But this formal relation is only required in order that we may be able to know that either the premiss is false or the conclusion is true. It is the truth of "not- or " that is required for the validity of the inference; what is required further is only required for the practical feasibility of the inference. Professor C. I. Lewis[37] has especially studied the narrower, formal relation which we may call "formal deducibility." He urges that the wider relation, that expressed by "not- or " should not be called "implication." That is, however, a matter of words. [Pg 153] Provided our use of words is consistent, it matters little how we define them. The essential point of difference between the theory which I advocate and the theory advocated by Professor Lewis is this: He maintains that, when one proposition is "formally deducible" from another , the relation which we perceive between them is one which he calls "strict implication," which is not the relation expressed by "not- or " but a narrower relation, holding only when there are certain formal connections between and . I maintain that, whether or not there be such a relation as he speaks of, it is in any case one that mathematics does not need, and therefore one that, on general grounds of economy, ought not to be admitted into our apparatus of fundamental notions; that, whenever the relation of "formal deducibility" holds between two propositions, it is the case that we can see that either the first is false or the second true, and that nothing beyond this fact is necessary to be admitted into our premisses; and that, finally, the reasons of detail which Professor Lewis adduces against the view which I advocate can all be met in detail, and depend for their plausibility upon a covert and unconscious assumption of the point of view which I reject. I conclude, therefore, that there is no need to admit as a fundamental notion any form of implication not expressible as a truth-function. [37]See Mind, vol. XXI., 1912, pp. 522-531; and vol. XXIII., 1914, pp. 240-247. [Pg 154] CHAPTER XV PROPOSITIONAL FUNCTIONS WHEN, in the preceding chapter, we were discussing propositions, we did not attempt to give a definition of the word "proposition." But although the word cannot be formally defined, it is necessary to say something as to its meaning, in order to avoid the very common confusion with "propositional functions," which are to be the topic of the present chapter.