This is a follow-up to question Completeness vs Compactness in logic 68788. One common theme was that compactness in logic is a purely semantic notion, so should have no need of completeness.

The definition of compactness seems to depend in an irreducible way on the concept of a sentence, which appears to be a syntactic notion. So my question is: Is there any purely semantic definition of a sentence in first-order logic?