Definition:Witness Property

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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:

$T \models (\exists v \phi(v)) \to \phi(c)$

that is, $(\exists v \phi(v)) \to \phi(c)$ is a semantic consequence of $T$.

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