When we use a variable, and speak of a propositional function, say, the process of applying general statements about to particular cases will consist in substituting a name for the letter "," assuming that is a function which has individuals for its arguments. Suppose, for example, that is "always true"; let it be, say, the "law of identity," . Then we may substitute for "" any name we choose, and we shall obtain a true proposition. Assuming for the moment that "Socrates," "Plato," and "Aristotle" are names (a very rash assumption), we can infer from the law of identity that Socrates is Socrates, Plato is Plato, and Aristotle is Aristotle. But we shall commit a fallacy if we attempt to infer, without further premisses, that the author of Waverley is the author of Waverley. This results [Pg 175] from what we have just proved, that, if we substitute a name for "the author of Waverley" in a proposition, the proposition we obtain is a different one. That is to say, applying the result to our present case: If "" is a name, "" is not the same proposition as "the author of Waverley is the author of Waverley," no matter what name "" may be. Thus from the fact that all propositions of the form "" are true we cannot infer, without more ado, that the author of Waverley is the author of Waverley. In fact, propositions of the form "the so-and-so is the so-and-so" are not always true: it is necessary that the so-and-so should exist (a term which will be explained shortly). It is false that the present King of France is the present King of France, or that the round square is the round square. When we substitute a description for a name, propositional functions which are "always true" may become false, if the description describes nothing. There is no mystery in this as soon as we realise (what was proved in the preceding paragraph) that when we substitute a description the result is not a value of the propositional function in question.