V. DISJUNCTIVE PROPOSITIONS…

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.

VI. THE INDIRECT METHOD OF INFERENCE

The forms of deductive reasoning as yet considered, are mostly cases of Direct Deduction as distinguished from those which we are now about to treat. The method of Indirect Deduction may be described as that which points out what a thing is, by showing that it cannot be anything else. We can define a certain space upon a map, either by colouring that space, or by colouring all except the space; the first mode is positive, the second negative. The difference, it will be readily seen, is exactly analogous to that between the direct and indirect modes of proof in geometry. Euclid often shows that two lines are equal, by showing that they cannot be unequal, and the proof rests upon the known number of alternatives, greater, equal or less, which are alone conceivable. In other cases, as for instance in the seventh proposition of the first book, he shows that two lines must meet in a particular point, by showing that they cannot meet elsewhere.

In logic we can always define with certainty the utmost number of alternatives which are conceivable. The Law of Duality (pp. 6, 74) enables us always to assert that any quality or circumstance whatsoever is either present or absent. Whatever may be the meaning of the terms A and B it is certainly true that

A = AB ꖌ A*b* B = AB ꖌ *a*B.

These are universal tacit premises which may be employed in the solution of every problem, and which are such invariable and necessary conditions of all thought, that they need not be specially laid down. The Law of Contradiction is a further condition of all thought and of all logical symbols; it enables, and in fact obliges, us to reject from further consideration all terms which imply the presence and absence of the same quality. Now, whenever we bring both these Laws of Thought into explicit action by the method of substitution, we employ the Indirect Method of Inference. It will be found that we can treat not only those arguments already exhibited according to the direct method, but we can include an infinite multitude of other arguments which are incapable of solution by any other means.

Some philosophers, especially those of France, have held that the Indirect Method of Proof has a certain inferiority to the direct method, which should prevent our using it except when obliged. But there are many truths which we can prove only indirectly. We can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which have divisors, and the remarkable process known as Eratosthenes’ Sieve is the only mode by which we can select the prime numbers.[72] It bears a strong analogy to the indirect method here to be described. We can prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing that the contrary supposition inevitably leads to contradiction.[73] Many other demonstrations in various branches of the mathematical sciences proceed upon a like method. Now, if there is only one important truth which must be, and can only be, proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its comparative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment.

[72] See Horsley, *Philosophical Transactions*, 1772; vol. lxii. p. 327. Montucla, *Histoire des Mathematiques*, vol. i. p. 239. *Penny Cyclopædia*, article “Eratosthenes.”

[73] Euclid, Book x. Prop. 117.

*Simple Illustrations.*

In tracing out the powers and results of this method, we will begin with the simplest possible instance. Let us take a proposition of the common form, A = AB, say,

*A Metal is an Element,*

and let us investigate its full meaning. Any person who has had the least logical training, is aware that we can draw from the above proposition an apparently different one, namely,

*A Not-element is a Not-metal.*

While some logicians, as for instance De Morgan,[74] have considered the relation of these two propositions to be purely self-evident, and neither needing nor allowing analysis, a great many more persons, as I have observed while teaching logic, are at first unable to perceive the close connection between them. I believe that a true and complete system of logic will furnish a clear analysis of this process, which has been called *Contrapositive Conversion*; the full process is as follows:--

[74] *Philosophical Magazine*, December 1852; Fourth Series, vol. iv. p. 435, “On Indirect Demonstration.”

Firstly, by the Law of Duality we know that

*Not-element is either Metal or Not-metal.*

If it be metal, we know that it is by the premise *an element*; we should thus be supposing that the same thing is an element and a not-element, which is in opposition to the Law of Contradiction. According to the only other alternative, then, the not-element must be a not-metal.

To represent this process of inference symbolically we take the premise in the form

A = AB. (1)

We observe that by the Law of Duality the term not-B is thus described

*b* = A*b* ꖌ *ab*. (2)

For A in this proposition we substitute its description as given in (1), obtaining

*b* = AB*b* ꖌ *ab*.

But according to the Law of Contradiction the term AB*b* must be excluded from thought, or

AB*b* = 0.

Hence it results that *b* is either nothing at all, or it is *ab*; and the conclusion is

*b* = *ab*.

As it will often be necessary to refer to a conclusion of this kind I shall call it, as is usual, the *Contrapositive Proposition* of the original. The reader need hardly be cautioned to observe that from all A’s are B’s it does not follow that all not-A’s are not-B’s. For by the Law of Duality we have

*a* = *a*B ꖌ *ab*,

and it will not be found possible to make any substitution in this by our original premise A = AB. It still remains doubtful, therefore, whether not-metal is element or not-element.

The proof of the Contrapositive Proposition given above is exactly the same as that which Euclid applies in the case of geometrical notions. De Morgan describes Euclid’s process as follows[75]:--“From every not-B is not-A he produces Every A is B, thus: If it be possible, let this A be not-B, but every not-B is not-A, therefore this A is not-A, which is absurd: whence every A is B.” Now De Morgan thinks that this proof is entirely needless, because common logic gives the inference without the use of any geometrical reasoning. I conceive however that logic gives the inference only by an indirect process. De Morgan claims “to see identity in Every A is B and every not-B is not-A, by a process of thought prior to syllogism.” Whether prior to syllogism or not, I claim that it is not prior to the laws of thought and the process of substitutive inference, by which it may be undoubtedly demonstrated.

[75] *Philosophical Magazine*, Dec. 1852; p. 437.

*Employment of the Contrapositive Proposition.*

We can frequently employ the contrapositive form of a proposition by the method of substitution; and certain moods of the ancient syllogism, which we have hitherto passed over, may thus be satisfactorily comprehended in our system. Take for instance the following syllogism in the mood Camestres:--

“Whales are not true fish; for they do not respire water, whereas true fish do respire water.”

Let us take

A = whale B = true fish C = respiring water

The premises are of the forms

A = A*c* (1) B = BC (2)

Now, by the process of contraposition we obtain from the second premise

*c* = *bc*

and we can substitute this expression for *c* in (1), obtaining

A = A*bc*

or “Whales are not true fish, not respiring water.”

The mood Cesare does not really differ from Camestres except in the order of the premises, and it could be exhibited in an exactly similar manner.

The mood Baroko gave much trouble to the old logicians, who could not *reduce* it to the first figure in the same manner as the other moods, and were obliged to invent, specially for it and for Bokardo, a method of Indirect Reduction closely analogous to the indirect proof of Euclid. Now these moods require no exceptional treatment in this system. Let us take as an instance of Baroko, the argument

All heated solids give continuous spectra (1) Some nebulæ do not give continuous spectra (2) Therefore, some nebulæ are not heated solids (3)

Treating the little word some as an indeterminate adjective of selection, to which we assign a symbol like any other adjective, let

A = some B = nebulæ C = giving continuous spectra D = heated solids

The premises then become

D = DC (1) AB = AB*c* (2)

Now from (1) we obtain by the indirect method the contrapositive proposition

*c* = *cd*

and if we substitute this expression for *c* in (2) we have

AB = AB*cd*

the full meaning of which is that “some nebulæ do not give continuous spectra and are not heated solids.”

We might similarly apply the contrapositive in many other instances. Take the argument, “All fixed stars are self-luminous; but some of the heavenly bodies are not self-luminous, and are therefore not fixed stars.” Taking our terms

A = fixed stars B = self-luminous C = some D = heavenly bodies

we have the premises

A = AB, (1) CD = *b*CD (2)

Now from (1) we can draw the contrapositive

*b* = *ab*

and substituting this expression for *b* in (2) we obtain

CD = *ab*CD

which expresses the conclusion of the argument that some heavenly bodies are not fixed stars.

*Contrapositive of a Simple Identity.*

The reader should carefully note that when we apply the process of Indirect Inference to a simple identity of the form

A = B

we may obtain further results. If we wish to know what is the term not-B, we have as before, by the Law of Duality,

*b* = A*b* ꖌ *ab*

and substituting for A we obtain

*b* = B*b* ꖌ *ab* = *ab*.

But we may now also draw a second contrapositive; for we have

*a* = *a*B ꖌ *ab*,

and substituting for B its equivalent A we have

*a* = *a*A ꖌ *ab* = *ab*.

Hence from the single identity A = B we can draw the two propositions

*a* = *ab* *b* = *ab*,

and observing that these propositions have a common term *ab* we can make a new substitution, getting

*a* = *b*.

This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it is not a self-evident result, independent of the steps of deduction by which we have reached it. For where two classes are coincident like A and B, whatever is true of the one is true of the other; what is excluded from the one must be excluded from the other similarly. Now as *a* bears to A exactly the same relation that *b* bears to B, the identity of either pair follows from the identity of the other pair. In every identity, equality, or similarity, we may argue from the negative of the one side to the negative of the other. Thus at ordinary temperatures

Mercury = liquid-metal,

hence obviously

Not-mercury = not liquid-metal;

or since

Sirius = brightest fixed star,

it follows that whatever star is not the brightest is not Sirius, and *vice versâ*. Every correct definition is of the form A = B, and may often require to be applied in the equivalent negative form.

Let us take as an illustration of the mode of using this result the argument following:

Vowels are letters which can be sounded alone, (1) The letter *w* cannot be sounded alone; (2) Therefore the letter *w* is not a vowel. (3)

Here we have a definition (1), and a comparison of a thing with that definition (2), leading to exclusion of the thing from the class defined.

Taking the terms

A = vowel, B = letter which can be sounded alone, C = letter *w*,

the premises are plainly of the forms

A = B, (1) C = *b*C. (2)

Now by the Indirect method we obtain from (1) the Contrapositive

*b* = *a*,

and inserting in (2) the equivalent for *b* we have

C = *a*C, (3)

or “the letter *w* is not a vowel.”

*Miscellaneous Examples of the Method.*

We can apply the Indirect Method of Inference however many may be the terms involved or the premises containing those terms. As the working of the method is best learnt from examples, I will take a case of two premises forming the syllogism Barbara: thus

Iron is metal (1) Metal is element. (2)

If we want to ascertain what inference is possible concerning the term *Iron*, we develop the term by the Law of Duality. Iron must be either metal or not-metal; iron which is metal must be either element or not-element; and similarly iron which is not-metal must be either element or not-element. There are then altogether four alternatives among which the description of iron must be contained; thus

Iron, metal, element, (α) Iron, metal, not-element, (β) Iron, not-metal, element, (γ) Iron, not-metal, not-element. (δ)

Our first premise informs us that iron is a metal, and if we substitute this description in (γ) and (δ) we shall have self-contradictory combinations. Our second premise likewise informs us that metal is element, and applying this description to (β) we again have self-contradiction, so that there remains only (α) as a description of iron--our inference is

Iron = iron, metal, element.

To represent this process of reasoning in general symbols, let

A = iron B = metal C = element,

The premises of the problem take the forms

A = AB (1) B = BC. (2)

By the Law of Duality we have

A = AB ꖌ A*b* (3) A = AC ꖌ A*c*. (4)

Now, if we insert for A in the second side of (3) its description in (4), we obtain what I shall call the *development of A with respect to B and C*, namely

A = ABC ꖌ AB*c* ꖌ A*b*C ꖌ A*bc*. (5)

Wherever the letters A or B appear in the second side of (5) substitute their equivalents given in (1) and (2), and the results stated at full length are

A = ABC ꖌ ABC*c* ꖌ AB*b*C ꖌ AB*b*C*c*.

The last three alternatives break the Law of Contradiction, so that

A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC.

This conclusion is, indeed, no more than we could obtain by the direct process of substitution, that is by substituting for B in (1), its description in (2) as in p. 55; it is the characteristic of the Indirect process that it gives all possible logical conclusions, both those which we have previously obtained, and an immense number of others or which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the class *not-element* or *c*. By the Law of Duality we can develop *c* into four alternatives, thus

*c* = AB*c* ꖌ A*bc* ꖌ *a*B*c* ꖌ *abc*.

If we substitute for A and B as before, we get

*c* = ABC*c* ꖌ AB*bc* ꖌ *a*BC*c* ꖌ *abc*,

and, striking out the terms which break the Law of Contradiction, there remains

*c* = *abc*,

or what is not element is also not iron and not metal. This Indirect Method of Inference thus furnishes a complete solution of the following problem--*Given any number of logical premises or conditions, required the description of any class of objects, or of any term, as governed by those conditions.*

The steps of the process of inference may thus be concisely stated--

1. By the Law of Duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises.

2. For each term in these alternatives substitute its description as given in the premises.

3. Strike out every alternative which is then found to break the Law of Contradiction.

4. The remaining terms may be equated to the term in question as the desired description.

*Mr. Venn’s Problem.*

The need of some logical method more powerful and comprehensive than the old logic of Aristotle is strikingly illustrated by Mr. Venn in his most interesting and able article on Boole’s logic.[76] An easy example, originally got, as he says, by the aid of my method as simply described in the *Elementary Lessons in Logic*, was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic. It was answered by, at most, five or six of them. It was afterwards set, as an example on Boole’s method, to a small class who had attended a few lectures on the nature of these symbolic methods. It was readily answered by half or more of their number.

[76] *Mind*; a Quarterly Review of Psychology and Philosophy; October, 1876, vol. i. p. 487.

The problem was as follows:--“The members of a board were all of them either bondholders, or shareholders, but not both; and the bondholders as it happened, were all on the board. What conclusion can be drawn?” The conclusion wanted is, “No shareholders are bondholders.” Now, as Mr. Venn says, nothing can look simpler than the following reasoning, *when stated*:--“There can be no bondholders who are shareholders; for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements; nor off it, by the second.” Yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem.

By symbolic statement the problem is instantly solved. Taking

A = member of board B = bondholder C = shareholder

the premises are evidently

A = AB*c* ꖌ A*b*C B = AB.

The class C or shareholders may in respect of A and B be developed into four alternatives,

C = ABC ꖌ A*b*C ꖌ *a*BC ꖌ *ab*C.

But substituting for A in the first and for B in the third alternative we get

C = ABC*c* ꖌ AB*b*C ꖌ A*b*C ꖌ *a*ABC ꖌ *ab*C.

The first, second, and fourth alternatives in the above are self-contradictory combinations, and only these; striking them out there remain

C = A*b*C ꖌ *ab*C = *b*C,

the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn’s reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method.

*Abbreviation of the Process.*

Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a thorough solution of a logical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are

AB *a*B A*b* *ab*.

The term A appears in the first and second; B in the first and third; *a* in the third and fourth; and *b* in the second and fourth. Now if we have any premise, say

A = B,

we must ascertain which of these combinations will be rendered self-contradictory by substitution; the second and third will have to be struck out, and there will remain only

AB *ba*.

Hence we draw the following inferences

A = AB, B = AB, *a* = *ab*, *b* = *ab*.

Exactly the same method must be followed when a question involves a greater number of terms. Thus by the Law of Duality the three terms A, B, C, give rise to eight conceivable combinations, namely

ABC (α) *a*BC (ε) AB*c* (β) *a*B*c* (ζ) A*b*C (γ) *ab*C (η) A*bc* (δ) *abc*. (θ)

The development of the term A is formed by the first four of these; for B we must select (α), (β), (ε), (ζ); C consists of (α), (γ), (ε), (η); *b* of (γ), (δ), (η), (θ), and so on.

Now if we want to investigate completely the meaning of the premises

A = AB (1) B = BC (2)

we examine each of the eight combinations as regards each premise; (γ) and (δ) are contradicted by (1), and (β) and (ζ) by (2), so that there remain only

ABC (α) *a*BC (ε) *ab*C (η) *abc*. (θ)

To describe any term under the conditions of the premises (1) and (2), we have simply to draw out the proper combinations from this list; thus, A is represented only by ABC, that is to say

A = ABC, similarly *c* = *abc*.

For B we have two alternatives thus stated,

B = ABC ꖌ *a*BC;

and for *b* we have

*b* = *ab*C ꖌ *abc*.

When we have a problem involving four distinct terms we need to double the number of combinations, and as we add each new term the combinations become twice as numerous. Thus

A, B produce four combinations A, B, C, " eight " A, B, C, D " sixteen " A, B, C, D, E " thirty-two " A, B, C, D, E, F " sixty-four "

and so on.

I propose to call any such series of combinations the *Logical Alphabet*. It holds in logical science a position the importance of which cannot be exaggerated, and as we proceed from logical to mathematical considerations, it will become apparent that there is a close connection between these combinations and the fundamental theorems of mathematical science. For the convenience of the reader who may wish to employ the *Alphabet* in logical questions, I have had printed on the next page a complete series of the combinations up to those of six terms. At the very commencement, in the first column, is placed a single letter X, which might seem to be superfluous. This letter serves to denote that it is always some higher class which is divided up. Thus the combination AB really means ABX, or that part of some larger class, say X, which has the qualities of A and B present. The letter X is omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described.

The reader ought to bear in mind that though the Logical Alphabet seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, A*b*, *a*B, *ab* really mean that any class X is described by the following proposition,

X = XAB ꖌ XA*b* ꖌ X*a*B ꖌ X*ab*.

If we select the A’s, we obtain the following proposition

AX = XAB ꖌ XA*b*.

Thus whatever group of combinations we treat must be conceived as part of a higher class, *summum genus* or universe symbolised in the term X; but, bearing this in mind, it is needless to complicate our formulæ by always introducing the letter. All inference consists in passing from propositions to propositions, and combinations *per se* have no meaning. They are consequently to be regarded in all cases as forming parts of propositions.

THE LOGICAL ALPHABET.

I. II. III. IV. V. VI. VII. X AX AB ABC ABCD ABCDE ABCDEF *a*X A*b* AB*c* ABC*d* ABCD*e* ABCDE*f* *a*B A*b*C AB*c*D ABC*d*E ABCD*e*F *ab* A*bc* AB*cd* ABC*de* ABCD*ef* *a*BC A*b*CD AB*c*DE ABC*d*EF *a*B*c* A*b*C*d* AB*c*D*e* ABC*d*E*f* *ab*C A*bc*D AB*cd*E ABC*de*F *abc* Ab*cd* AB*cde* ABC*def* *a*BCD A*b*CDE AB*c*DEF *a*BC*d* A*b*CD*e* AB*c*DE*f* *a*B*c*D A*b*C*d*E AB*c*D*e*F *a*B*cd* A*b*C*de* AB*c*D*ef* *ab*CD A*bc*DE AB*cd*EF *ab*C*d* A*bc*D*e* AB*cd*E*f* *abc*D A*bcd*E AB*cde*F *abcd* A*bcde* AB*cdef* *a*BCDE A*b*CDEF *a*BCD*e* A*b*CDE*f* *a*BC*d*E A*b*CD*e*F *a*BC*de* A*b*CD*ef* *a*B*c*DE A*b*C*d*EF *a*B*c*D*e* A*b*C*d*E*f* *a*B*cd*E A*b*C*de*F *a*B*cde* A*b*C*def* *ab*CDE A*bc*DEF *ab*CD*e* A*bc*DE*f* *ab*C*d*E A*bc*D*e*F *ab*Cd*e* A*bc*D*ef* *abc*DE A*bcd*EF *abc*D*e* A*bcd*E*f* *abcd*E A*bcde*F *abcde* A*bcdef* *a*BCDEF *a*BCDE*f* *a*BCD*e*F *a*BCD*ef* *a*BC*d*EF *a*BC*d*E*f* *a*BC*de*F *a*BC*def* *a*B*c*DEF *a*B*c*DE*f* *a*B*c*D*e*F *a*B*c*D*ef* *a*B*cd*EF *a*B*cd*E*f* *a*B*cde*F *a*B*cdef* *ab*CDEF *ab*CDE*f* *ab*CD*e*F *ab*CD*ef* *ab*C*d*EF *ab*C*d*E*f* *ab*C*de*F *ab*C*def* *abc*DEF *abc*DE*f* *abc*D*e*F *abc*D*ef* *abcd*EF *abcd*E*f* *abcde*F *abcdef*

In a theoretical point of view we may conceive that the Logical Alphabet is infinitely extended. Every new quality or circumstance which can belong to an object, subdivides each combination or class, so that the number of such combinations, when unrestricted by logical conditions, is represented by an infinitely high power of two. The extremely rapid increase in the number of subdivisions obliges us to confine our attention to a few qualities at a time.

When contemplating the properties of this Alphabet I am often inclined to think that Pythagoras perceived the deep logical importance of duality; for while unity was the symbol of identity and harmony, he described the number two as the origin of contrasts, or the symbol of diversity, division and separation. The number four, or the *Tetractys*, was also regarded by him as one of the chief elements of existence, for it represented the generating virtue whence come all combinations. In one of the golden verses ascribed to Pythagoras, he conjures his pupil to be virtuous:[77]

“By him who stampt *The Four* upon the Mind, *The Four*, the fount of Nature’s endless stream.”

[77] Whewell, *History of the Inductive Sciences*, vol. i. p. 222.

Now four and the higher powers of duality do represent in this logical system the numbers of combinations which can be generated in the absence of logical restrictions. The followers of Pythagoras may have shrouded their master’s doctrines in mysterious and superstitious notions, but in many points these doctrines seem to have some basis in logical philosophy.

*The Logical Slate.*

To a person who has once comprehended the extreme significance and utility of the Logical Alphabet the indirect process of inference becomes reduced to the repetition of a few uniform operations of classification, selection, and elimination of contradictories. Logical deduction, even in the most complicated questions, becomes a matter of mere routine, and the amount of labour required is the only impediment, when once the meaning of the premises is rendered clear. But the amount of labour is often found to be considerable. The mere writing down of sixty-four combinations of six letters each is no small task, and, if we had a problem of five premises, each of the sixty-four combinations would have to be examined in connection with each premise. The requisite comparison is often of a very tedious character, and considerable chance of error intervenes.

I have given much attention, therefore, to lessening both the manual and mental labour of the process, and I shall describe several devices which may be adopted for saving trouble and risk of mistake.

In the first place, as the same sets of combinations occur over and over again in different problems, we may avoid the labour of writing them out by having the sets of letters ready printed upon small sheets of writing-paper. It has also been suggested by a correspondent that, if any one series of combinations were marked upon the margin of a sheet of paper, and a slit cut between each pair of combinations, it would be easy to fold down any particular combination, and thus strike it out of view. The combinations consistent with the premises would then remain in a broken series. This method answers sufficiently well for occasional use.

A more convenient mode, however, is to have the series of letters shown on p. 94, engraved upon a common school writing slate, of such a size, that the letters may occupy only about a third of the space on the left hand side of the slate. The conditions of the problem can then be written down on the unoccupied part of the slate, and the proper series of combinations being chosen, the contradictory combinations can be struck out with the pencil. I have used a slate of this kind, which I call a *Logical Slate*, for more than twelve years, and it has saved me much trouble. It is hardly possible to apply this process to problems of more than six terms, owing to the large number of combinations which would require examination.

*Abstraction of Indifferent Circumstances.*

There is a simple but highly important process of inference which enables us to abstract, eliminate or disregard all circumstances indifferently present and absent. Thus if I were to state that “a triangle is a three-sided rectilinear figure, either large or not large,” these two alternatives would be superfluous, because, by the Law of Duality, I know that everything must be either large or not large. To add the qualification gives no new knowledge, since the existence of the two alternatives will be understood in the absence of any information to the contrary. Accordingly, when two alternatives differ only as regards a single component term which is positive in one and negative in the other, we may reduce them to one term by striking out their indifferent part. It is really a process of substitution which enables us to do this; for having any proposition of the form

A = ABC ꖌ AB*c*, (1)

we know by the Law of Duality that

AB = ABC ꖌ AB*c*. (2)

As the second member of this is identical with the second member of (1) we may substitute, obtaining

A = AB.

This process of reducing useless alternatives may be applied again and again; for it is plain that

A = AB (CD ꖌ C*d* ꖌ *c*D ꖌ *cd*)

communicates no more information than that A is B. Abstraction of indifferent terms is in fact the converse process to that of development described in p. 89; and it is one of the most important operations in the whole sphere of reasoning.

The reader should observe that in the proposition

AC = BC

we cannot abstract C and infer

A = B;

but from

AC ꖌ A*c* = BC ꖌ B*c*

we may abstract all reference to the term C.

It ought to be carefully remarked, however, that alternatives which seem to be without meaning often imply important knowledge. Thus if I say that “a triangle is a three-sided rectilinear figure, with or without three equal angles,” the last alternatives really express a property of triangles, namely, that some triangles have three equal angles, and some do not have them. If we put P = “Some,” meaning by the indefinite adjective “Some,” one or more of the undefined properties of triangles with three equal angles, and take

A = triangle B = three-sided rectilinear figure C = with three equal angles,

then the knowledge implied is expressed in the two propositions

PA = PBC *p*A = *p*B*c*.

These may also be thrown into the form of one proposition, namely,

A = PBC ꖌ *p*B*c*;

but these alternatives cannot be reduced, and the proposition is quite different from

A = BC ꖌ B*c*.

*Illustrations of the Indirect Method.*

A great variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the Indirect Method. We can treat either a single premise or a series of premises.

Take in the first place a simple definition, such as “a triangle is a three-sided rectilinear figure.” Let

A = triangle B = three-sided C = rectilinear figure,

then the definition is of the form

A = BC.

If we take the series of eight combinations of three letters in the Logical Alphabet (p. 94) and strike out those which are inconsistent with the definition, we have the following result:--

ABC *a*B*c* *ab*C *abc.*

For the description of the class C we have

C = ABC ꖌ *ab*C,

that is, “a rectilinear figure is either a triangle and three-sided, or not a triangle and not three-sided.”

For the class *b* we have

*b* = *ab*C ꖌ *abc*.

To the second side of this we may apply the process of simplification by abstraction described in the last section; for by the Law of Duality

*ab* = *ab*C ꖌ *abc*;

and as we have two propositions identical in the second side of each we may substitute, getting

*b* = *ab*,

or what is not three-sided is not a triangle (whether it be rectilinear or not).

*Second Example.*

Let us treat by this method the following argument:--

“Blende is not an elementary substance; elementary substances are those which are undecomposable; blende, therefore, is decomposable.”

Taking our letters thus--

A = blende, B = elementary substance, C = undecomposable,

the premises are of the forms

A = A*b*, (1) B = C. (2)

No immediate substitution can be made; but if we take the contrapositive of (2) (see p. 86), namely

*b* = *c*, (3)

we can substitute in (1) obtaining the conclusion

A = A*c*.

But the same result may be obtained by taking the eight combinations of A, B, C, of the Logical Alphabet; it will be found that only three combinations, namely,

A*bc* *a*BC *abc*,

are consistent with the premises, whence it results that

A = A*bc*,

or by the process of Ellipsis before described (p. 57)

A = A*c*.

*Third Example.*

As a somewhat more complex example I take the argument thus stated, one which could not be thrown into the syllogistic form:--

“All metals except gold and silver are opaque; therefore what is not opaque is either gold or silver or is not-metal.”

There is more implied in this statement than is distinctly asserted, the full meaning being as follows:

All metals not gold or silver are opaque, (1) Gold is not opaque but is a metal, (2) Silver is not opaque but is a metal, (3) Gold is not silver. (4)

Taking our letters thus--

A = metal C = silver B = gold D = opaque,

we may state the premises in the forms

A*bc* = A*bc*D (1) B = AB*d* (2) C = AC*d* (3) B = B*c*. (4)

To obtain a complete solution of the question we take the sixteen combinations of A, B, C, D, and striking out those which are inconsistent with the premises, there remain only

AB*cd* A*b*C*d* A*bc*D *abc*D *abcd*.

The expression for not-opaque things consists of the three combinations containing *d*, thus

*d* = AB*cd* ꖌ A*b*C*d* ꖌ *abcd*, or *d* = A*d* (B*c* ꖌ *b*C) ꖌ *abcd*.

In ordinary language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not-gold, or else it is not-metal and neither gold nor silver.

