Lesson 5

We shall be mostly studying sentences, that is, expressions with no free variables; our interest in other expressions is essentially limited to the extent that they may be subexpressions of sentences. Now, for sentences, satisfaction by a model does not depend on the values of the variables.

This follows from the more general statement. We leave its proof as an exercise.

Lemma. Suppose that \(\phi\) is an expression, and \(M\) and \(M'\) are two models appropriate to \(\phi\)'s vocabulary such that \(M\) and \(M'\) agree on everything except for the values they assign to the variables that are not free in \(\phi\). Then \(M \models \phi\) if and only if \(M' \models \phi\).

Thus, whether a model satisfies or fails to satisfy an expression does not depend on the values assigned to variables that are bound in the expression (or fail to appear in the expression). By “model appropriate to an expression” we shall henceforth mean the part of a model that deals with the functions, the relations, and the free variables of the expression, if any.