The principles of science

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

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.

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,

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,

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:‍—

Firstly, by the Law of Duality we know that

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

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

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

But according to the Law of Contradiction the term ABb must be excluded from thought, or

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

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

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 follows‍75:—“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.

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:‍—

Let us take

The premises are of the forms

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

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

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

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

The premises then become

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

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

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

we have the premises

Now from (1) we can draw the contrapositive

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

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

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,

and substituting for A we obtain

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

and substituting for B its equivalent A we have

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

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

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

hence obviously

or since

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:

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

the premises are plainly of the forms

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

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

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

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

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

To represent this process of reasoning in general symbols, let

The premises of the problem take the forms

By the Law of Duality we have

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

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

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

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

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

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

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.

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

the premises are evidently

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

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

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

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

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

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

Hence we draw the following inferences

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

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

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

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

For B we have two alternatives thus stated,

and for b we have

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

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, Ab, aB, ab really mean that any class X is described by the following proposition,

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

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.

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