I think the title describes the question clearly.
When I think of something being negated, I think of that something to have the property that it can't happen anymore.
Maybe the error is in my understanding of negation?
2026-04-12 22:52:27.1776034347
Regarding the logic on the meaning of negating "for all" to be there exists, can there exist then entail for all?
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No, "there exists" does not entail "for all". Entailment means that the second must be true in every structure in which the first one is true, and obviously there can be situations where "there is an x P(x)" is true but "for all x P(x)" is not.
However, "there exists" is consistent with "for all", that is, it is not contradictory for them to be true at the same time. So if "not for all x P(x)", which is equivalent to "there exists an x not P(x)", it may also be the case (but is not entailed) that even "for all x not P(x)", i.e. "there exists no x P(x)".