pengvado comments on Walkthrough of "Definability of Truth in Probabilistic Logic" - Less Wrong

11 Post author: So8res 09 December 2013 03:44AM

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Comment author: benkuhn 10 December 2013 12:40:18AM * 1 point [-]

I'm confused by a couple minor points here, also:

The paper asks for a "probability distribution over models of L". In fact, for many languages L, models of L form a proper class. Does this cause measure-theoretic difficulties? It seems like this might force mu to be zero on all sufficiently large models (otherwise you can do some sort of transfinite induction to get sets of unbounded measure) but I'm not very good at crazy set theory stuff.
At one point the authors state "We would like P(forall phi in L' <blah>)". I thought we were in a first-order language and therefore couldn't quantify over propositions?
It's not immediately clear to me that this actually constructs a measure on the set of theories: that is, if S is the set of all complete consistent theories, it's not clear to me that for the mu we construct by martingale, mu(S) = 1 (or even that mu(S) != 0). Mightn't additivity break when we take the limit and get a whole theory rather than just an incomplete bag of axioms?

Comment author: pengvado 11 December 2013 04:09:45AM 1 point [-]

Can we instead do "probability distribution over equivalence classes of models of L", where equivalence is determined by agreement on the truth-values of all first order sentences? There's only 2^ℵ₀ of those, and the paper never depends on any distinction within such an equivalence class.

Comment author: benkuhn 11 December 2013 07:14:06AM 1 point [-]

Yes, though we should just call it a "probability distribution over complete consistent theories" in that case (it's exactly the same).