66 NOTE ON THE CALCULUS OF LOGIC. [497 
Now taking X or X' and Z or Z' for the extreme terms, and F or Y' for the 
middle term, of a syllogism; the only combinations of premises are 
(1) 
XY= 
= 0, 
ZF= 
= 0. 
(2) 
XY = 
= 0, 
ZY 
not 
= 0, 
therefore 
X'Z not 
(3) 
XY 
not 
= 0, 
ZY 
not 
= 0. 
(4) 
XY = 
= 0, 
ZY’ 
= 0, 
therefore 
XZ= 0. 
(5) 
XY = 
= 0, 
ZY' 
not 
= 0. 
(6) 
XY 
not 
= 0, 
ZY' 
not 
= 0. 
And of these, there are (as shown by the third column) only two w T hich give rise to 
a conclusion (or relation between the extreme terms). As regards the negative cases, 
this is at once seen to be so; thus IF=0, ZY = 0 (no X’s are F’s, no Z’s are F’s) 
leads to no conclusion in regard to X, Z. As regards the positive cases, it is also at 
once seen that the conclusions do follow; but we may obtain the conclusions by 
symbolical reasoning, thus 
(2) Y=YX+YX',=YX 
therefore ZY — ZYX', not = 0; therefore ZX' not = 0. 
(4) XZ = XZY + XZY', where on the right-hand side each term (the first as 
containing XY, the second as containing ZX') is =0; that is, XZ=0] where the 
logical signification of each step is obvious.