Can you please explain what "Not to say" means in the context of the following sentence. "Not to say I might not be up for going tomorrow".
There's supposed to be a "that's" before "not to say," but people often omit it.
That's not to say (that)
used for adding a statement that corrects what you have just said or makes it less definite
This is used to disambiguate a sentence that may imply something that you don't mean to imply. Suppose I ask someone to go see a movie with me today, and they say no. I could easily infer from their reply that they don't want to see that movie at all, or that they don't want to see a movie with me specifically. If the person turning me down anticipates these false inferences, they may preempt them by saying something like, "I don't want to see a movie with you tonight, but that's not to say that I don't want to spend time with you," or something equivalent.
If you google this saying, you'll run into some difficulty because there's another, opposite meaning for the phrase "Not to say."
Not to say
used to introduce a stronger alternative or addition to something already said. - Google
It's my uneducated suspicion that this is a corruption of "to say nothing of" which is a way of pointing out a fact that complements the point that you're making (similar to "it goes without saying" or "not to mention"). Then again, I'm an American English speaker, and "not to say" is a British thing, so I may be completely off base.