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I read about Cox's theorem a long time ago in "Jaynes Probability Theory: The Logic of Science". It was used to justify the so-called "logical" interpretation of probability. My impression was that the theorem assumes an existent world where every proposition is either true of false, and our limited certainty about this existent world is represented by a system isomorphic to the calculus of probability theory.

So I guess that the three classic laws of thought

  • law of identity
  • law of noncontradiction
  • law of excluded middle

are implicitly assumed to hold in the proof of the theorem, even so they are never mentioned in the assumptions.

However, the resulting calculus seems to be quite robust, even in cases where the classic laws of thought are violated. Take for example the continuum hypothesis. We know that it is neither valid nor invalid, but we might still model our knowledge about its validity by using the prior probability 0.5. It doesn't look like we will be lead to any wrong conclusions because of this. What seems to be true however is that we haven't managed to accurately represent our actual knowledge, and that we might miss some important conclusions that we might have deduced from this knowledge.

But how misleading is Cox's theorem really? Are there any investigations of situations where it fails spectacularly? And if yes, do there exists modifications of the used calculus which better handle these situations?

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    It is not reasonable to assign a probability to the continuum hypothesis. – Ron Maimon Apr 28 '12 at 4:22
  • @RonMaimon Of course not. But a prior probability is different from a real probability. It just means to represent a subjective "certainty" or "bias". For example, I'm pretty biased to believe that "P is not equal to NP", so I might assign a prior probability of 0.99 to this proposition. I admit that these number don't mean much, especially when assigned to propositions where it is unlikely that I will receive further information that might change my bias. I just got the impression that the calculus is quite robust, even in cases where the classic laws of thought are violated. – Thomas Klimpel Apr 28 '12 at 9:34
  • It is meaningful to assing a probability to P!=NP. It is not meaningful to assing a probability to CH. The first is a question with a model-independent answer, the second is not. It's like asking "what is the probability that groups are Abelian?" There's no probability for that, because some groups are and some groups aren't. – Ron Maimon Apr 28 '12 at 14:41
  • @RonMaimon I would assign a prior probability of 10^-100 to the proposition that (all) groups are abelian. I simply may not know that the validity of that proposition is model dependent. Perhaps P!=NP wasn't a good example. At least Jaynes seemed to argue to apply prior probabilities even in contexts were neither classical logic nor frequentist probabilities were appropriate. Excluding propositions that are devoid of meaning - based on subjective judgement - sounds like a good idea. But what about the meaningless propositions that slip through that subjective judgement? – Thomas Klimpel Apr 28 '12 at 21:55
  • @Ron: You might be able to express the probability that CH can be derived from ZF or some such, no? I'm guessing this is what Thomas meant by it being "true." – Xodarap Apr 29 '12 at 1:34
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I don't think I fully understand your example about CH (it seems to me you could indeed go quite wrong assuming that CH is true with probability 1/2...) but I would encourage you to check out The Philosophical Significance of Cox's Theorem. It highlights some domains in which Cox's theorem could be plausibly said to fail. (One of which is, as you say, when the law of excluded middle fails.)

On the other side of things, Quantum probabilities as Bayesian probabilities demonstrates a link between Bayesian probability and quantum probability (a domain in which the distributive property doesn't hold). So this idea is quite robust, even in the bizarre world of QM.

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    Thanks for the links. The paper about QM actually doesn't mention Cox's theorem at all. I read a bit more about Cox' theorem now. There even exist counterexamples for cases where classic logic holds. My conclusion is that the main issues with this theorem are not of a philosophical nature, but that the used assumptions are not stated explicitly enough. The assumed structure of the space of proposition is simply a part of the assumptions, and should be stated explicitly (i.e. whether it's a σ-algebra, an orthomodular lattice, or ???). Cox's conjecture and Bayesian probability are not identical. – Thomas Klimpel Apr 29 '12 at 15:59
  • @Thomas: Certainly not identical, but one is often used to justify the other. I would agree with most of what you said, except to clarify that there's nothing wrong with the theorem itself - its just that people may use it in ways which aren't justified. – Xodarap Apr 29 '12 at 16:53
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It seems to me that there are two different questions here-- one regarding the reliance on classical logic, and the other about Cox's theorem in practice. The latter appears to me to be better suited for mathematicians or statisticians-- it is certainly beyond my ability to answer.

The former question, though, is a properly philosophical question.

It appears to me that Cox's theorem is implicitly reliant upon classical logic; but then again, so is most of mathematics. If we reject the three laws of classical logic, it becomes extremely difficult to argue for anything rationally; however, we have absolutely no way of grounding them, and are generally forced to take them as axiomatic.

It's certainly possible to propose non-classical or deviant logics, and there are those who have done so-- but these tend to have local applicability at best; most people believe that classical logic is most suitable for everyday operations.

  • My guess is that classical logic works fine as long as the interaction between the existent world referred to by the propositions and the logical reasoning including its conclusions can be neglected. However, probability theory was initially motivated by gambling, and there may be a strong interaction in this context if different players are anticipating the actions of the other players based on their assumed logical reasoning. But also the stock-market shows such interactions, as do placebo effects in medicine. So classical probability theory and statistics normally try to be careful... – Thomas Klimpel Apr 27 '12 at 14:32
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    Although I am not aware of this being done, you certainly could make a bootstrapping argument that rejects classical logic like so: start with classical logic and show that, given classical logic, logic system L is valid; find examples where L is valid but classical logic fails in some way; prove that under special circumstances, however, L does reduce to classical logic; show that the evaluation of the validity of L is such a circumstance. Then essentially one would have shown that "classical logic says to believe L instead and L agrees" which is about as good as one can ever manage. – Rex Kerr Apr 27 '12 at 19:20
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Cox's stated aim was to construct a logic of plausible inference. This was to be considered as an extension of the standard (classical) logic. One of Cox's axioms effectively demands that any such logic of plausible inference be compatible with standard logic. (In Van Horn's exposition of the theorem this is called R2). As has been mentioned in the comments, Cox isn't all that clear about exactly what is being assumed (especially, for instance about the space of propositions) but insofar as Cox does make his assumptions clear, his commitment to classical logic is clear.

So the answer to the question is no, there is nothing implicit about Cox's assumption of classical logic.

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