The problem is how classical logical statements work. The statement "If A then B" more properly translates as "~(A and ~B)".

Thus, we get valid logical statements that look bizarre to humans: "If Paris is the capital of France, then Rome is the capital of Italy" seems untrue in a causal sense (if we changed the capital of France, we would not change the capital of Italy, and vice versa) but it is true in a logical sense, because A is true, B is true, true and ~true is false, and ~false is true.

That example seems just silly, but the problem is the reverse example is disastrous. Notice that, because of the "and," if A is false then it doesn't matter what B is: false and X is false, ~false is true. If I choose the premise "Marseilles is the capital of France," then any B works. "If Marseilles is the capital of France, then I will receive infinite utility" is a true relationship under classical logic, but is clearly not a causal relationship: changing the capital will not grant me infinite utility, and as soon as the capital changes, the logical truth of the sentence will change.

If you have a reasoner that makes decisions, they need to use causal logic, not classical logic, or they'll get tripped up by the word "implication."

but it makes me uncomfortable when LWers say things like:

Edited! If that's poor phrasing, I want to fix it. My intended goal was "I need to reference the topic of this article in some manner, so that people will know why to read it." and from your post that wasn't getting across.

However, that is not the first critique I have gotten about phrasing, and in retrospect, I am concerned that I am more of a rationality pretender than an actual rationalist. I mean, I approve of rationality, and I try to follow the math (and can't when it starts getting hard, frequently because it would take too long and I am usually following Less Wrong intermittently while focusing on other things as well), but I have received multiple complaints that I feel like I can fairly sum up as "You're the rationalist equivalent of a annoying cheerleader yelling 'Go Team, Smash the Other Team', that's not what rationality is about, please stop."

I think it is safe to say that I really do have that as a problem (multiple different sources seem to indicate it to me.) And I would prefer to fix it, but I'm not sure how to fix it. If you or anyone else have thoughts on how to change, I am open to suggestion.

I believe that (mathematical) proofs aren't easily reducible to axiomatic proofs, and that proofs have been, and still are, profoundly social by their nature, although I don't know if that will continue indefinitely. I probably won't find the time to write a large post on this topic that I've been thinking of, so I want to quote here one observation that's been on my mind recently, in case someone finds it useful.

Eliezer quotes (in the post which I, with accordance to the above, see as wrong-headed in some respects) one definition of proof that he disagrees with: "A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing."

I think this is too vague as a definition of proof and doesn't really work, but it does capture the social/communication aspect of it I consider important. A thinks X is right and P is a proof of X. It isn't enough that P convinces A that X is right; when A communicates P to B, it should convince them too.

A few days ago, I was rereading Vladimir Uspensky's essay on the philosophy of mathematics that I read many years ago and forgot. In the section about proofs, Uspensky offers this informal definition:

A proof is a convincing argument that convinces us to such a degree that we can with its help convince others.

That is, it isn't enough that B is convinced by P that X is right; P should be such that B should be able to spread the gospel onwards. A proof doesn't just bridge a void between two minds; it's capable of leaping on and on. It's a virus with conviction as its payload: it instills conviction in its host and can spread itself (rather than merely conviction) to others.

I've been musing since then about this addition to what I'd thought of as an informal social definition of a proof. I've been going back and forth about how necessary and profound it is, but I think I'm converging on forth.

I found an article explaining Motivated Reasoning in The Atlantic and it seemed like a good fit for the Open Thread.

http://www.theatlantic.com/politics/archive/2012/11/how-partisans-fool-themselves-into-believing-their-own-spin/265336/

That being said, one of the core links inside the article (The one that links to the paper that it is using to draw some of it's conclusions) was broken. I've pasted the correct link below if you want to read the paper as well.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2071478

Can anyone point me toward work that's been done on the five-and-ten problem? Or does someone want to discuss it here? Specifically, I don't understand why it is a problem for probabilistic algorithms. I would reason:

There is a high probability that I prefer $10 to $5. Therefore I will decide to choose $5, with low probability.

And there's nowhere to go from there. If I try to use the fact that I chose $5 to prove that $5 was the better choice all along (because I'm rational), I get something like:

The probability that I prefer $5 to $10 is low. But I have very high confidence in my rationality, meaning that I assign high probability, a priori, to any choice I make being the choice I prefer. Therefore, given that I choose $5, the probability that I prefer $5 is high. So $5 doesn't seem like a bad choice, since I'll probably end up with what I prefer.

