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[The following passage comes from Alan Wood’s The Passionate Sceptic. Published in 1957 it is, I think, the earliest biography of Bertrand Russell, and a very readable one.]

[Russell] showed that his predecessors had over-estimated the capacity of logic to give us knowledge about the nature of the Universe. When people ask why Russell has been described as the greatest logician since Aristotle, the conventional answer is that he showed that there were many more forms of inference than Aristotle had found. The Greek logicians tried to guard against fallacious reasoning by making a complete list that might be called working rules of all the forms of deduction which are valid. Aristotle decided that these were nearly all syllogistic: for example ‘All men are mortal, Socrates is a man, therefore Socrates is mortal.’ Russell showed how much more there was to logic, and that syllogisms should have no such pre-eminence. But this is not all. I think that if we ask why Russell was a great logician, there is another important answer which is a somewhat paradoxical one; it was because he showed how little logic can do.

‘As logic improves,’ he said, ‘less and less can be proved.’ He pointed out that it was often the mark of a man lacking in logical capacity to think that one proposition implies another when it does not: from this point of view, he once remarked that ‘Logic is the art of not drawing conclusions.’ For instance, some of Aristotle’s syllogisms were not valid as they stood. Moreover, Russell insisted that the knowledge given by logic and mathematics is all hypothetical. It tells us that if something is true, then something else is true.

The above syllogism, for instance, should have been first stated in some such form as ‘If all men are mortal, and if Socrates is a man, then Socrates is mortal.’ We must think of logic as rather like the modern electronic brains, which can solve a problem with the necessary data to work on, but which cannot produce any results unless some facts are fed into them first. Logic can only work on premises supplied independently of logic; any proof must start with some premise which is unproved. This point, once it has been put plainly, appears simple and obvious, and by no means original. But though recognized in theory as early as Aristotle, it has continually been blurred over in the history of human thought.

To begin with, there is the natural human craving for certain knowledge. We have recorded Russell’s own disappointment, at the age of eleven, when he found that Euclid gave no proof of his axioms. His brother Frank did not tell him, as he might have done: ‘You have got to start from something which you accept without proof, and you might as well start here as anywhere.’ As it happened, if Frank had said this, he would have been wrong; for not all Euclid’s axioms are beyond question, and the beginning of the deductive system can be pushed much farther back. Russell was naturally inspired to see if, by pushing it back far enough, he could arrive at something absolutely certain; and it took all the labours of Principia Mathematica, and the continuation of the same work by Gödel, to show exactly what could not be proved in the foundations of mathematics, and why not.

Russell’s philosophical predecessors, like Kant, had assumed that Euclid’s theorems gave us knowledge about the actual world. It was not realized that, like any other deductive system, Euclidean geometry could not go further than saying that if certain premises were true, then certain conclusions followed. Russell’s insistence on this point had much more originality than may appear in retrospect, because when he first worked on geometry it was assumed that actual space was in fact Euclidean; the Theory of Relativity had not yet made scientists think of it as non-Euclidean.

One common reason for not seeing that any argument in logic or pure mathematics must be hypothetical could be a strong desire to prove some emotionally satisfying belief. Thus, again and again, philosophers thought they had succeeded in using logic to prove the existence of something they wanted to believe in, in spite of the impossibility of logic proving the existence of anything; just as countless inventors kept on imagining they had solved the secret of perpetual motion, in spite of it being a scientific impossibility.

Descartes thought he had proved his own existence by saying ‘I think, therefore I am’; and he then proceeded to deduce a philosophic system from this foundation. Many philosophers thought they could prove the existence of God by the ontological argument. As late as Russell’s own time, McTaggart believed that he had arrived at a logical demonstration of personal immortality. Even philosophers who realized that logic could not prove the existence of anything directly thought it could do so indirectly, by proving that all philosophies were logically impossible except their own. An example was the way in which Bradley, like Kant and Hegel, claimed to have found contradictions in the ordinary world of Appearance.

Some of these attempted proofs by logic depended on technical errors, some on mistakes about the use of words. Others depended on assuming that something which we cannot help believing must be true; one of Russell’s most important services was to disentangle logic from psychology, and to say that logic does not mean ‘the laws of thought.’

The implications of realizing the limitations of logic have only become obvious gradually, and Russell himself took some time to realize them all.

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