Definition:Witness Property

Definition
An $\mathcal{L}$-theory $T$ is said to have the witness property if for every $\mathcal{L}$-formula $\phi(v)$ with one free variable, there is a constant symbol $c$ in $\mathcal{L}$ such that $T \models (\exists v \phi(v)) \rightarrow \phi(c)$.

That is, every existential statement satisfied by $T$ is witnessed by a constant.