" 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.
We are now in a position to define propositions in which a definite description occurs. The only thing that distinguishes "the so-and-so" from "a so-and-so" is the implication of uniqueness. We cannot speak of "the inhabitant of London," because inhabiting London is an attribute which is not unique. We cannot speak about "the present King of France," because there is none; but we can speak about "the present King of England." Thus propositions about "the so-and-so" always imply the corresponding propositions about "a so-and-so," with the addendum that there is not more than one so-and-so. Such a proposition as "Scott is the author of Waverly" could not be true if Waverly had never been written, or if several people had written it; and no more could any other proposition resulting from a propositional function
by the substitution of "the author of Waverly" for "