Full Idea
Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence.
Gist of Idea
Logical consequence is intuitively semantic, and captured by model theory
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
A Reaction
If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation.