497] 
65 
497. 
NOTE ON THE CALCULUS OF LOGIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 282, 283.] 
It appears to me that the theory of the Syllogism, as given in Boole’s paper, 
“The Calculus of Logic,” Gamb. and Dubl. Math. Jour., t. III. (1848), pp. 183—198, 
may be presented in a more concise and compendious form as follows: 
We are concerned with complementary classes, X, X'; viz. these together make 
up the universe (of things under consideration), X + X' = 1; viz. X' is the class 
not-A, and X the class not-X'. 
Any kind whatever of simple relation between two classes (if we attend also to 
the complementary classes) can be expressed as a relation of total exclusion, ZF=0, 
or as a relation of partial (it may be total) inclusion, YX not = 0; viz. the relation 
X Y = 0 may be read in any of the forms 
No As are X’s, 
No Fs are As, 
All As are not-Fs, 
All Fs are not-A r ’s, 
and the relation AF not =0 in either of the forms 
Some As are Fs, 
Some F’s are A’s. 
I say the above are the only kinds of simple relations; it being understood that 
A"' may be substituted for A, or Y' for F; so that the example A I —0 (all I s 
are A’s) is the same kind of relation as A" 1=0; and A I not = 0 (some Is ai e 
not-A’s) the same kind of relation as AX not = 0. 
Q 
C. VIII. v