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I know that we use not to negate a verb and we use no to negate a noun!

  • he had no pencil

Why do we use not in this sentence?

  • He had a pen not a pencil

Why don’t we use “no” since no is used to negate or exclude noun?

  • He had a pen no pencil.

.............. Is there any grammatical explanation for this?

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The second example is a shortened version of a parallel construction:

2a. He had a pen not a pencil.

2b. He had a pen; he did not have a pencil.

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There are several uses of the word no. The UK version of the MacMillan dictionary has this definition

used for saying that there is not even one person or thing

with examples such as

There was no hospital in the town.

We frequently say

I have no idea

So I see

He had no pencil

As being in line with these examples, and as a UK native English speaker, would not have questioned such usage. I must admit now the usage is challenged the phrase does seem a little archaic. However I still believe the formulation is valid. We can see "no" as replacing a count when the count is zero.

He had 5 pencils. He had 1 pencil. He had no pencil.

I do wonder whether

He had no pencils

Would be better.

According to this answer "no" is not strictly a count, and may be used with singular or plural nouns.

In the second case we are correcting a previous incorrect statement

He had a pen.

He had a pencil not a pen

we are correcting the subject of had; the subject is not pen it is a pen so we include the full subject we are replacing.

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