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. FOOTNOTES: [44] _Md._ N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, _P. M._, p. 35. CHAPTER XVI THE COMMUTATIVE LAW Often the meaning of a sentence tacitly implies that the commutative law does not hold. We are all familiar with the passage in which Macaulay pointed out that, by using the commutative law because of exigencies of metre, Robert Montgomery unintentionally made Creation tremble at the Atheist's nod instead of the Almighty's. This use of the commutative law by writers of verse renders it doubtful whether, in the hymn-line: The humble poor believe, we are to understand a statement about the humble poor, or a doubtful maxim as to the attitude of our minds to statements made by the humble poor. The non-commutativity of English titles offers difficulties to some novelists and Americans who refer to Mary Lady So-and-So as Lady Mary So-and-So, and _vice versa_. CHAPTER XVII UNIVERSAL AND PARTICULAR PROPOSITIONS