However, where statistical evidence is obtainable, no one dreams of estimating probability by the quantity of his belief.  Insurance offices, dealing with fire, shipwreck, death, accident, etc., prepare elaborate statistics of these events, and regulate their rates accordingly.  Apart from statistics, at what rate ought the lives of men aged 40 to be insured, in order to leave a profit of 5 per cent. upon L1000 payable at each man’s death?  Is ‘quantity of belief’ a sufficient basis for doing this sum?

Sec. 4.  The ground of probability is experience, then, and, whenever possible, statistics; which are a kind of induction.  It has indeed been urged that induction is itself based upon probability; that the subtlety, complexity and secrecy of nature are such, that we are never quite sure that we fully know even what we have observed; and that, as for laws, the conditions of the universe at large may at any moment be completely changed; so that all imperfect inductions, including the law of causation itself, are only probable.  But, clearly, this doctrine turns upon another ambiguity in the word ‘probable.’  It may be used in the sense of ‘less than absolutely certain’; and such doubtless is the condition of all human knowledge, in comparison with the comprehensive intuition of arch-angels:  or it may mean ’less than certain according to our standard of certainty,’ that is, in comparison with the law of causation and its derivatives.

We may suppose some one to object that “by this relative standard even empirical laws cannot be called ‘only probable’ as long as we ’know no exception to them’; for that is all that can be said for the boasted law of causation; and that, accordingly, we can frame no fraction to represent their probability.  That ‘all swans are white’ was at one time, from this point of view, not probable but certain; though we now know it to be false.  It would have been an indecorum to call it only probable as long as no other-coloured swan had been discovered; not merely because the quantity of belief amounted to certainty, but because the number of events (seeing a swan) and the number of their happenings in a certain way (being white) were equal, and therefore the evidence amounted to 1 or certainty.”  But, in fact, such an empirical law is only probable; and the estimate of its probability must be based on the number of times that similar laws have been found liable to exceptions.  Albinism is of frequent occurrence; and it is common to find closely allied varieties of animals differing in colour.  Had the evidence been duly weighed, it could never have seemed more than probable that ‘all swans are white.’  But what law, approaching the comprehensiveness of the law of causation, presents any exceptions?