User:IssaRice/Computability and logic/Logic versus theory

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Many terms in mathematical logic apply to both logics and to theories. In these cases, the terms mean different things depending on what object it applies to. This distinction is similar to the distinction between truth-in-all-interpretations and truth-in-intended-interpretation.

Term Meaning for logic Meaning for theory
Provable For a logic, "provable" means "provable using the proof system and logical axioms". For a theory, "provable" means "provable using the proof system, logical axioms, and non-logical axioms". The non-logical axioms are specific to each theory.