Examples of verbs that can be negated with not instead of don't

There are some examples of verbs in English that can be negated with not:

I think not

However it seems that this statement must exist in isolation and it is incorrect in modern English to follow this with a statement:

I think not the shop is closed I don't think the shop is closed

Another common example is:

We need not call the police

This sounds natural in modern English. However in general this type of negation is not possible:

I drive not to work

Are there any rules on when this type of negation is possible in modern English? Can you give any more examples of verbs for which this is possible?

There are other verbs that follow this pattern, including:

• believe (Is he here yet? I believe not.)
• expect (Will he get here before eight? I expect not.)
• hope (Will he be out until midnight? I hope not.)
• guess (Will we have time for dinner? I guess not.)
• be afraid (Have you heard anything from him? I'm afraid not.)
• appear (Did he leave on time? It appears not.)
• assume (Did he get our message? I assume not.)
• imagine (Will he think to call us? I imagine not.)
• presume (Does he have a good excuse? I presume not.)
• seem (Does he care at all? It seems not.)
• suppose (Does it matter? I suppose not.)

(Here is an article that explains their use.)

However, I disagree that in "I think not" the "not" is negating "think" – rather, it's negating the (implicit) proposition that the speaker doesn't believe is true.

For one thing, the equivalent with "don't" is "I don't think so", where the so is clearly referring back to the proposition being discussed. This can be used in the positive, too, as "I think so"; the "not" replaces the "so" in the negative case. So, I'd say that the sentence

(Will he arrive on time?) I think not.

is not equivalent to

I don't think he will arrive on time.

but rather

I think he will not arrive on time.

In this case the meaning is almost identical, but for some of the other verbs it is not, and for example:

(Will he arrive on time?) I'm afraid not.

is definitely equivalent to

I'm afraid he will not arrive on time.

and certainly not

I'm not afraid he will arrive on time.

Similarly, "I hope not" would mean "I hope he will not arrive on time" rather than "I don't hope he will arrive on time".

There is no hard and fast rule, but the OED points out in its third definition of not that this construction was much more common prior to modern English, and certain words have maintained this construction since early modern English, while others have switched to the standard convention of using do not.

3. Following a full verb. Now chiefly arch. or literary and humorous.

In recent use, esp. in I kid you not and variants.

F. T. Visser ( Hist. Syntax Eng. Lang. (1969) III. §1441) observes that from the beginning of the modern English period certain verbs tended to continue to be used in this construction rather than the do not construction, the main ones being care, doubt, know, mistake, trow, and wot.

This list provided in the OED note is not comprehensive as far as the construction is used in modern English. Afterall, it doesn't include think, as pointed out in the question, or kid, as pointed out in their example of the recent phrase "I kid you not." But the note does show that this construction appears to have remained in use for certain words, while other constructions switched to use the do not construction in modern English.

That brings us to another relevant point, which is that this construction of not was more common in archaic use. A few examples from the OED citations show how it was applied earlier:

As longe as the significacion bode, it hurted not.

• William Tyndale · An answere vnto Sir Thomas Mores dialoge · 1st edition, 1531 (1 vol.).

With holy father sits not with such thinges to mell.

• Edmund Spenser · The faerie queene · 1st edition, 1590 (1 vol.).

Short answer

In sentences such as I think not, the word not is a proform standing in for a negated subordinate clause. It is not negating the verb think. Rather it is the complement of that verb. We use not in the same places where we use so to represent positive subordinate clauses.

• Has Bob eaten yet?
• I think [Bob hasn't eaten yet].
• I think [not].
• I think [Bob has eaten].
• I think [so].

Because not represents a subordinate clause here, it does not make the larger sentence negative.

Full answer

1. Verbal negation in finite clauses:

The word not is often used to negate clauses. When occurring in finite clauses, it occurs after the auxiliary verb (apart from in some types of question, where the auxiliary moves to the front of the clause leaving the word not in situ).

If there is no auxiliary verb, we need to insert the dummy auxiliary, do:

1. The elephants are swimming.
2. The elephants are not swimming.
3. My elephants swim.
4. My elephants do not swim.
5. *My elephants not swim. (ungrammatical).

We can show that the word not belongs with the auxiliary here by 'contracting' not with the auxiliary verb:

6. The elephants aren't swimming.

The reason that we can use not in the Original Poster's sentence We need not tell the police is that the verb need is an auxiliary verb here. We can show this by using a contraction:

• We needn't tell the police.

Notice that this whole sentence is therefore negative. We can show this by using a question tag. Negative sentences always take positive tags:

• We needn't tell the police, need we?
• *We needn't tell the police, needn't we? (ungrammatical)

2. Proforms

We often use proforms in English to refer back to a previously mentioned phrase or clause (what's really happening of course, is that we're referring back to a previously mentioned entity or idea). For example, we often use pronouns to refer back to a previously mentioned noun phrase, where the pronoun replaces the whole noun phrase , not just the noun:

• See that man in the red t-shirt? He is my Maths teacher.

In the example above the pronoun he is standing in for the previously mentioned phrase that man in the red t-shirt. We also have proforms, such as the word there, for example, which stand in for whole preposition phrases:

• We were at the bottom of the valley. It was very beautiful there.

In the example above the proform there is standing in for the at-preposition phrase at the bottom of the valley.

If we want to use a proform for a whole clause, we often use the proform so.

7. Has Bob finished his work?
8. Well, he said so.
9. Well, he said he has finished his work.

In the examples above we can see that the word so is replacing a clause as the complement of the verb said. The clause that has been replaced means something like he has finished his work.

Notice however, that so can't replace a negative clause. If we want to replace a negative clause with a proform, we have to use the proform not, which stands in for negative clauses:

11. He didn't eat it, did he?
12. I think so. (= "I think he ate it")
13. I think not. (= "I think he didn't eat it)

Notice that although not represents a negative subordinate clause here, it doesn't make the larger sentence negative. We can show this by putting a negative tag at the end of the sentence:

14. You think not, don't you?

Conclusion

In the Original Poster's sentence, I think not, the word not is a pro-clause item standing in for a negative clause. It represents a subordinate clause functioning as the complement of the verb think. It does not negate the verb think or make the larger sentence negative. We can use not or so as a proform with many other verbs that take subordinate clauses.

Below is an excerpt on pro-form so and not from The Cambridge Grammar of the English Language (Huddleston & Pullum, 2002 p.849).

Excerpt from the Cambridge Grammar of the English Language:

Relevant: "I don't think so" vs. "I think not"

I would say that the usual form of negation is to include the do as an auxiliary verb.

"I think not" is a colloquialism, but one that implies British upper classes and has grown from there into acceptable usage but usually in an amusing way. Unless you actually are posh, in which case you may use it unironically.

"Need" is different, but I see it as part of the group of verbs like should, would, could, do, and have. This must form a closed group of verbs of some description. The other verbs are often auxiliary verbs while need is not, so maybe that is not it afterall.

Edit: a comment from @Shoe to the original post pointed out that these are modal and semi-modal forms.