Definition:Instance

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.