Geoffrey Hunter.
Metalogic: an introduction to the metatheory of standard first order logic.
University of California Press. 1971
[p10]
Metatheory is the theory of formal languages and systems and their interpretations.
It takes formal languages and systems and their interpretations as its objects of study,
and consists in a body of truths and conjectures about these objects.
Among its main problems are problems about the consistency,
completeness (in various senses), decidability […]
and independence of sets of formulas.
Both model theory and proof theory belong to metatheory.
The metatheory of logic is the theory of those formal languages and systems that for one reason or another matter to the logician. Usually the logician is interested in a formal language because it has formulas that can be interpreted as expressing logical truths; and usually he is interested in a formal system because its theorems can be interpreted as expressing logical truths or because its transformation rules can be interpreted as logically valid rules of inference.
The metatheory of logic is the theory of those formal languages and systems that for one reason or another matter to the logician. Usually the logician is interested in a formal language because it has formulas that can be interpreted as expressing logical truths; and usually he is interested in a formal system because its theorems can be interpreted as expressing logical truths or because its transformation rules can be interpreted as logically valid rules of inference.