*Fourth Example.*

A good example for the illustration of the Indirect Method is to be found in De Morgan’s *Formal Logic* (p. 123), the premises being substantially as follows:--

From A follows B, and from C follows D; but B and D are inconsistent with each other; therefore A and C are inconsistent.

The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly every C is a D; but B and D cannot occur together. The premises therefore appear to be of the forms

A = AB, (1) C = CD, (2) B = B*d*. (3)

On examining the series of sixteen combinations, only five are found to be consistent with the above conditions, namely,

AB*cd* *a*B*cd* *ab*CD *abc*D *abcd*.

In these combinations the only A which appears is joined to *c*, and similarly C is joined to *a*, or A is inconsistent with C.

*Fifth Example.*

A more complex argument, also given by De Morgan,[78] contains five terms, and is as stated below, except that the letters are altered.

Every A is one only of the two B or C; D is both B and C, except when B is E, and then it is neither; therefore no A is D.

[78] *Formal Logic*, p. 124. As Professor Croom Robertson has pointed out to me, the second and third premises may be thrown into a single proposition, D = D*e*BC ꖌ DE*bc*.

The meaning of the above premises is difficult to interpret, but seems to be capable of expression in the following symbolic forms--

A = AB*c* ꖌ A*b*C, (1) De = D*e*BC, (2) DE = DE*bc*. (3)

As five terms enter into these premises it is requisite to treat their thirty-two combinations, and it will be found that fourteen of them remain consistent with the premises, namely

AB*cd*E *a*BCD*e* *ab*C*d*E AB*cde* *a*BC*d*E *ab*C*de* A*b*C*d*E *a*BC*de* *abc*DE A*b*C*de* *a*B*cd*E *abcd*E *a*B*cde* *abcde*.

If we examine the first four combinations, all of which contain A, we find that they none of them contain D; or again, if we select those which contain D, we have only two, thus--

D = *a*BCD*e* ꖌ *abc*DE.

Hence it is clear that no A is D, and *vice versâ* no D is A. We might draw many other conclusions from the same premises; for instance--

DE = *abc*DE,

or D and E never meet but in the absence of A, B, and C.

*Fallacies analysed by the Indirect Method.*

It has been sufficiently shown, perhaps, that we can by the Indirect Method of Inference extract the whole truth from a series of propositions, and exhibit it anew in any required form of conclusion. But it may also need to be shown by examples that so long as we follow correctly the almost mechanical rules of the method, we cannot fall into any of the fallacies or paralogisms which are often committed in ordinary discussion. Let us take the example of a fallacious argument, previously treated by the Method of Direct Inference (p. 62),

Granite is not a sedimentary rock, (1) Basalt is not a sedimentary rock, (2)

and let us ascertain whether any precise conclusion can be drawn concerning the relation of granite and basalt. Taking as before

A = granite, B = sedimentary rock, C = basalt,

the premises become

A = A*b*, (1) C = C*b*. (2)

Of the eight conceivable combinations of A, B, C, five agree with these conditions, namely

A*b*C *a*B*c* A*bc* *ab*C *abc*.

Selecting the combinations which contain A, we find the description of granite to be

A = A*b*C ꖌ A*bc* = A*b*(C ꖌ *c*),

that is, granite is not a sedimentary rock, and is either basalt or not-basalt. If we want a description of basalt the answer is of like form

C = A*b*C ꖌ *ab*C = *b*C(A ꖌ *a*),

that is basalt is not a sedimentary rock, and is either granite or not-granite. As it is already perfectly evident that basalt must be either granite or not, and *vice versâ*, the premises fail to give us any information on the point, that is to say the Method of Indirect Inference saves us from falling into any fallacious conclusions. This example sufficiently illustrates both the fallacy of Negative premises and that of Undistributed Middle of the old logic.

The fallacy called the Illicit Process of the Major Term is also incapable of commission in following the rules of the method. Our example was (p. 65)

All planets are subject to gravity, (1) Fixed stars are not planets. (2)

The false conclusion is that “fixed stars are not subject to gravity.” The terms are

A = planet B = fixed star C = subject to gravity.

And the premises are A = AC, (1) B = *a*B. (2)

The combinations which remain uncontradicted on comparison with these premises are

A*b*C *a*B*c* *a*BC *ab*C *abc*.

For fixed star we have the description

B = *a*BC ꖌ *a*B*c*,

that is, “a fixed star is not a planet, but is either subject or not, as the case may be, to gravity.” Here we have no conclusion concerning the connection of fixed stars and gravity.

*The Logical Abacus.*

The Indirect Method of Inference has now been sufficiently described, and a careful examination of its powers will show that it is capable of giving a full analysis and solution of every question involving only logical relations. The chief difficulty of the method consists in the great number of combinations which may have to be examined; not only may the requisite labour become formidable, but a considerable chance of mistake arises. I have therefore given much attention to modes of facilitating the work, and have succeeded in reducing the method to an almost mechanical form. It soon appeared obvious that if the conceivable combinations of the Logical Alphabet, for any number of letters, instead of being printed in fixed order on a piece of paper or slate, were marked upon light movable pieces of wood, mechanical arrangements could readily be devised for selecting any required class of the combinations. The labour of comparison and rejection might thus be immensely reduced. This idea was first carried out in the Logical Abacus, which I have found useful in the lecture-room for exhibiting the complete solution of logical problems. A minute description of the construction and use of the Abacus, together with figures of the parts, has already been given in my essay called *The Substitution of Similars*,[79] and I will here give only a general description.

[79] Pp. 55–59, 81–86.

The Logical Abacus consists of a common school black-board placed in a sloping position and furnished with four horizontal and equi-distant ledges. The combinations of the letters shown in the first four columns of the Logical Alphabet are printed in somewhat large type, so that each letter is about an inch from the neighbouring one, but the letters are placed one above the other instead of being in horizontal lines as in p. 94. Each combination of letters is separately fixed to the surface of a thin slip of wood one inch broad and about one-eighth inch thick. Short steel pins are then driven in an inclined position into the wood. When a letter is a large capital representing a positive term, the pin is fixed in the upper part of its space; when the letter is a small italic representing a negative term, the pin is fixed in the lower part of the space. Now, if one of the series of combinations be ranged upon a ledge of the black-board, the sharp edge of a flat rule can be inserted beneath the pins belonging to any one letter--say A, so that all the combinations marked A can be lifted out and placed upon a separate ledge. Thus we have represented the act of thought which separates the class A from what is not-A. The operation can be repeated; out of the A’s we can in like manner select those which are B’s, obtaining the AB’s; and in like manner we may select any other classes such as the *a*B’s, the *ab*’s, or the *abc*’s.

If now we take the series of eight combinations of the letters A, B, C, *a*, *b*, *c*, and wish to analyse the argument anciently called Barbara, having the premises

A = AB (1) B = BC, (2)

we proceed as follows--We raise the combinations marked *a*, leaving the A’s behind; out of these A’s we move to a lower ledge such as are *b*’s, and to the remaining AB’s we join the *a*’s which have been raised. The result is that we have divided all the combinations into two classes, namely, the A*b*’s which are incapable of existing consistently with premise (1), and the combinations which are consistent with the premise. Turning now to the second premise, we raise out of those which agree with (1) the *b*’s, then we lower the B*c*’s; lastly we join the *b*’s to the BC’s. We now find our combinations arranged as below.

+---+-----+-----+-----+-----+-----+-----+-----+ | A | | | | *a* | | *a* | *a* | | B | | | | B | | *b* | *b* | | C | | | | C | | C | *c* | +---+-----+-----+-----+-----+-----+-----+-----+ | | A | A | A | | *a* | | | | | B | *b* | *b* | | B | | | | | *c* | C | *c* | | *c* | | | +---+-----+-----+-----+-----+-----+-----+-----+

The lower line contains all the combinations which are inconsistent with either premise; we have carried out in a mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or upon paper. Accordingly, from the combinations remaining in the upper line we can draw any inference which the premises yield. If we raise the A’s we find only one, and that is C, so that A must be C. If we select the *c*’s we again find only one, which is *a* and also *b*; thus we prove that not-C is not-A and not-B.

When a disjunctive proposition occurs among the premises the requisite movements become rather more complicated. Take the disjunctive argument

A is either B or C or D, A is not C and not D, Therefore A is B.

The premises are represented accurately as follows:--

A = AB ꖌ AC ꖌ AD (1) A = A*c* (2) A = A*d*. (3)

As there are four terms, we choose the series of sixteen combinations and place them on the highest ledge of the board but one. We raise the *a*’s and out of the A’s, which remain, we lower the *b*’s. But we are not to reject all the A*b*’s as contradictory, because by the first premise A’s may be either B’s or C’s or D’s. Accordingly out of the A*b*’s we must select the *c*’s, and out of these again the *d*’s, so that only A*bcd* will remain to be rejected finally. Joining all the other fifteen combinations together again, and proceeding to premise (2), we raise the *a*’s and lower the AC’s, and thus reject the combinations inconsistent with (2); similarly we reject the AD’s which are inconsistent with (3). It will be found that there remain, in addition to all the eight combinations containing *a*, only one containing A, namely

AB*cd*,

whence it is apparent that A must be B, the ordinary conclusion of the argument.

In my “Substitution of Similars” (pp. 56–59) I have described the working upon the Abacus of two other logical problems, which it would be tedious to repeat in this place.

*The Logical Machine.*

Although the Logical Abacus considerably reduced the labour of using the Indirect Method, it was not free from the possibility of error. I thought moreover that it would afford a conspicuous proof of the generality and power of the method if I could reduce it to a purely mechanical form. Logicians had long been accustomed to speak of Logic as an Organon or Instrument, and even Lord Bacon, while he rejected the old syllogistic logic, had insisted, in the second aphorism of his “New Instrument,” that the mind required some kind of systematic aid. In the kindred science of mathematics mechanical assistance of one kind or another had long been employed. Orreries, globes, mechanical clocks, and such like instruments, are really aids to calculation and are of considerable antiquity. The Arithmetical Abacus is still in common use in Russia and China. The calculating machine of Pascal is more than two centuries old, having been constructed in 1642–45. M. Thomas of Colmar manufactures an arithmetical machine on Pascal’s principles which is employed by engineers and others who need frequently to multiply or divide. To Babbage and Scheutz is due the merit of embodying the Calculus of Differences in a machine, which thus became capable of calculating the most complicated tables of figures. It seemed strange that in the more intricate science of quantity mechanism should be applicable, whereas in the simple science of qualitative reasoning, the syllogism was only called an instrument by a figure of speech. It is true that Swift satirically described the Professors of Laputa as in possession of a thinking machine, and in 1851 Mr. Alfred Smee actually proposed the construction of a Relational machine and a Differential machine, the first of which would be a mechanical dictionary and the second a mode of comparing ideas; but with these exceptions I have not yet met with so much as a suggestion of a reasoning machine. It may be added that Mr. Smee’s designs, though highly ingenious, appear to be impracticable, and in any case they do not attempt the performance of logical inference.[80]

[80] See his work called *The Process of Thought adapted to Words and Language, together with a Description of the Relational and Differential Machines*. Also *Philosophical Transactions*, [1870] vol. 160, p. 518.

The Logical Abacus soon suggested the notion of a Logical Machine, which, after two unsuccessful attempts, I succeeded in constructing in a comparatively simple and effective form. The details of the Logical Machine have been fully described by the aid of plates in the Philosophical Transactions,[81] and it would be needless to repeat the account of the somewhat intricate movements of the machine in this place.

[81] *Philosophical Transactions* [1870], vol. 160, p. 497. *Proceedings of the Royal Society*, vol. xviii. p. 166, Jan. 20, 1870. *Nature*, vol, i. p. 343.

The general appearance of the machine is shown in a plate facing the title-page of this volume. It somewhat resembles a very small upright piano or organ, and has a keyboard containing twenty-one keys. These keys are of two kinds, sixteen of them representing the terms or letters A, *a*, B, *b*, C, *c*, D, *d*, which have so often been employed in our logical notation. When letters occur on the left-hand side of a proposition, formerly called the subject, each is represented by a key on the left-hand half of the keyboard; but when they occur on the right-hand side, or as it used to be called the predicate of the proposition, the letter-keys on the right-hand side of the keyboard are the proper representatives. The five other keys may be called operation keys, to distinguish them from the letter or term keys. They stand for the stops, copula, and disjunctive conjunctions of a proposition. The middle key of all is the copula, to be pressed when the verb *is* or the sign = is met. The key to the extreme right-hand is called the Full Stop, because it should be pressed when a proposition is completed, in fact in the proper place of the full stop. The key to the extreme left-hand is used to terminate an argument or to restore the machine to its initial condition; it is called the Finis key. The last keys but one on the right and left complete the whole series, and represent the conjunction *or* in its unexclusive meaning, or the sign ꖌ which I have employed, according as it occurs in the right or left hand side of the proposition. The whole keyboard is arranged as shown on the next page--

+-+-----------------------------------+-+-----------------------------------+---+ | | |C| | | |F| Left-hand side of Proposition. |o| Right-hand side of Proposition. |F S| |i| |p| |u t| |n+---+---+---+---+---+---+---+---+---+u+---+---+---+---+---+---+---+---+---+l o| |i| | | | | | | | | |l| | | | | | | | | |l p| |s|ꖌ|*d*| D |*c*| C |*b*| B |*a*| A |a| A |*a*| B |*b*| C |*c*| D |*d*|ꖌ| .| |.|Or | | | | | | | | |.| | | | | | | | | Or| | +-+---+---+---+---+---+---+---+---+---+-+---+---+---+---+---+---+---+---+---+---+

To work the machine it is only requisite to press the keys in succession as indicated by the letters and signs of a symbolical proposition. All the premises of an argument are supposed to be reduced to the simple notation which has been employed in the previous pages. Taking then such a simple proposition as

A = AB,

we press the keys A (left), copula, A (right), B (right), and full stop.

If there be a second premise, for instance

B = BC,

we press in like manner the keys--

B (left), copula, B (right), C (right), full stop.

