CHAPTER V.

DISJUNCTIVE PROPOSITIONS.

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--

A planet is either Mercury or Venus or the Earth or ... or Neptune.

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 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.

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.[64] 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 Hamilton, as well as Boole, took the exclusive side. But there are authorities to the opposite effect. Whately, Mansel, and J. S. Mill have all pointed out that we may often treat alternatives as *Compossible*, or true at the same time. Whately gives us an example,[65] “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,[66] “*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.” Mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive. “If we assert,” he says,[67] “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? Such a construction would be ridiculous.”

[64] Mansel’s *Aldrich*, p. 103, and *Prolegomena Logica*, p. 221.

[65] *Elements of Logic*, Book II. chap. iv. sect. 4.

[66] Aldrich, *Artis Logicæ Rudimenta*, p. 104.

[67] *Examination of Sir W. Hamilton’s Philosophy*, pp. 452–454.

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.[68] 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.

[68] *Pure Logic*, pp 76, 77.

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 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

A ꖌ B = B ꖌ A.

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 ꖌ 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.

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 versâ*. 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.

Expressed symbolically, we may say that the negative of

ABC is not-A or not-B or not-C; that is, *a* ꖌ *b* ꖌ *c*.

Reciprocally the negative of

P ꖌ Q ꖌ R is *pqr*.

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

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

*a* ꖌ *a* ꖌ *a*.

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

A = AB ꖌ A*b*.

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

A = B ꖌ *b*.

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 B ꖌ *b* is *b*B, or a self-contradictory term; thus if A were identical with B ꖌ *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.[69] It follows that any proposition of the form A = B ꖌ *b* is just as self-contradictory as one of the form A = B*b*.

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

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

Law of Identity A = A. Law of Contradiction A*a* = 0. Law of Duality A = AB ꖌ A*b*.

*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

Solids or liquids or gases are electrics or conductors of electricity

is an example of the doubly disjunctive form. The meaning of such a proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side. From what has been said before, it is apparent that the proposition

A ꖌ B = C ꖌ D

will correspond to

*ab* = *cd*,

each member of the latter being the negative of a member of the former proposition.

As an instance of a complex disjunctive proposition I may give Senior’s definition of wealth, which, briefly stated, amounts to the proposition “Wealth is what is transferable, limited in supply, and either productive of pleasure or preventive of pain.”[70]

[70] Boole’s *Laws of Thought*, p. 106. Jevons’ *Pure Logic*, p. 69.

Let A = wealth B = transferable C = limited in supply D = productive of pleasure E = preventive of pain.

The definition takes the form

A = BC(D ꖌ E);

but if we develop the alternatives by a method to be afterwards more fully considered, it becomes

A = BCDE ꖌ BCD*e* ꖌ BC*d*E.

An example of a still more complex proposition is found in De Morgan’s writings,[71] as follows:--“He must have been rich, and if not absolutely mad was weakness itself, subjected either to bad advice or to most unfavourable circumstances.”

[71] *On the Syllogism*, No. iii. p. 12. Camb. Phil. Trans. vol. x, part i.

If we assign the letters of the alphabet in succession, thus,

A = he B = rich C = absolutely mad D = weakness itself E = subjected to bad advice F = subjected to most unfavourable circumstances, the proposition will take the form

A = AB{C ꖌ D (E ꖌ F)},

and if we develop the alternatives, expressing some of the different cases which may happen, we obtain

A = ABC ꖌ AB*c*DEF ꖌ AB*c*DE*f* ꖌ AB*c*D*e*F.

The above gives the strict logical interpretation of the sentence, and the first alternative ABC is capable of development into eight cases, according as D, E and F are or are not present. Although from our knowledge of the matter, we may infer that weakness of character cannot be asserted of a person absolutely mad, there is no explicit statement to this effect.

*Inference by Disjunctive Propositions.*

Before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated, this process of combination is as follows,

A(B ꖌ C) = AB ꖌ AC.

