The principles of science

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:

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,

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

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

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

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 = Ef.

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.

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—

In a similar manner we ascertain that

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

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

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

or

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:

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

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

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:‍—

Our premises will be

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

and by substitution as before we shall obtain

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

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—

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:

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:‍—

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—

It has already been pointed out that a proposition of the form A = AB destroys one combination, Ab, 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—

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 = Ab and B = aB 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 Ab and aB, 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

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 2⁸ 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 aB = aBC, Ab = AbC, AB = ABc, ab = abC, 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 Ac = Abc or Bc = Bca. 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 Ac = Abc, so we find that ab = abC is equivalent to ac = acB.

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 Abc 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.

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, aB = aBc; 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, aB = aBC and A = AC, B = A ꖌ aBC. 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 = Bc ꖌ bC. 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,