176 Prof. Boole on certain Propositions in Algebra 



which st, s(l — t), &c. occur as the constituents, properly so called, 

 of a logical development, having therein a logical, and not a 

 quantitative interpretation. That the conditions of possible 

 experience may be deduced from such a development has been 

 shown in my last paper on the Theory of Probabilities, published 

 in this Journal (Dec. 1854, note). It is most convenient, indeed, 

 to deduce them from the immediate data. This is the course 

 which I have adopted in my paper 'On the Conditions, &c.' 

 (Phil. Mag. August 1854). But they may also be deduced from 

 the final logical development ; and the results are necessarily 

 equivalent, inasmuch as the final development and the initial 

 data are connected by a chain of analysis founded upon those 

 laws of thought which are the basis of ordinary deductive rea- 

 soning. 



And hence it will suffice to show, that the required conditions 

 of algebraic solution of the system (22) are the same as the con- 

 ditions of possible experience determined from the logical deve- 

 lopment from which the algebraic system is derived. 



To determine the conditions of algebraic solution, let us first 

 place the system in the form 



V * V < Urn 



and then assuming 



s t 



we obtain a result of the form 



\-t 



* x ' y e 



agreeing with the form considered in Prop. II. 

 Thus, if we have 



V=stu + st(l-u)+s{l-t)u+(l-s)tu + {l-s){l-t){l-u),(23) 



in which case the system (22) becomes 



stu + st(l — u)+s(l — t)u_ ~ 



stu-\-st(l — u) + (l — s)tu_ 



- y — -q 



stu + st(l — t)u+ (l—s)tu_ 



And if we therein make 



1-s 



= .r, 



t 

 l—t 



(24) 



■■y> . 



l-M 



