Given x and y could be any phrase, do these phrases always mean the same thing? If not, what's the difference?
- I believe x does not equal y
- I don't believe x equals y
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No, strictly, they do not convey the same meaning. In practice, your second sentence is often used to mean the first.
I believe x does not equal y means that you actually hold a belief about the inequality of x and y.
I don't believe that x equals y simply means that a belief about the equality exists, but you do not share that belief.
If you substitute another verb for believe, the difference may be clearer:
I have actual knowledge that x and y are not equal. Quite possibly I can show you facts to support this.
I have no knowledge about x being equal to y. Actually, I probably have no knowledge to the contrary. This sentence is in many cases equivalent to:
That said, in practice language is not mathematics. As Edwin Ashworth points out, there is a lot more to these kind of constructions than meets the eye (and a lot more than I would be willing to summarize in an answer here). For further reading, I suggest the article that Edwin linked to (Just in case the comments deteriorate, I include the link here as well).
Indeed, in actual usage, many people will use
I don't believe x equals y.
I believe x does not equal y.
While this is readily understood by most, if not all, speakers, I would like to note that this usage is a common ground for misunderstanding. In particular in theological discussions, it is common that the claim:
I don't believe in the existence of deity X. (1)
is wilfully (mis)interpreted as
I believe that deity X does not exist. (2)
in which case it can become the basis of a straw man argument if the speaker actually meant to make a distinction between agnosticism or so-called "weak atheism" (1) and so-called "strong atheism" (2).
So, depending on context, the two sentences may mean the same, but be aware of situations where a strict interpretation is better suited - in which case one can make a very clear distinction in meaning between the two phrases.
I consider the use in English to be ambiguous enough in the minds of the average reader that alternative meanings must be considered and analysed, and the following enumerates those meanings and reasons about them...
I believe x does not equal y
This is ambiguous, as - using symbolic notation to help show the difference - it may mean x != y or !(x = y)
I don't believe x equals y
This is also ambiguous, it may mean you actively believe !(x = y) or that you admit to not knowing whether x = y. To clarify the difference, imagine we replace "x equals y" with "there is a god": "I don't believe there is a god" might be someone's less-confrontational way of saying they're an atheist, or it may mean they're agnostic - they don't actively believe there's a god, but they acknowledge it's possible.
So, we now have on the table:
x != y
!(x = y)
I don't know if x = y... which only tells us about the person's knowledge and asserts nothing about x's relationship to y, so let's focus on the interesting comparison....
Can we say x != y is the same as !(x = y)?
for most x, y and senses of equals and unequals, !(x = y) and x != y are equivalent, but there can be exceptions...
there's a class of logic where assertions are categorised as True, False, or other states like Unknown, Unknowable, Irrelevant etc., in which the above doesn't hold. As example: say x is the assertion that I'll die aged 100+, and y that you'll die aged 100+ - the truth of each is currently Unknown (I'm less than 100, I'll assume you are too). "x equals y" may be asking "do we know that their eventual truth or falsehood is the same", i.e. will be either both live to be 100, or both die beforehand - that's Unknown too, so you might say you "don't believe x equals y", but that doesn't mean you actively believe x != y (i.e. that one of us will live to 100 and the other not).
So, if both phrases are intended and taken to mean !(x = y), then they're equivalent. If the first is x != y and the second !(x = y), it depends on the nature of x and y - whether they're e.g. numbers or assertions with uncertain states etc.. If the second phrase is just disavowing knowledge, then it's clearly not equivalent to any intent or interpretation of the first phrase.
The fact is, they are both very commonly used to mean exactly the same thing.
English is packed with many (slightly confusing) double-negatives, triple-negatives, and other messy constructions. (And then you have stuff like "it's awfully nice.")
The problem with what Ork. is saying, is, Ork. is talking as a mathematician and a logician. Unfortunately, almost everyone is very stupid. Very, very few people would understand the difference between an inequality and an equality. (I doubt 1 person in 100, in say the USA, has ever used the word "inequality.")
The fact is, it is 100% commonplace in English - all regions as far as I know - to say "I don't believe FFF" instead of saying "I believe FFF is false."
So, people say "I don't believe there's a train at 7" when they mean "I believe, there is no train, at 7."
So in answer to your literal question, what do they mean, the fact is the person speaking meant exactly the same thing both times.
Think, believe, seem, appear, likely, and many other predicates involving probability judgements
are in the class of predicates subject to what's called Negative-Raising.
