Editing Cox's theorem (section)

==Interpretation and further discussion==
Cox's theorem has come to be used as one of the [[theory of justification|justification]]s for the use of [[Bayesian probability theory]].  For example, in Jaynes it is discussed in detail in chapters 1 and 2 and is a cornerstone for the rest of the book.<ref name="Jaynes2003" />  Probability is interpreted as a [[formal system]] of [[logic]], the natural extension of [[Aristotelian logic]] (in which every statement is either true or false) into the realm of reasoning in the presence of uncertainty.

It has been debated to what degree the theorem excludes alternative models for reasoning about [[uncertainty]].  For example, if certain "unintuitive" mathematical assumptions were dropped then alternatives could be devised, e.g., an example provided by Halpern.<ref name="H99a" /> However Arnborg and Sjödin<ref name="AS1999" /><ref name="AS2000a" /><ref name="AS2000b" /> suggest additional
"common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example. Other approaches were devised by Hardy<ref>Michael Hardy, "Scaled Boolean algebras", ''[http://www.sciencedirect.com/science/journal/01968858 Advances in Applied Mathematics]'', August 2002, pages 243&ndash;292 (or  [https://arxiv.org/abs/math.PR/0203249 preprint]); Hardy has said, "I assert there that I think Cox's assumptions are too strong, although I don't really say why. I do say what I would replace them with." (The quote is from a Wikipedia discussion page, not from the article.)</ref> or Dupré and Tipler.<ref name="rbp">Dupré, Maurice J. & Tipler, Frank J. (2009). [http://projecteuclid.org/download/pdf_1/euclid.ba/1340369856 "New Axioms for Rigorous Bayesian Probability"], ''Bayesian Analysis'', '''4'''(3): 599-606.</ref>

The original formulation of Cox's theorem is in {{Harvtxt|Cox|1946}}, which is extended with additional results and more discussion in {{Harvtxt|Cox|1961}}. Jaynes<ref name="Jaynes2003" /> cites Abel<ref>[[Niels Henrik Abel]] "Untersuchung der Functionen zweier unabhängig veränderlichen Gröszen ''x'' und ''y'', wie ''f''(''x'', ''y''), welche die Eigenschaft haben, dasz ''f''[''z'', ''f''(''x'',''y'')] eine symmetrische Function von ''z'', ''x'' und ''y'' ist.", ''Jour. Reine u. angew. Math.'' (Crelle's Jour.), 1, 11&ndash;15, (1826).</ref> for the first known use of the associativity functional equation. [[János Aczél (mathematician)|János Aczél]]<ref>[[János Aczél (mathematician)|János Aczél]], ''Lectures on Functional Equations and their Applications,'' Academic Press, New York, (1966).</ref> provides a long proof of the "associativity equation" (pages 256-267). Jaynes<ref name="Jaynes2003" />{{rp|27}} reproduces the shorter proof by Cox in which differentiability is assumed. A guide to Cox's theorem by Van Horn aims at comprehensively introducing the reader to all these references.<ref>{{Cite journal | last1 = Van Horn | first1 = K. S. | doi = 10.1016/S0888-613X(03)00051-3 | title = Constructing a logic of plausible inference: A guide to Cox's theorem | journal = International Journal of Approximate Reasoning | volume = 34 | pages = 3–24 | year = 2003 | doi-access =  }}</ref>

Baoding Liu, the founder of uncertainty theory, criticizes Cox's theorem for presuming that the [[truth value]] of [[Logical conjunction|conjunction]] <math>P \land Q</math> is a [[Second derivative|twice differentiable]] [[Function (mathematics)|function]] <math>f</math> of truth values of the two [[Proposition|propositions]] <math>P</math> and <math>Q</math>, i.e., <math>T(P \land Q) = f(T(P), T(Q))</math>, which excludes uncertainty theory's "uncertain measure" from its start, because the function <math>f(x, y) = x \land y</math>,{{refn|group=lower-alpha|Liu uses the symbol ∧ as the "minimum operator", most likely referring to a binary operation that takes two numbers and returns the smaller (or minimum) of the two.}} used in uncertainty theory, is not differentiable with respect to <math>x</math> and <math>y</math>.<ref name=":0">{{Cite book |last=Liu |first=Baoding |title=Uncertainty Theory |date=2015 |publisher=Springer Berlin Heidelberg : Imprint: Springer |isbn=978-3-662-44354-5 |edition=4th ed. 2015 |series=Springer Uncertainty Research |location=Berlin, Heidelberg |pages=459–460}}</ref> According to Liu, "there does not exist any evidence that the truth value of conjunction is completely determined by the truth values of individual propositions, let alone a twice differentiable function."<ref name=":0" />