REVIEWS 515 



The expansion of a logical function is interesting and important. 

 Let/(.r) be the given function. 



Assume /(at) = ax + b(i - x), where a and b are to be determined. 

 Since x may be regarded as a quantity susceptible only of the values o and 1, 

 give it those values successively. 



Then/(o) = t>,f(i) = a, 



.-./(*) =/(i).t+/(o)(i -x). 



f(xy) can be developed in a similar manner by expanding it first as a function of 

 v and then developing the expanded function as a function of x. The result is : 



A*y) =/(!> *)*y + /(',°Wi -y) + /(o, i)(i - x) y +/(o,o)(i - x){i - y). 



The following is a simple illustration of Boole's method : 



Taking the biblical definition of clean beasts as those which chew the cud and 

 divide the hoof, find a description of unclean beasts. 



Let x = clean beasts, 

 y = chewing the cud, 

 Z = dividing the hoof, 

 then x = yz, 

 I — x = 1 - ys. 



Expanding the right-hand side by the formula we get — 



1 - x =y(i - s) + (1 - y)z + (1 - y){i - z). 



Unclean beasts, therefore, are those which chew the cud and do not divide the 

 hoof, those which divide the hoof and do not chew the cud, and those which 

 neither chew the cud nor divide the hoof. 



In translating a proposition into an equation it will naturally occur to the 

 logician to inquire what becomes of the Quantification of the Predicate. If 

 x = men,_y = mortal beings, we cannot express that all men are mortal by x =y, 

 because all mortal beings are not men. Boole meets the difficulty by introducing 

 a symbol to denote an indefinite class. He expresses the proposition " all men 

 are mortal " by x = vy, where v represents a class about which we know nothing 

 except that some of its members are/. 



The indefinite symbol v is easily eliminated, for in logic we can eliminate any 

 class symbol from a single equation with the help of the Law of Duality. The 

 result of eliminating x from the equation /(.r) = o is 



/(o)/(i) = o. 



To prove this expand /(.r) = o. 

 We have/(i)r +/(o) (1 - x) = o. 



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vy and to 



