Wouldn't the rule be something more like:

((P(H|E) > P(H)) if and only if (P(H) > P(H|~E))) and ((P(H|E) = P(H)) if and only if (P(H) = P(H|~E)))

So, if some statement is evidence of a hypothesis, its negation must be evidence against. And if some statement's truth value is independent of a hypothesis, then so is that statements negation.

This is implied by the expectation of posterior probabilities version. Since P(E) + P(~E) = 1, that means that P(H|E) and P(H|~E) are either equal, or one is greater than P(H) and one is less than. If they were both less than P(H), then P(H|E)P(E)+P(H|~E)P(~E) would have a lesser value than the largest conditional probability in that formula; suppose P(H|E) is the greater one, then P(H|E)P(E)+P(H|~E)P(~E) < P(H|E) and P(H|E) < P(H), so P(H|E)P(E)+P(H|~E)P(~E) ≠ P(H). If they are both larger than P(H), then P(H|E)P(E)+P(H|~E)P(~E) must be larger than the smallest conditional probability in that formula; suppose that P(H|E) is the smaller one, then we have P(H|E)P(E)+P(H|~E)P(~E) > P(H|E), and P(H|E) > P(H), so P(H) ≠ P(H|E)P(E)+P(H|~E)P(~E). And if both posterior probabilities are equal, then P(H|E)P(E)+P(H|~E)P(~E) = P(H|E), and both posteriors must eqaul the prior. Q.e.d.

I think that the formula that expresses the prior as the average of the posterior probability weighted by the probabilities of observing that evidence and not observing that evidence, is a great way to express the point of this article. But it might not be trivial for everyone to get:

((P(H|E) > P(H)) if and only if (P(H) > P(H|~E))) and ((P(H|E) = P(H)) if and only if (P(H) = P(H|~E)))

from

P(H) = P(H|E)P(E) + P(H|~E)P(~E)

That something is evidence in favor if and only if its negation is evidence against, and that some result is independent of some hypothesis if and only if not observing that result is independent of that hypothesis, are the take home messages of this post as far as i can tell. The law that "P(H) = P(H|E)P(E) + P(H|~E)P(~E)" says more than that, it also tells you how to get P(H|~E) from P(H|E), P(H) and P(E). But adding the boolean statement and its proof from the weighted average statement to the post, or at least to a comment on this post, not even necessarily using the boolean symbols or formalisms, might help a lot of students that come across this long after algebra class. I know it would have helped me.

Hello I am a philosophy student in north Jersey. I'm 20 years old, and am very familiar with LW and the sequences. I've been reading LW now for about a year, and it has completely changed my life. I am very grateful to Eliezer and all of you for letting me have my Bayesian enlightenment at 20. When I first read the twelve virtues my life changed forever. I am definitely one of those that considers the sequences to be one of the most important works i have read, at least as far as having a personal influence.

I want to work on the hard questions of philosophy, grue and induction, cognition and consciousness, nominalism v.s. realism, Bayesian epistemology, philosophy of probability and mathematics in general, and even meta-physics, though I would like to positivize the field a bit. What I want to do as a philosopher is find problems/paradoxes/questions which fascinate me, and use rationality to solve them. "Solve" being the key word there. I think LW has done a lot to pursue many those goals, which seem strictly like philosophical goals. It seems to me, that LW should go full force and treat itself as a philosophical movement, conveniently primarily concerned with systematically becoming less wrong. Yes, there are mathematicians, and AI designers, and physicists, and psychologists among us, but that is how it should be in any modern philosophical movement.

I have given myself some primer time to become familiar with your terminology, content, and techniques. I now want to use these techniques to solve problems on paper and share the solutions with you. I am doing this because I expect that this will let me know how I am doing so far, and where I need to improve.

Lastly, I would like to ask, how does less wrong see itself? I mean what is the general LW opinion of what LW is? Is it a blog? An open source research institute? A philosophical movement? A non-philosophical movement? A self-help movement? I am curious.