[42] The discussion is purposely made as formal and exact as possible. If indeed the doctrine of inverse variation cannot be treated with precision, it is better not to attempt to deal with it at all.
A. (1) Let connotation be supposed arbitrarily fixed, and used to determine denotation in some assigned universe of discourse. Then it will not be true that connotation and denotation will necessarily vary inversely. For suppose the connotation of a name, i.e., the attributes signified by it, to be a, b, c. It may happen that in fact wherever the attributes a and b are present, the attributes c and d are also present. 37 In this case, if c is dropped from the connotation, or d added to it, the denotation of the name will remain unaffected. We have concrete examples of this, if we suppose equiangularity added to the connotation of equilateral triangle, or cloven-hoofed to that of ruminant, or having jaws opening up and down to that of vertebrate, or if we suppose invalid dropped from the connotation of invalid syllogism with undistributed middle. It is clear, however, that if any alteration in denotation takes place when connotation is altered, it must necessarily be in the opposite direction. Some individuals possessing the attributes a and b may lack the attribute c or the attribute d ; but no individuals possessing the attributes a, b, c, or a, b, c, d can fail to possess the attributes a, b, or a, b, c. For example, if to the connotation of metal we add fusible, it makes no difference to the denotation; but if we add having great weight, we exclude potassium, sodium, &c.
The law of variation of denotation with connotation may then be stated as follows:—If the connotation of a term is arbitrarily enlarged or restricted, the denotation in an assigned universe of discourse will either remain unaltered or will change in the opposite direction.[43]
[43] Since reference is here made to the actual denotation of a term in some assigned universe of discourse, the above law may be said to turn partly on material, and not on purely formal, considerations. It should, therefore, be added that although an alteration in the connotation of a term will not always alter its actual denotation in an assigned universe of discourse, it will always affect potentially its subjective extension. If, for example, the connotation of a term X is a, b, c, and we add d ; then the (real or imaginary) class of X’s that are not d is necessarily excluded from, while it was previously included in, the subjective extension of the term X. Hence, if the connotation of a term is arbitrarily enlarged or restricted, the subjective extension will be potentially restricted or enlarged accordingly. Cf. Jevons, Principles of Science, 30, § 13.
(2) Let exemplification be supposed arbitrarily fixed, and used to determine comprehension. It is unnecessary to shew in detail that a corresponding law of variation of comprehension with exemplification will hold good, namely:—If the exemplification (extensive definition) of a term is arbitrarily enlarged or restricted, the comprehension in an assigned universe of discourse will either remain unaltered or will change in the opposite direction. 38
B. We may now consider the relation between the comprehension and the denotation of a term. Let P1, P2, … Px be the totality of attributes possessed by the class X, and Q1, Q2, … Qy the totality of objects included in the class X. Both these groups are objectively, not arbitrarily,[44] determined; and the relation between them is reciprocal. P1, P2, … Px are the only attributes possessed in common by the objects Q1, Q2, … Qy ; and Q1, Q2, … Qy are the only objects possessing all the attributes P1, P2, … Px.
[44] What may be arbitrary is the intensive definition (P1, P2, … Pm) or the extensive definition (Q1, Q2, … Qn) which determines them both.
We cannot suppose any direct arbitrary alteration either in comprehension or in denotation. We can, however, establish the following law of inverse variation, namely, that any arbitrary alteration in either intensive definition or extensive definition which results in an alteration of either denotation or comprehension will also result in an alteration in the opposite direction of the other.
Let X and Y be two terms which are so related that the definition (either intensive or extensive, as the case may be) of Y includes all that is included in the definition of X and more besides. We have to shew that either the denotations and comprehensions of X and Y will be identical or if the denotation of one includes more than the denotation of the other then its comprehension will include less, and vice versâ.
(a) Let X and Y be determined by connotation or intensive definition. Thus, let X be determined by the set of properties P1 … Pm and Y by the set P1 … Pm+1, which includes the additional property Pm+1.
Then Pm+1 either does or does not always accompany P1 … Pm.
If the former, no object included in the denotation of X is excluded from that of Y, so that the denotations of X and Y are the same; and it follows that the comprehensions of X and Y are also the same.
If the latter, then the denotation of Y is less than that of X by all those objects that possess P1 … Pm without also possessing Pm+1. At the same time, the comprehension of Y includes at 39 least Pm+1 in addition to the properties included in the comprehension of X.