalently chosen to measure probabilities from one to infinity, with infinity representing certain falsehood.) With these conventions, we obtain the laws of probability in a more familiar form:
- Certain truth is represented by Pr(A|B) = 1, and certain falsehood by Pr(A|B) = 0.
- Pr(A|B) + Pr(AC|B) = 1
- Pr(A, B|C) = Pr(A|C) Pr(B|A, C) = Pr(B|C) Pr(A|B, C).
Rule 2 is a rule for negation, and rule 3 is a rule for conjunction. Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone (the conjunctive normal form), we can now handle any compound proposition.
The laws thus derived yield finite additivity of probability, but not countable additivity. The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive. This slightly stronger condition is necessary for the proof of certain theorems.
Read more about this topic: Cox's Theorem
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