# Definition:Instance

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## 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 $\mathcal M$ be a structure for predicate logic of type $\mathcal P$ 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:

- $\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right)$

Thus $\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right) \in SENT \left({\mathcal P, M}\right)$.

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

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.4$