The subject of the present chapter must not be confused with the assertion of ordinary life. Commonly, an unasserted proposition is synonymous with a probably false statement, while an asserted proposition is synonymous with one that is certainly false. But in logic we apply assertion also to true propositions, and, as Lewis Carroll showed in his version of “What the Tortoise said to Achilles,”[44] usually pass over unconsciously an infinite series of implications in so doing. If p and q are propositions, p is true, and p implies q, then, at first sight, one would think that one might assert q. But, from (A) p is true, and (B) p implies q, we must, in order to deduce (Z) q is true, accept the hypothetical: (C) If A and B are true, Z must be true. And then, in order to deduce Z from A, B, and C, we must accept another hypothetical: (D) If A, B, and C are true, Z must be true; and so on ad infinitum. Thus, in deducing Z, we pass over an infinite series of hypotheticals which increase in complexity. Thus we need a new principle to be able to assert q.

Frege was the first logician sharply to distinguish between an asserted proposition, like “A is greater than B,” and one which is merely considered, like “A’s being greater than B,” although an analogous distinction had been made in our common discourse on certain psychological grounds, for long previously. In fact, soon after the invention of speech, the necessity of distinguishing between a considered proposition and an asserted one became evident, on account of the state of things referred to at the beginning of this chapter.

CHAPTER XVI