Definition:Instance

Definition
Let $\mathbf C$ be a plain WFF in the language of predicate logic.

Let $x_1, x_2, \ldots, x_n$ be the free variables of $\mathbf C$.

Let $\MM$ be a structure for predicate logic of type $\PP$ whose universe set is $M$.

Then an instance of $\mathbf C$ in $M$ is the sentence with parameters from $M$ formed by choosing $a_1, a_2, \ldots, a_n \in M$ and replacing all free occurrences of $x_k$ in $\mathbf C$ by $a_k$ for $k = 1, \ldots, n$.

The resulting sentence is denoted:
 * $\map {\mathbf C} {x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}$

Thus $\map {\mathbf C} {x_1, \ldots, x_n \,//\, a_1, \ldots, a_n} \in \map {SENT} {\PP, M}$.

If $\mathbf C$ is a plain sentence, then no parameters are needed, and $\mathbf C$ is already an instance of itself.