Another set of notions as to which philosophy has allowed itself to fall into hopeless confusions through not sufficiently separating propositions and propositional functions are the notions of "modality": necessary, possible, and impossible. (Sometimes contingent or assertoric is used instead of possible.) The traditional view was that, among true propositions, some were necessary, while others were merely contingent or assertoric; while among false propositions some were impossible, namely, those whose contradictories were necessary, while others merely happened not to be true. In fact, however, there was never any clear account of what was added to truth by the conception of necessity. In the case of propositional functions, the three-fold division is obvious. If "" is an undetermined value of a certain propositional function, it will be necessary if the function is always true, possible if it is sometimes true, and impossible if it is never true. This sort of situation arises in regard to probability, for example. Suppose a ball is drawn from a bag which contains a number of balls: if all the balls are white, " is white" is necessary; if some are white, it is possible; if none, it is impossible. Here all that is known about is that it satisfies a certain propositional function, namely, " was a ball in the bag." This is a situation which is general in probability problems and not uncommon in practical life—e.g. when a person calls of whom we know nothing except that he brings a letter of introduction from our friend so-and-so. In all such [Pg 165] cases, as in regard to modality in general, the propositional function is relevant. For clear thinking, in many very diverse directions, the habit of keeping propositional functions sharply separated from propositions is of the utmost importance, and the failure to do so in the past has been a disgrace to philosophy. [Pg 166] CHAPTER XVI DESCRIPTIONS We dealt in the preceding chapter with the words all and some; in this chapter we shall consider the word the in the singular, and in the next chapter we shall consider the word the in the plural. It may be thought excessive to devote two chapters to one word, but to the philosophical mathematician it is a word of very great importance: like Browning's Grammarian with the enclitic , I would give the doctrine of this word if I were "dead from the waist down" and not merely in a prison. We have already had occasion to mention "descriptive functions," i.e. such expressions as "the father of " or "the sine of ." These are to be defined by first defining "descriptions." A "description" may be of two sorts, definite and indefinite (or ambiguous). An indefinite description is a phrase of the form "a so-and-so," and a definite description is a phrase of the form "the so-and-so" (in the singular). Let us begin with the former.