o
accepted the same theories, but followed them out to their legitimate
conclusion--a substantially atheistic one.

Hamilton in this was himself but a follower of Kant, who brought this
law to support his celebrated "antinomies of the human understanding,"
warnings set up to all metaphysical explorers to keep off of holy
ground.

On another construction of it, one which sought to escape the dilemma of
the contradictories by confining them to matters of the understanding,
Hegel and Schelling believed they had gained the open field. They taught
that in the highest domain of thought, there where it deals with
questions of pure reason, the unity and limits which must be observed in
matters of the understanding and which give validity to this third law,
do not obtain. This view has been closely criticized, and, I think, with
justice. Pretending to deal with matters of pure reason, it constantly
though surreptitiously proceeds on the methods of applied logic; its
conclusions are as fallacious logically as they are experimentally. The
laws of thought are formal, and are as binding in transcendental
subjects as in those which concern phenomena.

The real bearing of this law can, it appears to me, best be derived from
a study of its mathematical expression. This is, according to the
notation of Professor Boole, _x_^{2}=_x_. As such, it presents a
fundamental equation of thought, and it is because it is of the second
degree that we classify in pairs or opposites. This equation can only be
satisfied by assigning to _x_ the value of 1 or 0. The "universal type
of form" is therefore _x_(1-_x_)=0.

This algebraic notation shows that there is, not two, but only one
thought in the antithesis; that it is made up of a thought and its
expressed limit; and, therefore, that the so-called "law of
contradictories" does not concern contradictories at all, in pure logic.
This result was seen, though not clearly, by Dr. Thompson, who indicated
the proper relation of the members of the formula as a positive and a
privative. He, however, retained Hamilton's doctrine that "privative
conceptions enter into and assist the higher processes of the reason in
all that it can know of the absolute and infinite;" that we must, "from
the seen realize an unseen world, not by extending to the latter the
properties of the former, but by assigning to it attributes entirely
opposite."[31-1]

The error that vitiates all such reasoning is the assumption that