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 the 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.

There can be no doubt that in a great many cases, perhaps the greater number of cases, alternatives are exclusive as a matter of fact. Any one number is incompatible with any other; one point of time or place is exclusive of all others. Roger Bacon died either in 1284 or 1292; it is certain that he could not die in both years. Henry Fielding was born either in Dublin or Somersetshire; he could not be born in both places. There is so much more precision and clearness in the use of exclusive alternatives that we ought doubtless to select them when possible. Old works on logic accordingly contained a rule directing that the Membra dividentia, the parts of a division or the constituent species of a genus should be exclusive of each other.

It is no doubt owing to the great prevalence and convenience of exclusive divisions that the majority of logicians have held it necessary to make every alternative in a disjunctive proposition exclusive of every other one. Aquinas considered that when this was not the case the proposition was actually false, and Kant adopted the same opinion. A multitude of statements to the same effect might readily be quoted, and if the question were to be determined by the weight of historical evidence, it would certainly go against my view. Among recent logicians Sir W. Hamilton, as well as Dr. Boole, took the exclusive side. But there are authorities to the opposite effect. Whately, Mansel, and J. S. Mill, have all pointed out that

a Mansel's 'Aldrich,' p. 103, and ‘Prolegomena Logica,' p. 221.

we may often treat alternatives as Compossible, or true at the same time. Whately gives as an exampleb, 'Virtue tends to procure us either the esteem of mankind, or the favour of God,' and he adds, Here both members are true, and consequently from one being affirmed we are not authorized to deny the other. Of course we are left to conjecture in each case, from the context, whether it is meant to be implied that the members are or are not exclusive.' Mansel says, We may happen to know that two alternatives cannot be true together, so that the affirmation of the second necessitates the denial of the first; but this, as Boethius observes, is a material, not a formal consequence.' Mr. J. S. Mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive. If we assert,' he says, 'that a man who has acted in some particular way must be either a knave or a fool, we by no means assert, or intend to assert, that he cannot be both.' Again, to make an entirely unselfish use of despotic power, a man must be either a saint or a philosopher.. ...... Does the disjunctive premise necessarily imply, or must it be construed as supposing, that the same person cannot be both a saint and a philosopher? 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 the late Dr. 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

bElements of Logic,' Book II. chap. iv. sect. 4.

e Aldrich, 'Artis Logica Rudimenta,' p. 104.

d Examination of Sir W. Hamilton's Philosophy,' pp. 452-454.

dispute. In the ordinary use of these conjunctions we do not necessarily 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 they are distinct. And when 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.

Take, for instance, the proposition 'A peer is either a duke, or a marquis, or an earl, or a viscount, or a baron.' If expressed in Professor Boole's symbols, it would be implied that a peer cannot be at once a duke and marquis, or marquis and earl. Yet many peers do possess two or more titles, and the Prince of Wales is Duke of Cornwall, Earl of Dublin, and Baron Renfrew. If it were enacted by parliament that no peer should have more than one title, this would be the tacit premise which Professor Boole assumes to exist. Nor is the restriction true of more common terms.

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,' 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 disjunctively. 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 manner, although such cases may be comparatively very 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 expressions, it is because our subject demands it. The matter, not the form of an expression, points out whether terms are exclusive e. The question, as we shall afterwards see, is one of the greatest theoretical importance, because it furnishes the true distinction between the sciences of Logic and Mathematics. It is the very foundation of number that every unit shall be distinct from every other unit; but Dr. 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. 39), we found that certain laws, those of Simplicity and Commutativeness, must be observed. In uniting 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 say generally

A B
+

B =

BA.

Pure Logic,' pp. 76, 77.

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

A + A = A.

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.

I am not aware that logicians have in any adequate way 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 metallic nature. Wherever any one or more of the