Definition:Instance

This page has been identified as a candidate for refactoring of medium complexity. In particular: move to, or become supplement to, either Definition:Assignment for Formula or Definition:Value of Formula under Assignment Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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.

Sources

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