Sentence (First-Order Logic)

first-order-logic

Definition

Sentence (First-Order Logic)

A sentence (or closed formula) is a formula with no free variables.

Satisfaction

Theorem

If $\phi$ is a sentence, then for any structure $\mathcal{M}$ and any variable assignments $\ell ,{\ell }^{\prime }$:

$$(\mathcal{M},\ell )\models \phi ⟺(\mathcal{M},{\ell }^{\prime })\models \phi$$

In this case we simply write $\mathcal{M}\models \phi$ and call $\mathcal{M}$ a model of $\phi$.