Is this a double negation: "doesn't exist for neither X nor Y"

Question

I'm having doubts about how to best put this in English, I'm torn between 3 options. Which of them are legal, and which is stylistically the best?

I want to convey that a solution does not exist for A and it does not exist for B (in logic gates terms NOR)

1. A solution does not exist for either A or B
2. A solution does not exist for neither A nor B
3. A solution exists for neither A nor B

My feeling is that not...neither...nor (2) is a double negation corresponding to OR

Regarding (1): not...either...or could mean XOR or NAND, not sure which (this English SO question suggests XOR, meaning this would not be correct)

Only for (3) neither...nor am I sure it means NOR

But, how do I emphasise it? I'd prefer the negation to be clear from the start, stylistically. If I start "A solution exists", it's a big change of expectation if suddenly "neither nor" arrives. It seems like there's ambiguity here between two possible uses of "does (not)", it can be used for emphasis, and it can be used for negation purposes.

The solution could be

4. A solution does not exist for A or B.

Answer 1

In Standard English, options 1 and 3 are grammatical; option 2 is not.

In "standard" dialects of English, neither is only used in affirmative constructions.* You cannot say, "I don't like neither this nor that." You must say, "I don't like either this or that" or "I like neither this nor that."

And this starts to show why the statement "either or means XOR" is not necessarily true.

If either or were truly XOR, "I don't like either A or B" would mean either I like both A and B, or I dislike both A and B. But no one would ever use this sentence to mean I like both A and B.

And so, we might be tempted to say not either or means NOR (not A and not B), but I suspect this is also not always true. The fact is, outside of certain technical contexts, the words or and and cannot be consistently mapped to logical operations.

Fortunately, this also means that we potentially have more ways of expressing the point. For example, in addition to options 1 and 3, and other contributors' suggestions ("There is no solution for either A or B)", you could try:

• There is no solution for A. There is no solution for B.
• There is no solution for A and there is no solution for B.
• Neither A nor B has a solution.
• Neither one has a solution. Not A. Not B.
• Neither one has a solution. Not A and not B.
• There are no solutions to these problems.
• A has no solution and B has no solution.
• and on and on and on...

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*There are "nonstandard" dialects of English where this kind of double negative is used with a negative meaning. That is, in dialectical English "I don't like neither of them" means the same thing as Standard English's "I don't like either of them." This dialectical feature is called negative concord. It's not a mistake, nor is it ungrammatical in the dialects that use it.

Answer 2

Negatives can be confusing, especially as some other languages do use double negatives for emphasis.

To understand what's valid and what's not, it can help to reorder the sentence.

• A solution does not exist for either A or B ... can be rearranged as: For either A or B, a solution does not exist. This makes sense.
• A solution does not exist for neither A nor B ... This rearranges to For neither A nor B, a solution does not exist. In English, this does not make sense.

Perhaps the issue is the attempt to equate these linguistic constructions with formal Boolean logical operators. I confess that I'm not clear on NAND, XOR, etc., but what I can see is that the logic breaks down when trying to combine what we might call the "logic gate" represented by "either" or "neither" with the additional negation represented by "not." In practice, instead of figuring out what they all add up to, we can look at them functionally:

• "either __ or __" means "whatever is applied to this clause is true of one of these things" (I guess that's XOR).*
• "neither __ nor __" means "whatever is applied to this is not-true of both of these things." (I think that's NOR? perhaps no coincidence that it actually contains the word nor.)
• When you add the additional negation, you're adding a NOT gate as you add the actual word not. "I do not want either eggs or pancakes" means "the negation ('not want') is applied truthfully to one of these things." (This would be a confusing and ambiguous sentence, by the way; maybe I want both eggs and pancakes!)
• The construction "A solution does not exist for neither A nor B" could be parsed logically, but the result is the opposite of what you mean, and as linguistic communication, it's confusing. The logic is actually saying "The negation is not-true of both these things."

Meanwhile, if the point is that you want to emphasize negation and get to it right away, Ronald's suggestion is sound:

There is no solution for either A or B.

* This might lead to some confusion as we apply it. Technically—technically—that's an "exclusive-OR" logical operator. The statement is true of one and only one of these things. The way we use it in language can sometimes be broader, though. Yes, "I would like either pancakes or eggs" means I expect to be served one food, although either would be okay. But "There is no solution for either A or B" doesn't mean that the lack of solution is true for only one of them.