What do these differences mean? To begin with, is (1) monomial and (2) binomial? No; in spite of the fact that there is only one symbol in (1), this equation is binomial in precisely the same sense as (2) is binomial; for it means that a certain attitude toward _m_, symbolized by the minus sign, transforms _m_ into something _distinguishable from_ _m_. If equation (1) expressed an identity, it would not represent the relation of convexity to concavity, which are not identical but distinguishable. But what is thus expressed in (1) by difference of sign is expressed in (2) by difference of coefficient; for (2) means that a certain attitude toward the entity symbolized by _x_ (an attitude symbolized by the phrase “divide by _a_”) transforms _x_ into _y_. In short, the connotation differs, on the two sides, _in both equations alike_. But on the other hand, the denotation is the same on both sides in each equation, for such is the nature of all equations, whether binomial or any other kind. Thus we have identity of denotation with difference of connotation in each of these equations, and they are so far homogeneous with each other. Now connotation is aspect, which is determined by subjective attitude; and attitudes are interrelated in determinate and accurately expressible ways; as, for instance, by antagonism or mutual exclusion, or by any of an indefinite number of forms of implication. The difference of attitude called antipodal oppositeness, or polarity, is the specific difference expressed in equation (1); whereas the coefficient _a_, in (2), expresses _mere_ difference of attitude, difference in general, including, therefore, that specific difference which is expressed by opposition of sign. Thus equation (1) is a case of equation (2). To sum up: The objection, stated in these algebraic symbols, was this: _m_ implies -_m_; _x_ does not imply _y_. Express the fact of relief in terms of _m_ and you have the correlative fact in -_m_ implied in the very definition of _m_; while if you express _x_ in terms of _y_, you have _y_ values, and nothing but _y_. In short, _x_ and _y_ exclude each other; _m_ and -_m_ imply each other. Our answer is that _x_ implies _y_ just as _m_ implies -_m_; for _ay_ is an aspect of the same denotation as _x_; and, since the specificity of every aspect of a given denotation is determinable or definable by relation to all other aspects of the same denotation, any one of such aspects, as _x_, implies, in its definition, every other, and so _y_, instead of excluding _y_.