saint and a philosopher? Such a construction would be ridiculous."

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.1 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 ...

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.

Laws of the Disjunctive Relation.

In considering the combination or synthesis of terms (p. 30), we found that certain laws, those of Simplicity 1 Pure Logic, pp 76, 77.

In uniting

and Commutativeness, must be observed. terms by the disjunctive symbol we shall find that the same or closely similar laws hold true. The alternatives of either member of a disjunctive proposition are certainly commutative. Just as we cannot properly distinguish between rich and rare gems and rare and rich gems, so we must consider as identical the expression rich or rare gems, and rare or rich gems. In our symbolic language we may

The order of statement, in short, has no effect upon the meaning of an aggregate of alternatives, so that the Law of Commutativeness holds true of the disjunctive symbol.

As we have admitted the possibility of joining as alternatives terms which are not really different, the question arises, How shall we treat two or more alternatives when they are clearly shown to be the same? If we have it asserted that P is Q or R, and it is afterwards proved that Q is but another name for R, the result is that P is either R or R. How shall we interpret such a statement? What would be the meaning, for instance, of "wreath or anadem" if, on referring to a dictionary, we found anadem described as a wreath? I take it to be self-evident that the meaning would then become simply "wreath." Accordingly we may affirm the general law

AAA.

Any number of identical alternatives may always be reduced to, and are logically equivalent to, any one of those alternatives. This is a law which distinguishes mathematical terms from logical terms, because it obviously does not apply to the former. I propose to call it the Law of Unity, because it must really be involved in any definition of a mathematical unit. This law is closely analogous to the Law of Simplicity, AA = A; and the nature of the connection is worthy of attention.

Few or no logicians except De Morgan have adequately noticed the close relation between combined and disjunctive terms, namely, that every disjunctive term is the negative of a corresponding combined term, and vice versa. Consider

the term

Malleable dense metal.

How shall we describe the class of things which are not malleable-dense-metals? Whatever is included under that term must have all the qualities of malleability, denseness, and metallicity. Wherever any one or more of the qualities is wanting, the combined term will not apply. Hence the negative of the whole term is

Not-malleable or not-dense or not-metallic. In the above the conjunction or must clearly be interpreted as unexclusive; for there may readily be objects which are both not-malleable, and not-dense, and perhaps not-metallic at the same time. If in fact we were required to use or in a strictly exclusive manner, it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three terms. The negatives of four or five terms would consist of fifteen or thirty-one alternatives. This consideration alone is sufficient to prove that the meaning of or cannot be always exclusive in common language.

Every disjunctive term, then, is the negative of a combined term, and vice versa.

Apply this result to the combined term AAA, and its negative is

aaa.

Since AAA is by the Law of Simplicity equivalent to A, so a ta a must be equivalent to a, and the Law of Unity holds true. Each law thus necessarily presupposes he other.

Symbolic expression of the Law of Duality.

We may now employ our symbol of alternation to express in a clear and formal manner the third Fundamental Law of Thought, which I have called the Law of Duality (p. 6). Taking A to represent any class or

object or quality, and B any other class, object or quality, we may always assert that A either agrees with B, or does not agree. Thus we may say

Α

=

AB Ab.

This is a formula which will henceforth be constantly employed, and it lies at the basis of reasoning.

The reader may perhaps wish to know why A is inserted in both alternatives of the second member of the identity, and why the law is not stated in the form

But if he will consider the contents of the last section (p. 73), he will see that the latter expression cannot be correct, otherwise no term could have a corresponding negative term. For the negative of Bb is bB, or a selfcontradictory term; thus if A were identical with Bj b its negative a would be non-existent. To say the least, this result would in most cases be an absurd one, and I see much reason to think that in a strictly logical point of view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that every term has its negative in thought. We cannot think at all without separating what we think about from other things, and these things necessarily form the negative notion.' It follows that any proposition of the form A = Bb is just as self-contradictory as one of the form A = Bb.

It is convenient to recapitulate in this place the three Laws of Thought in their symbolic form, thus

Law of Identity

Law of Contradiction
Law of Duality

Various Forms of the Disjunctive Proposition.

Disjunctive propositions may occur in a great variety of forms, of which the old logicians took insufficient notice. There may be any number of alternatives, each of which may be a combination of any number of simple terms. A proposition, again, may be disjunctive in one or both. members. The proposition

1 Pure Logic, p. 65. See also the criticism of this point by De Morgan in the Athenæum, No. 1892, 30th January, 1864; p. 155.