Neg-raising occurs when asserting ¬Fp (or denying Fp) tends to communicate F¬p. For example, 'John doesn't believe that he will win' tends to communicate that John believes that he won't win.
There appears to be no consensus on why this happens. Some think ¬Fp really does entail F¬p. Others think the effect is an implicature. Still others think it's caused by a presupposition of opinionatedness or "settledness": when we talk about whether Fp holds, we presuppose that F holds either for p or for an alternative to p, denying Fp therefore commits us to F¬p.
I know little about this debate, but here are a few thoughts and observations that cast doubt on the views just mentioned.
The neg-raising effect seems to go away if, instead of simply saying ¬Fp, we say other things that entail ¬Fp. Example:
There's a lottery with 5 tickets. If John thinks that ticket #1 will win then he will buy ticket #1. If he thinks that ticket #2 will win then he will buy ticket #2. And so on up to ticket #5. As it turns out, John doesn't buy ticket #1. He also doesn't buy ticket #2, or #3, or #4. Can we infer that he buys ticket #5?
Clearly not. So 'If John thinks p then q' and 'not q' don't convey that John thinks ¬p, even though together they entail that John doesn't think p (which, by neg-raising, tends to convery that John thinks ¬p).
The observation extends beyond lottery cases. Imagine John's friend Jones is about to arrive by train, and there are 5 stations where Jones might arrive. If John thinks Jones will arrive at station i, he will go to station i. It doesn't follow that he will go to one of the stations.
In some respects, the examples are even nicer if they concern the past. For each i, assume that if Jones had thought that ticket #i would win then he would have bought it. It doesn't follow that Jones bought a ticket.
The observation appears to be incompatible with views (such as the one proposed in Hawthorne, Rothschild, and Spectre (2016)?) on which neg-raising is semantic entailment. It also appears to be incompatible with the presupposition view. It might be compatible with an implicature view, because what's implicated by an assertion can depend on the exact words that are used.
The neg-raising effect also seems to go away if the relevant expression F is replaced by a more complicated, but equivalent, expression. Suppose to believe p is to be disposed to assent to 'p'. 'John doesn't believe that he will win' is then equivalent to 'John is disposed to assent to 'I will win'', and that doesn't convey that John is disposed to assent to 'I will not win'. This analysis of belief is of course overly simplistic, but I think the observation remains true for more complex and more credible proposals.
This also seems to cast doubt on the presuppositional view. If believing is being disposed to assent, then a presupposition that John believes either p or some alternative to p would be a presupposition that John is disposed to assent to either p or an alternative to p. And that presupposition would predict that neg-raising still occurs with the more complicated construction.
The neg-raising effect often goes away when focus is on the relevant expression F. 'John doesn't BELIEVE that he will win' doesn't suggest that John believes that he won't win.
This seems to cast doubt on views that make neg-raising a semantic entailment. (It is probably compatible with implicature and presupposition accounts.)
Sometimes, but not always, neg-raising is triggered in quantified environments. 'Nobody believes that Biden will win again' suggests that everybody believes that Biden will lose. But in German, I think, 'Niemand hofft, dass Biden verliert' ('nobody hopes that Biden loses') doesn't suggest that everyone hopes that Biden doesn't lose, even though 'Ich hoffe nicht, dass Biden verliert' ('I don't hope that Biden loses') does suggest that I hope that Biden doesn't lose.
This seems to cast doubt on semantic and presuppositional accounts.
It's tempting to think the neg-raising effect is a scalar implicature: if you know that John is unopinionated about whether he will win, you should say so, rather than assert the (literally) less informative 'he doesn't believe that he will win'. However, this assumes that 'John is unopinionated about whether he will win' is a scalar alternative to 'John doesn't believe that he will win' while 'John believes that he will not win' is not. That's odd.
My hunch is that neg-raising doesn't have any nice, rational explanation. It's a semi-conventionalisation of a processing error.
When our semantics module is tasked to compute what is communicated by ¬Fp, for some reason it is prone to mistakenly assume that F holds of either p or of an alternative to p. The mistake is so widespread that we have to take it into account even if we are not prone to making it ourselves, since we would otherwise misunderstand what other people are trying to say. That is, we will have to take care to assert ¬Fp only if we're willing to commit to F¬p, and we'll assume that other people do the same. Even speakers who don't make the mistake will therefore behave as if they do.
(Compare the widespread mistake people make when they say 'I could care less'. In some sense, this means that the speaker doesn't care – that's how it is used, and known to be used – although in another ("literal"?) sense it means the opposite.)
On this proposal, 'John doesn't believe that he will win' literally means that John lacks a relevant belief. But due to some quirk in our language faculty, some people are prone to use the sentence to express something stronger. This phenomenon is widely known, must be taken into account, and thereby reinforces itself.
In the ticket example (point 1 above), the phenomenon doesn't arise because our language faculty is not asked to compute what is said by ¬Fp. The semi-conventionalised phenomenon only pertains to utterances of '¬Fb', and closely related speech acts.
We can also expect that the "mistake" isn't made (or widely assumed to be made) when F is replaced by a more complicated but synonymous expression (point 2 above).
When we put focus on F, the mistake goes away because attention is drawn to the fact that the alternatives to Fp include not only F¬p but also various possibilities of type F'p, where F' is an alternative to F (e.g., suspension of judgement). That explains point 3.
The oddities for 'hoffen' in German (point 4), and the fact that 'hoffen' behaves differently from 'hope', are also not too surprising. Neg-raising only occurs if the "mistaken" use becomes so widespread that speakers and hearers have to factor it in. In German, the neg-raising use of 'hoffen' is common, but not very common, and it's only common for direct negations of 'hoffen'.