§5. The Meaning of an Argument
175. We have already seen 1) some reason to hold that the idea of meaning is such as to involve some reference to a purpose. But Meaning is attributed to representamens alone, and the only kind of representamen which has a definite professed purpose is an »argument.« The professed purpose of an argument is to determine an acceptance of its conclusion, and it quite accords with general usage to call the conclusion of an argument its meaning. But I may remark that the word meaning has not hitherto been recognized as a technical term of logic, and in proposing it as such (which I have a right to do since I have a new conception to express, that of the conclusion of an argument as its intended interpretant) I should have a recognized right slightly to warp the acceptation of the word »meaning,« so as to fit it for the expression of a scientific conception. It seems natural to use the word meaning to denote the intended interpretant of a symbol.
176. I may presume that you are all familiar with Kant's reiterated insistence that necessary reasoning does nothing but explicate the meaning of its premisses.2) Now Kant's conception of the nature of necessary reasoning is clearly shown by the logic of relations to be utterly mistaken, and his distinction between analytic and synthetic judgments, which he otherwise and better terms explicatory ( erläuternde) and ampliative ( erweiternde) judgments, which is based on that conception, is so utterly confused that it is difficult or impossible to do anything with it. But, nevertheless, I think we shall do very well to accept Kant's dictum that necessary reasoning is merely explicatory of the meaning of the terms of the premisses, only reversing the use to be made of it. Namely instead of adopting the conception of meaning from the Wolffian logicians, as he does, and making use of this dictum to express what necessary reasoning can do, about which he was utterly mistaken, we shall do well to understand necessary reasoning as mathematics and the logic of relations compels us to understand it, and to use the dictum, that necessary reasoning only explicates the meanings of the terms of the premisses, to fix our ideas as to what we shall understand by the meaning of a term.
177. Kant and the logicians with whose writings he was alone acquainted — he was far from being a thorough student of logic, notwithstanding his great natural power as a logician — consistently neglected the logic of relations; and the consequence was that the only account they were in condition to give of the meaning of a term, its »signification« as they called it, was that it was composed of all the terms which could be essentially predicated of that term. Consequently, either the analysis of the signification must be capable of [being] pushed on further and further, without limit — an opinion which Kant 1) expresses in a well-known passage but which he did not develop, or, what was more usual, one ultimately reached certain absolutely simple conceptions such as Being, Quality, Relation, Agency, Freedom, etc., which were regarded as absolutely incapable of definition and of being in the highest degree luminous and clear. It is marvellous what a following this opinion, that those excessively abstracted conceptions were in themselves in the highest degree simple and facile, obtained, notwithstanding its repugnancy to good sense. One of the many important services which the logic of relations has rendered has been that of showing that these so-called simple conceptions, notwithstanding their being unaffected by the particular kind of combination recognized in non-relative logic, are nevertheless capable of analysis in consequence of their implying various modes of relationship. For example, no conceptions are simpler than those of Firstness, Secondness, and Thirdness; but this has not prevented my defining them, and that in a most effective manner, since all the assertions I have made concerning them have been deduced from those definitions.
178. Another effect of the neglect of the logic of relations was that Kant imagined that all necessary reasoning was of the type of a syllogism in Barbara. Nothing could be more ridiculously in conflict with well-known facts.2) For had that been the case, any person with a good logical head would be able instantly to see whether a given conclusion followed from given premisses or not; and moreover the number of conclusions from a small number of premisses would be very moderate. Now it is true that when Kant wrote, Legendre and Gauss had not shown what a countless multitude of theorems are deducible from the very few premisses of arithmetic. I suppose we must excuse him, therefore, for not knowing this. But it is difficult to understand what the state of mind on this point could have been of logicians who were at the same time mathematicians, such as Euler, Lambert, and Ploucquet. Euler invented the logical diagrams which go under his name; for the claims that have been made in favor of predecessors may be set down as baseless;1) and Lambert used an equivalent system.2) Now I need not say that both of these men were mathematicians of great power. One is simply astounded that they should seem to say that all the reasonings of mathematics could be represented in any such ways. One may suppose that Euler never paid much attention to logic. But Lambert wrote a large book in two volumes on the subject, and a pretty superficial affair it is. One has a difficulty in realizing that the author of it was the same man who came so near to the discovery of the non-Euclidean geometry. The logic of relatives is now able to exhibit in strict logical form the reasoning of mathematics. You will find an example of it — although too simple a one to put all the features into prominence — in that chapter 3) of Schröder's logic in which he remodels the reasoning of Dedekind in his brochure Was sind und was sollen die Zahlen; and if it be objected that this analysis was chiefly the work of Dedekind who did not employ the machinery of the logic of relations, I reply that Dedekind's whole book is nothing but an elaboration of a paper published by me several years previously in the American Journal of Mathematics4) which paper was the direct result of my logical studies. These analyses show that although most of the steps of the reasoning have considerable resemblance to Barbara, yet the difference of effect is very great indeed.
179. On the whole, then, if by the meaning of a term, proposition, or argument, we understand the entire general intended interpretant, then the meaning of an argument is explicit. It is its conclusion; while the meaning of a proposition or term is all that that proposition or term could contribute to the conclusion of a demonstrative argument. But while this analysis will be found useful, it is by no means sufficient to cut off all nonsense or to enable us to judge of the maxim of pragmatism. What we need is an account of the ultimate meaning of a term. To this problem we have to address ourselves.