The Principles of Mathematics (1903)

§ 37

In the preceding chapter I endeavoured to present, briefly and uncritically, all the data, in the shape of formally fundamental ideas and propositions, that pure mathematics requires. In subsequent Parts I shall show that these are all the data by giving definitions of the various mathematical concepts--number, infinity, continuity, the various spaces of geometry, and motion. In the remainder of Part I, I shall give indications, as best I can, of the philosophical problems arising in the analysis of the data, and of the directions in which I imagine these problems to be probably soluble. Some logical notions will be elicited which, though they seem quite fundamental to logic, are not commonly discussed in works on the subject; and thus problems no longer clothed in mathematical symbolism will be presented for the consideration of philosophical logicians.(§ 37 ¶ 1)

Two kinds of implication, the material and the formal, were found to be essential in every kind of deduction. In the present chapter I wish to examine and distinguish these two kinds, and to discuss some methods of attempting to analyze the second of them.(§ 37 ¶ 2)

In the discussion of inference, it is common to permit the intrusion of a psychological element, and to consider our acquisition of new knowledge by its means. But it is plain that where we validly infer one proposition from another, we do so in virtue of a relation which holds between two propositions whether we perceive it or not: the mind, in fact, is as purely receptive in inference as common sense supposes it to be in perception of sensible objects. The relation in virtue of which it is possible for us validly to infer is what I call material implication. We have already seen that it would be a vicious circle to define this relation as meaning that if one proposition is true, then another is true, for if and then already involve implication. The relation holds, in fact, when it does hold, without any reference to the truth or falsehood of the propositions involved.(§ 37 ¶ 3)

But in developing the consequences of our assumptions as to implication, we were led to conclusions which do not by any means agree with what is commonly held concerning implication, for we found that any false proposition implies every proposition and any true proposition is implied by every proposition. Thus propositions are formally like a set of lengths each of which is one inch or two, and implication is like the relation equal to or less than among such lengths. It would certainly not be commonly maintained that 2 + 2 = 4 can be deduced from Socrates is a man, or that both are implied by Socrates is a triangle. But the reluctance to admit such implications is chiefly due, I think, to preoccupation with formal implication, hwich is a much more familiar notion, and is really before the mind, as a rule, even where material implication is what is explicitly mentioned. In inferences from Socrates is a man, it is customary not to consider the philosopher who vexed the Athenians, but to regard Socrates merely as a symbol, capable of being replaced by any other man; and only a vulgar prejudice in favour of true propositions stands in the way of replacing Socrates by a number, a table, or a plum-pudding. Nevertheless, wherever, as in Euclid, one particular proposition is deduced from another, material implication is involved, though as a rule the material implication may be regarded as a particular instance of some formal implication, obtained by giving some constant value to the variable or variables involved in the said formal implication. And although, while relations are still regarded with the awe caused by unfamiliarity, it is natural to doubt whether any such relation as implication is to be found, yet, in virtue of the general principles laid down in Section C of the preceding chapter, there must be a relation holding between nothing except propositions, and holding between any two propositions of which either the first is false or the second true. Of the various equivalent relations satisfying these conditions, one is to be called implication, and if such a notion seems unfamiliar, that does not suffice to prove that it is illusory.(§ 37 ¶ 4)