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The best advice is tailored to individuals, and the best explanations are targeted at avoiding or uninstalling specific confusions, instead of just pointing at the concept. But here I think the right call is giving evidence for 'a' reader instead of TurnTrout. So, a general case for rationality:

First, by rationality I mean a focus on cognitive process rather than a specific body of conclusions or thoughts. The Way is the art that pursues a goal, not my best guess at how to achieve that goal.

Why care about cognitive process? A few factors come to mind:

1) You're stuck doing cognition, and you might want to do it better. Using your process to focus on your process can actually stabilize and improve things; see Where Recursive Justification Hits Bottom and The Lens that Sees Its Own Flaws.

2) Studying the creation of tools lets you know what tool to use when. Rather than reflexively bouncing from moment to moment, you can be deliberately doing things that you expect to help.

3) As a special case of 2, sometimes, it's important to get things right on the first try. This means we can't rely on processes that require lots of samples (like empiricism) and have to instead figure out what's going on in a way that lets us do the right thing, which often also involves figuring out what sorts of cognitive work would validly give us that hope.

4) Process benefits tend to accumulate. If I expend effort and acquire food for myself today, I will be in approximately the same position tomorrow; if I expend effort and establish a system that provides me with food, I will be in a different, hopefully better position tomorrow.

Who shouldn't care about rationality? First, for any task where the correct strategy to employ is either 'obvious' or 'unintuitive but known to a tradition', then the benefits of thinking it through yourself are much lower. Second, to the extent that most rationality techniques that we know route through "think about it," the more expensive thinking is, the less useful the rationality techniques become.

Not so long ago I hit upon a definition of rationality which better captured what everything people actually use the term to refer to.

I think a solid definition of rationality is: trying to do better on purpose.

Instead of following your default procedure, you pull back and do something different to get a better result. This might be thinking in ways other than the default or doing things in ways other than the default.

A natural consequence of tying to do better on purpose is that you look for higher-level improvements rather than purely immediate, domain-specific concrete ones. Many people think to practice and train skills, but the rationalist seeks to reflect on how they train and process. Many people work towards goals, the rationalist pauses to reflect on their selection goals.

So why care about rationality? Because you want to do better.

If you believe that it is possible to do better, and that doing better results in more of what you want - then surely you would want that.

Vaniver’s answer covers instrumental rationality. That is probably the easiest way to persuade the average person that rationality is worth considering.

However, for some people, improving their models of the word is its own reward. If this is true for one, then epistemic rationality may appeal as much as instrumental rationality, if not more.

This requires a particular viewpoint on curiosity. If you are only looking for a rush of understanding, then mysticism may be just as effective. This requires you to value not just the [simplicity? wideness?] of your model but also its truth. One possible motivation for this value could be that true models are both interesting and useful, while false models are only interesting. (That is, a complex goal system [one with many subgoals] is more likely to produce this kind of outward desire.)

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(whimsical answer) Perhaps because pure rationality is so elusive that once you start caring about it, you start to notice how things which you want to fit the "rationality-shaped hole" don't really do it, and then keep looking; but you can always say "this is my stop" when you see something you can do about the world with the knowledge you have found, and you will have some experience of judging things in a more... solid way then.

It's like alchemical gold - in a way, more precious than the true substance.

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I would be interested to see the best explained answer to this.