In the previous chapter I have exhibited various cases of deductive reasoning by the process of substitution, avoiding the introduction of disjunctive propositions; but we cannot long defer the consideration of this more complex class of identities. General terms arise, as we have seen (p. 24), from classifying or mentally uniting together all objects which agree in certain qualities, the value of this union consisting in the fact that the power of knowledge is multiplied thereby. In forming such classes or general notions, we overlook or abstract the points of difference which exist between the objects joined together, and fix our attention only on the points of agreement. But every process of thought may be said to have its inverse process, which consists in undoing the effects of the direct process. Just as division undoes multiplication, and evolution undoes involution, so we must have a process which undoes generalization, or the operation of forming general notions. This inverse process will consist in distinguishing the separate objects or minor classes which are the constituent parts of any wider class. If we mentally unite together certain objects visible in the sky and call them planets, we shall afterwards need to distinguish the contents of this general notion, which we do in the disjunctive proposition—
Having formed the very wide class “vertebrate animal,” we may specify its subordinate classes thus:—“A vertebrate animal is either a mammal, bird, reptile, or fish.” Nor is there any limit to the number of possible alternatives. “An exogenous plant is either a ranunculus, a poppy, a crucifer, a rose, or it belongs to some one of the other seventy natural orders of exogens at present recognized by botanists.” A cathedral church in England must be either that of London, Canterbury, Winchester, Salisbury, Manchester, or of one of about twenty-four cities possessing such churches. And if we were to attempt to specify the meaning of the term “star,” we should require to enumerate as alternatives, not only the many thousands of stars recorded in catalogues, but the many millions unnamed.
Whenever we thus distinguish the parts of a general notion we employ a disjunctive proposition, in at least one side of which are several alternatives joined by the so-called disjunctive conjunction or, a contracted form of other. There must be some relation between the parts thus connected in one proposition; we may call it the disjunctive or alternative relation, and we must carefully inquire into its nature. This relation is that of ignorance and doubt, giving rise to choice. Whenever we classify and abstract we must open the way to such uncertainty. By fixing our attention on certain attributes to the exclusion of others, we necessarily leave it doubtful what those other attributes are. The term “molar tooth” bears upon the face of it that it is a part of the wider term “tooth.” But if we meet with the simple term “tooth” there is nothing to indicate whether it is an incisor, a canine, or a molar tooth. This doubt, however, may be resolved by further information, and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise.
Expression of the Alternative Relation.
In order to represent disjunctive propositions with convenience we require a sign of the alternative relation, equivalent to one meaning at least of the little conjunction or so frequently used in common language. I