The Principles of Mathematics (1903)

§ 52

It remains to discuss the verb, and to find marks by which it is distinguished from the adjective. In regard to verbs also, there is a twofold grammatical form corresponding to a difference in merely external relations. There is the verb in the form which it has as verb (the various inflexions of this form may be left out of account), and there is the veral noun, indicated by the infinitive or (in English) the present participle. The distinction is that between Felton killed Buckhingham nad Killing no murder. By analyzing this difference, the nature and function of the verb will appear.(§ 52 ¶ 1)

It is plain, to begin with, that the concept which occurs in the verbal noun is the very same as that which occurs as verb. this results from the previous argument, that every constituent of every proposition must, on pain of self-contradiction, be capable of being made a logical subject. If we say kills does not mean the same as to kill, we have already made kills a subject, and we cannot say that the concept expressed by the word kills cannot be made a subject. Thus the very verb which occurs as verb can occur also as subject. The question is: What logical difference is expressed by the difference of grammaticl form? And it is plain that the difference must be one of exteral relations. But in regard to verbs, there is a further point. By transforming the verb, as it occurs in a proposition, into a verbal noun, the whole proposition can be turned into a single logical subject, no longer asserted, and no longer containing in itself truth or falsehood. But here too, there seems to be no possibility of maintaining that the logical subject which results is a different entity from the proposition. Caesar died and the death of Caesar will illustrate this point. If we ask: What is asserted in the proposition Caesar died? the answer must be the death of Caesar is asserted. In that case, it would seem, it is the death of Caesar which is true or false; and yet neither truth nor falsity belongs to a mere logical subject. The answer here seems to be that the death of Caesar has an external relation to truth or falsehood (as the case may be), whereas Caesar died in some way or other contains its own truth or falsehood as an element. But if this is the correct analysis, it is difficult to see how Caesar died differs from the truth of Caesar's death in the case where it is true, or the falsehood of Caesar's death in the other case. Yet it is quite plain that the latter, at any rate, is never equivalent to Caesar died. There appears to be an ultimate notion of assertion, given by theverb, which is lost as soon as we substitute a verbal noun, and is lost when the proposition in question is made the subject of some other proposition. This does not depend upon grammatical form; for if I say Caesar died is a proposition, I do not assert that Caesar did die, and an element which is present in Caesar died has disappeared. Thus the contradiction which was to have been avoided, of an entity which cannot be made a logical subject, appears to have here become inevitable. This difficulty, which seems to be inherent in the very nature of truth and falsehood, is one with which I do not know how to deal satisfactorily. The most obvious course would be to say that the difference between an asserted and an unasserted proposition is not logical, but psychological. In the sense in which false propositions may be asserted, this is doubtless true. But there is another sense of assertion, very difficult to bring clearly before the mind, and yet quite undeniable, in which only true propositions are asserted. True and false propositions alike are in some sense entities, and are in some sense capable of being logical subjects; but when a proposition happens to be true, it has a further quality, over and above that which it shares with false propositions, and it is this further quality which is what I mean by assertion in a logical as opposed to a psychological sense. The nature of truth, however, belongs no more to the principles of mathematics than to the principles of everything else. I therefore leave this question to the logicians with the above brief indication of a difficulty.(§ 52 ¶ 2)