I tried applying the proof of the theorem to the problem, as P(q) = P(p) for equivalent statements p,q is clear from the claim P(q) = mu({M: M|= q}) so our desired result should be part of the proof of Theorem 1. However it seems that our desired result is implicitly assumed in the statement “Axiom 1 implies”. Setting P(A|B) = P(B && A)/P(B) (note the reversal of order, without being allowed to replace equivalent statements inside P the mess is even greater if we pick the natural order. Also we want P(B) =/= 0) and writing psi for phi j we can write the claim as:
P(Ti && phi)/P(Ti) = P(Ti && psi && phi)/P(Ti && psi) P(Ti && psi)/P(Ti) + P(Ti && ~psi && phi)/P(Ti && ~psi) P(Ti && ~psi)/P(Ti)
(Here I ignored some annoying brackets, actually there’s more of a mess as its not clear if P((A && B) && C) = P(A && (B && C))). Multiplying both sides by P(Ti) and simplifying now gives:
P(Ti && phi) = P(Ti && psi && phi) + P(Ti && ~psi && phi)
which does not follow from Axiom 1 as the added statements are in the middle of our statement. I am convinced that our claim (P(p) = P(q) for equivalent statements p,q) is required for the proof in the paper and should be added as an extra axiom.
On a sidenote:unde the assumption above (equivalent statements get equal probabilities) axiom 3 is a consequence of axioms 1 and 2. This can be seen as follows: Let q be a contradiction, so ~q is a tautology. By axioms 1 and 2 we have P(q) = P(q && q) + P(q && ~q). But ~q is a tautology, so p && ~q and p are equivalent for every p. In particular we have P(q && ~q) = P(q). But q and (q && q) are also equivalent, so the above can be written as P(q) = P(q) + P(q) so P(q) = 0.
Took the survey, and continued to finally make an account. Some questions were ambiguous though (as some other people partially pointed out). I had most problems with:
Having more children. Over which period of time? As an adolescent I’m not really keen on having children just yet, but I might be in 15 or 20 years.
Time on LW: I’ve recently finished reading almost all posts on LW, which meant I spent several hours a day on LW. But now that I have finished reading all those I am only reading new posts, which takes no more than 5-10 minutes a day on average. So there is a large difference between a best estimate of the amount of time I spent on LW any previous day and the best estimate of the time I will spend tomorrow. Which of these is the average day?
Hear about: I had problems interpreting the question. Taking the wording literally the category specified is extremely broad, including even casual comments by colleagues along the lines of: ‘Try checking the batteries more frequently.’ (which is a technique to improve your productivity, provided batteries are important in your line of work).
Akrasia: meditation. I’ve meditated after sporting frequently in the past, which had nothing to do with akrasia. I decided not to mention the meditation (contrary to Keller, whose comment I only noticed after filling in the survey).