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A currently existing social norm basically says that everyone has the right to an opinion on anything, no matter how little they happen to know about the subject.

...what if we had a social norm saying that by default, people do not have the right to an opinion on anything?

I get what you're saying, but I don't think that's quite the problem.

The real problem is the social norm that says "you aren't allowed to be critical of someone else's view because everyone has the right to an opinion".

The italicized bit is the problem. I think everyone should have the right to an opinion, but also that everyone should have the right to be able to express criticisms of other's opinions.

(I think the "you can't criticise other's views" thing stems from relativism).

If we had in place a social norm demanding an adequate amount of background knowledge on the topic before anyone voiced an opinion they expected to be taken seriously, the signal/noise ratio might be somewhat improved.

Unfortunately that'd skew things towards the status quo.

Advances in knowledge often come from taking a very different angle on a problem, by someone who isn't immersed -- and thus not necessarily knowledgeable in -- the existing viewpoints about the problem (e.g. by amateurs or people from different fields).

Ultimately a person's view should be judged just on its own merits.

I think a norm is likely to be a product of the solution, not the solution itself.

So the problem is we have a lot of people who don't appreciate what constitutes a reasonable foundation for an opinion. They think they can just say what they feel. To put it one way, they have a poor understanding of the nature of evidence.

I don't think a norm like you describe could have any effect on anyone like that who had a poor understanding of evidence. Those people would just think the norm was wrong or ridiculous.

If they were to come to better understand the nature of evidence, they would be more receptive to the norm. But if they were to undestand evidence better then simply from this fact you'd get the desired result of people not mouthing off as much with "ignorant opinions".

So it the solution has to involve getting people to better understand the nature of evidence (or however you want to describe what is missing from their mental toolkit).

If you were to get enough people to understand the nature of evidence, that could lead to the creation of such a norm. I doubt it could happen the other way around.

Caveat: I'm not 100% confident the above story is true, but I think there's at least and element of truth in it.

"Everyone has a right to his/her opinion" is a social standard because it helps people get along. The total solution to this problem is not telling people that they aren't entitled to their opinions, so much as making it so that when you someone's opinion is ill-founded, they say, "Hey, you're right. I should change my opinion!"

Given that, in reality, that almost never happens, the expression exists as a way of maintaining civility. That way the person pointing out that no, the earth is not 6000 years old becomes the "bad guy" if he keeps pushing the point after this defense is invoked. Not good for truth, but good for short-term social stability.

On the note of what an opinion is, the expression is totally accurate with respect to matters of taste, e.g. "Chocolate is the best ice cream flavor." But with fact-entangled should statements, e.g "The federal government should raise taxes on the top income bracket," the speaker is actually (usually) expressing a factual claim, such as this being an effective way to reduce the budget deficit. To the extent that these are factual claims, you are not entitled to them.

I agree that the end result would be valuable, but I think that changing norms for a whole society would be very hard.

Although it might still be easier than the converse of raising the rationality of a whole society: being informed has higher status in society than being rational. It is more related to being a professor, journalist or talking head, whereas rationality is more associated to being a nerd, scientist or economist.

I think the causality has to run: X-Rationalists raise the standards for ordinary rationalists and scientists-> People connected to the scientists raise their standards-> Everyone else

Sort of top down by osmosis rather than decree. Everyone gets slightly better, but most ordinary people won't have to unrealistically become X-Rationalists.