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The proof of correctness and completeness of a logical calculus w.r.t. a given semantics can be read as telling us that the tautologies (or, more gen erally, the relation of consequence) specified in a model theoretic way can be equally well specified in a proof theoretic way, by means of the calculus (as the theorems, resp. the relation of inferability of the calculus). Thus we know that both for the classical propositional calculus and for the clas sical predicate calculus theorems and tautologies represent two sides of the same coin. We also know that the relation of inference as instituted by any of the common axiom systems of the classical propositional calculus coin cides with the relation of consequence defined in terms of the truth tables; whereas the situation is a little bit more complicated w.r.t. the classical predicate calculus (the coincidence occurs if we restrict ourselves to closed ∀xFx is inferable from Fx without being its conse formulas; otherwise..
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References found in this work BETA

Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - Cambridge University Press.
Structural Proof Theory. [REVIEW]Harold T. Hodes - 2006 - Philosophical Review 115 (2):255-258.
Meaning as an Inferential Role.Jaroslav Peregrin - 2006 - Erkenntnis 64 (1):1-35.
Completeness and Super-Valuations.Gary M. Hardegree - 2005 - Journal of Philosophical Logic 34 (1):81 - 95.
Language and its Models: Is Model Theory a Theory of Semantics?Jaroslav Peregrin - 1997 - Nordic Journal of Philosophical Logic 2 (1):1-23.

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