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?