There is a sense in which Cox’s theorem and related formalizations of probability assume that the plausibility of (A|B) is some function F(A,B). But what they end up showing is not that F is some specific function, just that it must obey certain rules (the laws of probability).
So the objectivity is not in the results of the theorem, it’s more like there’s an assumption of some kind of objectivity (or at least self-consistency) that goes into what formalizers of probability are willing to think of as a “plausibility” in the first place.
Thinking about again, I am not sure if the assumption that such a function F exists is as intuitive as I first thought. We are trying to formalise the intuitive concept of the plausibility of a A given B, i.e. “how true the proposition A is given that we know that the proposition B is true”, and this assumption seems to contradict some of our, at least my, intuitions about plausibility.
For example, suppose A is some proposition suppose B is a proposition which tells us absolutely nothing about A. Maybe B = “1+2=3” and A = “The earth is not flat 🌎”. Intuitively, B being true tells us nothing about “how true” A so we should not be able to assign a plausibility but the existence of F contradicts this and implies that there is a number x which represents how plausible A is given B.
Someone might say that the plausibility of A given B is 0.5 applying Professor Jaynes’ principle of transformation-groups/indifference but this is a result of the theory and the existence of a function F s.t. [(A|B) = F(A, B) for any A and B] is an axiom of the theory. You can’t make an assumption, prove something using that assumption, and use your result to justify the assumption right?
I think the idea of imprecise probability was conceived to solve this exact problem. Instead of talking about the degree of plausibility (i.e. how true something is), imprecise probability talks about the upper bound and the lower bound of the degree of plausibility and represent that with real numbers, ¯P(A|B) and P––(A|B) respectively. So instead of postulating the existence of a function F s.t. (A|B) = F(A|B), they postulate the existence of functions ¯¯¯¯F and F–– s.t.¯P(A|B)=¯F(A,B) and P––(A|B)=F––(A,B). I think this is a weaker and philosophically better axiom because it will hold even in the example that I gave above: B = “1+2=3” and A = “The earth is not flat 🌎”. Intuitively even though B tells us nothing about A, we can still assign an upper bound and lower bound of plausibility: 1 (true) and 0 (false) so the new axiom is in line with our intuitions. How much our upper and lower bounds change of the plausibility of A when we condition on B from 1 and 0 respectively tells us how informative B is about A. In the example above ⬆️, B is not informative at all about A because the upper bound is still 1 and the lower bound is still 0.
Perhaps we can impose objectiveness using some other assumption instead of resorting to imprecise probability?
It is also controversial what the (fixed-up) mathematical result means philosophically. Whereas in 1946, when Cox published his Theorem, there clearly was nothing else like probability theory, there are now a variety of related mathematical systems for reasoning about uncertainty.
These share a common motivation. Probability theory doesn’t work when you have inadequate information. Implicitly, it demands that you always have complete confidence in your probability estimate,6like maybe 0.5279371, whereas in fact often you just have no clue. Or you might say “well, it seems the probability is at least 0.3, and not more than 0.8, but any guess more definite than that would be meaningless.”
So various systems try to capture this intuition: sometimes a specific numerical probability is unavailable, but you can still do some reasoning anyway. These systems coincide with probability theory in cases where you are confident of your probability estimate, and extend it to handle cases where you aren’t.
Please let me know what you think about this. Thanks!
For example, suppose A is some proposition suppose B is a proposition which tells us absolutely nothing about A. Maybe B = “1+2=3” and A = “The earth is not flat 🌎”. Intuitively, B being true tells us nothing about “how true” A so we should not be able to assign a plausibility but the existence of F contradicts this and implies that there is a number x which represents how plausible A is given B.
I’m not sure what your objection is here. You appear to be using propositions A and B that we believe to be almost certainly true, in which case the plausibility of A given B must be very close to 1 by definition. Maybe you meant to negate B?
In general though, yes Cox’s result is an existence proof, not a uniqueness one. It means that under given reasonable conditions you can use probability theory, but doesn’t tell you what probabilities to assign to which propositions.
Jaynes’ extension of this to objective probabilities is much more controversial and does not have anything like a mathematical proof.
I’m not sure what your objection is here. You appear to be using propositions A and B that we believe to be almost certainly true, in which case the plausibility of A given B must be very close to 1 by definition. Maybe you meant to negate B?
Sorry about that. Think it would have been clearer if chose a different A and B. But I believe the argument still holds because by (A|B), I did not mean (A|B, I) where I is the background information (I should have been clearer about this as well as I am going against the convention). So basically, we have are not conditioning on any background information so (A|B) is not close to one by definition.
In general though, yes Cox’s result is an existence proof, not a uniqueness one. It means that under given reasonable conditions you can use probability theory, but doesn’t tell you what probabilities to assign to which propositions.
Well what does it prove the existence of? Are you saying that Cox’s theorem implies the existence of a function F such that [the plausibility (A|B) = F(A, B) for any propositions A and B]?
