1. Most people not only don't need milk, they can't tolerate it.
2. Not only don't I want to sleep with you...I've never even fantasized about it.
3. Not only can't I swing that high but there's no way I would hold on with only one hand.
In all the three sentences, "only" grammatically modifies "don't" and "can't" ?
In (1) and (2), I would say that "not only" modifies "don't need" and "don't want" respectively. In (3), I would say that "not only" modifies "can't swing that high". I wouldn't split out "only" separately as a modifier.
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In (1) and (2), I would say that "not only" modifies "don't need" and "don't want" respectively.
In (3), I would say that "not only" modifies "can't swing that high".
I wouldn't split out "only" separately as a modifier.