Is there a functional difference between "not believing" and "believing not"?

Solution 1:

It is a fact known to millions of hardworking English language students all over the world that native English speakers strongly prefer negating the verbs think, believe and want, amongst others, to negating the complement clauses that they license. So, for instance, all other things being equal, we prefer:

(1). I don’t believe that the Yeti exists.

to:

(2). I believe that the Yeti doesn’t exist.

We also would tend to prefer:

• I don’t think I’m going to find it.

to:

• I think I’m not going to find it.

and there is absolutely no doubt that:

• I don't want to go.

is far more customary than the rather stilted:

• I want to not go.

Notice that what is implied by (1) is the same as what is literally encoded in (2). However, (1) does not in fact strictly semantically encode the same information as (2) at all. If we made no further pragmatic assumptions about what the speaker of (1) intended to convey, then the maximum we should be entitled to decode is that the speaker does not possess a positive creedal attitude about the existence of Yetis. It is entirely possible that the speaker may have no definite opinion about the existence or non-existence of Yetis, in which case they would not be able to truthfully commit to either a belief or disbelief in them. This might be due to an agnostic state of mind, or it may be merely because the speaker has never even thought about it. To commit the speaker of (1) to a belief in (2) is potentially doing them a great disservice.

Be that as it may, most listeners would understand (1) as conveying the same as (2), and they are indeed entitled to, because most speakers - unless they were wishing to be very explicitly technical about it - would prefer the former to the latter to convey the very same information. What is interesting here is that speakers are modifying the verb denoting the action of belief in order to manipulate the listener’s understanding of the object of the belief - the information in the complement clause. More specifically they are negating the verb denoting the believing, but implying a negation of the complement clause.

This phenomenon is known as SUBORDINATE NEGATION IMPLICATION. Verbs that tend to generate such implicatures seem to be verbs that denote states of intention, epistemic stance or opinion, or those which can be used performatively for advice. Dynamic verbs which denote actions, changes of mental states and so forth do not tend to generate these implicatures. Compare the following sentences with the dynamic verb say:

He didn't say that she danced.

He said that she didn't dance.

Here the two sentences do not convey the same information at all. We are not likely to infer the information in the second sentence when we read the first.

One more factor comes into play here. Verbs that generate subordinate negation implicatures, tend to be what are described in the CaGEL as medium strength verbs. They contrast for example 'stronger' know with 'medium strength' believe. The reason that these verbs tend to generate such implicatures is merely that, pragmatically, it does not seem very informative to tell somebody that you don't have a medium strength stance about something. We tacitly infer, on this basis, the more informative proposition that the speaker has a stance about a negative idea.

However, with so-called stronger verbs, on the other hand, it is informative to convey that your confidence in a stance is not 100%, or contrastingly with weak verbs to convey that that not even the slightest positive attitude is given to the proposition in the complement clause. The strong and weak usages of the following verbs do not, therefore, generate subordinate negation implication:

I don't know that she went. ≠ I know that she didn't go.

I don't suspect her of stealing. ≠ I suspect her of not stealing.

In relation to the original poster's question, it is fair to say that when we are not speaking in a very technical fashion indeed, that if we understand something as having a negative subordination implication, it probably has one. In other words the speaker was intending the listener to understand precisely that the content of the subordinate clause should be read as being negated. However, the original poster's debating partner was technically correct that when we negate verbs such as think, believe, want, what we say does not semantically entail the same thing as the negation of the proposition in the complement clause.

As to why speakers actually prefer to negate verbs such as want and believe rather than to negate their complement clauses, I do not believe that anybody knows (- by which I want you to infer that I believe that nobody knows).

It is worth noting, very much to the benefit of the OP's argument, that linguistic communication relies on us making inferences about what is meant - without these things being actually logically entailed by the language. If we did not do this, we would not be able to communicate at all.

Solution 2:

There actually is a discernible difference. When saying I don't believe x, you're not stating your personal consideration of the matter, but just your lack thereof. However, when you say I believe not (with regards to x), you're stating that of the two possible choices, you've actually determined which you've settled on.

Solution 3:

Belief can be characterized by degrees. There is belief, disbelief, and an in-between where the person is not sure if they believe in something. Saying "I don't believe X" can be interpreted to mean either of the categories that are not belief.

Ultimately, the person you're talking to can say "I don't believe X" and it mean something different from "I believe that X is not so," but will probably have to clarify it every time. It would be simpler to clearly state the ambiguity.