In what conditions should the negative of a " that-clause" move to the main clause?

There is a syntactic rule called Negative-Raising that operates on a subset of English verbs having to do with perception, thought, and belief. This rule, with these verbs only, allows equivalence between a negative in a complement clause and a negative in the main clause. Since think is one of these verbs, the following two sentences are equivalent in meaning:

• Bill thinks that the Orioles won't win the Series.
• Bill doesn't think that the Orioles will win the Series.

The effect of the rule is that the negation from the that-clause seems to rise up to the main clause, where in fact it doesn't apply -- Bill is thinking, but that they'll win is not what he's thinking. The literal interpretation some try to push -- that Bill has no opinions -- is not fluent English; nobody talks or writes that way, because that's not what think normally means.

However, claim is not one of these verbs, and thus the following two sentences do not mean the same thing:

• Bill doesn't claim the Orioles will win the Series.
• Bill claims the Orioles won't win the Series.

These do have separate senses -- in the first, Bill's belief is unknown, but his claim is denied. In the second, Bill's claim is asserted. But then claim is a different verb from think -- a claim is public and can be witnessed, but a thought is private and can be denied or imagined. That's what allows Neg-Raising in the first place. Every verb has a different grammar.