Benya_Fallenstein comments on Reflective probabilistic logic cannot assign positive probability to its own coherence and an inner reflection principle

[Benya_Fallenstein]

If you replace the inner “$=1$” by “$>1-ϵ$”, then the literal thing you wrote follows from the reflection principle: Suppose that the outer probability is $<1$. Then

$$P\left[\right(a<P\left[\phi \right]<b)\wedge P[a-ϵ<P\left[\phi \right]<b+\epsilon ]\le 1-ϵ]>0.$$

Now, $P[a<P[\phi ]<b]>0$ implies $P[a\le P[\phi ]\le b]>0$, which by the converse of the outer reflection principle yields $a\le P\left[\phi \right]\le b$, whence $a-ϵ<P\left[\phi \right]<b+ϵ$. Now, by the forward direction of the outer reflection principle, we have

$$P[a-ϵ<P[\phi ]<b+ϵ]=1>1-ϵ,$$

which, by the outer reflection principle again, implies

$$P\left[P\right[a-ϵ<P\left[\phi \right]<b+ϵ]>1-ϵ]=1,$$

a contradiction to the assumption that $\cdots \le 1-ϵ$ had outer probability $>0$.

However, what we’d really like is an inner reflection principle that assigns probability one to the statement *quantified over all $a$, $b$, and Gödel numbers $┌\phi ┐$. I think Paul Christiano has a proof that this is impossible for small enough $ϵ$, but I don’t remember how the details worked.

[paulfchristiano]

Here is the basic problem. I think that you can construct an appropriate liar’s sentence by using a Lipschitz function without an approximate fixed point. But someone might want to check that more carefully and write it up, to make sure and to see where the possible loopholes are. I think that it may not have ruled out this particular principle, just something slightly stronger (but the two were equivalent for the kinds of proof techniqeus we were considering).