Negation – Is ‘Doesn’t Exist for Neither X nor Y’ a Double Negation?

logicmultiple-negationnegationnegative-concordneither-nor

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

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

Best Answer

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...

*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.

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