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De Morgan's law isn't the right one for this. You want the identity ∀x.P(x) ⟺ ¬∃x.¬P(x).

∀x.P(x) is obviously true when the universe is empty -- no counterexamples exist. ∃x.P(x) is obviously not true in that case -- no examples exist.




That is De Morgan if you expand the quantifiers.


You have to make some assumptions if you want to "expand the quantifiers". In particular, you have to assume that the universe is non-empty, which is a really bad idea if you're commenting in this thread.




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