Since AAA is by the Law of Simplicity equivalent to A, so *a* ꖌ *a* ꖌ *a* must be equivalent to *a*, and the Law of Unity holds true. Each law thus necessarily presupposes the other. *Symbolic expression of the Law of Duality.* We may now employ our symbol of alternation to express in a clear and formal manner the third Fundamental Law of Thought, which I have called the Law of Duality (p. 6). Taking A to represent any class or object or quality, and B any other class, object or quality, we may always assert that A either agrees with B, or does not agree. Thus we may say A = AB ꖌ A*b*. This is a formula which will henceforth be constantly employed, and it lies at the basis of reasoning. The reader may perhaps wish to know why A is inserted in both alternatives of the second member of the identity, and why the law is not stated in the form A = B ꖌ *b*. But if he will consider the contents of the last section (p. 73), he will see that the latter expression cannot be correct, otherwise no term could have a corresponding negative term. For the negative of B ꖌ *b* is *b*B, or a self-contradictory term; thus if A were identical with B ꖌ *b*, its negative *a* would be non-existent. To say the least, this result would in most cases be an absurd one, and I see much reason to think that in a strictly logical point of view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that *every term has its negative in thought*. We cannot think at all without separating what we think about from other things, and these things necessarily form the negative notion.[69] It follows that any proposition of the form A = B ꖌ *b* is just as self-contradictory as one of the form A = B*b*. [69] *Pure Logic*, p. 65. See also the criticism of this point by De Morgan in the *Athenæum*, No. 1892, 30th January, 1864; p. 155. It is convenient to recapitulate in this place the three Laws of Thought in their symbolic form, thus Law of Identity A = A. Law of Contradiction A*a* = 0. Law of Duality A = AB ꖌ A*b*. *Various Forms of the Disjunctive Proposition.* Disjunctive propositions may occur in a great variety of forms, of which the old logicians took insufficient notice. There may be any number of alternatives, each of which may be a combination of any number of simple terms. A proposition, again, may be disjunctive in one or both members. The proposition Solids or liquids or gases are electrics or conductors of electricity is an example of the doubly disjunctive form. The meaning of such a proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side. From what has been said before, it is apparent that the proposition A ꖌ B = C ꖌ D will correspond to *ab* = *cd*, each member of the latter being the negative of a member of the former proposition.