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tickli comments on Reflection in Probabilistic Logic - Less Wrong
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Comments (171)
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)<1/n, we deduce that with probability one, "Pr("G")<1/n" is true for all n. On the other hand, if, with probability greater than 0, "Pr("G")=0", by the construction of G, 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".
This is right. (I think we point out in the paper that the model necessarily contains non-standard infinitesimals.) But an interesting question is: can this distribution assign any model a positive probability? I would guess not (e.g. because P("G") is uniformly random for some "G").
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.