the Theory of Probabilities. 48f 



simple events, being connected by the relation 



xyst-\-x{l-y)s{l-t) + (l—x)y{ls)t-{-xy{ls){l—t) 

 4-^(l-2/)(l-5)(l-/) + (l-%(l-^)(l-/) 

 + {l-a:){l-y){l-s){l-t) = l; ..... (4) 

 and the event whose probability w is sought being under the same 

 conditions 



a!yst-^w{l—y)s{l'-t)-{-{l—x)y{l—s)t. ... (5) 

 These results (4) and (5) are both given by the development 

 (Laws of Thought, p. 322). The probabilities (3), together with 

 the relation (4), are equivalent to the data of the problem as 

 expressed in terms of a?, y^ and z in the section referred to, 

 Now I remark that the mere probabilities (3) do not of them- 

 selves furnish any relations connecting sc, y, s and t. The whole 

 of the relation connecting those elements is given by (4), and it 

 is given in the form of a logical equation, i. e. of an equation 

 interpretable into a proposition. We possess of that relation an 

 eocplicit and available knowledge. But it is not so with the rela- 

 tion connecting the elements x, y, z, when, as in the primary 

 statement of the problem, these are assumed as simple events. 

 We are explicitly informed that these elements are connected by 

 the relation 2'(1— <a7)(l— 2/)=0; but beside this, they are con- 

 nected with each other in a complex manner through the data 



Prob. w=c^f Prob. y=c^i Prob. xz = c-^p^j Prob. yz^Cc^p^^. 

 These data exhibit both x and y as connected with z, and thereby 

 also connected with each other. But that connexion is not of a 

 kind which can be exhibited in an explicit form by means of pro- 

 positions. And our consequent inability to express by any di- 

 stinct and intelligible formula the implied relations among the 

 elements x, y, z, renders it difficult to judge of the "reasonable- 

 ness '' or of the " anomalous " character of results in the expres- 

 sion of which these elements are employed*. 



* I need scarcely remark, that the statement of the problem furnished 

 by (3), (4), and (5) will lead, and h§r the same method, to the conditions 

 connecting Cj, c^, Cipi, c^Pi, and w, investigated in my former paper 

 (vol. viii. p. 91). If we assume 



Prob. a?2/s^=X Vvoh. x{\—y)s{\—t)=.yL 



Yvoh. {\—x)y{\-s)t=v Prob. xy{\—s){\—t)=p 



Vroh. x{\-y){\-s){\—t)=(T Prob. {\-x)y{\-s){\-t)^T 

 Vvoh.{\-x){\-y){\-s){\-t)=v, 

 we shall have the following equations : 

 X+/x+p+o-=Ci 

 X + i/+p+r=C2 



\-\-v :=C2p2 



X-|-/x + i/4-p + o-+r+i;=l, 

 whence the conditions in question may be deduced. 



