tickli

• tickli 27 Mar 2013 21:55 UTC

  0 points

  in reply to: paulfchristiano’s comment on: 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.

• tickli 27 Mar 2013 14:37 UTC

  8 points

  on: 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”.

• tickli 26 Mar 2013 14:56 UTC

  27 points

  on: 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 $mathbb\left\{P\right\}$ 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 $mathbb\left\{P\right\}$ to only have value 0 or 1. Note, that the third reflection principle imposes no constraint on the value of $mathbb\left\{P\right\}$ 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.