Editing Cox's theorem (section)

==Implications of Cox's postulates==
The laws of probability derivable from these postulates are the following.<ref name="Jaynes2003">[[Edwin Thompson Jaynes]], ''Probability Theory: The Logic of Science,'' Cambridge University Press (2003). &mdash;  preprint version (1996) at {{cite web
 |url=http://omega.albany.edu:8008/JaynesBook.html 
 |title=Archived copy 
 |access-date=2016-01-19 
 |url-status=dead 
 |archive-url=https://web.archive.org/web/20160119131820/http://omega.albany.edu:8008/JaynesBook.html 
 |archive-date=2016-01-19 
}}; Chapters 1 to 3 of published version at http://bayes.wustl.edu/etj/prob/book.pdf
</ref> Let <math>A\mid B</math> be the plausibility of the proposition <math>A</math> given <math>B</math> satisfying Cox's postulates. Then there is a function <math>w</math> mapping plausibilities to interval [0,1] and a positive number <math>m</math> such that

# Certainty is represented by <math>w(A\mid B)=1.</math>
# <math>w^m(A|B)+w^m(\text{not }A\mid B)=1.</math>
# <math>w(AB\mid C)=w(A\mid C)w(B\mid AC)=w(B\mid C)w(A\mid BC).</math>

It is important to note that the postulates imply only these general properties. We may recover the usual laws of probability by setting a new function, conventionally denoted <math>P</math> or <math>\Pr</math>, equal to <math>w^m</math>. Then we obtain the laws of probability in a more familiar form:

# Certain truth is represented by <math>\Pr(A\mid B)=1</math>, and certain falsehood by <math>\Pr(A\mid B)=0.</math>
# <math>\Pr(A\mid B)+\Pr(\text{not }A\mid B)=1.</math>
# <math>\Pr(AB\mid C)=\Pr(A\mid C)\Pr(B\mid AC)=\Pr(B\mid C)\Pr(A\mid BC).</math>

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 [[Measure (mathematics)|finite additivity]] of probability, but not [[Measure (mathematics)|countable additivity]]. The [[Probability axioms|measure-theoretic formulation of Kolmogorov]] assumes that a probability measure is countably additive. This slightly stronger condition is necessary for certain results. An elementary example (in which this assumption merely simplifies the calculation rather than being necessary for it) is that the probability of seeing heads for the first time after an even number of flips in a sequence of coin flips is <math>\tfrac13</math>.<ref>{{citation
 | last = Price | first = David T.
 | doi = 10.2307/2319450
 | journal = American Mathematical Monthly
 | jstor = 2319450
 | mr = 350798
 | pages = 886–889
 | title = Countable additivity for probability measures
 | volume = 81
 | year = 1974| issue = 8
 }}</ref>