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 propose to use for this purpose the symbol ꖌ. In my first logical essay I followed the practice of Boole and adopted the sign +; but this sign should not be employed unless there exists exact analogy between mathematical addition and logical alternation. We shall find that the analogy is imperfect, and that there is such profound difference between logical and mathematical terms as should prevent our uniting them by the same symbol. Accordingly I have chosen a sign ꖌ, which seems aptly to suggest whatever degree of analogy may exist without implying more. The exact meaning of the symbol we will now proceed to investigate. *Nature of the Alternative Relation.* Before treating disjunctive propositions it is indispensable to decide whether the alternatives must be considered exclusive or unexclusive. By *exclusive alternatives* we mean those which cannot contain the same things. If we say “Arches are circular or pointed,” it is certainly to be understood that the same arch cannot be described as both circular and pointed. Many examples, on the other hand, can readily be suggested in which two or more alternatives may hold true of the same object. Thus Luminous bodies are self-luminous or luminous by reflection. It is undoubtedly possible, by the laws of optics, that the same surface may at one and the same moment give off light of its own and reflect light from other bodies. We speak familiarly of *deaf or dumb* persons, knowing that the majority of those who are deaf from birth are also dumb.