I discuss this subject fully because it is really the point which separates my logical system from that of Boole. In his Laws of Thought (p. 32) he expressly says, “In strictness, the words ‘and,’ ‘or,’ interposed between the terms descriptive of two or more classes of objects, imply that those classes are quite distinct, so that no member of one is found in another.” This I altogether dispute. In the ordinary use of these conjunctions we do not join distinct terms only; and when terms so joined do prove to be logically distinct, it is by virtue of a tacit premise, something in the meaning of the names and our knowledge of them, which teaches us that they are distinct. If our knowledge of the meanings of the words joined is defective it will often be impossible to decide whether terms joined by conjunctions are exclusive or not.
In the sentence “Repentance is not a single act, but a habit or virtue,” it cannot be implied that a virtue is not a habit; by Aristotle’s definition it is. Milton has the expression in one of his sonnets, “Unstain’d by gold or fee,” where it is obvious that if the fee is not always gold, the gold is meant to be a fee or bribe. Tennyson has the expression “wreath or anadem.” Most readers would be quite uncertain whether a wreath may be an anadem, or an anadem a wreath, or whether they are quite distinct or quite the same. From Darwin’s Origin of Species, I take the expression, “When we see any part or organ developed in a remarkable degree or manner.” In this, or is used twice, and neither time exclusively. For if part and organ are not synonymous, at any rate an organ is a part. And it is obvious that a part may be developed at the same time both in an extraordinary degree and an extraordinary manner, although such cases may be comparatively rare.
From a careful examination of ordinary writings, it will thus be found that the meanings of terms joined by “and,” “or” vary from absolute identity up to absolute contrariety. There is no logical condition of distinctness at all, and when we do choose exclusive alternatives, it is because our subject demands it. The matter, not the form of an expression, points out whether terms are exclusive or not.5 In bills, policies, and other kinds of legal documents, it is sometimes necessary to express very distinctly that alternatives are not exclusive. The form and/or is then used, and, as Mr. J. J. Murphy has remarked, this form coincides exactly in meaning with the symbol ꖌ.
In the first edition of this work (vol. i., p. 81), I took the disjunctive proposition “Matter is solid, or liquid, or gaseous,” and treated it as an instance of exclusive alternatives, remarking that the same portion of matter cannot be at once solid and liquid, properly speaking, and that still less can we suppose it to be solid and gaseous, or solid, liquid, and gaseous all at the same time. But the experiments of Professor Andrews show that, under certain conditions of temperature and pressure, there is no abrupt change from the liquid to the gaseous state. The same substance may be in such a state as to be indifferently described as liquid and gaseous. In many cases, too, the transition from solid to liquid is gradual, so that the properties of solidity are at least partially joined with those of liquidity. The proposition then, instead of being an instance of exclusive alternatives, seems to afford an excellent instance to the opposite effect. When such doubts can arise, it is evidently impossible to treat alternatives as absolutely exclusive by the logical nature of the relation. It becomes purely a question of the matter of the proposition.
The question, as we shall afterwards see more fully, is one of the greatest theoretical importance, because it concerns the true distinction between the sciences of Logic and Mathematics. It is the foundation of number that every unit shall be distinct from every other unit; but Boole imported the conditions of number into the science of Logic, and produced a system which, though wonderful in its results, was not a system of logic at all.