When descriptions occur in propositions, it is necessary to distinguish what may be called "primary" and "secondary" occurrences. The abstract distinction is as follows. A description has a "primary" occurrence when the proposition in which it occurs results from substituting the description for "" in some propositional function ; a description has a "secondary" occurrence when the result of substituting the description for in gives only part of the proposition concerned. An instance will make this clearer. Consider "the present King of France is bald." Here "the present King of France" has a primary occurrence, and the proposition is false. Every proposition in which a description which describes nothing has a primary occurrence is false. But now consider "the present King of France is not bald." This is ambiguous. If we are first to take " is bald," then substitute "the present King of France" for "" and then deny the result, the occurrence of "the present King of France" is secondary and our proposition is true; but if we are to take " is not bald" and substitute "the present King of France" for "" then "the present King of France" has a primary occurrence and the proposition is false. Confusion of primary and secondary occurrences is a ready source of fallacies where descriptions are concerned. [Pg 179] Descriptions occur in mathematics chiefly in the form of descriptive functions, i.e. "the term having the relation to ," or "the of " as we may say, on the analogy of "the father of " and similar phrases. To say "the father of is rich," for example, is to say that the following propositional function of : " is rich, and ' begat ' is always equivalent to 'is ,'" is "sometimes true," i.e. is true for at least one value of . It obviously cannot be true for more than one value. The theory of descriptions, briefly outlined in the present chapter, is of the utmost importance both in logic and in theory of knowledge. But for purposes of mathematics, the more philosophical parts of the theory are not essential, and have therefore been omitted in the above account, which has confined itself to the barest mathematical requisites. [Pg 180] CHAPTER XVII CLASSES