Certain modern logicians, such as Frege, have found it necessary so to extend the meaning of implication of q by p that it holds when p is not a proposition at all. Hitherto, politicians, finding that either identical or false propositions are sufficient for their needs, have made no use of this principle; but it is obvious that their stock of arguments would be vastly increased thereby.
Logical implication is often an enemy of dignity and eloquence. De Morgan[54] relates “a tradition of a Cambridge professor who was once asked in a mathematical discussion, ‘I suppose you will admit that the whole is greater than its part?’ and who answered, ‘Not I, until I see what use you are going to make of it.’” And the care displayed by cautious mathematicians like Poincaré, Schoenflies, Borel, Hobson, and Baire in abstaining from pushing their arguments to their logical conclusions is probably founded on the unconscious—but no less well-grounded—fear of appearing ridiculous if they dealt with such extreme cases as “the series of all ordinal numbers.”[55] They are, probably, as unconscious of implication as Gibbon, when he remarked that he always had a copy of Horace in his pocket, and often in his hand, was of the necessary implication of these propositions that his hand was sometimes in his pocket.
[51] Md., N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W. E. Johnson (ibid., p. 583) and Russell (P. M., p. 18, note, and Md., N. S., vol. xiv., 1905, pp. 400-1).
[52] The inhabitants of “Erewhon” punished invalids more severely than criminals. In modern times, one frequently hears the statement that crime is a disease; and if so, it is surely false that criminals ought to be punished.
[53] Irrelevant in a popular sense; one would not say, speaking loosely, that the fact that Brutus killed Cæsar implies that the sea is salt; and yet this conclusion is implied both by the above premiss, and the premiss that Cæsar killed Brutus. Cf. on such questions Venn, S. L., 2nd ed., pp. 240-4.
[54] F. L., p. 264.
[55] Cf. Chapters XXIX and XXXVII.