174 PROFESSOR Boole's mathematical theory 



/(*',y,^)=/(l. 1. l)xyz -^f{\, \,(i)xy{\ -^z-) 



+ /(1, 0,1)0^.^(1 _y)+/(i,0,0^a:(l -y){\ - s) 

 + /(0, 1, 1) yz (1 - x) +/(0, 1, 0) y (1 - x) (1 - z) 

 + /(0, 0, 1)«(1 -^) (1 -y) 

 + /(0, 0, 0) (1 -X) (I ^y) (I ~^) (12) 



As the object of the expansion of logical symbols may not be evi- 

 dent at first sight, and as the process may consequently be regarded 

 by some as barbarous, we may observe that not only is there a defi- 

 nite aim in the development, but the thing aimed at, has, in our 

 opinion, been most felicitously accomplished. Of this our readers 

 will probably be satisfied when they are introduced to some speci- 

 mens of the use which is made of the formtilae obtained ; in the 

 meantime it may throw some light on the character of these formulse 

 if we notice that the constituents of an expansion represent the 

 several exclusive divisions of what our author terms the universe of 

 discourse, formed by the predication and denial in every possible 

 way of the qualitie's denoted by the literal symbols. In the simplest 

 case, that in which the function is one of a single concept, it will be 

 seen by a glance at (10) that there are only two such possible ways, 

 X and 1 — X. In the case of a function of two symbols, there are 

 [see (11)] four such ways, xy, x {I — y), y (1 — x), (1 — cc) (I — y). 

 In a function of three symbols there are eight such ways ; and so on. 

 A development in which the constituents are of this kind prepares 

 the way for ascertaining all the possible conclusions, in the way 

 either of affirmation or denial, that can be deduced, regarding any 

 concept, from any given relations between it and the other concepts. 



If S be the sum of the constituents of an expansion, and P the 

 product of any two of them, then 



*S= 1, ; (13) 



andP = (14) 



The truth of these beautiful and important propositions will easily 

 be gathered by an intelligent reader from an inspection of the for- 

 mulae, (10), (11), (12). Another important proposition is involved 

 in (14), namely, that, if /"(a;) = 0, either the constituent or the co- 

 efficient in every term of the expansion of / (^x) must be zero. For, 

 let 



/(x) = Q + AX+A,X^ + +A^»; 



where A, ^j, &c., are the coefficients which are not zero, their corres- 

 ponding constituents being X, X^, ^c. ; while Q represents the sum 



