s are

s. To the above scheme, which is that of Aristotle, we might annex the four categorical propositions

1 - y = vx, All not-Ys are Xs,
1 - y = v(1 - x), All not-Ys are not-Xs,
v(1 - y) = v'x, Some not-Ys are Xs,
v(1 - y) = v'(1 - x), Some not-Ys are not-Xs,

the two first of which are similarly convertible into

1 - x = v'y, All not-Xs are Ys,
x = v'y, All Xs are Ys,
or No not-Xs are Ys,

If now the two premises of any syllogism are expressed by equations of the above forms, the elimination of the common symbol

will lead us to an equation expressive of the conclusion.