negation - verb+"to not"+infinitive OR verb+"not to"+infinitive

I believe the negative of a sentence, in which an infinitive is used, is made in two different ways.

For example, "I know what to say." can be negated as follows;

1. I know what not to say.


2. I don’t know what to say.

However, I often come across the swapped version of "not to" as follows;

3. I know what to not say.

Other examples follow;

• Tom had enough sense to not go out by himself after dark.


• I want you to not go to Boston with Tom.


• Why did you decide to not go to Boston with Tom?

Grammarly finds the third way to be fine.

Google Books Ngram Viewer prefers not to over to not in general.

I need to know if the third way of negating an infinitive-sentence is grammatical. Please, explain!

Answer

First of all "I don't know what to say" and "I know what not to say" mean different things. "I don't know what to say" means "It's not true that I know what to say". "I know what to say" means "There are things that I know I shouldn't say".

As to your question, "to not say" is known as "split infinitive". The "to say" phrase is called the "full infinitive", and putting words between "to" and "say" is known as "splitting the infinitive. There are some who say that infinitives "shouldn't" be split. However, this rule based more on an attachment to Latin (in which infinitives are a single word, and thus it is impossible to split an infinitive), rather than actual grammar. There is nothing ungrammatical about split infinitive, and they are arguably more logical.