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Reflection in Probabilistic Logic

I agree with your guess and I think that I see a way of proving it.

Let us make the same assumptions as in Theo. 2 and assume that T contains PA.

Let G be any sentence of L'. We can build a second sentence G2 such that "G2 <=> ( G and (Pr(G and G2)< p))" for P(G)/3< p <2P(G)/3, using diagonalization. From this and the reflection scheme, it should be possible to prove that P(G and G2) and P(G and not G2) are both smaller than (2/3)P(G).

Repeating the argument above, we can show that any complete theory in L' must have vanishing probability and therefore every model must also have vanishing probability.

Reflection in Probabilistic Logic

I think that Theo. 2 of your paper, in a sense, produces only non-standard models. Am I correct?

Here is my reasoning:

If we apply Theo. 2 of the paper to a theory T containing PA, we can find a sentence G such that "G <=> Pr("G")=0" is a consequence of T. ("Pr" is the probability predicate inside L'.)

If P(G) is greater than 1/n (I use P to speak about probability over the distribution over models of L'), using the reflection scheme, we get that with probability one, "Pr("G")>1/n" is true. This means that with probability one, G is false, a contradiction. We deduce that P(G)=0.

Using the reflection scheme and that, for all n, P(G)0, which as shown above is false.

From the last two sentences, with probability one, inside of the model, there is a strictly positive number smaller than 1/n for any standard integer n: the value of Pr("G").

It might be interesting to see if we can use this to show that Theo. 2 cannot be done constructively, for some meaning of "constructively".

Reflection in Probabilistic Logic

The main theorem of the paper (Theo. 2) does not seem to me to accomplish the goals stated in the introduction of the paper. I think that it might sneakily introduce a meta-language and that this is what "solves" the problem.

What I find unsatisfactory is that the assignment of probabilities to sentences is not shown to be definable in L. This might be too much to ask, but if nothing of the kind is required, the reflection principles lack teeth. In particular, I would guess that Theo. 2 as stated is trivial, in the sense that you can simply take to only have value 0 or 1. Note, that the third reflection principle imposes no constraint on the value of on sentences of L' \ L.

Your application of the diagonalisation argument to refute the reflection scheme (2) also seems suspect, since the scheme only quantifies over sentences of L and you apply it to a sentence G which might not be in L. To be exact, you do not claim that it refutes the scheme, only that it seems to refute it.