Of course, the sentence "No!" may be used to signify:
  • Refusal to carry out a request or order. This may seem to imply a verb; but viewing it as a proposition of the form 'it is not the case that I will do the washing up,' for example, misses the point. If we undertake this kind of analysis, how do we know we should not render it with a different verb, viz.: 'it is not the case that I will follow your order to do the washing up,' or 'it is the case that I refuse to follow your order ... ?'

    The fact that the 'implied verb' is indeterminate argues that it's some kind of philosopher's or linguist's fiction, I think. (This kind of confusion often happens when we conceive language as merely a mechanism for the transfer of information.)

  • General horror at a situation. If we take 'no' to be shorthand for some proposition, such as 'I refuse to believe it.' or 'I don't like it at all' we are again forced to forge some definite propositional sense from a sentence which may be more or less 'non-cognitive' (as proponents of the boo-hurrah theory of ethics, such as A.J. Ayer, might say.)

  • A response to another's actions, rather than to their words; a negative imperative. And in this case it seems easy to see that the sentence may well require, or admit of, no further explication, and still be perfectly clear in its meaning.

  • Disagreement on general, but unspecified, grounds. Examples are left as an exercise for the reader.

That it is a sentence can be seen from this: it might be the only word ever spoken to us by someone, even someone that we had never ourselves spoken to -- and yet, given the right circumstances, we would understand them perfectly.

This is a reasonable test of sentencehood: whether, isolated like this, it can be understood. Often this is a case of chasing dangling parts of speech, as in the grammar-school idea of a sentence given below by Phinslit, but this task is subservient to the idea of semantic completeness. Sometimes, understanding can occur without the kind of specificity required by the grammarians.

Regards the corresponding question of whether 'Yes' must always imply some propositional scaffolding in order to make sense, a close reading of the final chapter of Ulysses is advised.