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 "." We may say that "the author of Waverly" means "the value of for which ' wrote [Pg 176] Waverly' is true." Thus the proposition "the author of Waverly was Scotch," for example, involves: (1) " wrote Waverly" is not always false; (2) "if and wrote Waverly, and are identical" is always true; (3) "if wrote Waverly, was Scotch" is always true. These three propositions, translated into ordinary language, state: (1) at least one person wrote Waverly; (2) at most one person wrote Waverly; (3) whoever wrote Waverly was Scotch. All these three are implied by "the author of Waverly was Scotch." Conversely, the three together (but no two of them) imply that the author of Waverly was Scotch. Hence the three together may be taken as defining what is meant by the proposition "the author of Waverly was Scotch." We may somewhat simplify these three propositions. The first and second together are equivalent to: "There is a term such that ' wrote Waverly' is true when is and is false when is not ." In other words, "There is a term such that ' wrote Waverly' is always equivalent to ' is .'" (Two propositions are "equivalent" when both are true or both are false.) We have here, to begin with, two functions of , " wrote Waverly" and " is ," and we form a function of by considering the equivalence of these two functions of for all values of ; we then proceed to assert that the resulting function of is "sometimes true," i.e. that it is true for at least one value of . (It obviously cannot be true for more than one value of .) These two conditions together are defined as giving the meaning of "the author of Waverly exists." We may now define "the term satisfying the function exists." This is the general form of which the above is a particular case. "The author of Waverly" is "the term satisfying the function ' wrote Waverly.'" And "the so-and-so" will [Pg 177] always involve reference to some propositional function, namely, that which defines the property that makes a thing a so-and-so. Our definition is as follows:—