the Theory of Probabilities. 4a4S 



If the problem be a real one, the system (I.) will furnish one set, 

 and only one set, of positive fractional values of s, t, ^c., which, 

 substituted in (II.), will determine Prob. w. 



The interpretation of c, when it appears in the solution, is 



Prob. Cw 

 Prob. C ' 



and it indicates the new experience requisite to complete the 

 solution of the problem. 



If the system (I.) does not furnish a single ' system of positive 

 fractional values of s, t, &c., the problem is not a real one, and 

 does not in its statement represent a possible experience. 



The passages in italics contain the additions to the rule as it 

 is presented in the Laws of Thought. 



In concluding this paper, I shall briefly consider the only two 

 objections which have at any time occurred to my own mind as 

 likely to occasion a difficulty in the reception of the above results. 



1st. It may be, and indeed it has been, urged that the logical 

 calculus upon which the investigation proceeds does not consti- 

 tute a science or represent " reality,^^ being only based upon a 

 system of '^ substituted ratios.^^ 



To this it is replied, that pure science, as such, is concerned 

 only with ratios or relations. To know things as they are in 

 themselves, is the professed but unattainable object of a so-called 

 philosophy proper. It is, however, here maintained that the 

 logical calculus does represent reality and constitute science, in- 

 asmuch, — 1st, as the laws of thought upon which it is founded, 

 and which it expresses by the fundamental equations xy=yx, 

 x^=-x, &c. are not fictitious, but are derived from a real analysis 

 of the intellectual operations ; 2nd, as it is a fact, and not an 

 assumption, that the laws thus determined are formally identical 

 with the laws of a certain properly defined species of arithmetic ; 

 3rd, as it accords with the catholic objects of science to avail 

 itself of all discovered laws and relations, without regard to the 

 fashion of the schools or the prescription of ancient usage. 



2nd. It may be objected, that, although in the representative 

 problem of the urn we can readily pass in thought from a system 

 of balls having an actual physical nexus to the same system free 

 from that nexus, \Ye cannot, in the represented problem in which 

 the events s, t, &c. are subject to the logical and therefore ne- 

 cessary connexion D = 0, interpret to ourselves the same events 

 as freed from that connexion ; and therefore the problem of the 

 urn does not completely and adequately represent the problem 

 for which it is substituted, inasmuch as in the one case the nexus 

 or condition implied by the equation D = is merely actual, 

 while in the other case it is not only actual but necessary. 



2G2 



