Welcome to Less Wrong!
My short answer to the conundrum is that if the first thing your tool does is destroy itself, the tool is defective. That doesn't make "rationality" defective any more than crashing your first attempt at building a car implies that "The Car" is defective.
Designing foundations for human intelligence is rather like designing foundations for artificial (general) intelligence in this respect. (I don't know if you've looked at The Sequences yet, but it has a lot of material on the common fallacies the latter enterprise has often fallen into, fallacies that apply to everyday thinking as well.) That people, on the whole, do not go crazy — at least, not as crazy as the tool that blows itself up as soon as you turn it on — is a proof by example that not going crazy is possible. If your hypothetical system of thought immediately goes crazy, the design is wrong. The idea is to do better at thinking than the general run of what we can see around us. Again, we have a proof by example that this is possible: some people do think better than the general run.
Well, it sounds right. But which mistake in rationality was done in that described situation, and how can it be improved? My first idea was that there are things we shouldn't doubt... But it is kind of dogmatic and feels wrong. So should it maybe be like "Before doubting X think of what will you become if you succeed, and take it into consideration before actually trying to doubt X". But this still implies "There are cases when you shouldn't doubt", which is still suspicious and doesn't sound "rational". I mean, doesn't sound like making the map reflect the territory.
A few notes about the site mechanics
A few notes about the community
If English is not your first language, don't let that make you afraid to post or comment. You can get English help on Discussion- or Main-level posts by sending a PM to one of the following users (use the "send message" link on the upper right of their user page). Either put the text of the post in the PM, or just say that you'd like English help and you'll get a response with an email address.
* Normal_Anomaly
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A note for theists: you will find the Less Wrong community to be predominantly atheist, though not completely so, and most of us are genuinely respectful of religious people who keep the usual community norms. It's worth saying that we might think religion is off-topic in some places where you think it's on-topic, so be thoughtful about where and how you start explicitly talking about it; some of us are happy to talk about religion, some of us aren't interested. Bear in mind that many of us really, truly have given full consideration to theistic claims and found them to be false, so starting with the most common arguments is pretty likely just to annoy people. Anyhow, it's absolutely OK to mention that you're religious in your welcome post and to invite a discussion there.
A list of some posts that are pretty awesome
I recommend the major sequences to everybody, but I realize how daunting they look at first. So for purposes of immediate gratification, the following posts are particularly interesting/illuminating/provocative and don't require any previous reading:
More suggestions are welcome! Or just check out the top-rated posts from the history of Less Wrong. Most posts at +50 or more are well worth your time.
Welcome to Less Wrong, and we look forward to hearing from you throughout the site!
Once a post gets over 500 comments, the site stops showing them all by default. If this post has 500 comments and you have 20 karma, please do start the next welcome post; a new post is a good perennial way to encourage newcomers and lurkers to introduce themselves. (Step-by-step, foolproof instructions here; takes <180seconds.)
If there's anything I should add or update on this post (especially broken links), please send me a private message—I may not notice a comment on the post.
Finally, a big thank you to everyone that helped write this post via its predecessors!