Talk:Cox's theorem

TODO

The variables y and z are not defined. Falk (talk) 15:25, 7 June 2011 (UTC)
I've added a section about the laws of probability that are derived from Cox's postulates, following Jaynes's exposition. I make a remark about finite and countable additivity that someone else may wish to review. Wile E. Heresiarch 17:39, 8 Jan 2004 (UTC)

• As noted by Mike Hardy, lines with both TeX and plain text don't look so good. One possibility to fix that up would be to format the lists entirely as equations (using matrix since eqnarray isn't supported). I think I'd prefer that to trying to make it all work out in plain text. Just a matter of taste. FWIW, Wile E. Heresiarch 23:45, 8 Jan 2004 (UTC)
  • I've fixed up the equations to use only plain text (not TeX). Wile E. Heresiarch 18:00, 9 Jan 2004 (UTC)
    

I may be missing something here, but "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." seems strange. Shouldn't this be "decreasing" operations? Multiplication in the interval [0, 1] is not increasing (e.g. 0.5*0.5 = 0.25) and neither is P(A and B) ("both A and B are true" is never more plausible than "A is true"). 82.103.195.37 21:00, 15 June 2006 (UTC)

It's an increasing function of each variable separately, i.e., xy increases as x increases (with y fixed). Michael Hardy 01:08, 16 June 2006 (UTC)

... and Pr(A and B) increases as the event A increases, i.e., if

$A_{1}\subseteq A_{2}\,$

then

$\Pr(A_{1}\cup B)\leq \Pr(A_{2}\cup B).\,$

Michael Hardy 01:10, 16 June 2006 (UTC)

So I'm trying to figure this out on my own. In what way is "isomorphic" meant here? If I take it to mean "bijective algebraic homomorphism," it doesn't seem correct. Take the addition operation $+$ with the isomorphism given by $f$. Then we would require:

$f(a+b)=f(a)*f(b)$

From which it necessarily follows that

$f(0)=1$

But then we have much of $\mathbb {R}$ being mapped outside of $[0,1]$. In fact, if continuity is required of $f$, then $f$ must necessarily be an exponential function, whose range is either all positive or negative reals. I must be misinterpreting something here, so any help would be appreciated. — Preceding unsigned comment added by Nathan.s.chappell (talk • contribs) 14:07, 27 October 2019 (UTC)

Indeed, Nathan.s.chappell you found out that there were some parts of the proof missing. I added the parts and hope the text looks better now. Ladislav Mecir (talk) 08:29, 28 October 2019 (UTC)

In the section "Implications of Cox's postulates", it would nice if someone could clarify: A) What is the significance of the w(x) notation, compared to the standard p(x), and B) what the "exponential" notation w^m(x) means. Also, is the use of A^C to represent negation of A a wikipedia standard? If not, I wonder if something else would be less confusing. --Ryguasu (talk) 02:37, 26 May 2010 (UTC)

A^C represents the absolute complement, or the relative complement of any set A relative to the currently-defined universe U. Since the operation of relative complement results in the set of all elements outside of the set whose relative complement was taken but inside the set B relevant to which this relative complement was taken, then one can state that the absolute complement of any set A – that is to say, the relative complement of this set A relative to the current universe U – is equivalent to the logical negation of the set A in question.

— RandomDSdevel (talk) 17:05, 8 March 2013 (UTC)

P.S.: Before I go, can I ask why the function $f\!\left(x\right)$ used early on in the article isn't clearly defined as either the absolute complement x^C or the logical negation $\neg \,x$ outside of the paragraph preceding its use? Shouldn't the other functions be clearly defined outside of text prior to their use as well?

What actually is the theorem?

Is "Cox's theorem" supposed to be a theorem in the mathematical sense? Currently the page doesn't contain anything that looks like a statement of a theorem. Jowa fan (talk) 12:12, 9 October 2012 (UTC)

I think this is a valid point. I have never seen a mathematically correct derivation of Cox's result without extraneous assumptions like differentiability. Kevin Van Horn made a commendable effort but his presentation relies on mathematically undefined entities (`states of information') and thus cannot be considered a mathematical theorem. I have not seen Jeff Paris' version though.93.156.32.168 (talk) 12:02, 19 April 2014 (UTC)

The paragraph that start with "Cox's axioms and functional equations are" seems complete gibberish to me. Simple type analysis: f maps proposition to [0, 1] on real line. Does it also map [0, 1] to proposition? Is this relationship injective? That all reads like a load of bs some old physicist would spout when (s)he realized (s)he'll die soon, and rush to create something reminiscient of a legacy. Only to fail because of frailness of old age. Sad.

I seriously question this page should be in Wikipedia in the first place. Or at least the explanation must be inaccurately framed. This certainly does not seem like a mathematical theorem in usual sense. --Sigmundur (talk) 07:32, 23 February 2017 (UTC)

The Cox Theorem has a lot of academic literature about it, so of course it deserves an article on Wikipedia. That academic literature outweighs your personal emotional reaction. MartinPoulter (talk) 14:22, 23 February 2017 (UTC)

Agreed, the Cox's theorem is notable and there are many academic sources discussing it. Ladislav Mecir (talk) 19:26, 23 February 2017 (UTC)

Jupp. examples: http://biasandbelief.pbworks.com/w/page/6537213/References%20on%20the%20Cox%20Proof 79.253.49.28 (talk) 14:21, 11 May 2017 (UTC)

