Karel Lambert and Jaakko Hintikka showed that classical quantifier interdefinability fails in free logic systems designed to handle non-denoting terms and empty universes of discourse.
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Non-denoting terms(as a logical problem that free logic addresses)
Words or names that seem to refer to something but actually don't point to anything real—like 'unicorn' or 'the present King of France.'
Quantifier(in formal logic)
Words in logic that specify how many things you're talking about, like 'all' (universal quantifier) or 'some/at least one' (existential quantifier).
Universe of discourse(as a foundational concept in logic)
The set of all things you're talking about in a particular conversation or logical system—basically, the boundaries of what counts as 'real' for your purposes.
classical logic(Contrasted with Hegel's dialectical approach that accepts contradictions)
Aristotelian logic that dominated during Hegel's lifetime
free logic(Contrasted with standard first-order predicate logic)
A logical system in which the existential generalization '∃x φ(x)' cannot in general be derived from 'φ(t)' for a singular term 't', meaning singular terms do not automatically carry existential import