The Problems of Philosophy Bertrand Russell (chrome ebook reader .TXT) 📖

(This is obvious, because if all men are mortal, so is Socrates; but if Socrates is mortal, it does not follow that all men are mortal.) Hence we shall reach the conclusion that Socrates is mortal with a greater approach to certainty if we make our argument purely inductive than if we go by way of “all men are mortal” and then use deduction.

This illustrates the difference between general propositions known a priori such as “two and two are four,” and empirical generalizations such as “all men are mortal.” In regard to the former, deduction is the right mode of argument, whereas in regard to the latter, induction is always theoretically preferable, and warrants a greater confidence in the truth of our conclusion, because all empirical generalizations are more uncertain than the instances of them.

We have now seen that there are propositions known a priori, and that among them are the propositions of logic and pure mathematics, as well as the fundamental propositions of ethics. The question which must next occupy us is this: How is it possible that there should be such knowledge? And more particularly, how can there be knowledge of general propositions in cases where we have not examined all the instances, and indeed never can examine them all, because their number is infinite? These questions, which were first brought prominently forward by the German philosopher Kant (1724⁠–⁠1804), are very difficult, and historically very important.

VIII How a Priori Knowledge Is Possible

Immanuel Kant is generally regarded as the greatest of the modern philosophers. Though he lived through the Seven Years War and the French Revolution, he never interrupted his teaching of philosophy at Königsberg in East Prussia. His most distinctive contribution was the invention of what he called the “critical” philosophy, which, assuming as a datum that there is knowledge of various kinds, inquired how such knowledge comes to be possible, and deduced, from the answer to this inquiry, many metaphysical results as to the nature of the world. Whether these results were valid may well be doubted. But Kant undoubtedly deserves credit for two things: first, for having perceived that we have a priori knowledge which is not purely “analytic,” i.e. such that the opposite would be self-contradictory, and secondly, for having made evident the philosophical importance of the theory of knowledge.

Before the time of Kant, it was generally held that whatever knowledge was a priori must be “analytic.” What this word means will be best illustrated by examples. If I say, “A bald man is a man,” “A plane figure is a figure,” “A bad poet is a poet,” I make a purely analytic judgement: the subject spoken about is given as having at least two properties, of which one is singled out to be asserted of it. Such propositions as the above are trivial, and would never be enunciated in real life except by an orator preparing the way for a piece of sophistry. They are called “analytic” because the predicate is obtained by merely analysing the subject. Before the time of Kant it was thought that all judgements of which we could be certain a priori were of this kind: that in all of them there was a predicate which was only part of the subject of which it was asserted. If this were so, we should be involved in a definite contradiction if we attempted to deny anything that could be known a priori. “A bald man is not bald” would assert and deny baldness of the same man, and would therefore contradict itself. Thus according to the philosophers before Kant, the law of contradiction, which asserts that nothing can at the same time have and not have a certain property, sufficed to establish the truth of all a priori knowledge.

Hume (1711⁠–⁠76), who preceded Kant, accepting the usual view as to what makes knowledge a priori, discovered that, in many cases which had previously been supposed analytic, and notably in the case of cause and effect, the connection was really synthetic. Before Hume, rationalists at least had supposed that the effect could be logically deduced from the cause, if only we had sufficient knowledge. Hume argued⁠—correctly, as would now be generally admitted⁠—that this could not be done. Hence he inferred the far more doubtful proposition that nothing could be known a priori about the connection of cause and effect. Kant, who had been educated in the rationalist tradition, was much perturbed by Hume’s scepticism, and endeavoured to find an answer to it. He perceived that not only the connection of cause and effect, but all the propositions of arithmetic and geometry, are “synthetic,” i.e. not analytic: in all these propositions, no analysis of the subject will reveal the predicate. His stock instance was the proposition 7 + 5 = 12. He pointed out, quite truly, that 7 and 5 have to be put together to give 12: the idea of 12 is not contained in them, nor even in the idea of adding them together. Thus he was led to the conclusion that all pure mathematics, though a priori, is synthetic; and this conclusion raised a new problem of which he endeavoured to find the solution.

The question which Kant put at the beginning of his philosophy, namely “How is pure mathematics possible?” is an interesting and difficult one, to which every philosophy which is not purely sceptical must find some answer. The answer of the pure empiricists, that our mathematical knowledge is derived by induction from particular instances, we have already seen to be inadequate, for two reasons: first, that the validity of the inductive principle itself cannot be proved by induction; secondly, that the general propositions of mathematics, such as “two and two always make four,” can obviously be known with certainty by consideration of a single instance, and gain nothing by