I agree that it shouldn't be called a theorem. But it is an important work that deserves to be in Wikipedia under RT Cox's name. What it actually is, and what does not seem to be stated explicitly in the article, is this: a probability is a probability of a conditioned proposition, ie p(A|B) is the probability that binary proposition A is true suppposing that binary proposition B is true, according to relations known between the referents of A and B. Now, binary propositions obey an algebra (Boolean algebra), and this implies an algebra of probabilities since probabilities have propositions as their arguments. Cox showed that this algebra is just the sum and product rules of probability. (The main property of Boolean algebra that is involved is associativity of the logical sum and logical product.) Some further minor assumptions might be involved in reaching this result, but that is the conceptual core of Cox's work. Even though the laws of probability were known 300 years earlier, some researchers consider this to be the deepest derivation of them. What is remarkable is that Cox's 1946 paper came out of a blue sky: he was patently well read in logic but he had published nothing significant other than papers in applied physics before that. He gives no hint of how he reached it in his 8-page 'autobiographical narrative' lodged in the Bohr Library of the American Institute of Physics. 80.41.139.242 (talk) 20:54, 28 January 2023 (UTC)

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Is Jaynes a reliable source?

Jaynes is referenced in four places in the current article, but I've seen repeated commentary indicating his books are filled with "basic errors"; for example, a quote from a blog: "In Probability Theory: The Logic of Science, p. 4, he claims to explain what a syllogism is, but his explanation is actually of modus ponens! Modus ponens is an operation of propositional logic, whereas the syllogism requires Aristotelian logic, i.e. a single universal quantifier. Jaynes did not see the distinction between the two; a very basic error." Given this status of this particular book, I don't see how it can provide a suitable reference for this article, much less four of them ... 67.198.37.16 (talk) 19:27, 25 October 2018 (UTC)

Looking more carefully, some of the references seem uncontroversial. So that's OK. But for some of the other references, caution is suggested: yes, WP is not a place for book reviews, but if a book is controversial, then is some caution in order? 67.198.37.16 (talk) 19:51, 25 October 2018 (UTC)

Jaynes' book was published by the Cambridge University Press, so there is no doubt that the publishing process was reliable. As opposed to that, "a blog" is not likely to qualify as reliably published. And since the facts confirmed by the citations are uncontroversial, as you noted, there is no need to revise any of them. Ladislav Mecir (talk) 05:33, 26 October 2018 (UTC)

Coming to the above discussion late, but Jaynes was an eminent statistical physicist whose book is hugely influential. Since he's not a logician by background it's not surprising that he would make a mistake with logical terminology and we shouldn't use his book as a reliable source for Aristotelian logic. His core argument (or the way he explains Cox's argument) involves the mathematics of functional equations, within his expertise in mathematical statistics; it's not dependent on Aristotelian logic. MartinPoulter (talk) 13:12, 16 April 2024 (UTC)

Baoding Liu's criticism

@Thiagovscoelho Thanks for adding a paragraph about uncertainty theory to the article. A couple of things you have written do not make sense and I wonder if you can check them or clarify them. "the function $f(x,y)=x\land y$, used in uncertainty theory" This kind of function is completely different from those considered by Cox. Cox considered functions from pairs of propositions to numbers, not from propositions to other propositions. So the criticism that the functions are not necessarily differentiable makes no sense, because how would a function whose values are propositions even be differentiable? You quote Baoding Liu "there does not exist any evidence that the truth value of conjunction is completely determined by the truth values of individual propositions," but if we're talking about the truth value of propositions (not the values that a numerical function attaches to the propositions) then this is straightforwardly false. It's not possible for A and B both to be true while A and B is not true. It's not possible for A or B to be false while A and B is not false. So Baoding Liu's point must be different from what the summary makes it appear to be. MartinPoulter (talk) 13:07, 16 April 2024 (UTC)

I have added a footnote about Liu's use of ∧ as a "minimum operator", which seems to mean that it does return a number after all. Also, I think Liu uses "truth value" analogically, to mean the "probability measure" $Pr\{A\}$, or "uncertainty measure" ${\mathcal {M}}\{A\}$, as the case may be. Thiagovscoelho (talk) 13:46, 16 April 2024 (UTC)

Okay, that makes a lot of sense. It sounds like Liu is using the terminology in a completely non-standard way. In the usual terminology, truth value and probability are completely different things. So it's worth spelling out Liu's point in terminology that the reader is more likely to be used to. I'm not sure I understand your new footnote. I understand the predicate which takes a minimum of two values, but in x ∧ y, aren't x and y supposed to be propositions, not numbers? There isn't such a thing as the minimum of a pair of propositions. MartinPoulter (talk) 19:37, 16 April 2024 (UTC)

No, it's a function of the truth-value (probability/uncertainty measure), not of the propositions themselves. Liu describes Cox as saying that $T(P\land Q)=f(T(P),T(Q))$ as a motivation for thinking that $Pr\{P\cap Q\}=f(Pr\{P\},Pr\{Q\})=Pr\{P\}\times Pr\{Q\}$. So by analogy, he is saying that in uncertainty theory ${\mathcal {M}}\{P\cap Q\}=f({\mathcal {M}}\{P\},{\mathcal {M}}\{Q\})={\mathcal {M}}\{P\}\land {\mathcal {M}}\{Q\}$ —which is to say, $min({\mathcal {M}}\{P\},{\mathcal {M}}\{Q\})$—, for two independent events. Thiagovscoelho (talk) 14:50, 20 April 2024 (UTC)

(I'm not sure how best to improve the article itself to make this clearer.) Thiagovscoelho (talk) 15:00, 20 April 2024 (UTC)