The process is exactly the same however numerous the premises may be. When they are completed the operator will see indicated on the face of the machine the exact combinations of letters which are consistent with the premises according to the principles of thought.

As shown in the figure opposite the title-page, the machine exhibits in front a Logical Alphabet of sixteen combinations, exactly like that of the Abacus, except that the letters of each combination are separated by a certain interval. After the above problem has been worked upon the machine the Logical Alphabet will have been modified so as to present the following appearance--

+-------------------------------------------------------+ | | +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | A | A | | | | | | |*a*|*a*| | |*a*|*a*|*a*|*a*| +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | | +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | B | B | | | | | | | B | B | | |*b*|*b*|*b*|*b*| +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | | +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | C | C | | | | | | | C | C | | | C | C |*c*|*c*| +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | | +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | D |*d*| | | | | | | D |*d*| | | D |*d*| D |*d*| +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ | | +-------------------------------------------------------+

The operator will readily collect the various conclusions in the manner described in previous pages, as, for instance that A is always C, that not-C is not-B and not-A; and not-B is not-A but either C or not-C. The results are thus to be read off exactly as in the case of the Logical Slate, or the Logical Abacus.

Disjunctive propositions are to be treated in an exactly similar manner. Thus, to work the premises

A = AB ꖌ AC B ꖌ C = BD ꖌ CD,

it is only necessary to press in succession the keys

A (left), copula, A (right), B, ꖌ, A, C, full stop. B (left), ꖌ, C, copula, B (right), D, ꖌ, C, D, full stop.

The combinations then remaining will be as follows

ABCD *a*BCD *abc*D AB*c*D *a*B*c*D *abcd.* A*c*CD *ab*CD

On pressing the left-hand key A, all the possible combinations which do not contain A will disappear, and the description of A may be gathered from what remain, namely that it is always D. The full-stop key restores all combinations consistent with the premises and any other selection may be made, as say not-D, which will be found to be always not-A, not-B, and not-C.

At the end of every problem, when no further questions need be addressed to the machine, we press the Finis key, which has the effect of bringing into view the whole of the conceivable combinations of the alphabet. This key in fact obliterates the conditions impressed upon the machine by moving back into their ordinary places those combinations which had been rejected as inconsistent with the premises. Before beginning any new problem it is requisite to observe that the whole sixteen combinations are visible. After the Finis key has been used the machine represents a mind endowed with powers of thought, but wholly devoid of knowledge. It would not in that condition give any answer but such as would consist in the primary laws of thought themselves. But when any proposition is worked upon the keys, the machine analyses and digests the meaning of it and becomes charged with the knowledge embodied in that proposition. Accordingly it is able to return as an answer any description of a term or class so far as furnished by that proposition in accordance with the Laws of Thought. The machine is thus the embodiment of a true logical system. The combinations are classified, selected or rejected, just as they should be by a reasoning mind, so that at each step in a problem, the Logical Alphabet represents the proper condition of a mind exempt from mistake. It cannot be asserted indeed that the machine entirely supersedes the agency of conscious thought; mental labour is required in interpreting the meaning of grammatical expressions, and in correctly impressing that meaning on the machine; it is further required in gathering the conclusion from the remaining combinations. Nevertheless the true process of logical inference is really accomplished in a purely mechanical manner.

It is worthy of remark that the machine can detect any self-contradiction existing between the premises presented to it; should the premises be self-contradictory it will be found that one or more of the letter-terms disappears entirely from the Logical Alphabet. Thus if we work the two propositions, A is B, and A is not-B, and then inquire for a description of A, the machine will refuse to give it by exhibiting no combination at all containing A. This result is in agreement with the law, which I have explained, that every term must have its negative (p. 74). Accordingly, whenever any one of the letters A, B, C, D, *a*, *b*, *c*, *d*, wholly disappears from the alphabet, it may be safely inferred that some act of self-contradiction has been committed.

It ought to be carefully observed that the logical machine cannot receive a simple identity of the form A = B except in the double form of A = B and B = A. To work the proposition A = B, it is therefore necessary to press the keys--

A (left), copula, B (right), full stop; B (left), copula, A (right), full stop.

The same double operation will be necessary whenever the proposition is not of the kind called a partial identity (p. 40). Thus AB = CD, AB = AC, A = B ꖌ C, A ꖌ B = C ꖌ D, all require to be read from both ends separately.

The proper rule for using the machine may in fact be given in the following way:--(1) *Read each proposition as it stands, and play the corresponding keys*: (2) *Convert the proposition and read and play the keys again in the transposed order of the terms.* So long as this rule is observed the true result must always be obtained. There can be no mistake. But it will be found that in the case of partial identities, and some other similar forms of propositions, the transposed reading has no effect upon the combinations of the Logical Alphabet. One reading is in such cases all that is practically needful. After some experience has been gained in the use of the machine, the worker naturally saves himself the trouble of the second reading when possible.

It is no doubt a remarkable fact that a simple identity cannot be impressed upon the machine except in the form of two partial identities, and this may be thought by some logicians to militate against the equational mode of representing propositions.

Before leaving the subject I may remark that these mechanical devices are not likely to possess much practical utility. We do not require in common life to be constantly solving complex logical questions. Even in mathematical calculation the ordinary rules of arithmetic are generally sufficient, and a calculating machine can only be used with advantage in peculiar cases. But the machine and abacus have nevertheless two important uses.

In the first place I hope that the time is not very far distant when the predominance of the ancient Aristotelian Logic will be a matter of history only, and when the teaching of logic will be placed on a footing more worthy of its supreme importance. It will then be found that the solution of logical questions is an exercise of mind at least as valuable and necessary as mathematical calculation. I believe that these mechanical devices, or something of the same kind, will then become useful for exhibiting to a class of students a clear and visible analysis of logical problems of any degree of complexity, the nature of each step being rendered plain to the eyes of the students. I often used the machine or abacus for this purpose in my class lectures while I was Professor of Logic at Owens College.

Secondly, the more immediate importance of the machine seems to consist in the unquestionable proof which it affords that correct views of the fundamental principles of reasoning have now been attained, although they were unknown to Aristotle and his followers. The time must come when the inevitable results of the admirable investigations of the late Dr. Boole must be recognised at their true value, and the plain and palpable form in which the machine presents those results will, I hope, hasten the time. Undoubtedly Boole’s life marks an era in the science of human reason. It may seem strange that it had remained for him first to set forth in its full extent the problem of logic, but I am not aware that anyone before him had treated logic as a symbolic method for evolving from any premises the description of any class whatsoever as defined by those premises. In spite of several serious errors into which he fell, it will probably be allowed that Boole discovered the true and general form of logic, and put the science substantially into the form which it must hold for evermore. He thus effected a reform with which there is hardly anything comparable in the history of logic between his time and the remote age of Aristotle.

Nevertheless, Boole’s quasi-mathematical system could hardly be regarded as a final and unexceptionable solution of the problem. Not only did it require the manipulation of mathematical symbols in a very intricate and perplexing manner, but the results when obtained were devoid of demonstrative force, because they turned upon the employment of unintelligible symbols, acquiring meaning only by analogy. I have also pointed out that he imported into his system a condition concerning the exclusive nature of alternatives (p. 70), which is not necessarily true of logical terms. I shall have to show in the next chapter that logic is really the basis of the whole science of mathematical reasoning, so that Boole inverted the true order of proof when he proposed to infer logical truths by algebraic processes. It is wonderful evidence of his mental power that by methods fundamentally false he should have succeeded in reaching true conclusions and widening the sphere of reason.

The mechanical performance of logical inference affords a demonstration both of the truth of Boole’s results and of the mistaken nature of his mode of deducing them. Conclusions which he could obtain only by pages of intricate calculation, are exhibited by the machine after one or two minutes of manipulation. And not only are those conclusions easily reached, but they are demonstratively true, because every step of the process involves nothing more obscure than the three fundamental Laws of Thought.

*The Order of Premises.*

Before quitting the subject of deductive reasoning, I may remark that the order in which the premises of an argument are placed is a matter of logical indifference. Much discussion has taken place at various times concerning the arrangement of the premises of a syllogism; and it has been generally held, in accordance with the opinion of Aristotle, that the so-called major premise, containing the major term, or the predicate of the conclusion, should stand first. This distinction however falls to the ground in our system, since the proposition is reduced to an identical form, in which there is no distinction of subject and predicate. In a strictly logical point of view the order of statement is wholly devoid of significance. The premises are simultaneously coexistent, and are not related to each other according to the properties of space and time. Just as the qualities of the same object are neither before nor after each other in nature (p. 33), and are only thought of in some one order owing to the limited capacity of mind, so the premises of an argument are neither before nor after each other, and are only thought of in succession because the mind cannot grasp many ideas at once. The combinations of the logical alphabet are exactly the same in whatever order the premises be treated on the logical slate or machine. Some difference may doubtless exist as regards convenience to human memory. The mind may take in the results of an argument more easily in one mode of statement than another, although there is no real difference in the logical results. But in this point of view I think that Aristotle and the old logicians were clearly wrong. It is more easy to gather the conclusion that “all A’s are C’s” from “all A’s are B’s and all B’s are C’s,” than from the same propositions in inverted order, “all B’s are C’s and all A’s are B’s.”

*The Equivalence of Propositions*.

One great advantage which arises from the study of this Indirect Method of Inference consists in the clear notion which we gain of the Equivalence of Propositions. The older logicians showed how from certain simple premises we might draw an inference, but they failed to point out whether that inference contained the whole, or only a part, of the information embodied in the premises. Any one proposition or group of propositions may be classed with respect to another proposition or group of propositions, as

1. Equivalent, 2. Inferrible, 3. Consistent, 4. Contradictory.

Taking the proposition “All men are mortals” as the original, then “All immortals are not men” is its equivalent; “Some mortals are men” is inferrible, or capable of inference, but is not equivalent; “All not-men are not mortals” cannot be inferred, but is consistent, that is, may be true at the same time; “All men are immortals” is of course contradictory.

One sufficient test of equivalence is capability of mutual inference. Thus from

All electrics = all non-conductors,

I can infer

All non-electrics = all conductors,

and *vice versâ* from the latter I can pass back to the former. In short, A = B is equivalent to *a* = *b*. Again, from the union of the two propositions, A = AB and B = AB, I get A = B, and from this I might as easily deduce the two with which I started. In this case one proposition is equivalent to two other propositions. There are in fact no less than four modes in which we may express the identity of two classes A and B, namely,

FIRST MODE. SECOND MODE. THIRD MODE. FOURTH MODE.

A = B *a* = *b* A = AB } *a* = *ab* } B = AB } *b* = *ab* }

The Indirect Method of Inference furnishes a universal and clear criterion as to the relationship of propositions. The import of a statement is always to be measured by the combinations of terms which it destroys. Hence two propositions are equivalent when they remove the same combinations from the Logical Alphabet, and neither more nor less. A proposition is inferrible but not equivalent to another when it removes some but not all the combinations which the other removes, and none except what this other removes. Again, propositions are consistent provided that they jointly allow each term and the negative of each term to remain somewhere in the Logical Alphabet. If after all the combinations inconsistent with two propositions are struck out, there still appears each of the letters A, *a*, B, *b*, C, *c*, D, *d*, which were there before, then no inconsistency between the propositions exists, although they may not be equivalent or even inferrible. Finally, contradictory propositions are those which taken together remove any one or more letter-terms from the Logical Alphabet.

What is true of single propositions applies also to groups of propositions, however large or complicated; that is to say, one group may be equivalent, inferrible, consistent, or contradictory as regards another, and we may similarly compare one proposition with a group of propositions.

To give in this place illustrations of all the four kinds of relation would require much space: as the examples given in previous sections or chapters may serve more or less to explain the relations of inference, consistency, and contradiction, I will only add a few instances of equivalent propositions or groups.

In the following list each proposition or group of propositions is exactly equivalent in meaning to the corresponding one in the other column, and the truth of this statement may be tested by working out the combinations of the alphabet, which ought to be found exactly the same in the case of each pair of equivalents.

A = A*b* . . . . . . . B = *a*B A = *b* . . . . . . . . *a* = B A = BC . . . . . . . . *a* = *b* ꖌ *c* A = AB ꖌ AC . . . . . . *b* = *ab* ꖌ A*b*C A ꖌB = C ꖌ D . . . . . . . *ab* = *cd* A ꖌ *c* = B ꖌ *d* . . . . . . *a*C = *b*D A = AB*c* ꖌ A*b*C . . .{ A = AB ꖌ AC { AB = AB*c*

A = B } { A = B B = C } . . . . . . . . . { A = C

A = AB } { A = AC B = BC }. . . . . . . . . { B = A ꖌ *a*BC

Although in these and many other cases the equivalents of certain propositions can readily be given, yet I believe that no uniform and infallible process can be pointed out by which the exact equivalents of premises can be ascertained. Ordinary deductive inference usually gives us only a portion of the contained information. It is true that the combinations consistent with a set of premises may always be thrown into the form of a proposition which must be logically equivalent to those premises; but the difficulty consists in detecting the other forms of propositions which will be equivalent to the premises. The task is here of a different character from any which we have yet attempted. It is in reality an inverse process, and is just as much more troublesome and uncertain than the direct process, as seeking is compared with hiding. Not only may several different answers equally apply, but there is no method of discovering any of those answers except by repeated trial. The problem which we have here met is really that of induction, the inverse of deduction; and, as I shall soon show, induction is always tentative, and, unless conducted with peculiar skill and insight, must be exceedingly laborious in cases of complexity.

De Morgan was unfortunately led by this equivalence of propositions into the most serious error of his ingenious system of Logic. He held that because the proposition “All A’s are all B’s,” is but another expression for the two propositions “All A’s are B’s” and “All B’s are A’s,” it must be a composite and not really an elementary form of proposition.[82] But on taking a general view of the equivalence of propositions such an objection seems to have no weight. Logicians have, with few exceptions, persistently upheld the original error of Aristotle in rejecting from their science the one simple relation of identity on which all more complex logical relations must really rest.

