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A logic used in the discussion of a formal axiomatic theory within the frame of some [[Meta-theory|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|abstraction of actual infinity]], the abstraction of reasoning corresponding to the appearance of antinomies (cf. [[Antinomy|Antinomy]]), etc. This, in general, leads to the use of a logic different from the classical one, e.g. [[Modal logic|modal logic]], or intuitionistic logic if the meta-theory is built within the frame of [[Intuitionism|intuitionism]].

On the other hand, in [[Proof theory|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.