alex_zag_al comments on A Limited But Better Than Nothing Way To Assign Probabilities to Statements of Logic, Arithmetic, etc.

[alex_zag_al]

the problem is that, in probability theory as usually formalized and discussed, we assign the same probabilities, though we shouldn’t… do you see?

EDIT: and it’s a problem because she can’t calculate her probability without proving whether or not it’s prime.

[Watercressed]

One of his “desiderata”, his principles of construction, is that the robot gives equal plausibility assignments to logically equivalent statements

I don’t see this desiderata. The consistency requirement is that if there are multiple ways of calculating something, then all of them yield the same result. A few minutes of thought didn’t lead to any way of leveraging a non 1 or zero probability for Prime(53) into two different results.

If I try to do anything with P(Prime(53)|PA), I get stuff like P(PA|Prime(53)), and I don’t have any idea how to interpret that. Since PA is a set of axioms, it doesn’t really have a truth value that we can do probability with. Technically speaking, Prime(N) means that the PA axioms imply that 53 has two factors. Since the axioms are in the predicate, any mechanism that forces P(Prime(53)) to be one must do so for all priors.

One final thing: Isn’t it wrong to assign a probability of zero to Prime(4), i.e. PA implies that 4 has two factors, since PA could be inconsistent and imply everything?

[alex_zag_al]

I now think you’re right that logical uncertainty doesn’t violate any of Jaynes’s desiderata. Which means I should probably try to follow them more closely, if they don’t create problems like I thought they would.

An Aspiring Rationalist’s Ramble has a post asserting the same thing, that nothing in the desiderata implies logical omniscience.