is that of mere tautologies, they are true, because they assert nothing of any fact, they are purely analytic. If a man tells me that he owns 7+5 acres of land, and I say to him: "ah! — you own 12 acres!" I have not told him anything new (even he should not happen to be able to add 5 to 7). I have simply repeated his own statement in different words. "5 + 7 = 12" is no proposition at all, it is a rule, which permits us to transform a proposition in which the signs 5+7 occur into an equivalent one in which the sign 12 occurs. It is a rule about the use of signs and therefore does not depend on any experience, but only on the arbitrary definitions of the signs. An arithmetical formula never expresses a real fact, but it is always applicable to real facts in the sense that it is applicable to propositions which express real facts by means of numbers, as is shown in the above example. (Another example : the arithmetical rule 1+1+1=3 teaches me that the proposition "he called me once, and once more, and once more" has the same meaning as the proposition "he called me three times".)
I repeat: arithmetical rules have tautological character; they do not express any knowledge in the sense in which we used this term. The same is true of all logical rules (no matter wether arithmetic is just a part of logic — as Bertrand Russell will have it — or not); it really would have been quite consistent of Kant if he had declared the logical principles (e.g. the Law of Contradiction) to be synthetic and a priori propositions; but it is evidently due to his sound instinct that such a nonsensical idea never occurred to him. In reality the logical principles are no propositions either, they do not express any knowledge, but are rules for the transformation of propositions into one another. A deductive inferinference is nothing but such a purely analytical transformation.
The application of logic to reality consists in its application to propositions about reality — but in applying the logical rules in this way we are not asserting anything about reality. I may, for instance, consider it an application of the Law of the Excluded Middle when I say, "to-morrow it will either rain or not rain"; here I have made a statement which is, undoubtedly absolutely true, and it appears to be a statement about a future fact. It does speak of the future, beyond doubt, but it does not assert anything about it, for evidently I know absolutely no more about to-morrow if I am told that it will either be raining or not raining, than if I had not been told anything at all.
