OF ARTS AND SCIENCES: MARCH 12/1867. 259 



Let 1 === the questions put to both witnesses, 



p == those which the first answers truly, 



q == those which the second answers truly, 



r ■==. those wherein they disagree, 



w == those which both answer truly, 



w 1 = those which both answer falsely. 

 w=p,q w'=p,q r==p -{rq — w=p -\rq — w'. 



Now by (28.) 



p -ir q=p -\- q — w p -\r q = p — p-\-i — q — w> - 



Substituting and transposing, 



2w =p -\- q — r 2^tf' = 2 — p — q — r. 



Now w 1 _ T = —^ J - but w , (1 — r) = w. 



w\_ T = —^ '- but w 1 , (1 — r) = w' 



2 — p — q—r 

 •'• »*-* = 2 (1 - r) ^-r) - 2(1 _ r) • 



The differences of Boole's system, as given by himself, from the 

 modification of it given here, are three. 



1st. Boole does not make use of the operations here termed logical 

 addition and subtraction. The advantages obtained by the introduc- 

 tion of them are three, viz. they give unity to the system ; they 

 greatly abbreviate the labor of working with it ; and they enable us 

 to express particular propositions. This last point requires illustra- 

 tion. Let i be a class only determined to be such that only some one 

 individual of the class a comes under it. Then a — * , a is the ex- 

 pression for some a. Boole cannot properly express some a. 



2d. Boole uses the ordinary sign of multiplication for logical 

 multiplication. This debars him from converting every logical iden- 

 tity into an equality of probabilities. Before the transformation can 

 be made the equation has to be brought into a particular form, and 

 much labor is wasted in bringing it to that form. 



3d. Boole has no such function as a b . This involves him in two 



