CHAPTER I.

INTRODUCTION.

Science arises from the discovery of Identity amidst Diversity. The process may be described in different words, but our language must always imply the presence of one common and necessary element. In every act of inference or scientific method we are engaged about a certain identity, sameness, similarity, likeness, resemblance, analogy, equivalence or equality apparent between two objects. It is doubtful whether an entirely isolated phenomenon could present itself to our notice, since there must always be some points of similarity between object and object. But in any case an isolated phenomenon could be studied to no useful purpose. The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account.

Nature is a spectacle continually exhibited to our senses, in which phenomena are mingled in combinations of endless variety and novelty. Wonder fixes the mind’s attention; memory stores up a record of each distinct impression; the powers of association bring forth the record when the like is felt again. By the higher faculties of judgment and reasoning the mind compares the new with the old, recognises essential identity, even when disguised by diverse circumstances, and expects to find again what was before experienced. It must be the ground of all reasoning and inference that *what is true of one thing will be true of its equivalent*, and that under carefully ascertained conditions *Nature repeats herself*.

Were this indeed a Chaotic Universe, the powers of mind employed in science would be useless to us. Did Chance wholly take the place of order, and did all phenomena come out of an *Infinite Lottery*, to use Condorcet’s expression, there could be no reason to expect the like result in like circumstances. It is possible to conceive a world in which no two things should be associated more often, in the long run, than any other two things. The frequent conjunction of any two events would then be purely fortuitous, and if we expected conjunctions to recur continually, we should be disappointed. In such a world we might recognise the same kind of phenomenon as it appeared from time to time, just as we might recognise a marked ball as it was occasionally drawn and re-drawn from a ballot-box; but the approach of any phenomenon would be in no way indicated by what had gone before, nor would it be a sign of what was to come after. In such a world knowledge would be no more than the memory of past coincidences, and the reasoning powers, if they existed at all, would give no clue to the nature of the present, and no presage of the future.

Happily the Universe in which we dwell is not the result of chance, and where chance seems to work it is our own deficient faculties which prevent us from recognising the operation of Law and of Design. In the material framework of this world, substances and forces present themselves in definite and stable combinations. Things are not in perpetual flux, as ancient philosophers held. Element remains element; iron changes not into gold. With suitable precautions we can calculate upon finding the same thing again endowed with the same properties. The constituents of the globe, indeed, appear in almost endless combinations; but each combination bears its fixed character, and when resolved is found to be the compound of definite substances. Misapprehensions must continually occur, owing to the limited extent of our experience. We can never have examined and registered possible existences so thoroughly as to be sure that no new ones will occur and frustrate our calculations. The same outward appearances may cover any amount of hidden differences which we have not yet suspected. To the variety of substances and powers diffused through nature at its creation, we should not suppose that our brief experience can assign a limit, and the necessary imperfection of our knowledge must be ever borne in mind.

Yet there is much to give us confidence in Science. The wider our experience, the more minute our examination of the globe, the greater the accumulation of well-reasoned knowledge,--the fewer in all probability will be the failures of inference compared with the successes. Exceptions to the prevalence of Law are gradually reduced to Law themselves. Certain deep similarities have been detected among the objects around us, and have never yet been found wanting. As the means of examining distant parts of the universe have been acquired, those similarities have been traced there as here. Other worlds and stellar systems may be almost incomprehensively different from ours in magnitude, condition and disposition of parts, and yet we detect there the same elements of which our own limbs are composed. The same natural laws can be detected in operation in every part of the universe within the scope of our instruments; and doubtless these laws are obeyed irrespective of distance, time, and circumstance.

It is the prerogative of Intellect to discover what is uniform and unchanging in the phenomena around us. So far as object is different from object, knowledge is useless and inference impossible. But so far as object resembles object, we can pass from one to the other. In proportion as resemblance is deeper and more general, the commanding powers of knowledge become more wonderful. Identity in one or other of its phases is thus always the bridge by which we pass in inference from case to case; and it is my purpose in this treatise to trace out the various forms in which the one same process of reasoning presents itself in the ever-growing achievements of Scientific Method.

*The Powers of Mind concerned in the Creation of Science.*

It is no part of the purpose of this work to investigate the nature of mind. People not uncommonly suppose that logic is a branch of psychology, because reasoning is a mental operation. On the same ground, however, we might argue that all the sciences are branches of psychology. As will be further explained, I adopt the opinion of Mr. Herbert Spencer, that logic is really an objective science, like mathematics or mechanics. Only in an incidental manner, then, need I point out that the mental powers employed in the acquisition of knowledge are probably three in number. They are substantially as Professor Bain has stated them[24]:--

[24] *The Senses and the Intellect*, Second Ed., pp. 5, 325, &c.

1. The Power of Discrimination. 2. The Power of Detecting Identity. 3. The Power of Retention.

We exert the first power in every act of perception. Hardly can we have a sensation or feeling unless we discriminate it from something else which preceded. Consciousness would almost seem to consist in the break between one state of mind and the next, just as an induced current of electricity arises from the beginning or the ending of the primary current. We are always engaged in discrimination; and the rudiment of thought which exists in the lower animals probably consists in their power of feeling difference and being agitated by it.

