400 PROFESSOR DE MORGAN', ON THE STRUCTURE OF THE SYLLOGISM, 



im-alidities of the arguments for and against are in the proportion of the testimonies of authority 

 for and against, the same thing occurs; or the alteration of testimony in the above transition is 



exactly transferred to the probability of the conclusion. When lies between 1 and , 



1 — O /I 



more than the alteration of testimony is transferred ; in other cases less. The greatest transference 



is when ,r = V/ ( ]■> 'i which case the amount of probability transferred is 



2 



It appears from what precedes that in the formulae, the invalidity of the argument against, 

 1 - 6, enters for the conclusion, and the invalidity of the argument for, 1 — a, enters against the 

 conclusion, precisely in the same manner as the testimony for it, fi, and that against it, 1 - «. If 



then we call the testimony of ateument fur the conclusion, and that 



I_6+l-a V J 6 J l_fc+l_„ 



against it, just as we call /j. and 1 — /i the testimonies of authority for and against : and if also we call 



the relative testimony of the arguments : then we may express the result of Problem 4 by 



saying that the joint relative testimony of the combined arguments and authorities is the product of 

 all the separate relative testimonies, both of arguments and authorities. 



It must be observed that the mode of entrance of the testimonies of argument makes it follow 

 that if, after obtaining a result from certain arguments and authorities, we use the probability 

 obtained as a new authority, in combination with additional data, — the final result will be the same 

 as if we had collected all the arguments separately and all the authorities, and then proceeded as in 

 Problem 4. This follows from the property of the functions p-~(,p + p) and p'-i-{p + p), which 

 contain a mode of composition in which the order of the processes is indifferent, and their partial 

 collection allowable. If we denote the preceding functions by [p] and [p'], we have 



[ [P] M J = [P9]> [ [P?]"-] = [??'■] &c. 

 When there are any number of arguments for, of validities a, a', a", &c., the chance that one or 

 more are valid is 1 - (1 - a) (1 - a) (1 - n") ..., and the testimony of argument against the conclu- 

 sion is (1 - a) (l - a') (1 - a") divided by (1 - a) (1 - a') ... + (l - b) (I - 6') + ... Hence, the 

 arguments against having the validities 6, h', &c., and the authorities for and against being n, 

 ft', &c., and 1 — /I, 1 — iM, &c., and J being the probability for the conclusion derived from the 

 whole of the data, the principle of relative testimonies may be expressed thus : 



A 1 - b J - b' \ - b" M /u' m" 



1 — A I — a 1 — a' I — a" I — /i 1 - ft 1 — im" 



or as follows ; — let the probabilities of the conclusion, derived from the several arguments backed 

 by no authority, be considered as testimonies of authority to the conclusion, and used as in Pro- 

 blem 1. 



It may happen that, besides the validity a, obtained directly from the premises, there is sepa- 

 rate testimony of authority to the validity of an argument. Let it be ^ : then instead of a must be 



used —3 ;r- TT r^ • 



of + (1 -")(l -0 



I now return to the question of the dismissal of authority, which was partially entered on at the 



beginning of this paper. I assume that the mind will form an opinion upon any proposition which 



is laid before it. Even if the assertion were in a scaled packet, with no reason whatever to suppose 



it one rather than another of all that could possibly be made, an opinion would be formed as to its 



truth namely, that it is an even chance whether it be true or false. And this opinion is a just one ; 



