404 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM, 



(l- 6)(l-c)...M ( l-»')(l-a .-. (l-a)(l-c). . .(l-/x);-(l-0... (l-a)(l-6)...(l-;u)(l-.>)^... 



This problem contains all that have gone before, except the second. But this may not be appa- 

 rent at first. In fact, if I had commenced this paper with the general case now in hand, and had 

 then descended to the particular cases, the method of descending might have appeared exceptionable, 

 requiring tlie authority of an independent consideration of the particular results arrived at. Suppose 

 a dilemma of two horns, such as a proposition and its contradiction. If the testimony of authority 

 for the proposition be ju, there is in this case the testimony 1 - /x implied for the contradiction. But 

 this does not enter the formula : it is only the form belonging to the case of what is virtually repre- 

 sented in the general formula above, namely, that there is the testimony I — n implied in favour of 

 one or other of the horns following the first, because there is the testimony fi given for the first. 

 No express testimony is given to the contradiction : so that it enters with the testimony i. And if 

 there be only two horns, and the testimonies be fx and ^, it will be found that the preceding expres- 

 sions agree with the answer to Problem 4. There was no need in that case to suppose testimonies ^ 

 and 1/, because, as the testimony to each liorn is a definite testimony to the other, they would but 

 have amounted to a joint testimony for the proposition. 



If we want the case of the last problem, we have to take three horns, making c = and ^ = ^■ 

 Or we may if we like suppose argument and testimony offered for the third case, namely, that both 

 the proposition and its contrary are false. 



If we wish to construct the general case upon the supposition that no one need be true, all we 

 have to do is to add one more horn with an argument and a testimony i. 



The easiest way of representing the result of the general case is as follows. Let A„ represent 

 the probability of the m"' horn from argument only, and M,„ the same from authority only. We 

 have then (using Oj a„ he. and /u, ix^ S:c.), 



I 



M„ = 





1 - a,n 



A M 



and the probability of the to"" horn is „ , "' "' , . 



2 (A^ M,„) 



The term A,„M„, or "' may be called the exponent of probability of the ;»"' case : 



1 - Cm 1 - M,„ 



and the probability of that case is its exponent divided by the sum of all the exponents. This 

 exponent is proportional to the number of balls in the urn the exits of which are favourable to the 

 case. It is the product of two relative testimonies, that of the authority, and that of the argument 

 alone, to establish the conclusion against its contradictory, that is against everything opposed to it. 



Now suppose a complex dilemma of this kind, namely, that to of the horns, neither more nor less, 

 must be true, and the rest false. An examination of this problem leads to the following result. 

 The product of the exponents of any m cases, divided by the sum of the products of all the 

 exponents, m and m together, is the probability that the to cases chosen are the true ones. Hence 

 can be readily found the probability that any one case is among the true ones. If there be four 

 cases, for instance, of which two must be true, and if ei, e.~, e^, e,, be the exponents, the probability 

 that the first case is true is 



61(^2 + 6;, + ej 



e,e2 + 6163 + 6,64 -I- 6263 + 6264 + 636, 



If it should be that m cases or fewer, but not more, may be true, then the probability that any 

 m - p cases and no others, shall be true, is the product of the exponents of those m — p cases, divided 