[82] *Syllabus of a proposed system of Logic*, §§ 57, 121, &c. *Formal Logic*, p. 66.

*The Nature of Inference.*

The question, What is Inference? is involved, even to the present day, in as much uncertainty as that ancient question, What is Truth? I shall in more than one part of this work endeavour to show that inference never does more than explicate, unfold, or develop the information contained in certain premises or facts. Neither in deductive nor inductive reasoning can we add a tittle to our implicit knowledge, which is like that contained in an unread book or a sealed letter. Sir W. Hamilton has well said, “Reasoning is the showing out explicitly that a proposition not granted or supposed, is implicitly contained in something different, which is granted or supposed.”[83]

[83] Lectures on Metaphysics, vol. iv. p. 369.

Professor Bowen has explained[84] with much clearness that the conclusion of an argument states explicitly what is virtually or implicitly thought. “The process of reasoning is not so much a mode of evolving a new truth, as it is of establishing or proving an old one, by showing how much was admitted in the concession of the two premises taken together.” It is true that the whole meaning of these statements rests upon that of such words as “explicit,” “implicit,” “virtual.” That is implicit which is wrapped up, and we render it explicit when we unfold it. Just as the conception of a circle involves a hundred important geometrical properties, all following from what we know, if we have acuteness to unfold the results, so every fact and statement involves more meaning than seems at first sight. Reasoning explicates or brings to conscious possession what was before unconscious. It does not create, nor does it destroy, but it transmutes and throws the same matter into a new form.

[84] Bowen, *Treatise on Logic*, Cambridge, U.S., 1866; p. 362.

The difficult question still remains, Where does novelty of form begin? Is it a case of inference when we pass from “Sincerity is the parent of truth” to “The parent of truth is sincerity?” The old logicians would have called this change *conversion*, one case of immediate inference. But as all identity is necessarily reciprocal, and the very meaning of such a proposition is that the two terms are identical in their signification, I fail to see any difference between the statements whatever. As well might we say that *x* = *y* and *y* = *x* are different equations.

Another point of difficulty is to decide when a change is merely grammatical and when it involves a real logical transformation. Between a *table of wood* and a *wooden table* there is no logical difference (p. 31), the adjective being merely a convenient substitute for the prepositional phrase. But it is uncertain to my mind whether the change from “All men are mortal” to “No men are not mortal” is purely grammatical. Logical change may perhaps be best described as consisting in the determination of a relation between certain classes of objects from a relation between certain other classes. Thus I consider it a truly logical inference when we pass from “All men are mortal” to “All immortals are not-men,” because the classes *immortals* and *not-men* are different from *mortals* and *men*, and yet the propositions contain at the bottom the very same truth, as shown in the combinations of the Logical Alphabet.

The passage from the qualitative to the quantitative mode of expressing a proposition is another kind of change which we must discriminate from true logical inference. We state the same truth when we say that “mortality belongs to all men,” as when we assert that “all men are mortals.” Here we do not pass from class to class, but from one kind of term, the abstract, to another kind, the concrete. But inference probably enters when we pass from either of the above propositions to the assertion that the class of immortal men is zero, or contains no objects.

It is of course a question of words to what processes we shall or shall not apply the name “inference,” and I have no wish to continue the trifling discussions which have already taken place upon the subject. What we need to do is to define accurately the sense in which we use the word “inference,” and to distinguish the relation of inferrible propositions from other possible relations. It seems to be sufficient to recognise four modes in which two apparently different propositions may be related. Thus two propositions may be--

1. *Tautologous* or *identical*, involving the same relation between the same terms and classes, and only differing in the order of statement; thus “Victoria is the Queen of England” is tautologous with “The Queen of England is Victoria.”

2. *Grammatically related*, when the classes or objects are the same and similarly related, and the only difference is in the words; thus “Victoria is the Queen of England” is grammatically equivalent to “Victoria is England’s Queen.”

3. *Equivalents* in qualitative and quantitative form, the classes being the same, but viewed in a different manner.

4. *Logically inferrible*, one from the other, or it may be *equivalent*, when the classes and relations are different, but involve the same knowledge of the possible combinations.

VII. INDUCTION

We enter in this chapter upon the second great department of logical method, that of Induction or the Inference of general from particular truths. It cannot be said that the Inductive process is of greater importance than the Deductive process already considered, because the latter process is absolutely essential to the existence of the former. Each is the complement and counterpart of the other. The principles of thought and existence which underlie them are at the bottom the same, just as subtraction of numbers necessarily rests upon the same principles as addition. Induction is, in fact, the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation, so that the question of relative importance cannot arise. Who thinks of asking whether addition or subtraction is the more important process in arithmetic? But at the same time much difference in difficulty may exist between a direct and inverse operation; the integral calculus, for instance, is infinitely more difficult than the differential calculus of which it is the inverse. Similarly, it must be allowed that inductive investigations are of a far higher degree of difficulty and complexity than any questions of deduction; and it is this fact no doubt which led some logicians, such as Francis Bacon, Locke, and J. S. Mill, to erroneous opinions concerning the exclusive importance of induction.

Hitherto we have been engaged in considering how from certain conditions, laws, or identities governing the combinations of qualities, we may deduce the nature of the combinations agreeing with those conditions. Our work has been to unfold the results of what is contained in any statements, and the process has been one of *Synthesis*. The terms or combinations of which the character has been determined have usually, though by no means always, involved more qualities, and therefore, by the relation of extension and intension, fewer objects than the terms in which they were described. The truths inferred were thus usually less general than the truths from which they were inferred.

In induction all is inverted. The truths to be ascertained are more general than the data from which they are drawn. The process by which they are reached is *analytical*, and consists in separating the complex combinations in which natural phenomena are presented to us, and determining the relations of separate qualities. Given events obeying certain unknown laws, we have to discover the laws obeyed. Instead of the comparatively easy task of finding what effects will follow from a given law, the effects are now given and the law is required. We have to interpret the will by which the conditions of creation were laid down.

*Induction an Inverse Operation*

I have already asserted that induction is the inverse operation of deduction, but the difference is one of such great importance that I must dwell upon it. There are many cases in which we can easily and infallibly do a certain thing but may have much trouble in undoing it. A person may walk into the most complicated labyrinth or the most extensive catacombs, and turn hither and thither at his will; it is when he wishes to return that doubt and difficulty commence. In entering, any path served him; in leaving, he must select certain definite paths, and in this selection he must either trust to memory of the way he entered or else make an exhaustive trial of all possible ways. The explorer entering a new country makes sure his line of return by barking the trees.

The same difficulty arises in many scientific processes. Given any two numbers, we may by a simple and infallible process obtain their product; but when a large number is given it is quite another matter to determine its factors. Can the reader say what two numbers multiplied together will produce the number 8,616,460,799? I think it unlikely that anyone but myself will ever know; for they are two large prime numbers, and can only be rediscovered by trying in succession a long series of prime divisors until the right one be fallen upon. The work would probably occupy a good computer for many weeks, but it did not occupy me many minutes to multiply the two factors together. Similarly there is no direct process for discovering whether any number is a prime or not; it is only by exhaustively trying all inferior numbers which could be divisors, that we can show there is none, and the labour of the process would be intolerable were it not performed systematically once for all in the process known as the Sieve of Eratosthenes, the results being registered in tables of prime numbers.

The immense difficulties which are encountered in the solution of algebraic equations afford another illustration. Given any algebraic factors, we can easily and infallibly arrive at the product; but given a product it is a matter of infinite difficulty to resolve it into factors. Given any series of quantities however numerous, there is very little trouble in making an equation which shall have those quantities as roots. Let *a*, *b*, *c*, *d*, &c., be the quantities; then

(*x* - *a*)(*x* - *b*)(*x* - *c*)(*x* - d) ... = 0

is the equation required, and we only need to multiply out the expression on the left hand by ordinary rules. But having given a complex algebraic expression equated to zero, it is a matter of exceeding difficulty to discover all the roots. Mathematicians have exhausted their highest powers in carrying the complete solution up to the fourth degree. In every other mathematical operation the inverse process is far more difficult than the direct process, subtraction than addition, division than multiplication, evolution than involution; but the difficulty increases vastly as the process becomes more complex. Differentiation, the direct process, is always capable of performance by fixed rules, but as these rules produce considerable variety of results, the inverse process of integration presents immense difficulties, and in an infinite majority of cases surpasses the present resources of mathematicians. There are no infallible and general rules for its accomplishment; it must be done by trial, by guesswork, or by remembering the results of differentiation, and using them as a guide.

Coming more nearly to our own immediate subject, exactly the same difficulty exists in determining the law which certain things obey. Given a general mathematical expression, we can infallibly ascertain its value for any required value of the variable. But I am not aware that mathematicians have ever attempted to lay down the rules of a process by which, having given certain numbers, one might discover a rational or precise formula from which they proceed. The reader may test his power of detecting a law, by contemplation of its results, if he, not being a mathematician, will attempt to point out the law obeyed by the following numbers:

1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, 3617/510, 43867/798, etc.

These numbers are sometimes in low terms, but unexpectedly spring up to high terms; in absolute magnitude they are very variable. They seem to set all regularity and method at defiance, and it is hardly to be supposed that anyone could, from contemplation of the numbers, have detected the relations between them. Yet they are derived from the most regular and symmetrical laws of relation, and are of the highest importance in mathematical analysis, being known as the numbers of Bernoulli.

Compare again the difficulty of decyphering with that of cyphering. Anyone can invent a secret language, and with a little steady labour can translate the longest letter into the character. But to decypher the letter, having no key to the signs adopted, is a wholly different matter. As the possible modes of secret writing are infinite in number and exceedingly various in kind, there is no direct mode of discovery whatever. Repeated trial, guided more or less by knowledge of the customary form of cypher, and resting entirely on the principles of probability and logical induction, is the only resource. A peculiar tact or skill is requisite for the process, and a few men, such as Wallis or Wheatstone, have attained great success.

Induction is the decyphering of the hidden meaning of natural phenomena. Given events which happen in certain definite combinations, we are required to point out the laws which govern those combinations. Any laws being supposed, we can, with ease and certainty, decide whether the phenomena obey those laws. But the laws which may exist are infinite in variety, so that the chances are immensely against mere random guessing. The difficulty is much increased by the fact that several laws will usually be in operation at the same time, the effects of which are complicated together. The only modes of discovery consist either in exhaustively trying a great number of supposed laws, a process which is exhaustive in more senses than one, or else in carefully contemplating the effects, endeavouring to remember cases in which like effects followed from known laws. In whatever manner we accomplish the discovery, it must be done by the more or less conscious application of the direct process of deduction.

The Logical Alphabet illustrates induction as well as deduction. In considering the Indirect Process of Inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premises. The inductive problem is just the inverse. Having given certain combinations of terms, we need to ascertain the propositions with which the combinations are consistent, and from which they may have proceeded. Now, if the reader contemplates the following combinations,

ABC *ab*C *a*BC *abc*,

he will probably remember at once that they belong to the premises A = AB, B = BC (p. 92). If not, he will require a few trials before he meets with the right answer, and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data. To test the facility with which he can solve this inductive problem, let him casually strike out any of the combinations of the fourth column of the Logical Alphabet, (p. 94), and say what laws the remaining combinations obey, observing that every one of the letter-terms and their negatives ought to appear in order to avoid self-contradiction in the premises (pp. 74, 111). Let him say, for instance, what laws are embodied in the combinations

ABC *a*BC A*bc* *ab*C.

The difficulty becomes much greater when more terms enter into the combinations. It would require some little examination to ascertain the complete conditions fulfilled in the combinations

AC*e* *ab*C*e* *a*BC*e* *abc*E. *a*B*cd*E

The reader may discover easily enough that the principal laws are C = *e*, and A = A*e*; but he would hardly discover without some trouble the remaining law, namely, that BD = BD*e*.

The difficulties encountered in the inductive investigations of nature, are of an exactly similar kind. We seldom observe any law in uninterrupted and undisguised operation. The acuteness of Aristotle and the ancient Greeks did not enable them to detect that all terrestrial bodies tend to fall towards the centre of the earth. A few nights of observation might have convinced an astronomer viewing the solar system from its centre, that the planets travelled round the sun; but the fact that our place of observation is one of the travelling planets, so complicates the apparent motions of the other bodies, that it required all the sagacity of Copernicus to prove the real simplicity of the planetary system. It is the same throughout nature; the laws may be simple, but their combined effects are not simple, and we have no clue to guide us through their intricacies. “It is the glory of God,” said Solomon, “to conceal a thing, but the glory of a king to search it out.” The laws of nature are the invaluable secrets which God has hidden, and it is the kingly prerogative of the philosopher to search them out by industry and sagacity.

*Inductive Problems for Solution by the Reader.*

In the first edition (vol. ii. p. 370) I gave a logical problem involving six terms, and requested readers to discover the laws governing the combinations given. I received satisfactory replies from readers both in the United States and in England. I formed the combinations deductively from four laws of correction, but my correspondents found that three simpler laws, equivalent to the four more complex ones, were the best answer; these laws are as follows: *a* = *ac*, *b* = *cd*, *d* = E*f*.

In case other readers should like to test their skill in the inductive or inverse problem, I give below several series of combinations forming problems of graduated difficulty.

PROBLEM I.

A B *c* A *b* C *a* B C

PROBLEM II.

A B C A *b* C *a* B C *a* B *c*

PROBLEM III.

A B C A *b* C *a* B C *a* B *c* *a* *b* *c*

PROBLEM IV.

A B C D A *b* *c* D *a* B *c* *d* *a* *b* C *d*

PROBLEM V.

A B C D A B C *d* A B *c* *d* A *b* C D A *b* *c* D *a* B C D *a* B *c* D *a* B *c* *d* *a* *b* C *d*

PROBLEM VI.

A B C D E A B C *d* *e* A B *c* D E A B *c* *d* *e* A *b* C D E *a* B C D E *a* B C *d* *e* *a* *b* C D E *a* *b* *c* *d* *e*

PROBLEM VII.

A *b* *c* D *e* *a* B C *d* E *a* *b* C *d* E

PROBLEM VIII.

