These infamous logical quantifiers are considered part of the foundations of mathematics. But they are not at all fundamental and (frankly) have no business there. They are based on collection review methods and on pattern matching, as follows.
In the context of a collection, you need a mechanism for reviewing the elements of the collection.
In case of "for all", you have a pattern and look for a pattern mismatch during the review. If you find one you stop early with failure.
In case of "there exists", you have a pattern and look for a pattern match during the review. If you find one you stop early with success.
In other words, the logical quantifiers are not as fundamental as: reviewing a collection, pattern matching, and stopping early with a result. Some patterns have "opposites" and pattern matching one is mismatching the other. Absent such symmetry, matching is different from mismatching. If people wanted to think about foundations, they would think about that.
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