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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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Yes, but I know that x does not equal y does not mean the same thing as I don't know that x equals y. The context could also change even your example. Fro instance I don't believe x equals y. I know it! –  bib Jul 16 at 14:00
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As a side note about mathematical logic, consider the following statements: (A) x does not equal y; (B) it's not true that x equals y. In Aristotelian logic, every statement is either true or false, so A and B are equivalent. But there are nonaristotelian logical systems in which A and B are not logically equivalent. –  Ben Crowell Jul 16 at 17:46
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This isn't the right stack exchange for this. I think there is a logic one. This problem is not unique to the English language, and is purely logical. –  JFA Jul 16 at 20:33
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@JFA I disagree that the question is purely logical. Answers will probably involve some (fairly trivial) logic, but the real issue here is the translation of natural language into logical constructs, which is subtle and tricky. That's something that sits at the boundary of logic and language; it wouldn't be out of place on a logic-oriented SE but it's fine here too. Not sure what not being unique to the English language has to do with anything. Neither is, say, verb conjugation. –  Curtis H. Jul 16 at 21:16

10 Answers 10

up vote 40 down vote accepted

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 know that x doesn't equal y.

I have actual knowledge that x and y are not equal. Quite possibly I can show you facts to support this.

  • I don't know that x equals y.

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:

  • I don't know if x equals y.

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.

to mean

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.

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This is not the whole story of how the constructions are interpreted. See Neg Raising in Lexical Resource Semantics: Manfred Sailer: eg ' ... (1a) can either mean that it is not the case that John thinks Peter will come, or it can be seen as expressing the same idea as in (1b): (1a) John doesn't think Peter will come. (1b) John thinks Peter will not come.' >> Colloquially, (1a) would probably be more frequently used even for the (1b) sense. –  Edwin Ashworth Jul 16 at 14:51
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Yes, they do convey the same meaning in English. Negative syntax is complicated and one can't depend on inner and outer negative scope with certain predicates. The reason why know behaves this way is because it's a factive predicate, which is certainly not true of believe. –  John Lawler Jul 16 at 16:50
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@JohnLawler While I rarely disagree with you, I would say that I could assert both "I don't believe that x equals y" and "I don't believe that x does not equal y" without contradiction. I simply might not have enough information to form a belief, in which case I believe neither of the two available options. Thus, saying "I don't believe" with respect to either option is valid. That said, I have a background in philosophy, mathematical logic, and computer science, so I may be in the 0.0001% here. :) –  Two-Bit Alchemist Jul 16 at 19:42
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@Two-BitAlchemist: You may, in fact, be unique. Many people believe that they ALWAYS say 'Z` in only one way, or ALWAYS mean X when they say S, or NEVER use P. This is usually because they confuse writing with speech, and rarely if ever pay attention to what they actually say in ordinary speech. Generally they are completely wrong, because unmonitored speech is, well, unmonitored, and most people have a lot of unrealistic beliefs about themselves. Especially when it comes to language, as it turns out. –  John Lawler Jul 16 at 20:13
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@JohnLawler If "I don't believe X" is the same as "I believe not-X" then taking X to be "god exists" proves that agnosticism and atheism are the same thing, which they clearly aren't. –  David Richerby Jul 16 at 22:51

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.

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The use of logic symbols (and I'm not sure that '!' is what you actually intended) is off-topic. –  Edwin Ashworth Jul 16 at 16:41
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@EdwinAshworth: the exclamation sign is used in some programming languages to indicate negation, and I think its use here is perfectly appropriate as long as it's explained. –  Marthaª Jul 16 at 16:49
    
The usual symbol for Negation in logic is ¬ (UTF8 C2 AC) in standard notation, or N in Polish notation. Bang is used in C-derived languages, but not, for instance, in regular expressions. –  John Lawler Jul 16 at 16:54
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The use of logic to explain English usage is not always going to work. 'John doesn't think Peter will come' shouldn't logically mean 'John thinks that Peter will not come', but that's the way it's usually used. And I personally have found that this answer does the opposite of clarify. –  Edwin Ashworth Jul 16 at 16:56
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+1 for acknowledging that the construction is ambiguous. (Something that should be apparent from the fact that the various answers attempting to ascribe exactly one meaning completely contradict each other.) –  Curtis H. Jul 16 at 21:25

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.

