Can the empty-set be used for a model that satisfies an axiom system?

Question

I'm reading some notes on Model Theory and wondering if axiom systems that make no existential claims are trivially satisfied by the empty set. For instance, if you just have the axiom that all points are on some line, would taking the model $\mathbb{P}=\emptyset=\mathbb{L}$ prove the consistency of the axioms?

Answer 1

No, usually an interpretation is not allowed to be empty.

In some sense this is just an arbitrary choice, but it turns out that proof systems that are sound and complete with respect to model theory without empty models can be simpler than when one needs to account for empty models.

For example, if $p$ is some predicate, then $p(x)\lor\neg p(x)$ is usually provable as an instance of a tautology, and the most natural rule for introducing existential quantifiers would then allow one to prove $\exists x.p(x)\lor\neg p(x)$. This rule would not be sound if empty domains were allowed.

Since the empty structure is generally a quite uninteresting case, a priori we might either restrict the model theory to forbid empty structures, or change our proof theory to be sound with respect to the empty structure. In practice it turns out that assuming that structures are non-empty is a lot less cumbersome than the necessary changes to the proof system would be, so this is what is usually done.