Can an existentially quantified formula be a tautology

Question

I was wondering if a formula of the form $\exists xP(x)$, where $P(x)$ is any formula, can be a tautology. To me the domain could be empty, and therefore it is not satisfiable in every structure. But from an intuitive standpoint this is hard to grasp, especially if there are several quantifiers nested and implications involved, where the antecedent might then be false.

Answer 1

Ultimately this depends on exactly what semantics we use. Most presentations of first-order logic do not allow the domain to be empty; consequently, sentences like "$\exists x(x=x)$" are tautologies. On the other hand, if we do allow the empty domain, then no sentence of the form "$\exists xP$" can be a tautology since such a sentence cannot hold in the empty structure (similarly, no sentence of the form "$\forall x P$" can be a contradiction since such a sentence cannot fail in the empty structure).