= 27d567bc2d48d9f40e9ccc20ea370b15)
tickli comments on Reflection in Probabilistic Logic - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (171)
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.
You are completely right, and thanks for the correction! A new version with this problem fixed should be up in a bit.
(The actual proof of theorem 2 establishes the correct thing, there is just a ' missing from the statement.)
Actually, I believe you've found a bug in the paper which everyone else seems to have missed so far, but fortunately it's just a typo in the statement of the theorem! The quantification should be over L', not over L, and the proof does prove this much stronger statement. The statement in the paper is indeed trivial for the reason you say.
Given the stronger statement, the reason you can't just have P have value 0 or 1 is sentences like G <=> P('G') < 0.5: if P(G) = 0, by reflection it would follow that P(G) = 1, and if P(G) = 1, then P(not G) = 0, and by reflection it would follow that P(not G) = 1.