  As a special case of this decision problem, Hilbert considered the validity problem for first-order logic. First-order logic is a mathematical language in which most mathematical statements can be formulated. Every statement in first-order logic has a precise meaning in every appropriate logical structure, i.e., it is true or false in each such structure. Those statements that are true in every appropriate structure are called valid. Those statements that are true in some structure are called satisfiable. Notice that a formula, \(\varphi\), is valid iff its negation, \(\neg \varphi\), is not satisfiable.