It is plain that any positive term and its corresponding negative divide between them the whole universe of thought: whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Supposing that the things represented by A and B are found to differ, we may indicate (see p. 17) the result of the judgment by the notation A ~ B. We may now represent the same judgment by the assertion that A agrees with those things which differ from B, or that A agrees with the not-B’s. Using our notation for negative terms (see p. 14), we obtain A = A*b* as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition:-- Quicksilver = Quicksilver not-solid. There may also be several other classes of negative propositions, of which no notice was taken in the old logic. We may have cases where all A’s are not-B’s, and at the same time all not-B’s are A’s; there may, in short, be a simple identity between A and not-B, which may be expressed in the form A = *b*. An example of this form would be Conductors of electricity = non-electrics. We shall also frequently have to deal as results of deduction, with simple, partial, or limited identities between negative terms, as in the forms *a* = *b*, *a* = *a**b*, *a*C = *b*C, etc. It would be possible to represent affirmative propositions in the negative form. Thus “Iron is solid,” might be expressed as “Iron is not not-solid,” or “Iron is not fluid;” or, taking A and *b* for the terms “iron,” and “not-solid,” the form would be A ~ *b*. But there are very strong reasons why we should employ all propositions in their affirmative form. All inference proceeds by the substitution of equivalents, and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner. As will be more fully shown, we can infer *in* a negative proposition, but not *by* it. Difference is incapable of becoming the ground of inference; it is only the implied agreement with other differing objects which admits of deductive reasoning; and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements. *Conversion of Propositions.*