A B C D E A B C D *e* A B C *d* *e* A B *c* *d* *e* A *b* C D E A *b* *c* *d* E A *b* *c* *d* *e* *a* B C D *e* *a* B C *d* *e* *a* B *c* D *e* *a* *b* C D *e* *a* *b* C *d* E *a* *b* *c* D *e* *a* *b* *c* *d* E

PROBLEM IX.

A B *c* D E F A B *c* D *e* F A *b* C D *e* *f* A *b* *c* D E *f* A *b* *c* D *e* *f* A *b* *c* *d* E F A *b* *c* *d* *e* F *a* B *c* D E F *a* B *c* D *e* F *a* B *c* *d* E F *a* *b* C D E F *a* *b* C D *e* F *a* *b* C D *e* *f* *a* *b* *c* D *e* *f* *a* *b* *c* D E *f* *a* *b* *c* *d* *e* F

PROBLEM X.

A B C D *e* F A B *c* D E *f* A *b* C D E F A *b* C D *e* F A *b* *c* D *e* F *a* B C D E *f* *a* B *c* D E *f* *a* *b* C D *e* F *a* *b* C *d* *e* F *a* *b* *c* D *e* *f* *a* *b* *c* *d* *e* *f*

*Induction of Simple Identities*.

Many important laws of nature are expressible in the form of simple identities, and I can at once adduce them as examples to illustrate what I have said of the difficulty of the inverse process of induction. Two phenomena are conjoined. Thus all gravitating matter is exactly coincident with all matter possessing inertia; where one property appears, the other likewise appears. All crystals of the cubical system, are all the crystals which do not doubly refract light. All exogenous plants are, with some exceptions, those which have two cotyledons or seed-leaves.

A little reflection will show that there is no direct and infallible process by which such complete coincidences may be discovered. Natural objects are aggregates of many qualities, and any one of those qualities may prove to be in close connection with some others. If each of a numerous group of objects is endowed with a hundred distinct physical or chemical qualities, there will be no less than (1/2)(100 × 99) or 4950 pairs of qualities, which may be connected, and it will evidently be a matter of great intricacy and labour to ascertain exactly which qualities are connected by any simple law.

One principal source of difficulty is that the finite powers of the human mind are not sufficient to compare by a single act any large group of objects with another large group. We cannot hold in the conscious possession of the mind at any one moment more than five or six different ideas. Hence we must treat any more complex group by successive acts of attention. The reader will perceive by an almost individual act of comparison that the words *Roma* and *Mora* contain the same letters. He may perhaps see at a glance whether the same is true of *Causal* and *Casual*, and of *Logica* and *Caligo*. To assure himself that the letters in *Astronomers* make *No more stars*, that *Serpens in akuleo* is an anagram of *Joannes Keplerus*, or *Great gun do us a sum* an anagram of *Augustus de Morgan*, it will certainly be necessary to break up the act of comparison into several successive acts. The process will acquire a double character, and will consist in ascertaining that each letter of the first group is among the letters of the second group, and *vice versâ*, that each letter of the second is among those of the first group. In the same way we can only prove that two long lists of names are identical, by showing that each name in one list occurs in the other, and *vice versâ*.

This process of comparison really consists in establishing two partial identities, which are, as already shown (p. 58), equivalent in conjunction to one simple identity. We first ascertain the truth of the two propositions A = AB, B = AB, and we then rise by substitution to the single law A = B.

There is another process, it is true, by which we may get to exactly the same result; for the two propositions A = AB, *a* = *ab* are also equivalent to the simple identity A = B. If then we can show that all objects included under A are included under B, and also that all objects not included under A are not included under B, our purpose is effected. By this process we should usually compare two lists if we are allowed to mark them. For each name in the first list we should strike off one in the second, and if, when the first list is exhausted, the second list is also exhausted, it follows that all names absent from the first must be absent from the second, and the coincidence must be complete.

These two modes of proving an identity are so closely allied that it is doubtful how far we can detect any difference in their powers and instances of application. The first method is perhaps more convenient when the phenomena to be compared are rare. Thus we prove that all the musical concords coincide with all the more simple numerical ratios, by showing that each concord arises from a simple ratio of undulations, and then showing that each simple ratio gives rise to one of the concords. To examine all the possible cases of discord or complex ratio of undulation would be impossible. By a happy stroke of induction Sir John Herschel discovered that all crystals of quartz which cause the plane of polarization of light to rotate are precisely those crystals which have plagihedral faces, that is, oblique faces on the corners of the prism unsymmetrical with the ordinary faces. This singular relation would be proved by observing that all plagihedral crystals possessed the power of rotation, and *vice versâ* all crystals possessing this power were plagihedral. But it might at the same time be noticed that all ordinary crystals were devoid of the power. There is no reason why we should not detect any of the four propositions A = AB, B = AB, *a* = *ab*, *b* = *ab*, all of which follow from A = B (p. 115).

Sometimes the terms of the identity may be singular objects; thus we observe that diamond is a combustible gem, and being unable to discover any other that is, we affirm--

Diamond = combustible gem.

In a similar manner we ascertain that

Mercury = metal liquid at ordinary temperatures, Substance of least density = substance of least atomic weight.

Two or three objects may occasionally enter into the induction, as when we learn that

Sodium ꖌ potassium = metal of less density than water,

Venus ꖌ Mercury ꖌ Mars = major planet devoid of satellites.

*Induction of Partial Identities*.

We found in the last section that the complete identity of two classes is almost always discovered not by direct observation of the fact, but by first establishing two partial identities. There are also a multitude of cases in which the partial identity of one class with another is the only relation to be discovered. Thus the most common of all inductive inferences consists in establishing the fact that all objects having the properties of A have also those of B, or that A = AB. To ascertain the truth of a proposition of this kind it is merely necessary to assemble together, mentally or physically, all the objects included under A, and then observe whether B is present in each of them, or, which is the same, whether it would be impossible to select from among them any not-B. Thus, if we mentally assemble together all the heavenly bodies which move with apparent rapidity, that is to say, the planets, we find that they all possess the property of not scintillating. We cannot analyse any vegetable substance without discovering that it contains carbon and hydrogen, but it is not true that all substances containing carbon and hydrogen are vegetable substances.

The great mass of scientific truths consists of propositions of this form A = AB. Thus in astronomy we learn that all the planets are spheroidal bodies; that they all revolve in one direction round the sun; that they all shine by reflected light; that they all obey the law of gravitation. But of course it is not to be asserted that all bodies obeying the law of gravitation, or shining by reflected light, or revolving in a particular direction, or being spheroidal in form, are planets. In other sciences we have immense numbers of propositions of the same form, as, for instance, all substances in becoming gaseous absorb heat; all metals are elements; they are all good conductors of heat and electricity; all the alkaline metals are monad elements; all foraminifera are marine organisms; all parasitic animals are non-mammalian; lightning never issues from stratous clouds; pumice never occurs where only Labrador felspar is present; milkmaids do not suffer from small-pox; and, in the works of Darwin, scientific importance may attach even to such an apparently trifling observation as that “white tom-cats having blue eyes are deaf.”

The process of inference by which all such truths are obtained may readily be exhibited in a precise symbolic form. We must have one premise specifying in a disjunctive form all the possible individuals which belong to a class; we resolve the class, in short, into its constituents. We then need a number of propositions, each of which affirms that one of the individuals possesses a certain property. Thus the premises must be of the forms

A = B ꖌ C ꖌ D ꖌ .... ꖌ P ꖌ Q B = BX C = CX ... ... ... ... Q = QX.

Now, if we substitute for each alternative of the first premise its description as found among the succeeding premises, we obtain

A = BX ꖌ CX ꖌ .... ꖌ PX ꖌ QX

or

A = (B ꖌ C ꖌ .... ꖌ Q)X

But for the aggregate of alternatives we may now substitute their equivalent as given in the first premise, namely A, so that we get the required result:

A = AX.

We should have reached the same result if the first premise had been of the form

A = AB ꖌ AC ꖌ .... ꖌ AQ.

We can always prove a proposition, if we find it more convenient, by proving its equivalent. To assert that all not-B’s are not-A’s, is exactly the same as to assert that all A’s are B’s. Accordingly we may ascertain that A = AB by first ascertaining that *b* = *ab*. If we observe, for instance, that all substances which are not solids are also not capable of double refraction, it follows necessarily that all double refracting substances are solids. We may convince ourselves that all electric substances are nonconductors of electricity, by reflecting that all good conductors do not, and in fact cannot, retain electric excitation. When we come to questions of probability it will be found desirable to prove, as far as possible, both the original proposition and its equivalent, as there is then an increased area of observation.

The number of alternatives which may arise in the division of a class varies greatly, and may be any number from two upwards. Thus it is probable that every substance is either magnetic or diamagnetic, and no substance can be both at the same time. The division then must be made in the form

A = AB*c* ꖌ A*b*C.

If now we can prove that all magnetic substances are capable of polarity, say B = BD, and also that all diamagnetic substances are capable of polarity, C = CD, it follows by substitution that all substances are capable of polarity, or A = AD. We commonly divide the class substance into the three subclasses, solid, liquid, and gas; and if we can show that in each of these forms it obeys Carnot’s thermodynamic law, it follows that all substances obey that law. Similarly we may show that all vertebrate animals possess red blood, if we can show separately that fish, reptiles, birds, marsupials, and mammals possess red blood, there being, as far as is known, only five principal subclasses of vertebrata.

Our inductions will often be embarrassed by exceptions, real or apparent. We might affirm that all gems are incombustible were not diamonds undoubtedly combustible. Nothing seems more evident than that all the metals are opaque until we examine them in fine films, when gold and silver are found to be transparent. All plants absorb carbonic acid except certain fungi; all the bodies of the planetary system have a progressive motion from west to east, except the satellites of Uranus and Neptune. Even some of the profoundest laws of matter are not quite universal; all solids expand by heat except india-rubber, and possibly a few other substances; all liquids which have been tested expand by heat except water below 4° C. and fused bismuth; all gases have a coefficient of expansion increasing with the temperature, except hydrogen. In a later chapter I shall consider how such anomalous cases may be regarded and classified; here we have only to express them in a consistent manner by our notation.

Let us take the case of the transparency of metals, and assign the terms thus:--

A = metal D = iron B = gold E, F, &c. = copper, lead, &c. C = silver X = opaque.

Our premises will be

A = B ꖌ C ꖌ D ꖌ E, &c. B = B*x* C = C*x* D = DX E = EX,

and so on for the rest of the metals. Now evidently

A*bc* = (D ꖌ E ꖌ F ꖌ ...)*bc*,

and by substitution as before we shall obtain

A*bc* = A*bc*X,

or in words, “All metals not gold nor silver are opaque;” at the same time we have

A(B ꖌ C) = AB ꖌ AC = AB*x* ꖌ AC*x* = A(B ꖌ C)*x*,

or “Metals which are either gold or silver are not opaque.”

In some cases the problem of induction assumes a much higher degree of complexity. If we examine the properties of crystallized substances we may find some properties which are common to all, as cleavage or fracture in definite planes; but it would soon become requisite to break up the class into several minor ones. We should divide crystals according to the seven accepted systems--and we should then find that crystals of each system possess many common properties. Thus crystals of the Regular or Cubical system expand equally by heat, conduct heat and electricity with uniform rapidity, and are of like elasticity in all directions; they have but one index of refraction for light; and every facet is repeated in like relation to each of the three axes. Crystals of the system having one principal axis will be found to possess the various physical powers of conduction, refraction, elasticity, &c., uniformly in directions perpendicular to the principal axis; in other directions their properties vary according to complicated laws. The remaining systems in which the crystals possess three unequal axes, or have inclined axes, exhibit still more complicated results, the effects of the crystal upon light, heat, electricity, &c., varying in all directions. But when we pursue induction into the intricacies of its application to nature we really enter upon the subject of classification, which we must take up again in a later part of this work.

*Solution of the Inverse or Inductive Problem, involving Two Classes*.

It is now plain that Induction consists in passing back from a series of combinations to the laws by which such combinations are governed. The natural law that all metals are conductors of electricity really means that in nature we find three classes of objects, namely--

1. Metals, conductors; 2. Not-metals, conductors; 3. Not-metals, not-conductors.

It comes to the same thing if we say that it excludes the existence of the class, “metals not-conductors.” In the same way every other law or group of laws will really mean the exclusion from existence of certain combinations of the things, circumstances or phenomena governed by those laws. Now in logic, strictly speaking, we treat not the phenomena, nor the laws, but the general forms of the laws; and a little consideration will show that for a finite number of things the possible number of forms or kinds of law governing them must also be finite. Using general terms, we know that A and B can be present or absent in four ways and no more--thus:

AB, A*b*, *a*B, *ab*;

therefore every possible law which can exist concerning the relation of A and B must be marked by the exclusion of one or more of the above combinations. The number of possible laws then cannot exceed the number of selections which we can make from these four combinations. Since each combination may be present or absent, the number of cases to be considered is 2 × 2 × 2 × 2, or sixteen; and these cases are all shown in the following table, in which the sign 0 indicates absence or non-existence of the combination shown at the left-hand column in the same line, and the mark 1 its presence:--

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 * * * * * * * AB 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 A*b* 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 *a*B 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 *ab* 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Thus in column sixteen we find that all the conceivable combinations are present, which means that there are no special laws in existence in such a case, and that the combinations are governed only by the universal Laws of Identity and Difference. The example of metals and conductors of electricity would be represented by the twelfth column; and every other mode in which two things or qualities might present themselves is shown in one or other of the columns. More than half the cases may indeed be at once rejected, because they involve the entire absence of a term or its negative. It has been shown to be a logical principle that every term must have its negative (p. 111), and when this is not the case, inconsistency between the conditions of combination must exist. Thus if we laid down the two following propositions, “Graphite conducts electricity,” and “Graphite does not conduct electricity,” it would amount to asserting the impossibility of graphite existing at all; or in general terms, A is B and A is not B result in destroying altogether the combinations containing A, a case shown in the fourth column of the above table. We therefore restrict our attention to those cases which may be represented in natural phenomena when at least two combinations are present, and which correspond to those columns of the table in which each of A, *a*, B, *b* appears. These cases are shown in the columns marked with an asterisk.

