" alone is not, since it asserts nothing definite unless we are further told, or led to suppose, that
and
are to have all possible values, or are to have such-and-such values. The former of these is tacitly assumed, as a rule, in the enunciation of mathematical formulæ, which thus become propositions; but if no such assumption were made, they would be "propositional functions." A "propositional function," in fact, is an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition. In other words, it is a function whose values are propositions. But this latter definition must be used with caution. A descriptive function, e.g. "the hardest proposition in
's mathematical treatise," will not be a propositional function, although its values are propositions. But in such a case the propositions are only described: in a propositional function, the values must actually enunciate propositions.
Examples of propositional functions are easy to give: "
