Full Idea
The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions.
Gist of Idea
The two standard explanations of consequence are semantic (in models) and deductive
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.222
A Reaction
Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions.