Negative clauses are best described in terms of modifications of canonical clauses, which by definition have positive polarity. And English has quite an array of different ways in which the modification can be done.
First, note the various syntactic tests that can be used to determine which clauses are negative clauses. It is a more subtle matter than you might have thought. For example, this is a negative clause:
We are in no way troubled.
But this is not:
We are in no end of trouble.
And this is a negative clause:
He has done not a single helpful thing.
But this is not:
He has done a not unhelpful thing.
Clearly, we need syntactic tests for negation. Some were suggested by MIT syntactician Ed Klima in the early 1960s (they are summarised nicely in the article "Negation" by John Payne, in the volume edited by Tim Shopen called Language Typology and Syntactic Fieldwork). Very briefly, Klima suggested these tests:
It also seems to be the case (for most speakers, anyway) that negative clauses cannot take negative tags even of the same-polarity type.
A surprising number of distinctions have to be drawn to understand how negation works. Here are the main four:
Let's look at each of them in turn.
1. Verbal versus Non-verbal negation
| Type | Example | Notes |
| Verbal negation: | I couldn't ever lie to her. | The negation morpheme is the verbal suffix n't. |
| Non-verbal negation: | I could never lie to her. | The negation morpheme is on a non-verb. |
2. Analytic versus synthetic negation
| Type | Example | Notes |
| Analytic negation: | This is not complete. | The negation morpheme is the separate word not. |
| Synthetic negation: | This isn't complete. | The negation morpheme is the suffix n't. |
3. Clausal versus sub-clausal negation
| Type | Example | Notes |
| Clausal negation: | She didn't have any substantial income. | The whole clause is negated. |
| Sub-clausal negation: | She had a not insubstantial income. | The negation only affects a phrase (here, the AdjP headed by insubstantial). |
4. Ordinary versus metalinguistic negation
| Type | Example | Notes |
| Ordinary negation: | He hasn't got four kids; he's got three. | The claim is that "has four kids" is actually not true. |
| Metalinguistic negation: | He hasn't got four kids; he's got five. | The claim that he has four kids is actually true (anyone who has five kids has four kids); what is being negated is the appropriateness of the word choice given that it's misleading. |
If you take in those four distinctions, a significant part of the work of understanding negation in English is done.
The notion of scope
The scope of a negation is the constituent that it has effect over. In these sentences, the negative marker (which in all these cases is the suffix -n't in the word didn't) is boldfaced (in red, if you're browsing) and the words contained in the part of the sentence whose meaning constitutes its scope is underlined:
Liz didn't delete the backup, because we found that on the other disk.
Liz didn't delete the backup because she wanted to be annoying; it was a genuine mistake.
Liz didn't delete the backup and the access log shows that.
Liz didn't delete the backup and the access log; I checked both of them last night.
Non-affirmative polarity items
Some items (words or idioms) cannot be used (or cannot be used with the same sense) in positive declarative environments. We call these non-affirmative polarity items. In donkey's years (meaning "since a very long time ago") is a particularly clear case. It never occurs positively, but any kind of clausal negation makes it fine.
*I have seen my old school friend Pete in donkey's years.
I haven't seen my old school friend Pete in donkey's years.
I have not seen my old school friend Pete in donkey's years.
Not in donkey's years have I seen my old school friend Pete.
I have seen none of my old school friends in donkey's years.
There are also plenty of items that occur either in negative clauses or interrogatives or various other contexts, but not in positive declaratives: any non-affirmative context will permit them. (This is covered on page 155 of the textbook.)
Finally, notice the Prescriptive Grammar Note on page 156 about the interesting non-standard phenomenon of negative concord (which just about every non-standard dialect still has, having preserved it from Middle English). It is not "double negation", it is multiple marking of a single negation at all relevant points in the negated clause — roughly, at all the indefinite Determiners and quantifiers, including quantificational adverbs:
I didn't never mean no harm to nobody or nothin'.
(Translation to colloquial Standard English: "I didn't ever mean any harm to anybody or anything.")
That example has quintuple negation. Under the rules of logic, five negations are equivalent to one: "It is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the butler did it" is equivalent in classical logic to "The butler didn't do it":
But of course that has no relevance either way to the analysis of non-standard dialects. It doesn't make Don't give me no trouble illogical in non-standard dialects, but it also doesn't make I didn't never mean no harm to nobody or nothing grammatical in Standard English. The term "double negation" is just a hopeless red herring in grammar. It has never been clear what it means.