We find that seven cases remain for examination, thus characterised--

Four cases exhibiting three combinations, Two cases exhibiting two combinations, One case exhibiting four combinations.

It has already been pointed out that a proposition of the form A = AB destroys one combination, A*b*, so that this is the form of law applying to the twelfth column. But by changing one or more of the terms in A = AB into its negative, or by interchanging A and B, *a* and *b*, we obtain no less than eight different varieties of the one form; thus--

12th case. 8th case. 15th case. 14th case. A = AB A = A*b* *a* = *a*B *a* = *ab* *b* = *ab* B = *a*B *b* = A*b* B = AB

The reader of the preceding sections will see that each proposition in the lower line is logically equivalent to, and is in fact the contrapositive of, that above it (p. 83). Thus the propositions A = A*b* and B = *a*B both give the same combinations, shown in the eighth column of the table, and trial shows that the twelfth, eighth, fifteenth and fourteenth columns are thus accounted for. We come to this conclusion then--*The general form of proposition* A = AB *admits of four logically distinct varieties, each capable of expression in two modes*.

In two columns of the table, namely the seventh and tenth, we observe that two combinations are missing. Now a simple identity A = B renders impossible both A*b* and *a*B, accounting for the tenth case; and if we change B into *b* the identity A = *b* accounts for the seventh case. There may indeed be two other varieties of the simple identity, namely *a* = *b* and *a* = B; but it has already been shown repeatedly that these are equivalent respectively to A = B and A = *b* (p. 115). As the sixteenth column has already been accounted for as governed by no special conditions, we come to the following general conclusion:--The laws governing the combinations of two terms must be capable of expression either in a partial identity or a simple identity; the partial identity is capable of only four logically distinct varieties, and the simple identity of two. Every logical relation between two terms must be expressed in one of these six forms of law, or must be logically equivalent to one of them.

In short, we may conclude that in treating of partial and complete identity, we have exhaustively treated the modes in which two terms or classes of objects can be related. Of any two classes it can be said that one must either be included in the other, or must be identical with it, or a like relation must exist between one class and the negative of the other. We have thus completely solved the inverse logical problem concerning two terms.[85]

[85] The contents of this and the following section nearly correspond with those of a paper read before the Manchester Literary and Philosophical Society on December 26th, 1871. See Proceedings of the Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. pp. 119–130.

*The Inverse Logical Problem involving Three Classes.*

No sooner do we introduce into the problem a third term C, than the investigation assumes a far more complex character, so that some readers may prefer to pass over this section. Three terms and their negatives may be combined, as we have frequently seen, in eight different combinations, and the effect of laws or logical conditions is to destroy any one or more of these combinations. Now we may make selections from eight things in 2^{8} or 256 ways; so that we have no less than 256 different cases to treat, and the complete solution is at least fifty times as troublesome as with two terms. Many series of combinations, indeed, are contradictory, as in the simpler problem, and may be passed over, the test of consistency being that each of the letters A, B, C, *a*, *b*, *c*, shall appear somewhere in the series of combinations.

My mode of solving the problem was as follows:--Having written out the whole of the 256 series of combinations, I examined them separately and struck out such as did not fulfil the test of consistency. I then chose some form of proposition involving two or three terms, and varied it in every possible manner, both by the circular interchange of letters (A, B, C into B, C, A and then into C, A, B), and by the substitution for any one or more of the terms of the corresponding negative terms. For instance, the proposition AB = ABC can be first varied by circular interchange so as to give BC = BCA and then CA = CAB. Each of these three can then be thrown into eight varieties by negative change. Thus AB = ABC gives *a*B = *a*BC, A*b* = A*b*C, AB = AB*c*, *ab* = *ab*C, and so on. Thus there may possibly exist no less than twenty-four varieties of the law having the general form AB = ABC, meaning that whatever has the properties of A and B has those also of C. It by no means follows that some of the varieties may not be equivalent to others; and trial shows, in fact, that AB = ABC is exactly the same in meaning as A*c* = A*bc* or B*c* = B*ca*. Thus the law in question has but eight varieties of distinct logical meaning. I now ascertain by actual deductive reasoning which of the 256 series of combinations result from each of these distinct laws, and mark them off as soon as found. I then proceed to some other form of law, for instance A = ABC, meaning that whatever has the qualities of A has those also of B and C. I find that it admits of twenty-four variations, all of which are found to be logically distinct; the combinations being worked out, I am able to mark off twenty-four more of the list of 256 series. I proceed in this way to work out the results of every form of law which I can find or invent. If in the course of this work I obtain any series of combinations which had been previously marked off, I learn at once that the law giving these combinations is logically equivalent to some law previously treated. It may be safely inferred that every variety of the apparently new law will coincide in meaning with some variety of the former expression of the same law. I have sufficiently verified this assumption in some cases, and have never found it lead to error. Thus as AB = ABC is equivalent to A*c* = A*bc*, so we find that *ab* = *ab*C is equivalent to *ac* = *ac*B.

Among the laws treated were the two A = AB and A = B which involve only two terms, because it may of course happen that among three things two only are in special logical relation, and the third independent; and the series of combinations representing such cases of relation are sure to occur in the complete enumeration. All single propositions which I could invent having been treated, pairs of propositions were next investigated. Thus we have the relations, “All A’s are B’s, and all B’s are C’s,” of which the old logical syllogism is the development. We may also have “all A’s are all B’s, and all B’s are C’s,” or even “all A’s are all B’s, and all B’s are all C’s.” All such premises admit of variations, greater or less in number, the logical distinctness of which can only be determined by trial in detail. Disjunctive propositions either singly or in pairs were also treated, but were often found to be equivalent to other propositions of a simpler form; thus A = ABC ꖌ A*bc* is exactly the same in meaning as AB = AC.

This mode of exhaustive trial bears some analogy to that ancient mathematical process called the Sieve of Eratosthenes. Having taken a long series of the natural numbers, Eratosthenes is said to have calculated out in succession all the multiples of every number, and to have marked them off, so that at last the prime numbers alone remained, and the factors of every number were exhaustively discovered. My problem of 256 series of combinations is the logical analogue, the chief points of difference being that there is a limit to the number of cases, and that prime numbers have no analogue in logic, since every series of combinations corresponds to a law or group of conditions. But the analogy is perfect in the point that they are both inverse processes. There is no mode of ascertaining that a number is prime but by showing that it is not the product of any assignable factors. So there is no mode of ascertaining what laws are embodied in any series of combinations but trying exhaustively the laws which would give them. Just as the results of Eratosthenes’ method have been worked out to a great extent and registered in tables for the convenience of other mathematicians, I have endeavoured to work out the inverse logical problem to the utmost extent which is at present practicable or useful.

I have thus found that there are altogether fifteen conditions or series of conditions which may govern the combinations of three terms, forming the premises of fifteen essentially different kinds of arguments. The following table contains a statement of these conditions, together with the numbers of combinations which are contradicted or destroyed by each, and the numbers of logically distinct variations of which the law is capable. There might be also added, as a sixteenth case, that case where no special logical condition exists, so that all the eight combinations remain.

+---------+-------------------------------+-----------+------------+ | | | Number of | Number of | |Reference| Propositions expressing the | distinct |combinations| | Number. | general type of the logical | logical |contradicted| | | conditions. |variations.| by each. | +---------+-------------------------------+-----------+------------+ | I. | A = B | 6 | 4 | | II. | A = AB | 12 | 2 | | III. | A = B, B = C | 4 | 6 | | IV. | A = B, B = BC | 24 | 5 | | V. | A = AB, B = BC | 24 | 4 | | VI. | A = BC | 24 | 4 | | VII. | A = ABC | 24 | 3 | | VIII. | AB = ABC | 8 | 1 | | IX. | A = AB, *a*B = *a*B*c* | 24 | 3 | | X. | A = ABC, *ab* = *ab*C | 8 | 4 | | XI. | AB = ABC, *ab* = *abc* | 4 | 2 | | XII. | AB = AC | 12 | 2 | | XIII. | A = BC ꖌ A*bc* | 8 | 3 | | XIV. | A = BC ꖌ *bc* | 2 | 4 | | XV. | A = ABC, *a* = B*c* ꖌ *b*C | 8 | 5 | +---------+-------------------------------+-----------+------------+

There are sixty-three series of combinations derived from self-contradictory premises, which with 192, the sum of the numbers of distinct logical variations stated in the third column of the table, and with the one case where there are no conditions or laws at all, make up the whole conceivable number of 256 series.

We learn from this table, for instance, that two propositions of the form A = AB, B = BC, which are such as constitute the premises of the old syllogism Barbara, exclude as impossible four of the eight combinations in which three terms may be united, and that these propositions are capable of taking twenty-four variations by transpositions of the terms or the introduction of negatives. This table then presents the results of a complete analysis of all the possible logical relations arising in the case of three terms, and the old syllogism forms but one out of fifteen typical forms. Generally speaking, every form can be converted into apparently different propositions; thus the fourth type A = B, B = BC may appear in the form A = ABC, *a* = *ab*, or again in the form of three propositions A = AB, B = BC, *a*B = *a*B*c*; but all these sets of premises yield identically the same series of combinations, and are therefore of equivalent logical meaning. The fifth type, or Barbara, can also be thrown into the equivalent forms A = ABC, *a*B = *a*BC and A = AC, B = A ꖌ *a*BC. In other cases I have obtained the very same logical conditions in four modes of statements. As regards mere appearance and form of statement, the number of possible premises would be very great, and difficult to exhibit exhaustively.

The most remarkable of all the types of logical condition is the fourteenth, namely, A = BC ꖌ *bc*. It is that which expresses the division of a genus into two doubly marked species, and might be illustrated by the example--“Component of the physical universe = matter, gravitating, or not-matter (ether), not-gravitating.” It is capable of only two distinct logical variations, namely, A = BC ꖌ *bc* and A = B*c* ꖌ *b*C. By transposition or negative change of the letters we can indeed obtain six different expressions of each of these propositions; but when their meanings are analysed, by working out the combinations, they are found to be logically equivalent to one or other of the above two. Thus the proposition A = BC ꖌ *bc* can be written in any of the following five other modes,

*a* = *b*C ꖌ B*c*, B = CA ꖌ *ca*, *b* = *c*A ꖌ C*a*, C = AB ꖌ *ab*, *c* = *a*B ꖌ A*b*.

I do not think it needful to publish at present the complete table of 193 series of combinations and the premises corresponding to each. Such a table enables us by mere inspection to learn the laws obeyed by any set of combinations of three things, and is to logic what a table of factors and prime numbers is to the theory of numbers, or a table of integrals to the higher mathematics. The table already given (p. 140) would enable a person with but little labour to discover the law of any combinations. If there be seven combinations (one contradicted) the law must be of the eighth type, and the proper variety will be apparent. If there be six combinations (two contradicted), either the second, eleventh, or twelfth type applies, and a certain number of trials will disclose the proper type and variety. If there be but two combinations the law must be of the third type, and so on.

The above investigations are complete as regards the possible logical relations of two or three terms. But when we attempt to apply the same kind of method to the relations of four or more terms, the labour becomes impracticably great. Four terms give sixteen combinations compatible with the laws of thought, and the number of possible selections of combinations is no less than 2^{16} or 65,536. The following table shows the extraordinary manner in which the number of possible logical relations increases with the number of terms involved.

+---------+-------------+---------------------------------------+ |Number of| Number of |Number of possible selections of combi-| | terms. | possible | nations corresponding to consistent | | |combinations.| or inconsistent logical relations. | +-----------------------+---------------------------------------+ | 2 | 4 | 16 | | 3 | 8 | 256 | | 4 | 16 | 65,536 | | 5 | 32 | 4,294,967,296 | | 6 | 64 | 18,446,744,073,709,551,616 | +---------+-------------+---------------------------------------+

Some years of continuous labour would be required to ascertain the types of laws which may govern the combinations of only four things, and but a small part of such laws would be exemplified or capable of practical application in science. The purely logical inverse problem, whereby we pass from combinations to their laws, is solved in the preceding pages, as far as it is likely to be for a long time to come; and it is almost impossible that it should ever be carried more than a single step further.

In the first edition, vol i. p. 158, I stated that I had not been able to discover any mode of calculating the number of cases in which inconsistency would be implied in the selection of combinations from the Logical Alphabet. The logical complexity of the problem appeared to be so great that the ordinary modes of calculating numbers of combinations failed, in my opinion, to give any aid, and exhaustive examination of the combinations in detail seemed to be the only method applicable. This opinion, however, was mistaken, for both Mr. R. B. Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers of inconsistent cases both for three and for four terms, without much difficulty. In the case of four terms they find that there are 1761 inconsistent selections and 63,774 consistent, which with one case where no condition exists, make up the total of 65,536 possible selections.

The inconsistent cases are distributed in the manner shown in the following table:--

+--------------+---------------------------------------------------+ | Number of | | | Combinations | 0 1 2 3 4 5 6 7 8 9 10, &c. | | remaining. | | +--------------+---------------------------------------------------+ | Number of | | | Inconsistent | 1 16 112 352 536 448 224 64 8 0 0, &c. | | Cases. | | +--------------+---------------------------------------------------+

When more than eight combinations of the Logical Alphabet (p. 94, column V.) remain unexcluded, there cannot be inconsistency. The whole numbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are given in the 17th line of the Arithmetical Triangle given further on in the Chapter on Combinations and Permutations, the sum of the numbers in that line being 65,536.

*Professor Clifford on the Types of Compound Statement involving Four Classes.*

In the first edition (vol. i. p. 163), I asserted that some years of labour would be required to ascertain even the precise number of types of law governing the combinations of four classes of things. Though I still believe that some years’ labour would be required to work out the types themselves, it is clearly a mistake to suppose that the *numbers* of such types cannot be calculated with a reasonable amount of labour, Professor W. K. Clifford having actually accomplished the task. His solution of the numerical problem involves the use of a complete new system of nomenclature and is far too intricate to be fully described here. I can only give a brief abstract of the results, and refer readers, who wish to follow out the reasoning, to the Proceedings of the Literary and Philosophical Society of Manchester, for the 9th January, 1877, vol. xvi., p. 88, where Professor Clifford’s paper is printed in full.

