Metalogic
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The metalogic of a system of logic is the formal theory of the formal logic. Results in metalogic will consist of such things as formal proofs demonstrating the soundness of the logic.
For example, the metalogic of any system of first-order predicate calculus is the demonstration of its completeness and consistency.
The deduction theorem is an important meta-theorem and a key step to proving completeness. Gödel's incompleteness theorems are major results in metalogic.
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