As many of you know, two years after Kant published the Critique of Pure Reason in 1781, he published a short work intended to help puzzled readers make their way into the difficult arguments of that monumental work. It was called Prolegomena to Any Future Metaphysics, and Kant hoped that the essay would make his arguments sufficiently accessible to encourage the German-speaking philosophical community to do the hard work of mastering his dramatically revolutionary doctrines. The brief work was not intended in any sense as a restatement of the proofs Kant had given in the Critique, but was simply a way of acquainting his readers with his new teaching so that they could make more sense of the Critique itself.
Accordingly, in the Prolegomena Kant assumed as true his central conclusion, that we can a priori have scientific and mathematical knowledge that is expressed in synthetic judgments, and then asked how that is possible, without in any way actually attempting to prove that it is. This was of course not intended as any sort of answer to Hume's scepticism, for although "yes we can" may be a powerful political slogan, it is not much of a philosophical argument.
Unfortunately for subsequent generations of puzzled readers, Kant was so taken with the formula he devised in the Prolegomena for posing that work's problem ["How is mathematics possible? How is natural science possible? etc.] that he lifted it from its original source, where it was quite appropriate, and stuck it wholesale into the revised Introduction to the second edition of the Critique, published in 1787.
In the Prolegomena, Kant had described himself as using the analytical or regressive method of exposition, ascending from his conclusion, which was what was to be proved, to the conditions or premises from which the conclusion follows. Now, any beginning student of logic knows that from the fact the p entails q, and q is true, it does not follow that p is true. p might quite well be false and yet entail q. [For example, from the premises Donald Trump is a New Yorker and All New Yorkers are despicable, it follows necessarily that Donald Trump is despicable. The conclusion is true but at least one of the premises is false.] Thus, even if we assume the conclusion, q, and show that it follows from p, that lends no weight whatsoever to p. So even if his audience was willing, at least provisionally, to grant that q, and was willing to grant as well that q follows from p, that by itself would give a reader no reason to suppose that p is true. Kant knew all of this, of course, so why on earth would he confuse his readers and undermine the cogency of his Introduction by introducing this "analytical or regressive" mode of exposition where it had no business being?
Well, there is in fact one circumstance under which the fact that p entails q and that q is true actually constitutes an argument for p, namely if q also entails p, which is to say, if p is a necessary condition of q. For somewhat counterintuitively, if p is the necessary condition of q, then that amounts to saying that q entails p. And that is the same as saying that p is the only condition [the conditio sine qua non] of q.
Now, I had a memory from long, long ago that somewhere in the Prolegomena Kant actually inserts the magical word "only," thereby making what he was saying true, however misleading it might be. So a few minutes ago, I took out my tattered copy of the Prolegomena, read and re-read fifty years and more ago until its cover fell off, and went looking. Within moments, I found the essential word, just where it ought to be, in a long footnote to subsection 5.
It was rather like finding the face of an old friend in a crowd.