Definition:Instance

Definition
Let $$\mathbf C$$ be a plain WFF of predicate calculus.

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

Let $$\mathcal M$$ be a model for predicate calculus 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.