By a *simple statement* Professor Clifford means the denial of the existence of any single combination or *cross-division*, of the classes, as in ABCD = 0, or A*b*C*d* = 0. The denial of two or more such combinations is called a *compound statement*, and is further said to be *twofold*, *threefold*, &c., according to the number denied. Thus ABC = 0 is a twofold compound statement in regard to four classes, because it involves both ABCD = 0 and ABC*d* = 0. When two compound statements can be converted into one another by interchange of the classes, A, B, C, D, with each other or with their complementary classes, *a*, *b*, *c*, *d*, they are called *similar*, and all similar statements are said to belong to the same *type*.

Two statements are called *complementary* when they deny between them all the sixteen combinations without both denying any one; or, which is the same thing, when each denies just those combinations which the other permits to exist. It is obvious that when two statements are similar, the complementary statements will also be similar, and consequently for every type of *n*-fold statement, there is a complementary type of (16--*n*)-fold statement. It follows that we need only enumerate the types as far as the eighth order; for the types of more-than-eight-fold statement will already have been given as complementary to types of lower orders.

One combination, ABCD, may be converted into another A*b*C*d* by interchanging one or more of the classes with the complementary classes. The number of such changes is called the *distance*, which in the above case is 2. In two similar compound statements the distances of the combinations denied must be the same; but it does not follow that when all the distances are the same, the statements are similar. There is, however, only one example of two dissimilar statements having the same distances. When the distance is 4, the two combinations are said to be *obverse* to one another, and the statements denying them are called *obverse statements*, as in ABCD = 0 and *abcd* = 0 or again A*b*C*d* = 0 and *a*B*c*D = 0. When any one combination is given, called the *origin*, all the others may be grouped in respect of their relations to it as follows:--Four are at distance *one* from it, and may be called *proximates*; six are at distance *two*, and may be called *mediates*; four are at distance *three*, and may be called *ultimates*; finally the obverse is at distance *four*.

Origin and Six Obverse and four proximates. mediates. four ultimates. *ab*CD | *a*BCD A*bc*D | A*b*C*d* A*bcd* | \ | / | | \ | / | | \|/ | ABC*d*--ABCD--A*b*CD + *abc*D--*abcd*--*a*B*cd* | /|\ | | / | \ | | / | \ | AB*c*D *a*B*c*D | *a*BC*d* *ab*C*d*. | AB*cd*

It will be seen that the four proximates are respectively obverse to the four ultimates, and that the mediates form three pairs of obverses. Every proximate or ultimate is distant 1 and 3 respectively from such a pair of mediates.

Aided by this system of nomenclature Professor Clifford proceeds to an exhaustive enumeration of types, in which it is impossible to follow him. The results are as follows:--

1-fold statements 1 type } 2 " " 4 types} 3 " " 6 " } 4 " " 19 " } 159 5 " " 27 " } 6 " " 47 " } 7 " " 55 " } 8-fold statements 78 "

Now as each seven-fold or less-than-seven-fold statement is complementary to a nine-fold or more-than-nine-fold statement, it follows that the complete number of types will be 159 × 2 + 78 = 396.

It appears then that the types of statement concerning four classes are only about 26 times as numerous as those concerning three classes, fifteen in number, although the number of possible combinations is 256 times as great.

Professor Clifford informs me that the knowledge of the possible groupings of subdivisions of classes which he obtained by this inquiry has been of service to him in some applications of hyper-elliptic functions to which he has subsequently been led. Professor Cayley has since expressed his opinion that this line of investigation should be followed out, owing to the bearing of the theory of compound combinations upon the higher geometry.[86] It seems likely that many unexpected points of connection will in time be disclosed between the sciences of logic and mathematics.

[86] *Proceedings of the Manchester Literary and Philosophical Society*, 6th February, 1877, vol. xvi., p. 113.

*Distinction between Perfect and Imperfect Induction.*

We cannot proceed with advantage before noticing the extreme difference which exists between cases of perfect and those of imperfect induction. We call an induction *perfect* when all the objects or events which can possibly come under the class treated have been examined. But in the majority of cases it is impossible to collect together, or in any way to investigate, the properties of all portions of a substance or of all the individuals of a race. The number of objects would often be practically infinite, and the greater part of them might be beyond our reach, in the interior of the earth, or in the most distant parts of the Universe. In all such cases induction is *imperfect*, and is affected by more or less uncertainty. As some writers have fallen into much error concerning the functions and relative importance of these two branches of reasoning, I shall have to point out that--

1. Perfect Induction is a process absolutely requisite, both in the performance of imperfect induction and in the treatment of large bodies of facts of which our knowledge is complete.

2. Imperfect Induction is founded on Perfect Induction, but involves another process of inference of a widely different character.

It is certain that if I can draw any inference at all concerning objects not examined, it must be done on the data afforded by the objects which have been examined. If I judge that a distant star obeys the law of gravity, it must be because all other material objects sufficiently known to me obey that law. If I venture to assert that all ruminant animals have cloven hoofs, it is because all ruminant animals which have come under my notice have cloven hoofs. On the other hand, I cannot safely say that all cryptogamous plants possess a purely cellular structure, because some cryptogamous plants, which have been examined by botanists, have a partially vascular structure. The probability that a new cryptogam will be cellular only can be estimated, if at all, on the ground of the comparative numbers of known cryptogams which are and are not cellular. Thus the first step in every induction will consist in accurately summing up the number of instances of a particular phenomenon which have fallen under our observation. Adams and Leverrier, for instance, must have inferred that the undiscovered planet Neptune would obey Bode’s law, because *all the planets known at that time obeyed it*. On what principles the passage from the known to the apparently unknown is warranted, must be carefully discussed in the next section, and in various parts of this work.

It would be a great mistake, however, to suppose that Perfect Induction is in itself useless. Even when the enumeration of objects belonging to any class is complete, and admits of no inference to unexamined objects, the statement of our knowledge in a general proposition is a process of so much importance that we may consider it necessary. In many cases we may render our investigations exhaustive; all the teeth or bones of an animal; all the cells in a minute vegetable organ; all the caves in a mountain side; all the strata in a geological section; all the coins in a newly found hoard, may be so completely scrutinized that we may make some general assertion concerning them without fear of mistake. Every bone might be proved to contain phosphate of lime; every cell to enclose a nucleus; every cave to hide remains of extinct animals; every stratum to exhibit signs of marine origin; every coin to be of Roman manufacture. These are cases where our investigation is limited to a definite portion of matter, or a definite area on the earth’s surface.

There is another class of cases where induction is naturally and necessarily limited to a definite number of alternatives. Of the regular solids we can say without the least doubt that no one has more than twenty faces, thirty edges, and twenty corners; for by the principles of geometry we learn that there cannot exist more than five regular solids, of each of which we easily observe that the above statements are true. In the theory of numbers, an endless variety of perfect inductions might be made; we can show that no number less than sixty possesses so many divisors, and the like is true of 360; for it does not require a great amount of labour to ascertain and count all the divisors of numbers up to sixty or 360. I can assert that between 60,041 and 60,077 no prime number occurs, because the exhaustive examination of those who have constructed tables of prime numbers proves it to be so.

In matters of human appointment or history, we can frequently have a complete limitation of the number of instances to be included in an induction. We might show that the propositions of the third book of Euclid treat only of circles; that no part of the works of Galen mentions the fourth figure of the syllogism; that none of the other kings of England reigned so long as George III.; that Magna Charta has not been repealed by any subsequent statute; that the price of corn in England has never been so high since 1847 as it was in that year; that the price of the English funds has never been lower than it was on the 23rd of January, 1798, when it fell to 47-1/4.

It has been urged against this process of Perfect Induction that it gives no new information, and is merely a summing up in a brief form of a multitude of particulars. But mere abbreviation of mental labour is one of the most important aids we can enjoy in the acquisition of knowledge. The powers of the human mind are so limited that multiplicity of detail is alone sufficient to prevent its progress in many directions. Thought would be practically impossible if every separate fact had to be separately thought and treated. Economy of mental power may be considered one of the main conditions on which our elevated intellectual position depends. Mathematical processes are for the most part but abbreviations of the simpler acts of addition and subtraction. The invention of logarithms was one of the most striking additions ever made to human power: yet it was a mere abbreviation of operations which could have been done before had a sufficient amount of labour been available. Similar additions to our power will, it is hoped, be made from time to time; for the number of mathematical problems hitherto solved is but an indefinitely small fraction of those which await solution, because the labour they have hitherto demanded renders them impracticable. So it is throughout all regions of thought. The amount of our knowledge depends upon our power of bringing it within practicable compass. Unless we arrange and classify facts and condense them into general truths, they soon surpass our powers of memory, and serve but to confuse. Hence Perfect Induction, even as a process of abbreviation, is absolutely essential to any high degree of mental achievement.

*Transition from Perfect to Imperfect Induction.*

It is a question of profound difficulty on what grounds we are warranted in inferring the future from the present, or the nature of undiscovered objects from those which we have examined with our senses. We pass from Perfect to Imperfect Induction when once we allow our conclusion to apply, at all events apparently, beyond the data on which it was founded. In making such a step we seem to gain a net addition to our knowledge; for we learn the nature of what was unknown. We reap where we have never sown. We appear to possess the divine power of creating knowledge, and reaching with our mental arms far beyond the sphere of our own observation. I shall have, indeed, to point out certain methods of reasoning in which we do pass altogether beyond the sphere of the senses, and acquire accurate knowledge which observation could never have given; but it is not imperfect induction that accomplishes such a task. Of imperfect induction itself, I venture to assert that it never makes any real addition to our knowledge, in the meaning of the expression sometimes accepted. As in other cases of inference, it merely unfolds the information contained in past observations; it merely renders explicit what was implicit in previous experience. It transmutes, but certainly does not create knowledge.

There is no fact which I shall more constantly keep before the reader’s mind in the following pages than that the results of imperfect induction, however well authenticated and verified, are never more than probable. We never can be sure that the future will be as the present. We hang ever upon the will of the Creator: and it is only so far as He has created two things alike, or maintains the framework of the world unchanged from moment to moment, that our most careful inferences can be fulfilled. All predictions, all inferences which reach beyond their data, are purely hypothetical, and proceed on the assumption that new events will conform to the conditions detected in our observation of past events. No experience of finite duration can give an exhaustive knowledge of the forces which are in operation. There is thus a double uncertainty; even supposing the Universe as a whole to proceed unchanged, we do not really know the Universe as a whole. We know only a point in its infinite extent, and a moment in its infinite duration. We cannot be sure, then, that our observations have not escaped some fact, which will cause the future to be apparently different from the past; nor can we be sure that the future really will be the outcome of the past. We proceed then in all our inferences to unexamined objects and times on the assumptions--

1. That our past observation gives us a complete knowledge of what exists.

2. That the conditions of things which did exist will continue to be the conditions which will exist.

We shall often need to illustrate the character of our knowledge of nature by the simile of a ballot-box, so often employed by mathematical writers in the theory of probability. Nature is to us like an infinite ballot-box, the contents of which are being continually drawn, ball after ball, and exhibited to us. Science is but the careful observation of the succession in which balls of various character present themselves; we register the combinations, notice those which seem to be excluded from occurrence, and from the proportional frequency of those which appear we infer the probable character of future drawings. But under such circumstances certainty of prediction depends on two conditions:--

1. That we acquire a perfect knowledge of the comparative numbers of balls of each kind within the box.

2. That the contents of the ballot-box remain unchanged.

Of the latter assumption, or rather that concerning the constitution of the world which it illustrates, the logician or physicist can have nothing to say. As the Creation of the Universe is necessarily an act passing all experience and all conception, so any change in that Universe, or, it may be, a termination of it, must likewise be infinitely beyond the bounds of our mental faculties. No science no reasoning upon the subject, can have any validity; for without experience we are without the basis and materials of knowledge. It is the fundamental postulate accordingly of all inference concerning the future, that there shall be no arbitrary change in the subject of inference; of the probability or improbability of such a change I conceive that our faculties can give no estimate.

The other condition of inductive inference--that we acquire an approximately complete knowledge of the combinations in which events do occur, is in some degree within our power. There are branches of science in which phenomena seem to be governed by conditions of a most fixed and general character. We have ground in such cases for believing that the future occurrence of such phenomena can be calculated and predicted. But the whole question now becomes one of probability and improbability. We seem to leave the region of logic to enter one in which the number of events is the ground of inference. We do not really leave the region of logic; we only leave that where certainty, affirmative or negative, is the result, and the agreement or disagreement of qualities the means of inference. For the future, number and quantity will commonly enter into our processes of reasoning; but then I hold that number and quantity are but portions of the great logical domain. I venture to assert that number is wholly logical, both in its fundamental nature and in its developments. Quantity in all its forms is but a development of number. That which is mathematical is not the less logical; if anything it is more logical, in the sense that it presents logical results in a higher degree of complexity and variety.

Before proceeding then from Perfect to Imperfect Induction I must devote a portion of this work to treating the logical conditions of number. I shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves, and the probability or improbability of their occurrence under definite circumstances. It is in later parts of the work that I must endeavour to establish the notions which I have set forth upon the subject of Imperfect Induction, as applied in the investigation of Nature, which notions maybe thus briefly stated:--

1. Imperfect Induction entirely rests upon Perfect Induction for its materials.

2. The logical process by which we seem to pass directly from examined to unexamined cases consists in an inverse application of deductive inference, so that all reasoning may be said to be either directly or inversely deductive.

3. The result is always of a hypothetical character, and is never more than probable.

4. No net addition is ever made to our knowledge by reasoning; what we know of future events or unexamined objects is only the unfolded contents of our previous knowledge, and it becomes less probable as it is more boldly extended to remote cases.