Some Comments on Cox's Theorem
Apparently Cox's original paper has some subtle issues, some assumptions that were not explicitly mentioned or even realized by Cox. In the interim many years, some of these technicalities have been patched up. Kevin S Van Horn has a great "guide" to Cox's theorem which I like a lot. He has some other interesting related stuff as well.
In his presentation of Cox's theorem, he relies on 5 requirements
- R1: plausibility is a real number
- R2-(1-4): plausibility agrees in the limiting case with propositional calculus (where everything is either known to be true, or known to be false).
- R3: plausibility of a proposition and of its negation are functionally related.
- R4: Universality: In essence plausibility values take on a dense subset of an interval of real numbers, and we can construct propositions that have certain plausibilities associated with them.
- R5: Conjunction: The conjunction function is a continuous strictly increasing function F such that p(A and B | X) = F(p(A|B,X),p(B|X)).
Kevin motivates each of these and gives some objections and discussion about why each one is required. For me, the point of the exercise is to generalize propositional calculus to real-number plausibility, so R1,R2,R3 are obvious to me, though I realize some people have objected (for example, there are 2-dimensional alternatives, where a proposition and its negation have different plausibilities unrelated by a functional equation)
I'm just not interested in such things, so when I look at Kevin's requirements, Universality and Conjunction are the ones I think you could attack. To me, the dense set of values, and that there must be more than one value (the endpoints of the interval are not the same) seem obvious.
The full version of R4 is:
R4: There exists a nonempty set of real number with the following two properties:
- is a dense subset of (F,T).
- For every there exists some consistent X with a basis of at least three atomic propositions such that
This is perhaps the least understandable part. It says basically that for any three plausibility values, there exists some consistent state of knowledge that assigns those plausibility values to certain propositions. A consistent state of knowledge is one where we can't prove a contradiction (ie. A and Not A).
Note that this isn't a claim about the actual world. We aren't restricted to "states of knowlege" that are say consistent with the laws of physics, or consistent with a historically accurate account of the Norman Conquest, etc. This is a mathematical statement about whether we can *assign* certain plausibilities to certain statements without causing a contradiction. So "A1 = Unicorns keep an extra horn at home for emergencies" and "A2 = The second sun of the planet Gorlab rises every thursday" and soforth are acceptable statements whose truth value we could assign certain plausibilities to to meet these logical requirements. This is called "universality" because it expresses the desire to be able to assign plausibilities at will to at least 3 statements regardless of what those statements are about (whether that's unicorns, or the position of fictional suns, or it's the efficacy of a real drug for treatment of diabetes, or the albedo of a real planet circling a remote star).
Van Horn points out that without R4 we can construct a counterexample, so something possibly weaker might be allowable, but we can't omit it entirely. Van Horn also points out that Halpern requires that the set of pairs of propositions be infinite. But note, there's nothing that requires us to restrict Cox's theorem to any one area of study. So, for example, while you may be using probability as logic to make Bayesian decision theory decisions about a finite set of objects whose measurements come from finite sets, someone else is using it to make decisions about continuous parameters over infinite domains. This is Van Horn's argument, basically that we're looking for a single universal system that works for all propositions, not just some finite set of propositions about your favorite subject. He goes further to point out that what Halpern describes as really should be where is a state of information about the statements. So long as you allow an infinite set of possible states of information, you can still apply Cox's theorem within a finite domain of finite objects.
Van Horn's article is worth a read, and he discusses many issues worth considering.