But things still turn out right, because:

However, the probability that I prefer $10, given that I choose $10, is even higher, because the probability that I prefer $10 was high to begin with. Therefore, $10 is a better choice than $5, because the probability that (I prefer $10 to $5 given that I choose $10) is higher than the probability that (I prefer $5 to $10 given that I choose $5).

So unless I'm missing something, the five-and-ten problem is just a problem of overconfidence.

If I 'work' for forty hours next week on reading LW

Reading Less Wrong is consumptive, not productive. You need to have something to show for your work, ex. a novel draft, a fitter body, a cleaner house.

If you want to accomplish anything in a post-forager society, you're going to need to learn how to plan, and how to follow through with those plans. How are you going to get anything done if you don't have the discipline to put in the hours?

And yes, self-discipline in one area is linked to self-discipline in another. You have a "tank" so to speak of self-control that gets depleted when you are doing something difficult, and gets renewed when you are resting or leasuring. Using your self-control in any area depletes the amount of self-control left for another. If you have small tank (low self-discipline) then you run out of fuel faster (you quit working sooner). In the long run though, you can increase the size of your tank by by doing difficult tasks, such as working for a specified number of hours each week.

"Even if you could rule out man-made and weather-related causes for some UFOs, that wouldn't imply that they were caused by an extra-terrestrial civilization either."

I agree. But in the cases of grey beings emerging from UFOs we can at least conclude that grey beings can occupy UFOs, if we trust primary evidence. This would be a massive discovery in itself, so why don't we hear about it? We don't have to conclude they come from outer space - who knows, they maybe live underground. Lets not speculate on that as we have plenty of interesting observations to delve into already - little gray men emerging from airborn thingies is HUGE in itself.

"So what is it that you think you know about these "Aliens"?"

It's not that I know anything about aliens. It's that more earthly explanations are completely implausible in many cases.

"That said, I don't think you can rule out weather and human craft."

In which cases? Just all cases, a priory? Or did you go through all previous sightings and came to that conclusion in every one case? Maybe others did the study for you, so you could provide a reference?

"I don't think that generic aliens should be considered especially improbable a priori - before the evidence is considered. I think that they are unlikely a posteriori - based on the fact that we don't see them"

Citation?

There's plenty of evidence for non-man made, non-hoaxed, non-astronomical, non-weatherrealated unidentified flying objects according to studies made by the US and French military:

http://en.wikipedia.org/wiki/Project_Blue_Book#Project_Blue_Book_Special_Report_No._14

most important highlights:

http://lesswrong.com/lw/ffd/struck_with_a_belief_in_alien_presence/7t4i

The black swan example was just a general pondering.

"I don't think that generic aliens should be considered especially improbable a priori - before the evidence is considered. I think that they are unlikely a posteriori - based on the fact that we don't see them. I think that any intelligent space-faring life would be busy building spheres around stars (if not outright disassembling the stars) as quickly as they spread out into the cosmos. So we'd notice them by the wake of solar systems going dark. At the very least, there's no reason to think that they would hide from us, which is what these scenarios tend to require"

This is very speculative to me. I don't think we can use it as evidence for or against.

If we're going to talk about how and why we should formulate priors, rather than what Bayes' rule says, this is what we're interested in.

But that's not what I'm talking about. I was specifically responding to your claim that:

"prior probability", by definition, means that we throw out all previous evidence.

So far as I can tell, that's not part of the accepted definition. For example, Jaynes' work on prior probabilities explicitly invokes prior information:

in two problems where we have the same prior information, we should assign the same prior probabilities.

I don't mean to come off as a dick for nit-picking about definitions. But rigorous mathematical definitions are really important, especially if you are claiming to argue something is true by definition - and you were.

Wait, what? Bayesians never assign 0 probability to anything, because it means the probability will always remain 0 regardless of future updates.

Yes. This name for this is Cromwell's rule.

And "prior probability", by definition, means that we throw out all previous evidence.

Not quite. The prior probability is the probability of the hypothesis and the background information, independent from the evidence we are updating on. This includes previous evidence. We usually write the "prior probability" as P(H), but it should really be written as P(H.B), where "H" is hypothesis and "B" is background information.

For example, let's say I am asking you to update your belief that Julius Caesar existed given a recently discovered, apparently first-hand account of Caesar's crossing the Rubicon. Your prior probability should NOT exclude all previous evidence on whether Caesar actually existed - e.g. official Roman documents and coins with his face. Ideally, your prior probability should be your posterior probability from your most recent update.