Completeness
Completeness
A logistic system (q.v.) may be called complete if there is no formula of the system which is not a theorem and which can be added to the list of primitive formulas (no other change being made) without rendering the system inconsistent, in one of the senses of consistency (q.v.). The pure propositional calculus — as explained under logic, formal, 1 — is complete in this sense.
Given the concept of semantical truth (q.v.), we may also define a logistic system as complete if every true formula of the system is a theorem. This sense of completeness is not, in general, equivalent to the other; and may be the weaker one if formulas containing free variables occur. See Logic, formal, 3, 6. — A.C.