# theory

If $L$ is a logical language for some logic $\mathcal{L}$, a set $T$ of formulas with no free variables is called a theory (of $\mathcal{L}$). If $\mathcal{L}$ is a first-order logic, then $T$ is called a first-order theory.

We write $T\vDash\phi$ for any formula $\phi$ if every model $\mathcal{M}$ of $\mathcal{L}$ such that $M\vDash T$, $M\vDash\phi$.

We write $T\vdash\phi$ is for there is a proof of $\phi$ from $T$.

Remark. Let $S$ be an $L$-structure for some signature $L$. The theory of $S$ is the set of formulas satisfied by $S$:

 $\{\varphi\mid S\models\varphi\},$

and is denoted by $\operatorname{Th}(S)$.

Title theory Theory 2013-03-22 13:00:12 2013-03-22 13:00:12 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 03B10 msc 03B05