User:IssaRice/Computability and logic/Semantic completeness

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

Semantic completeness is the completeness that is the topic of Godel's completeness theorem.

Semantic completeness differs from negation completeness.

Semantic completeness is about the completeness of a logic (not about the completeness of a theory).

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 $\mathrm{\Sigma }⊢\varphi$.

Alternative formulation

It is possible to formulate completeness by saying that a consistent set of sentences is satisfiable. In other words, the following are equivalent:

1. Let $\mathrm{\Gamma }$ be a set of sentences, and let $\varphi$ be a sentence. If $\mathrm{\Gamma }\models \varphi$, then $\mathrm{\Gamma }⊢\varphi$.
2. Let $\mathrm{\Gamma }$ be a set of sentences. If $\mathrm{\Gamma }$ is consistent, then $\mathrm{\Gamma }$ is satisfiable (has a model).