Property of a deductive system that every formula which is semantically valid (true in all models of the system's semantics) is provable within the system's proof calculus.

Principle

Semantic completeness requires alignment between model-theoretic consequence and syntactic derivability: if a sentence holds in all models, the formal rules of the system can derive it; completeness is typically proved by constructing a canonical model from consistent sets or by completeness theorems.

Demonstration

First-order logic is semantically complete in the sense of Gödel's completeness theorem: any sentence true in every model of a first-order theory is derivable from the theory's axioms using first-order proof rules.

Misapplication

Confusing semantic completeness with syntactic (or negation) completeness of a theory (every sentence or its negation being provable), or misreading Gödel's incompleteness theorems as contradicting the completeness theorem for first-order logic.

Consequence

A semantically complete system ensures that proof search is adequate for capturing all semantic truths of the language: if a formula is valid, some finite proof exists within the system (though it may be hard to find).

Reversal

Semantic incompleteness: there exist semantically valid sentences that are not provable in the system; syntactic completeness is orthogonal and concerns provability of each statement or its negation.

Boundary

Applies to a specified language and semantic framework; completeness does not entail decidability, nor does it guarantee the existence of effective proof search procedures for arbitrary theories expressed in the language.

Semantic Tension

Tension between semantic completeness and decidability or syntactic completeness: a system can be semantically complete but undecidable, and completeness does not imply every statement is decidable within the theory.

Synthesis

Semantic completeness connects truth across all models with provability: when a deductive system is semantically complete, model-theoretic validity and syntactic derivability coincide, enabling proofs to capture all universal semantic consequences.