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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.
| 19058 | Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples [Dummett] |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| 10691 | Logical consequence needs either proofs, or absence of counterexamples [Beall/Restall] |
| 13253 | There are several different consequence relations [Beall/Restall] |
| 10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |