All of Ebthgidr's Comments + Replies

My suspicion about the thin-tailed risk here is that either congress or the SEC passes landmark regulation about SPACs (which is potentially plausible) and those stocks go to 0, very quickly, as the initial investors who IPOed the SPAC pull their money out.  See, ICOs (though those were obviously higher risk)

Ohhh, thanks. That explains it. I feel like there should exist things for which provable(not(p)), but I can't think of any offhand, so that'll do for now.

0DanielFilan

To answer the below: I'm not saying that provable(X or notX) implies provable (not X). I'm saying...I'll just put it in lemma form(P(x) means provable(x):

If P( if x then Q) AND P(if not x then Q)

Then P(not x or Q) and P(x or Q): by rules of if then

Then P( (X and not X) or Q): by rules of distribution

Then P(Q): Rules of or statements

So my proof structure is as follows: Prove that both Provable(P) and not Provable(P) imply provable(P). Then, by the above lemma, Provable(P). I don't need to prove Provable(not(Provable(P))), that's not required by the ... (read more)

0DanielFilan
Breaking my no-comment commitment because I think I might know what you were thinking that I didn't realise that you were thinking (won't comment after this though): if you start with (provable(provable(P)) or provable(not(provable(P)))), then you can get your desired result, and indeed, provable(provable(P) or not(provable(P))). However, provable(Q or not(Q)) does not imply provable(Q) or provable(not(Q)), since there are undecideable questions in PA.
0DanielFilan
I agree that if you could prove that (if not(provable(P)) then provable(P)), then you could prove provable(P). That being said, I don't think that you can actually prove (if not(provable(P)) then provable(P)). A few times in this thread, I've shown what I think the problem is with your attempted proof - the second half of step 3 does not follow from the first half. You are assuming X, proving Y, then concluding provable(Y), which is false, because X itself might not have been provable. I am really tired of this thread, and will no longer comment.

is x or not x provable? Then use my proof structure again.

0DanielFilan
The whole point of this discussion is that I don't think that your proof structure is valid. To be honest, I'm not sure where your confusion lies here. Do you think that all statements that are true in PA are provable in PA? If not, how are you deriving provable(if x then q) from (if x then q)? In regards to your above comment, just because you have provable(x or not(x)) doesn't mean you have provable(not(x)), which is what you need to deduce provable(if x then q).

So then here's a smaller lemma: for all x and all q:

If(not(x))

Then provable(if x then q): by definition of if-then

So replace x by Provable(P) and q by p.

Where's the flaw?

0DanielFilan
The flaw is that you are correctly noticing that provable(if(not(x) then (if x then q)), and incorrectly concluding if(not(x)) then provable(if x then q). It is true that if(not(x)) then (if x then q), but if(not(x)) is not necessarily provable, so (if x then q) is also not necessarily provable.

Oh, that's what I've been failing to get across.

I'm not saying if not(p) then (if provable(p) then q). I'm saying if not provable(p) then (if provable(p) then q)

0DanielFilan
You aren't saying that though. In the post where you numbered your arguments, you said (bolding mine) which is different, because it has an extra 'provable'.

So the statement (if not(p) then (if p then q)) is not provable in PA? Doesn't it follow immediately from the definition of if-then in PA?

0DanielFilan
(if not(p) then (if p then q)) is provable. What I'm claiming isn't necessarily provable is (if not(p) then provable(if provable(p) then q)), which is a different statement.

That doesn't actually answer my original question--I'll try writing out the full proof.

Premises:

  1. P or not-P is true in PA

  2. Also, because of that, if p -> q and not(p)-> q then q--use rules of distribution over and/or

So:

  1. provable(P) or not(provable(P)) by premise 1

2: If provable(P), provable(P) by: switch if p then p to not p or p, premise 1

3: if not(provable(P)) Then provable( if provable(P) then P): since if p then q=not p or q and not(not(p))=p

4: therefore, if not(provable(P)) then provable(P): 3 and Lob's theorem

5: Therefore Prova... (read more)

0DanielFilan
I think step 3 is wrong. Expanding out your logic, you are saying that if not(provable(P)), then (if provable(P) then P), then provable(if provable(P) then P). The second step in this chain is wrong, because there are true facts about PA that we can prove, that PA cannot prove.

Well, there is, unless i misunderstand what meta level provable(not(provable(consistency))) is on.

0DanielFilan
I think you do misunderstand that, and that the proof of not(provable(consistency(PA))) is not in fact in PA (remember that the "provable()" function refers to provability in PA). Furthermore, regarding your comment before the one that I am responding to now, just because not(provable(C)) isn't provable in PA, doesn't mean that provable(C) is provable in PA: there are lots of statements P such that neither provable(P) nor provable(not(P)), since PA is incomplete (because it's consistent).

Your reasons were that not(provable(c)) isn't provable in PA, right? If so, then I will rebut thusly: the setup in my comment immediately above(I.e. either provable(c) or not provable(c)) gets rid of that.

0DanielFilan
I'm not claiming that there is no proposition C such that not(provable(C)), I'm saying that there is no proposition C such that provable(not(provable(C))) (again, where all of these 'provable's are with respect to PA, not our whole ability to prove things). I'm not seeing how you're getting from not(provable(not(provable(C)))) to provable(C), unless you're commuting 'not's and 'provable's, which I don't think you can do for reasons that I've stated in an ancestor to this comment.

I'll rephrase it this way:
For all C: Either provable(C) or not(provable(C)) If provable(C), then provable(C) If not provable(C), then use the above logic to prove provable C. Therefore all C are provable.

0DanielFilan
Which "above logic" are you referring to? If you mean your OP, I don't think that the logic holds, for reasons that I've explained in my replies.

Wait. Not(provable(consistency)) is provable in PA? Then run that through the above.

0DanielFilan
I'm not sure that this is true. I can't find anything that says either way, but there's a section on Godel's second incompleteness theorem in the book "Set theory and the continuum hypothesis" by Paul Cohen that implies that the theorem is not provable in the theory that it applies to.

Ok, thanks for clearing that up.

That's an interesting correlation, but I'm curious about the causal link: is it that a certain type of neural architecture causes both predisposition to rationality and asperger's, or the social awkwardness added on to the neural architecture creates the predisposition--i.e. I'm curious to see how much being social affects rationality. I shall need to look into this more closely.

1ilzolende
On the subject of potential causal linkages: I think that at least part of the reason us diagnosed autistic/Asperger's people are more prevalent on LessWrong is that those of us diagnosed as children spend a lot of time with adults who think that something's wrong with our mental processes, often without telling us why. I know that I picked up on this, and then when I heard about cognitive biases, I jumped to the conclusion "These are what's wrong with me, but if I read more about them, then I can try and correct for them." Then, I looked up cognitive biases, found the Overcoming Bias blog, decided it was more economics than I could handle, and then I ended up here, because it had less real-world economics. Test: See if more LWs were incorrectly given a psychiatric diagnosis as children than members of the general population were.

I forget the formal name for the theorem, but isn't (if X then Y) iff (not-x or Y) provable in PA? Because I was pretty sure that's a fundamental theorem in first order logic. Your solution is the one that looked best, but it still feels wrong. Here's why: Say P is provable. Then not-P is provably false. Then not(provable(not-P)) is provable. Not being able to prove not(provable(x)) means nothing is provable.

0DanielFilan
You're right that (if X then Y) is just fancy notation for (not(X) or Y). However, I think you're mixing up levels of where things are being proved. For the purposes of the rest of this comment, I'll use provable(X) to mean that PA or whatever proves X, and not that we can prove X. Now, suppose provable(P). Then provable(not(not(P))) is derivable in PA. You then claim that not(provable(not(P))) follows in PA, that is to say, that provable(not(Q)) -> not(provable(Q)). However, this is precisely the statement that PA is consistent, which is not provable in PA. Therefore, even though we can go on to prove not(provable(not(P))), PA can't, so that last step doesn't work.

A question about Lob's theorem: assume not provable(X). Then, by rules of If-then statements, if provable(X) then X is provable But then, by Lob's theorem, provable(X), which is a contradiction. What am I missing here?

3DanielFilan
I'm not sure how you're getting from not provable(X) to provable(provable(X) -> X), and I think you might be mixing meta levels. If you could prove not provable(X), then I think you could prove (provable(X) ->X), which then gives you provable(X). Perhaps the solution is that you can never prove not provable(X)? I'm not sure about this though.

I finished up to the first major plot twist/divergence in the rationalfic(well, sort of. I'll just call it an attempted rationalfic) I've been working on for 3 months or so, and it's now in the top 15 most followed fics in the fandom(Danganronpa). Link: light in despair's darkness

Not only that--the greater degree of neuroplasticity that I think 16-year olds still have(if I'm wrong about this, someone please correct me) makes it a good deal easier to learn skills/ingrain rationality techniques.

2dxu
As a fellow 16-year-old (there really seem to be a lot of us popping up around here recently), I concur. With that said, rationality skills are difficult for anyone to learn, because the human brain did not evolve to be rational, but rather to succeed socially. I would add that a good deal of rationality potential is ingrained in those who find themselves attracted to LW at a young age, particularly since surveys have shown that LW users tend to have a higher incidence rate of Asperger Syndrome, the symptoms of which include social awkwardness. This suggests to me that rational thinking comes more easily to people with certain personality types, which is arguably genetic. As a single data point, I suppose I'll add that I myself have been diagnosed with Asperger's when I was younger, although with how trigger-happy American doctors are with their diagnoses these days, that's not really saying much.

Nice to meet you--it's rather reassuring to see another member at my age.

Hello. I'm Leor Fishman, and also go by 'avret' on both reddit and ffn. I am currently 16. The path I took to get here isn't as...dramatic as some of the others I've seen, but I may as well record it: For as long as I can remember, I've been logically minded, preferring to base hypotheses on evidence than to rest them on blind faiths. However, for the majority of my life, that instinct was unguided and more often than not led to rationalizations rather than belief-updating. A few years back, I discovered MoR during a stumbleupon binge. I took to it lik... (read more)

5ilzolende
Welcome! I'm also 16. Welcome to the group of people who answer "no" to the "were you alive 20 years ago" question on a technicality. It's really great to know about risk assessment errors and whatnot when we're still teenagers, just because the bugs in our brains are even more dangerous when ignored than normal.
5Shmi
Oh how I wish I had access to this kind of material when I was 16.
1Gondolinian
Welcome, Leor! I'm also a 16 year old new member.

Is this a possible explanation or corollary to the sunk-costs fallacy of economics?

For number 3, I realize the implied point, and I assume that there is more to this argument, but that sentence was one big strawman. Also, I would respond by asking why someone following the 'true essence' but confirming to modern societal/ethical norms is any worse than someone who is following said norms for a different reason. For #4, those novels don't explicitly provide ethical direction-one can use a system of ethical precepts without it being absolute and unchangeable.

Isn't his usage of "switches flipping" basically another 'literary genre' switch--i.e. he had attached some sort of negative connotation to the phrase which he could not conceive of attaching to intelligence?