A logic used in the discussion of a formal axiomatic theory within the frame of some meta-theory. In the foundations of mathematics one often imposes specific demands on the meta-theory, related to the rejection of some common mathematical abstractions, with the aim of improving the philosophical acceptability of the meta-theory. Examples of such abstractions subject to criticism are the abstraction of actual infinity, the abstraction of reasoning corresponding to the appearance of antinomies (cf. Antinomy), etc. This, in general, leads to the use of a logic different from the classical one, e.g. modal logic, or intuitionistic logic if the meta-theory is built within the frame of intuitionism.
On the other hand, in proof theory intuitionistic and other non-classical logical theories are studied by traditional mathematical means without any restrictions, e.g. by means of set theory. In this case, classical logic acts as the meta-logic.
Meta-logic. A.G. Dragalin (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meta-logic&oldid=14952