Classical equivalences like ¬∃x[P(x) ∧ ¬D(x)] presuppose a non-empty domain, but free logic permits empty domains where ∀x[P(x) → D(x)] holds vacuously while ¬∃x[P(x) ∧ ¬D(x)] may differ in truth value.
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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
vacuously true
The status assigned to a universally quantified formula ∀xA in a model with an empty domain, on which it counts as true despite having no instances