There have been others explaining why "If P, then Q. Q. Therefore, P." isn't what is going on here, and why postdictions are not so bad.

But I'd like to address another issue: there are a lot of historical examples where all the major evidence for a hypothesis was a postdiction, but the hypothesis was so simple and fit the data so well that it became accepted mainly on the power of the postdictions. The most famous example was Kepler's model of the solar system using ellipses. Based to a large extent on their postdictive power, the basic elliptical model was largely accepted before Newton gave an explanation for why it worked. This example isn't perfect because the model did provide further confirmation by later astronomical observations.

Simplified & short;
If P, then Q. Q. Therefore, P.

That would be a logical fallacy. But, importantly, its probabilistic analogue is not a fallacy. It really is a mathematical fact that,

If P makes Q more likely, then Q makes P more likely.

In other words, it is a mathematical fact that p(Q|P) > p(Q) implies p(P|Q) > p(P).

I agree with you that predictions are better than postdictions in practice, but postdictions ain't nothing.

I observe a certain group of people in a culture doing something, then I postdict it with EvPsy or alien control. I observe many people dying around age 80. My theory is that if alines exist, they kill people around age 80.

First, see my final parenthetical remark in my previous comment. We already have causal accounts of why people die around 80. Alternative causal accounts (such as aliens) don't get much of a probability boost from explaining what we can already explain. In contrast, no competing theory predicts the "three brothers but not one" numbers specifically. If observations bore this out, the EvPysch explanation would not be competing with any alternative explanations.

Second, recall that I said that, when a theory says that an observation is likely, and the observation actually happens, then

that observation makes the probability of T increase by a very large factor.

That is true. Nonetheless, if T started out as very, very improbable, then even an increase by a "very large factor" will still leave T with a small probability. If epsilon is sufficiently small, then epsilon x 10^100 is still very small. Now, "Aliens kill people around age 80", starts out with a very low prior probability. So it will have to predict/postdict some very improbable observations to rise above a negligible probability.

Third, and most importantly, simply adding an improbable observation to a theory lowers the prior probability of that theory. Take the theory "Aliens kill people". Now augment the theory by adding the "around age 80" part to get "Aliens kill people around age 80". This addition lowers the probability of the theory. (Under the original theory, the aliens could be killing people at any age. Thus, the original theory would be true under a wider variety of circumstances, so it is more probable.)

In fact you can prove that the addition of "around age 80" exactly counteracts the boost that the augmented theory gets for successfully post-dicting that people die around age 80. You don't gain any probability for your theory by explicitly building post-dictions into it. In symbols, let T be the original theory, and let T&E be the augmented theory. It follows that, if p(E|T) = p(E), then p(T&E | E) = p(T). That is, if the original theory didn't make E any more likely, then observing E doesn't make the augmented theory any more probable than the original theory was prior to the observation of E. And "Aliens kill people" started out pretty improbable!

In contrast, the "three brothers but not one" numbers are not just added to EvPsych. They are deduced from simpler premises. So, if observations bear these numbers out, then that really is a big boost to EvPsych.

Simplified & short; If P, then Q. Q. Therefore, P.

While propositional logic may be a special case of Bayesian reasoning, the Bayes's theorem formalization of the scientific method cannot be usefully reduced to propositional logic.

Also, welcome to Less Wrong!. It sounds like you may want to check out Bayes' Theorem and/or Technical Explanation.

Actually, it's:

If P, then Q is plausible. Q. Therefore P is plausible.

And it's a valid argument in probability theory as extended logic; see the first chapter of Probability Theory: The Logic of Science, which is available on the linked webpage.