Secondly, to combine two disjunctive terms with each other, combine each alternative of one with each alternative of the other. Since flowering plants are either exogens or endogens, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives--namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form

(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE.

It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination, provided each alternative is combined with each alternative of the other terms, as in the algebraic process of multiplication.

Some processes of deduction may be at once exhibited. We may always, for instance, unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive. Thus let

A = B ꖌ C.

Now it is self-evident that

AD = AD,

and in one side of this identity we may for A substitute its equivalent B ꖌ C, obtaining

AD = BD ꖌ CD.

Since “a gaseous element is either hydrogen, or oxygen, or nitrogen, or chlorine, or fluorine,” it follows that “a free gaseous element is either free hydrogen, or free oxygen, or free nitrogen, or free chlorine, or free fluorine.”

This process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives. Since chlorine is a coloured gas, we may infer that “a colourless gaseous element is either (colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative chlorine disappears because colourless chlorine does not exist. Again, since “a tooth is either an incisor, canine, bicuspid, or molar,” it follows that “a not-incisor tooth is either canine, bicuspid, or molar.” The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substitution; for if we have the proposition

A = B ꖌ C ꖌ D,

and we insert this expression for A on one side of the self-evident identity

A*b* = A*b*,

we obtain A*b* = AB*b* ꖌ A*b*C ꖌ A*b*D;

and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction: there remains

A*b* = A*b*C ꖌ A*b*D.

Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the *modus tollendo ponens*.

But the reader must carefully observe that the Disjunctive Syllogism of the mood *ponendo tollens*, which affirms one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that

Water is either salt or fresh water,

it seems evident that “water which is salt is not fresh.” But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say

Gems are either rare stones or beautiful stones, (1)

it will obviously not follow that

A rare gem is not a beautiful stone, (2)

nor that

A beautiful gem is not a rare stone. (3)

Our symbolic method gives only true conclusions; for if we take

A = gem B = rare stone C = beautiful stone,

the proposition (1) is of the form

A = B ꖌ C hence AB = B ꖌ BC and AC = BC ꖌ C;

but these inferences are not equivalent to the false ones (2) and (3).

We can readily represent disjunctive reasoning by the *modus ponendo tollens*, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of

Water is either salt or fresh,

and take

A = Water B = salt C = fresh,

then the premise is apparently of the form

A = AB ꖌ AC;

but in reality there is an unexpressed condition that “what is salt is not fresh,” from which follows, by a process of inference to be afterwards described, that “what is fresh is not salt.” We have then, in letter-terms, the two propositions

B = B*c* C = *b*C.

If we substitute these descriptions in the original proposition, we obtain /* A = AB*c* ꖌ A*b*C; */

uniting B to each side we infer

AB = AB*c* ꖌ AB*b*C or AB = AB*c*;

that is,

Water which is salt is water salt and not fresh.

I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name *abscissio infiniti*. Take the case:

Red-coloured metal is either copper or gold (1) Copper is dissolved by nitric acid (2) This specimen is red-coloured metal (3) This specimen is not dissolved by nitric acid (4) Therefore, this specimen consists of gold (5)

Let us assign the letter-symbols thus--

A = this specimen B = red-coloured metal C = copper D = gold E = dissolved by nitric acid.

Assuming that the alternatives copper or gold are intended to be exclusive, as just explained in the case of fresh and salt water, the premises may be stated in the forms

B = BC*d* ꖌ B*c*D (1) C = CE (2) A = AB (3) A = A*e* (4)

Substituting for C in (1) by means of (2) we get

B = BC*d*E ꖌ B*c*D

From (3) and (4) we may infer likewise

A = AB*e*

and if in this we substitute for B its equivalent just stated, it follows that

A = ABC*d*E*e* ꖌ AB*c*D*e*

The first of the alternatives being contradictory the result is

A = AB*c*D*e*

which contains a full description of “this specimen,” as furnished in the premises, but by ellipsis asserts that it is gold. It will be observed that in the symbolic expression (1) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good.