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

Question

If you tell your friend some incredible story and they say, "I don't believe you!" It seems like they are pretty obviously trying to say that they believe that your story isn't true.

I have someone I am talking with trying to telling me that there is a difference between the sentences, "I don't believe that X is so" and "I believe that X is not so." This seems like semantic nit picking though to me.

Wouldn't it be much easier to say: "I lack belief that X is so" or "I'm not sure that X is so?"

The placement of the "not" seems trivial in relation to the "belief" but I am open to being wrong on that.

Araucaria - Not here any more. · Accepted Answer

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.

Oso · Answer

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.

Answered by Oso on September 9, 2020

PixPrefect · Answer

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.

heathenJesus · Answer

Absolutely. "I don't believe that X" has a subtly but discernibly different meaning than "I believe that not X".

In the first, the emphasis of negation is on your belief. In the latter, the emphasis of negation is on the subject of your belief. The former is also more passive—you deny the belief in X, but do not specify what you do believe. The latter is far more active. Not only do you not believe X, but you actively believe in not X.

That can be a significant difference in some contexts and negligible in others, but there is a difference.

Drew · Answer

An easy way to understand the difference is to think in terms of proof or persuasion.

He proved that the square root of two is not a rational number.
He did not prove that the square root of two is a rational number.
She convinced the jury that the defendant is not guilty.
She did not convince the jury that the defendant is guilty.

Merk · Answer

First, placement of 'not', or any negative idea, often matters greatly.  Consider the following:

It is not necessary for him to go.
It is necessary for him not to go.

These sentences obviously have very different meanings (non-obligation vs. prohibition).  Linguists might say that the word "not" does not commute.  (cf. http://en.wikipedia.org/wiki/Commutative_property)
The contrast in your example is not as strong, but still there.
"I believe that" introduces a propositional attitude -- an attitude of the speaker towards a following idea.  Introducing "not" in this part of the sentence denies that you have a specific propositional attitude (belief) towards that idea.
Introducing 'not' to a declarative statement (e.g., the weather is not nice) negates the idea itself.
There are really two consequences of this.  One is a change in focus.  The latter formulation sets the 'focal point' on the fact in question, the former formulation sets the 'focal point' on your feelings about it.
But more importantly, there can indeed be a subtle change in surface meaning, as the following examples show:

I don't believe that X is so.  But I suspect it may be.
I don't believe that X knows the truth, I know he does.
I don't believe that X is funny, only Jane does.

These additions cannot be attached to the other version:

I believe that X is not so.  But I suspect it may be.* (wrong)
I believe that X does not know the truth, I know he does.* (wrong)
I believe that X is not funny, only Jane does. (wrong)

barlop · Answer

There is a difference any those that have studied logic, are aware of it..
If you say you don't believe X, then maybe you are 50/50.. on the fence about it.  But if you say You believe ¬X then you are maybe 95% sure or 100% sure that X is false.
So somebody could say they don't believe X, but they don't believe ¬X either. Or somebody could say they don't believe X and they believe ¬X (which logically shortens to they believe ¬X!".
So if somebody says to you that somebody else's name is Paul.  But you know that the guy that said that, sometimes does pranks from time to time. So you might say you don't believe the guy's name is Paul, but you also don't believe his name isn't Paul.  However, if you knew that this prankster always lies about peoples names, then sure you could say that you don't believe his name is Paul. But you can also say, and this would be more accurate to your belief,  you can instead say that you believe his name is not Paul. (which means not only that you don't believe his name is Paul, but you believe his name is not Paul).
In logic, one may write his premises down if you "don't believe X" then you don't write down X in your list of premises.  If you believe ¬X, then you write ¬X in your list of premises. So it's dealt with differently.  If at some point during your reasoning, you conclude that X is true. So now you believe X, but you haven't contradicted yourself. Whereas if you believe ¬X then did some reasoning, and concluded X, then there is a contradiction in your reasoning.
Suppose it's the case that ¬X->H    where X=bob is home.   ¬X means "bob is not home".  H=you get £10 from somebody.    ¬X->H means that if ¬X, i.e. if it's false that bob is home, i.e. if bob is not home, then, you get £10 from somebody.  If you don't believe X, you get nothing. But if you believe ¬X, you get £10.
¬X is a premise, and deserves a burden of evidence/proof, as much as X.
A person could say (incorrectly), that you can't prove or provide evidence for a negative.. Well, you can. You can make a reasoned argument that there's no Tyrannosaurus Rex in your bedroom.  Or that there are no married bachelors.

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

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