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Are both these sentences correct?

  1. There isn’t a cat in the kitchen.
  2. There isn’t any cat in the kitchen.
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2 Answers

If you are looking for a yes–no answer, then it depends on what you mean by “correct”. Neither of your two sentences is particularly natural, but neither is ungrammatical, either.

In other words, they are “grammatical” but they may not be correct for your situation. In fact, I suspect that neither is a good match for the general case. That doesn’t make them ungrammatical, but being grammatical and being fitting or reasonable or desirable or correct, let alone being optimal, are all different things entirely.

To expand and explain, the following closely related sentences below are all grammatical — or at least can be under the right circumstances. They are not all equivalent to one another, however. Most differ in nuance, or perhaps in register.

  1. There isn’t a cat in the kitchen.

    This is probably the most marginal of the set, but could possibly be made to work with sufficient set-up. Compare:

    • There isn’t a clean dish in the whole damn house.

    See number 4.

  2. There isn’t any cat in the kitchen.

    Again, this sounds a bit odd, but this time it is because it seems to be construing cat to be a mass noun instead of a count noun. Compare:

    • There isn’t any milk in the fridge.
  3. There’s no cat in the kitchen.

    Unlike the earlier pair, this one is perfectly fine. It seems to answer the question,

    • Is there a cat in the kitchen?
  4. There’s not a cat in the kitchen.

    This is equivalent to #1, and so would not sound right under most but not all possible circumstances. It suggests questions like

    • Well, what is in the kitchen then?
    • You have a cat in every room of your house, don’t you now?
  5. There are no cats in the kitchen.

    This one is perfectly fine, and is fully equivalent to the next one, but for the slight possible register shift triggered by the contraction.

  6. There aren’t any cats in the kitchen.

    Again, sounds just fine. It appears to answer the question

    • Are there any cats in the kitchen?
  7. No cat’s in the kitchen.

    Here again is a rarer one. But perhaps you’ve got a cat in the den, and in the foyer, and in the pantry, but no cat’s in the kitchen.

  8. No cats are in the kitchen.

    Sounds like part of those arithmetic puzzles. Mostly equivalent to numbers 5 and 6.

  9. Not a cat was in the kitchen.

    It sounds a bit literary to begin a sentence with Not a — or with Nary a, for that matter, which means the same thing. This says that there wasn’t even one single cat in the whole kitchen. Compare:

    • Not a creature was stirring, not even a mouse.

    Kitchen seems unusual here; it is not a big place, after all, so perhaps farm or town would work better here. More natural, but still literary, equivalent formulations might include:

    • Not a cat was to be seen, despite the thundering herd of field mice.
    • Not a cat was to be seen in the kitchen that day.
    • Nary a cat was to be found, the whole hamlet through.
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+0.83 I strenuously disagree with your characterization of #1 as "marginal" - This strikes me as the most neutral of all. "Did you look in the kitchen?" -"#1." OR "There's a cat in the kitchen." -"No, #1". OR "#1—but there's one sleeping in the chandelier." –  StoneyB Nov 4 '12 at 16:16
    
The awkwardness of about all the examples comes from the sentence concentrating on the state of the kitchen regarding count of cats (who cares?), and not the location of the cat in respect to the kitchen (common interest). "The cat is not in the kitchen" would be the most expected expression I think. –  SF. Nov 4 '12 at 16:31
    
Since negation is the opposite of definitive -- This is not the Eiffel Tower* tells you practically nothing about what this is -- there are potentially an infinite number of ways to say negation. Most of them resolve down to the same situations, by various semantic and pragmatic means, but one can go on forever with subtle differences that most people won't notice, or care about if they do, or which don't arise in most contacts. Waste of time for language learners, for the most part, but harmless for native speakers, who can argue endlessly and entertainingly about them. –  John Lawler Nov 4 '12 at 17:28
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If you're asking if they're grammatical, then, yes, they are, but they'd be used on different occasions, depending on what had gone on before in the conversation.

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Hmmm ... In AE you'd say "There aren't any cats", but outside of jocular uses 1)"There isn't any cat" would have to employ cat to signify the absence of some abstract felinity. 2)"There's no cat in the kitchen" might be employed as an emphatic denial of a presence previously asserted. –  StoneyB Nov 4 '12 at 15:14
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I can imagine a conversation such as: 'There's a cat in the kitchen. How did that get there?' 'What do you mean? There isn't any cat in the kitchen.' Perhaps it's a BrEng peculiarity. –  Barrie England Nov 4 '12 at 16:00
    
@PeterShor We may talk in different circles. I'd expect "Whaddya mean? There isn't a cat in the kitchen!" or "There's not a cat in the kitchen!", or even (under the influence of Tom Hanks in A League of Their Own) "There's no cat in the kitchen!"; but "any cat" sounds strange to me. –  StoneyB Nov 4 '12 at 16:09
    
The nonstandard 'There ain't no cat in the kitchen' is a further possibility. –  Barrie England Nov 4 '12 at 18:03
    
Which opens up a whole new can of catfood. –  StoneyB Nov 4 '12 at 18:08
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