PROBLEM 3.-In a + b trials, a and b not being small numbers, and nothing whatever being known except what these trials tell, it is found that r happens a times, and Q happens b times; from which it ie ex ceedingly likely that the probabilities of r and q happening at a' single trial would have been nearly as a to b., if we had known enough to form an d priori opinion. This we cannot do, by hypothesis : but, judging from the preceding events, with what degree of probability may we infer that if we had been able to form an et priori opinion, the odds for r against Q would have appeared to us to lie between a-k to b+k and a+k to b-k1 Rom-Calculate k=a/(a + a let this beA (in the table) : then the corresponding B is the probability required.
ExAmmr..-In 600 drawings from an urn, the ball being replaced after each drawing, there were 418 white balls and 182 black ones. What is the probability that the proportion of white and black balls in meant an absolute indication, certainly inexplicable except by the sup position of insanity, the answer is complete, and the opinion shown to be untenable. But suppose the energies of many acute persons and the resources of a whole nation to fail in making the motive of a crime apparent, and that this is what is meant by there being no apparent motive; suppose moreover that by an indication of insanity is meant a circumstance which renders insanity so moderately probable that the hypothesis deserves to be weighed : the answer then is wholly irrele vant. The opinion, expanded into an argument, is—a crime committed absolutely without motive or object shows insanity ; a motive, if it exist, may almost certainly be discovered by proper exertions; con sequently the appearance of no motive, after all exertions to discover one have been tried, makes it most likely • that the crime was an act of insanity ; it is in fact as likely that the crime was an act of insanity, as it was unlikely that the exertionato discover a motive should have failed, if there had been a motive.
The application of this theory to pure logic is contained in the con sideration of testimony, argument, and their combination. The following rules t will be found demonstrated in De Morgan s' Formal Logic, ch. x.
Twstirnony, or authority—which means only testimony of great force, as the word is commonly used—speaks to the truth or falsehood of the assertion. But argurnent speaks, not to the truth or false hood of the conclusion, but to the validity of the mode of establish ing it. An argument may be invalid, either by false premises or bad logic : and yet the conclusion may be true. Consequently, the proba bility of an argument being valid is a totally different thing from that of the conclusion being true.
The effect of testimony, as also of argument, must depend upon the initial state of the receiver's mind with reference to the conclusion asserted. But though the receiver himself may stand in a very different position from the witnesses who bring him testimony, yet in the mathematical formuhe he ranks but as one of those witnesses : if his initial state be that it is 999 to 1 against the conclusion, the results of the theory aro the same as if he, being unbiassed, found an addi tional witness to deny the assertion for whose correctness it is in his miod, d priori, 999 to 1. A person without any bias on his mind begins with an even chance for the assertion. And a witness of credibility- p, who asserts, ranks as a witness of credibility 1—p, who denies. The following rules are given in mathematical form, for consistency with the rest of the article : the non-mathematical reader will finch them reduced to arithmetical rules in Do Morgan's Syllabus of a Proposed System of Logic; 1860.
Let a number of witnesses, of credibilities p, ,u', &c., affirm : let others, of credibilities v, v', , deny. By the credibility of a witness we mean the belief we have of an assertion of his being correct, before wo turn our attention to the particular character of that assertion.