CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 285 



and passing from Logic to Algebra, 



uzvy + u (1 - z) (1 - v) y _ uzvy + (1 - u) zvy + (1 - u) z (1 - 0)(1 -y) 

 cp c' 



uzvy + (1 - u) zvy 



p r ob. 



wherein 



F= wzvy + w(l -2)(1 -u)y + (1 -w) zvy + (l -u) z(l-v) (1 -^) 



+ (\-u)(l-z)(l-v)y + (l- u) (1 - z) (1 - *) (1 - y), 

 the solution of this system of equations gives 



Prob. w = Jpq + ac (1 - q\ 

 whence 



Prob. xy /i \ 



- ^ - =pq + a(l-q), 



C 



the value required. In this expression the arbitrary constant a 

 is the probability that if the proposition Z is true and Y false, X 

 is true. In other words, it is the probability, that if the minor 

 premiss is false, the conclusion is true. 



This investigation might have been greatly simplified by as- 

 suming the proposition ^"to be true, and then seeking the proba- 

 bility of X. The data would have been simply 



Prob. y = #, Prob. xy = pq ; 



whence we should have found Prob. x = pq + a (1 - q). It is 

 evident that under the circumstances this mode of procedure 

 would have been allowable, but I have preferred to deduce the 

 solution by the direct and unconditioned application of the 

 method. The result is one which ordinary reasoning verifies, 

 and which it does not indeed require a calculus to obtain. Ge- 

 neral methods are apt to appear most cumbrous when applied to 

 cases in which their aid is the least required. 



Let it be observed, that the above method is equally appli- 

 cable to the categorical syllogism, and not to the syllogism only, 



