This is a follow-up to question <http://mathoverflow.net/questions/68788/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?