I think this question on ELU is tricky, so I've created this chat room to provide a bit of "overflow space" for kicking it around.

kicks

I entirely agree, though.

@Mahnax hi. agree what? that it's tricky?

@FumbleFingers Yeah.

I might try to answer it later today.

@Mahnax wot? you mean you know you know the answer too?

Anyhow, I need to get myself gone, farewell.

@FumbleFingers No, I might try to find it. But yeah, bye.

So, the difference is between *I think him to be stupid and I think that he is stupid, right? And the question is why the first is ungrammatical and the second is OK? Or am I missing something?

That's about it. Or putting it another way - many/most native speakers are happy with "I believe him to be stupid", as well as "I think/believe he is stupid", but they don't like "I think him to be stupid". Also see my recent comment to Peter Shor's answer on the original question.

I'm not sure it's exactly a matter of "grammar". I think the answer may be as much semantics as grammar. But I don't know the answer, of course.

In fact, it's not semantics. It's exactly syntax, because it's arbitrary and unpredictable, and doesn't seem to accord with the expectations of semantics. That's syntax for you.

Well, let's start with the basics. Both sentences have object complement clauses; one is tensed, and uses the that complementizer, whereas the other is an infinitive and therefore tenseless, and uses the to complementizer. And both complements are governed by the verb think. OK so far?

...um... I think so (I guess I could have said "I believe so").

Ah! - is that it? I think I can say I believe that is true, equally I believe I can say I think that is true. So that doesn't seem to be the be all and end all here.

No, not at all. At this point it's a matter of what kinds of complements each verb can take, and what rules each verb allows, forbids, or requires with those complements. Mostly, "verbs of a feather flock together"; i.e, verbs that mean similar things will have similar affordances. But there are exceptions, and this is one.

In particular, verbs of mental process like think, believe, know, guess, understand, and discover commonly take that complements. It's the simplest way to indicate what was thought, believed, known, guessed, understood, or discovered. Make a declarative sentence, put that in front, put it in the object position, and you're done.

It's the infinitives that cause all the trouble.

You mean the core of what we think is a specific thought, whereas the core of what we believe is an authority, the source of a thought? Have I got that bit, and now need to be guided through how that's affected by infinitives?

I'm not sure what you mean by "core". The Proposition that represents the content of some mental (or communicational) event is the complement of these verbs. Normally that clauses are used to express propositions (i.e, declarative sentences). Core is a metaphor that indicates the central part -- but not the whole -- of something, and it's not a term in semantics.

Well, just to restate my starting position here. I don't believe the aversion to think [noun] to be [adjective] can be "just one of those things". Reason being that even believe [noun] to be [adjective] is relatively rare compared to think/believe [noun] is [adjective].

...so it simply isn't credible to me that people could "learn" we do say one of those, but not the other, purely on the basis of "frequency of use". We must be applying some principle that operates primarily at a level below conscious awareness, imho.