If "and so" connects predicates such that the second is the result of the first, do we need to repeat the same words in the second? For example,
It does not exist locally and so globally.
or
It does not exist locally and so does not exist globally.
If "and so" connects predicates such that the second is the result of the first, do we need to repeat the same words in the second? For example,
It does not exist locally and so globally.
or
It does not exist locally and so does not exist globally.
The inference in your example seems doubtful if not fallacious, but you asked about the grammar, specifically elliptical grammar. The second part of your second example leaves out only "It" but repeats everything else from the first part of the sentence, beyond substituting "globally" for "locally." That works. In your first example, the second part of the sentence leaves a good deal more out. That works fine too if you do not go too far with it; but you did go too far, in leaving out "not," which cannot well go without saying here: the clause's negativity does not survive ellipsis of the negative. I accordingly read the first example (before reading the second) as meaning the thing does not exist locally and so MUST exist globally, or at least elsewhere on the globe.
You do not need to restate words except where the result is unclear. Your first example is not clear, because it leads the reader to think because a thing does not exist locally it must exist globally. I don't think you can generalize a rule about "and so", because in some cases, the result is something completely different than the cause. Ex: He stressed too much and so ended up in the hospital. The second phrase there is a result of the first but nothing can be omitted if it is to be understood.
Why not just rephrase it? Here:
Since it does not exist locally, it cannot exist globally.
This sentence expresses the causal relationship much more clearly.
No, you don't need to repeat all the words except the auxiliary verb without the negative not. The sentence would be: It does not exist locally and so does globally. Look at more examples below:
- Men like travelling and so do women.
- Zachary was asleep when I came in and so was the puppy.
You can see how the auxiliaries do and was have been used in the sentences.
I assume from the content of the sentence and from your handle that the context is mathematical writing. The original is clear enough to a mathematician. It means "It does not exist locally, and so not globally either". But since you have doubts, I suggest you use the wording I just gave. You do not have to repeat "does not exist".