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You can't say this, either, without sufficient context. It could be oerkelens speaking. As the article cited in my comment above says, it can mean either (though I'd agree that your sense is the more likely). –  Edwin Ashworth Jul 16 at 15:12
    
By all means, there's a very slight chance (I'd say, 1:2500) the OP was quoting from a math lecture, or some similar context. For sure. BTW - it's surely the case that there has already been questions on here (over and over) about the whacky "I don't believe there's a train at 7" form in English. –  Joe Blow Jul 16 at 15:18
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I think they have somewhat different shades of meaning. In general, saying "I believe that florbles are not fnorbles" would imply that one has investigated potential equality and found it false. Saying "I don't believe that florbles are fnorbles" would not imply that one has particularly investigated the issue, but would imply that any claim that florbles were fnorbles would be regarded skeptically. By contrast, something like "I'm unaware of florbles being fnorbles" would suggest that one has no opinion on the matter and would not particularly disbelieve any claim related to it. –  supercat Jul 16 at 19:33

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

  • She thinks/believes that he won't get here on time.
  • It appears/seems/is likely that he won't get here on time.

or upstairs, in the matrix clause with the NR predicate

  • She doesn't think/believe that he'll get here on time.
  • It doesn't appear/seem/isn't likely that he'll get here on time.

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

  • She claimed/said that he was not late She didn't claim/say that he was late

and neither do possible or easy

  • It's possible/easy for him not to stay home. It's not possible/easy for him to stay home
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... that all said, it really depends on the context. When I say "I don't believe x equals y", I mean something different than if I say "I believe x does not equal y". However, if I say "I don't believe I like passion fruit", I mean pretty much the same thing as if I say "I believe I don't like passion fruit", which is to say "I detest passion fruit, but I'm trying to be polite". –  Marthaª Jul 16 at 16:59
    
@Marthaª: You may well mean that every time you say it. However, native speakers don't, usually; and therefore will not expect consistent usage like that in others. So don't be surprised if you're misunderstood. –  John Lawler Jul 16 at 17:04
    
@JohnLawler Your answer seems to imply consistent usage, though. (That is, that "I don't believe x equals y" and "I believe x does not equal y" consistently mean the same thing.) As a native speaker, I can attest that sometimes these mean the same thing for me, and sometimes they do not. As Martha said, it's context-dependent. –  Curtis H. Jul 16 at 21:21
    
Everything is context-dependent. And there are always contexts for which the most unlikely sentence is completely grammatical. But such contexts are rare, and not the norm. Grammars deal with norms, not exceptions. –  John Lawler Jul 16 at 21:38
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Sure, neg raising is a phenomenon that exists and is clearly described. It explains why the two sentences can and often are interpreted the same way. The fact that linguists are spending a lot of time trying to explain why and how such a strange thing is happening, does by no means constitute an argument for the idea that the opposite (the normal, logical, formerly prescribed) interpretation does not occur. If I investigate the reasons for speeding, it does not mean that nobody keeps to the speed limit! –  oerkelens Jul 17 at 18:16

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[1] 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.

[1] http://en.wikipedia.org/wiki/Modal_logic

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Interesting that there are essentially two sets of answers:

  • "Yes, they are the same in common usage"
  • "No, they are not the same as anyone with a basic understanding of logic knows"

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.

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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.

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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:

  1. In my opinion, x is definitely not equal to y. (Same as the first statement).
  2. I don't have sufficient evidence to support a claim that x equals y. (I have no opinion, or am unsure of it).
  3. Contrary to my expectations, x IS equal to y. (Equivalent to "I can't believe that x equals y").
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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

to

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.

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I dont believe stepping on a crack will break my mothers back.

I believe stepping on a crack will not break my mothers back.

Equivalent. Even tho language useage wise the former feels less commital

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