Introduction to Mathematical Philosophy

satisfies " is defined as meaning: "There is a term such that (1) is always equivalent to ' is ,' (2) is true." This is the definition of propositions in which descriptions occur. It is possible to have much knowledge concerning a term described, i.e. to know many propositions concerning "the so-and-so," without actually knowing what the so-and-so is, i.e. without knowing any proposition of the form " is the so-and-so," where "" is a name. In a detective story propositions about "the man who did the deed" are accumulated, in the hope that ultimately they will suffice to demonstrate that it was who did the deed. We may even go so far as to say that, in all such knowledge as can be expressed in words—with the exception of "this" and "that" and a few other words of which the meaning varies on different occasions—no names, in the strict sense, occur, but what seem like names are really descriptions. We may inquire significantly whether Homer existed, which we could not do if "Homer" were a name. The proposition "the so-and-so exists" is significant, whether true or false; but if is the so-and-so (where "" is a name), the words " exists" are meaningless. It is only of descriptions—definite [Pg 178] or indefinite—that existence can be significantly asserted; for, if "" is a name, it must name something: what does not name anything is not a name, and therefore, if intended to be a name, is a symbol devoid of meaning, whereas a description, like "the present King of France," does not become incapable of occurring significantly merely on the ground that it describes nothing, the reason being that it is a complex symbol, of which the meaning is derived from that of its constituent symbols. And so, when we ask whether Homer existed, we are using the word "Homer" as an abbreviated description: we may replace it by (say) "the author of the Iliad and the Odyssey." The same considerations apply to almost all uses of what look like proper names. 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]