User:IssaRice/Computability and logic/Semantic completeness

Semantic completeness is sometimes written as: if $\mathrm{\Sigma }\models \varphi$, then $\mathrm{\Sigma }⊢\varphi$.

Semantic completeness differs from negation completeness.

Definition

I want to make sure all these definitions are saying the same thing, so let me list some from several textbooks so I can explicitly compare.

Smith's definition: a logic is semantically complete iff for any set of wffs $\mathrm{\Sigma }$ and any sentence $\varphi$, if $\mathrm{\Sigma }\models \varphi$ then $\mathrm{\Sigma }⊢\varphi$.

Leary/Kristiansen's definition: A deductive system consisting of logical axioms $\mathrm{\Lambda }$ and a collection of rules of inference is said to be complete iff for every set of nonlogical axioms $\mathrm{\Sigma }$ and every $\mathcal{L}$-formula $\varphi$, if $\mathrm{\Sigma }\models \varphi$, then $\mathrm{\Sigma }⊢\varphi$.