OF TESTIMONIES OR JUDGMENTS. 6*09 



tions of a smaller number of symbols, except when the condition referred to in 

 (3) is fulfilled, in which case, the methods are identical. 



15. It remains to show how the conditions of possible experience as above 

 determined, restrict us in the choice of the hypothesis, by the aid of which the 

 final solution is to be obtained. 



Taking for example the above problem of Art. 13, let us inquire whether it 

 would be lawful to assume x, y, and z as the primary simple events of the 

 problem. 



If we make this assumption and then write 



Prob. x = a Prob. y = (3 Prob . z = y 

 we find 



Prob. xy = a(3 Prob. yz=fiy Prob. zcc=ya 



whence comparing with (1) Art. 13— 



a(3=p @y=q ya = r 

 solving which equations we have 



a = Jj P=Vj 7 = \/^7 •*• Prob - xy* = a fo = Vpq? • (4) 



Now, a, j9, 7. being by assumption probabilities, and therefore, lying numerically 

 between the limits and 1, we must have 



qr^p rp<.q pq<.r . . . . (5) 



as the conditions to which p, q, and r (beside being fractional) must be subject. 

 These conditions do not, however, agree with, and are not involved in, the condi- 

 tions of possible experience, determined in (6) Art. 13. We may conclude, there- 

 fore, that the hypothesis upon which our solution is founded, involves elements, 

 the introduction of which is unwarranted, and that the value of Prob. xyz deter- 

 mined is erroneous. 



We may show, in fact, that the conditions (5) imply the conditions of possible 

 experience, and something more. If qr <p then, a fortiori, qr <p + (1— q) (1— *■) 

 since (l — q) 0-—r) is essentially positive. Therefore, 



qr <p + l — q — r + qr 



whence p>q + r—l 



which is one of the conditions (6) Art 13. In the same way the other conditions 

 in that article may be deduced from (5). The reverse reduction is, however, 

 impossible. 



16. The hypothesis upon which the method developed in the Laws of Thought, 

 cap. xvii., for the solution of questions in the theory of probabilities whose elements 

 are logical, is founded, seems to be the only one which satisfies the requirement 

 referred to in Art. 12. It was not, however, upon such considerations as this, 



VOL. XXI. PART IV. 8 B 



