Given that devastating objections arise against their very foundations, one cannot, on their own premises, maintain either that these realities are confined within any conceivable set of definitions or that they are altogether free of definition. If they are not independent of a definition but in fact require one, and yet no definition is attainable, their very comprehension would hinge on the impossible—so defining them would be impossible, even though we plainly do possess knowledge of them. This alone invalidates their position.
Moreover, a person would then be left unable either to affirm that these realities have definitions or to deny that they do. But whatever precludes both affirmation and denial—or abolishes them altogether—is untenable. Every definition that is advanced encounters, on their principles, a refutation that undermines it, while they also insist that a definition is indispensable. It follows that a definition is both necessary and, by their own admission, impossible—a manifest contradiction.
This contradiction exists only because they take the aim of definition to be an exact portrayal of the thing defined. If, however, the aim of definition is said to be simply the differentiation of the defined from everything else, then both assertions become valid and affirmation no longer conflicts with denial. Indeed, those who have supplied definitions have succeeded in distinguishing the defined from what is not it, and such differentiation is achieved by a term that coincides with the defined in both generality and particularity, being concomitant with it from both sides—affirmation and negation.
Concomitants, however, may be broader than the definiendum, just as antecedents may be narrower. The antecedent can be more specific than its consequent, and the consequent can be broader than its antecedent: “human” entails “animal,” yet “human” is narrower than “animal,” and “animal” follows necessarily from “human” while being broader than it. This differs from two mutually entailing properties, which are equal in scope, such as “humanity” and “rationality,” or “equinity” and “horseness,” and the like.
Accordingly, legitimate definitions are permissible only by means of what is concomitant with the defined in both affirmation and negation—not by mere consequents, as some suppose, nor by mere antecedents. Differentiation is achieved through what is both matching and concomitant, although one may dispense with this differentiation when the name itself already suffices.