Theory

Remarks

Example: in the signature $\set{\cdot, e, {}^{-1}}$, the set of the three sentences $\forall x \, \forall y \, \forall z \, ((x \cdot y) \cdot z = x \cdot (y \cdot z))$, $\forall x \, (x \cdot e = x \land e \cdot x = x)$ and $\forall x \, (x \cdot x^{-1} = e \land x^{-1} \cdot x = e)$ is a theory, the theory of groups. Each of its elements is a formula without free variables, so it is a set of sentences.

Non-example: the set $\set{x \cdot e = x}$ is not a theory, because the variable $x$ is free in its only formula.

Some authors require a theory to be closed under logical consequence; a set of sentences is then called a set of axioms for the theory it generates. For an $L$-structure $\mathcal{M}$, the set $\mathrm{Th}(\mathcal{M})$ of all $L$-sentences true in $\mathcal{M}$ is the theory of $\mathcal{M}$.