Essentially, these verbs (or predicate adjectives) are transparent to negation, and it doesn't make any difference whether an overt negative appears downstairs, in their complement
or upstairs, in the matrix clause with the NR predicate
because they mean the same thing either way.
This is not true of most predicates. Claim and say, for instance, don't work that way
and neither do possible or easy
No. They do not mean the same thing in general.
This response may be more mathematical than what you are looking for, but I see others attempting to apply straight logic so here goes a answer using Modal Logic.
A lot of the answers are attempting to apply propositional logic to the analysis of these statements, however the problem is that 'belief' is not an expressible concept in plain propositional logic, you cannot qualify a proposition over a proposition. as in, you have the proposition x == y on one hand, then try to modify said proposition with the belief quantifier, propositional logic alone cannot express such a thing and be both consistent (only true things are proveable) and complete (all true things can be proven). You can never express something 'might be true' or express belief in something being true in a way that does not also imply it actually is true. There is no grammar for it.
There are a few ways to extend classic logic with quantifiers over propositions, the most common is 'universal quantification' which allows the form ∀x.X, read as 'for all x X is true'. this is very useful in math, where you want to prove some statement about all natural numbers. but not as useful in interpreting natural language statements.
The proper logic to examine such statements is Modal Logic which extends propositional logic with an explicit notion of belief. It adds two symbols to the logic □ which is read as "It is necessary that or i belive that" and ◇ which is "it is possible that"
so you have
□ x ≠ y (I believe that x ≠ y) ¬□ (x = y) (I do not believe x equals y) which can be rewritten ¬□¬(x ≠ y) and by the modal logic reduction this is equivalent to ◇(x ≠ y) (it is possible that x does not equal y)
note that these are assertions about your beliefs and not about x and y themselves. Whether x and y are actually equal, whether that is even decidable or whether the truth even depends on context or time of day is not relevant to analyzing the statements about belief like this.
Modal logic is handy stuff, another common place it can be used is distributed learning systems with different nodes working with incomplete information, such as cooperating robots as it actually can express things like "agent1 believes that things agent2 tells him are possible." It allows for a subjective view of the world where different agents come to different conclusions,or for reasoning about possible alternate worlds. This is something that has no ability to be expressed in classic logic where all expressible statements are true or not true everywhere for everyone. For instance you can express "It is possible for bigfoot to exist, even though he does not" in modal logic whereas classically it is not possible for things to be true because they happen to not actually be true.
By interpeting the two quantifiers (□,◇) differently, you find a lot of logical systems are just specializations of modal logic. Temporal logic is when you interpret them as saying whether a statement is sometimes true, or always true, denotic logic is when you interpet them as "you must" and "you may", Epistemic logic treats them as "you know that x is true" and "nothing you know contradicts x being true". Fun Stuff.
Interesting that there are essentially two sets of answers:
The mere fact that you have two sets of answers proves that the statements are not the same as asked by the question title.
The difference is that their interpretation may vary according the interpreter and the context.
Two mathematicians/logicians/philosophers (or maybe just pedants) have a professional discussion will certainly interpret the statements as quite different.
Two "normal people" (whatever that means) would more likely interpret the statements as the same.
Most mathematicians/logicians/philosophers know that "normal people" interpret them as the same; some will treat that simply as an interesting fact of the ambiguity of language and others will not.
In effect, they are holding one thing stable and making the other variable.
"I believe x is not y" is focused on the value of x and its relationship to y. The positive statement of "I believe" is not in question, we are making a statement about the value of the relationship.
"I do not believe that x is y" is focused on your belief. The positive statement "x is y" is not changing, we are making a statement about the value of your belief.
Even if we were to talk about how belief is a scale that is non-binary, and that people aren't usually precise about their concepts of probability, the difference in focus is enough to say that there is a different message being expressed.
I can only see one interpretation for the first statement: "I believe that x does not equal y" means that in my opinion, x is definitely not equal to y.
The second statement ("I don't believe that x equals y") could mean any of the following:
Tricky question. This isn't a question about logic, or equality, but about epistemology and belief. Compare:
I don't believe the Eiffel tower is tall
I believe the Eiffel tower is not tall
As statements about your beliefs, these sentences are not equivalent. The latter asserts a positive statement. The former is weaker. You are merely denying a positive statement. Strictly speaking, saying that you don't believe the Eiffel tower is tall only tells us what your belief is not.
So, the meaning of the statements comes down to how "belief" is interpreted. There are many epistemic theories on how one should do that. And, obviously, it varies by context.