Strong completeness establishes sufficiency for capturing logical consequence: whenever a sentence follows logically from a set of hypotheses, there is a proof of that sentence in the calculus
If φ is a semantic consequence of Γ (Γ ⊨ φ), then φ is provable from Γ (Γ ⊢ φ)
logical consequence(WL II, 391–395; noted as similar to Tarski 1956, 419)
A proposition s is a logical consequence of a set of premises σ if and only if s follows from σ with respect to the sequence of all extra-logical simple ideas contained in σ or s.
proof(Frege's formal system; the definition still used by logicians today)
Any finite sequence of statements such that each statement is either an axiom of the formal system or follows from previous members of the sequence by a valid rule of inference.
, determining validity, or equivalently, testing for satisfiability of given formulas) for many-sorted logic is undecidable. So, we are in the same situation encountered in one-sorted first-order logic. Of course, if a calculus is to be helpful it would never allow erroneous reasonings: it is not going to drive us from true hypotheses to false conclusions. It must be a sound calculus. Further, it is highly desirable that all the consequences of a set \(\Gamma\) of hypotheses could be derived fr