Having studied logical implication and rules of inference, we now move from whole statements to statements containing variables. Open statements and universe of discourse form the entry point to predicate logic, because a sentence such as $x+2$ is an even integer cannot be judged true or false until $x$ is chosen. The same expression may behave differently when the allowed values of the variable change. In this lesson, open statements and universe of discourse will be used to assign truth values, form truth sets, and evaluate compound open statements.

A declarative sentence containing one or more variables is called an \textbf{open statement} if its truth value cannot be determined until values are assigned to its variables. If a value from the allowed set is substituted for each variable, then the open statement becomes a statement and has a definite truth value.