410 Mb DE morgan, ON THE THEORY OP 



we mean that we carry r.ith us into the field of practice a true knowledge of equal facility of 

 positive and negative error, as to what its effects will be. What use we shall be able to make 

 of this knowledge experience alone can tell us: theory has nothing to do with the answer to 

 this question. 



Every part of exact science has a defined foundation, upon which it is the condition of 

 science that the superstructure shall entirely rest. The theory of probabilities postulates for 

 its foundation a + b equally probable — or to our minds similarly situated — cases, of which a 

 favour one event, and b the alternative. Assent cannot be claimed to any fraction as express- 

 ing a probability, unless this derivation of its terms can be substantiated. Nevertheless, as in 

 other branches of science, we find in ourselves a certain amount of rude, but not very inaccu- 

 rate, knowledge of those details which it is our business to deduce from first principles. 

 Geometry, for instance, does not give us more confidence in the proportion of the diagonals of 

 squares to their sides than we began with. But the mischief of natural knowledge is this : 

 with full confidence in a great deal of truth, we have also full confidence in a great deal of 

 falsehood. Many persons begin by believing that doubling the side of a square doubles the 

 area as well as the diagonal. And if we be liable to such mistake in judging of space, a 

 matter in which our most unbiassed thoughts and our keenest perceptions keep watch upon 

 one another, we must needs be in still greater danger in a subject of comparatively rough and 

 infrequent experiment, in which the instruments of the mind have been trained under bias 

 both of prejudice and self-love. The greatest stumblingblocks lie in the way when the argu- 

 ment is from the finite to the infinite, or from the infinite to the finite. I shall take an 

 example of each. 



From a sack of white and black beans, mixed and shaken, we take out a score, and find 

 13 white and 7 black. We naturally conclude that the sorts are in the proportion of 13 to 7, 

 or not far from it ; and also that we can have no reason to declare against that proportion on 

 one side rather than the other. Is it not just as likely beforehand that the selected portion 

 should belie the general average by excess as by defect .'' Before this is granted, we are 

 tempted to recal the story of a Cambridge professor whom some living persons remember, who 

 is said to have sturdily refused to concede that the whole is greater than its part until he saw 

 what use his opponent would make of the concession. Let a person be required to stake upon 

 his own statement of the proportion in such manner that the nearer he is to the truth the 

 more he is to receive. He will do wisely to name 13 to 7. What odds then shall he offer 

 that the next bean drawn is white.'' Surely, it will be answered, IS to 7: nevertheless, this 

 answer is wrong; he ought to offer 14 to 8. 



Next, let A and B be arranged in every possible order, in infinite sequence, but so that in 

 the long run A shall occur five times as often as B : that is, let the unending succession 

 AAAAAB, AAAAAB, &c. be made to take every possible variety of arrangement. Let an 

 arrangement be drawn at hazard ; what is the probability that its first letter shall be A. 

 Any one can see, if he take the point of view, that we merely ask, on the supposition that in 

 the long run A occurs five times as often as B, what is the chance of drawing A at the first 

 trial. And the true answer is, five to one in favour of A. But how are we to meet the fol- 

 lowing reasoning.? Let every one of the arrangements be made to have a duplicate; no two 

 of the original arrangements being entirely alike. Of each related pair let one be headed 