Jaynes’ extension of this to objective probabilities is much more controversial and does not have anything like a mathematical proof.
I do not think that Professor Jaynes’ theory necessarily warrants a mathematical proof as we are only trying to formalise our intuitions about plausibilities. My contention is that Professor Jaynes’ theory contradicts our intuitions about plausibilities and hence the necessity of an “imprecise” theory of plausibility which addresses this problem.
We cannot argue that Cox’s theorem justifies Professor Jaynes’ theory if we start with axioms which are not consistent with our understanding of plausibility. This is what David Chapman argues:
Cox’s Theorem says that there is no formal system other than probability theory that is very similar to it; so if you want something like that, you’ve only got one choice.5 This is irrelevant unless you are considering using one of the dubious alternatives, none of which seems to work as well in practice. - What probability can’t do
Sort of?
There is a sense in which Cox’s theorem and related formalizations of probability assume that the plausibility of (A|B) is some function F(A,B). But what they end up showing is not that F is some specific function, just that it must obey certain rules (the laws of probability).
So the objectivity is not in the results of the theorem, it’s more like there’s an assumption of some kind of objectivity (or at least self-consistency) that goes into what formalizers of probability are willing to think of as a “plausibility” in the first place.
Thinking about again, I am not sure if the assumption that such a function F exists is as intuitive as I first thought. We are trying to formalise the intuitive concept of the plausibility of a A given B, i.e. “how true the proposition A is given that we know that the proposition B is true”, and this assumption seems to contradict some of our, at least my, intuitions about plausibility.
For example, suppose A is some proposition suppose B is a proposition which tells us absolutely nothing about A. Maybe B = “1+2=3” and A = “The earth is not flat 🌎”. Intuitively, B being true tells us nothing about “how true” A so we should not be able to assign a plausibility but the existence of F contradicts this and implies that there is a number x which represents how plausible A is given B.
Someone might say that the plausibility of A given B is 0.5 applying Professor Jaynes’ principle of transformation-groups/indifference but this is a result of the theory and the existence of a function F s.t. [(A|B) = F(A, B) for any A and B] is an axiom of the theory. You can’t make an assumption, prove something using that assumption, and use your result to justify the assumption right?
I think the idea of imprecise probability was conceived to solve this exact problem. Instead of talking about the degree of plausibility (i.e. how true something is), imprecise probability talks about the upper bound and the lower bound of the degree of plausibility and represent that with real numbers, ¯P(A|B) and P––(A|B) respectively. So instead of postulating the existence of a function F s.t. (A|B) = F(A|B), they postulate the existence of functions ¯¯¯¯F and F–– s.t.¯P(A|B)=¯F(A,B) and P––(A|B)=F––(A,B). I think this is a weaker and philosophically better axiom because it will hold even in the example that I gave above: B = “1+2=3” and A = “The earth is not flat 🌎”. Intuitively even though B tells us nothing about A, we can still assign an upper bound and lower bound of plausibility: 1 (true) and 0 (false) so the new axiom is in line with our intuitions. How much our upper and lower bounds change of the plausibility of A when we condition on B from 1 and 0 respectively tells us how informative B is about A. In the example above ⬆️, B is not informative at all about A because the upper bound is still 1 and the lower bound is still 0.
Perhaps we can impose objectiveness using some other assumption instead of resorting to imprecise probability?
David Chapman, the metarationality guy, shares a similar critique here—Probability theory does not extend logic:
Please let me know what you think about this. Thanks!
I’m not sure what your objection is here. You appear to be using propositions A and B that we believe to be almost certainly true, in which case the plausibility of A given B must be very close to 1 by definition. Maybe you meant to negate B?
In general though, yes Cox’s result is an existence proof, not a uniqueness one. It means that under given reasonable conditions you can use probability theory, but doesn’t tell you what probabilities to assign to which propositions.
Jaynes’ extension of this to objective probabilities is much more controversial and does not have anything like a mathematical proof.
Thanks for the reply!
Sorry about that. Think it would have been clearer if chose a different A and B. But I believe the argument still holds because by (A|B), I did not mean (A|B, I) where I is the background information (I should have been clearer about this as well as I am going against the convention). So basically, we have are not conditioning on any background information so (A|B) is not close to one by definition.
Well what does it prove the existence of? Are you saying that Cox’s theorem implies the existence of a function F such that [the plausibility (A|B) = F(A, B) for any propositions A and B]?
I do not think that Professor Jaynes’ theory necessarily warrants a mathematical proof as we are only trying to formalise our intuitions about plausibilities. My contention is that Professor Jaynes’ theory contradicts our intuitions about plausibilities and hence the necessity of an “imprecise” theory of plausibility which addresses this problem.
We cannot argue that Cox’s theorem justifies Professor Jaynes’ theory if we start with axioms which are not consistent with our understanding of plausibility. This is what David Chapman argues:
Chapman discusses this in more detail here: Probability theory does not extend logic.
Fantastic answer! Thanks a lot—I really appreciate it