Chapter VI: The Indirect Method of Inference (1)

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*|
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| |
+-------------------------------------------------------+