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The difference between Leibniz and Schopenhauer is that the former is optimistic; the latter, pessimistic.

I don't think the part after the semicolon is an independent clause so how can a semicolon be used?

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Either a semicolon or a comma is acceptable here; and I, for one, prefer the semicolon.

What follows the semicolon is reduced, with the verb deleted by ordinary ellipsis. Restored to its full form it is clearly an independent clause:

The difference between Leibniz and Schopenhauer is that the former is optimistic; the latter is pessimistic.

A comma merely delays a sentence briefly; a semicolon acts as a 'hinge' on which the sentence can change direction.

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