Yet had we the power of discrimination only, Science could not be created. To know that one feeling differs from another gives purely negative information. It cannot teach us what will happen. In such a state of intellect each sensation would stand out distinct from every other; there would be no tie, no bridge of affinity between them. We want a unifying power by which the present and the future may be linked to the past; and this seems to be accomplished by a different power of mind. Lord Bacon has pointed out that different men possess in very different degrees the powers of discrimination and identification. It may be said indeed that discrimination necessarily implies the action of the opposite process of identification; and so it doubtless does in negative points. But there is a rare property of mind which consists in penetrating the disguise of variety and seizing the common elements of sameness; and it is this property which furnishes the true measure of intellect. The name of “intellect” expresses the interlacing of the general and the single, which is the peculiar province of mind.[25] To *cogitate* is the Latin *coagitare*, resting on a like metaphor. Logic, also, is but another name for the same process, the peculiar work of reason; for λογος is derived from λεγειν, which like the Latin *legere* meant originally to gather. Plato said of this unifying power, that if he met the man who could detect *the one in the many*, he would follow him as a god.

[25] Max Müller, *Lectures on the Science of Language*, Second Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view of the etymological meaning of “intellect” is given above on the authority of Professor Max Müller. It seems to be opposed to the ordinary opinion, according to which the Latin *intelligere* means to choose between, to see a difference between, to discriminate, instead of to unite.

*Laws of Identity and Difference.*

At the base of all thought and science must lie the laws which express the very nature and conditions of the discriminating and identifying powers of mind. These are the so-called Fundamental Laws of Thought, usually stated as follows:--

1. The Law of Identity. *Whatever is, is.*

2. The Law of Contradiction. *A thing cannot both be and not be.*

3. The Law of Duality. *A thing must either be or not be.*

The first of these statements may perhaps be regarded as a description of identity itself, if so fundamental a notion can admit of description. A thing at any moment is perfectly identical with itself, and, if any person were unaware of the meaning of the word “identity,” we could not better describe it than by such an example.

The second law points out that contradictory attributes can never be joined together. The same object may vary in its different parts; here it may be black, and there white; at one time it may be hard and at another time soft; but at the same time and place an attribute cannot be both present and absent. Aristotle truly described this law as the first of all axioms--one of which we need not seek for any demonstration. All truths cannot be proved, otherwise there would be an endless chain of demonstration; and it is in self-evident truths like this that we find the simplest foundations.

The third of these laws completes the other two. It asserts that at every step there are two possible alternatives--presence or absence, affirmation or negation. Hence I propose to name this law the Law of Duality, for it gives to all the formulæ of reasoning a dual character. It asserts also that between presence and absence, existence and non-existence, affirmation and negation, there is no third alternative. As Aristotle said, there can be no mean between opposite assertions: we must either affirm or deny. Hence the inconvenient name by which it has been known--The Law of Excluded Middle.

It may be allowed that these laws are not three independent and distinct laws; they rather express three different aspects of the same truth, and each law doubtless presupposes and implies the other two. But it has not hitherto been found possible to state these characters of identity and difference in less than the threefold formula. The reader may perhaps desire some information as to the mode in which these laws have been stated, or the way in which they have been regarded, by philosophers in different ages of the world. Abundant information on this and many other points of logical history will be found in Ueberweg’s *System of Logic*, of which an excellent translation has been published by Professor T. M. Lindsay (see pp. 228–281).

*The Nature of the Laws of Identity and Difference.*

I must at least allude to the profoundly difficult question concerning the nature and authority of these Laws of Identity and Difference. Are they Laws of Thought or Laws of Things? Do they belong to mind or to material nature? On the one hand it may be said that science is a purely mental existence, and must therefore conform to the laws of that which formed it. Science is in the mind and not in the things, and the properties of mind are therefore all important. It is true that these laws are verified in the observation of the exterior world; and it would seem that they might have been gathered and proved by generalisation, had they not already been in our possession. But on the other hand, it may well be urged that we cannot prove these laws by any process of reasoning or observation, because the laws themselves are presupposed, as Leibnitz acutely remarked, in the very notion of a proof. They are the prior conditions of all thought and all knowledge, and even to question their truth is to allow them true. Hartley ingeniously refined upon this argument, remarking that if the fundamental laws of logic be not certain, there must exist a logic of a second order whereby we may determine the degree of uncertainty: if the second logic be not certain, there must be a third; and so on *ad infinitum*. Thus we must suppose either that absolutely certain laws of thought exist, or that there is no such thing as certainty whatever.[26]

[26] Hartley on Man, vol. i. p. 359.

Logicians, indeed, appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always possible, and of not unfrequent occurrence. The Laws of Thought are often called necessary laws, that is, laws which cannot but be obeyed. Yet as a matter of fact, who is there that does not often fail to obey them? They are the laws which the mind ought to obey rather than what it always does obey. Our thoughts cannot be the criterion of truth, for we often have to acknowledge mistakes in arguments of moderate complexity, and we sometimes only discover our mistakes by collision between our expectations and the events of objective nature.

Mr. Herbert Spencer holds that the laws of logic are objective laws,[27] and he regards the mind as being in a state of constant education, each act of false reasoning or miscalculation leading to results which are likely to prevent similar mistakes from being again committed. I am quite inclined to accept such ingenious views; but at the same time it is necessary to distinguish between the accumulation of knowledge, and the constitution of the mind which allows of the acquisition of knowledge. Before the mind can perceive or reason at all it must have the conditions of thought impressed upon it. Before a mistake can be committed, the mind must clearly distinguish the mistaken conclusion from all other assertions. Are not the Laws of Identity and Difference the prior conditions of all consciousness and all existence? Must they not hold true, alike of things material and immaterial? and if so, can we say that they are only subjectively true or objectively true? I am inclined, in short, to regard them as true both “in the nature of thought and things,” as I expressed it in my first logical essay;[28] and I hold that they belong to the common basis of all existence. But this is one of the most difficult questions of psychology and metaphysics which can be raised, and it is hardly one for the logician to decide. As the mathematician does not inquire into the nature of unity and plurality, but develops the formal laws of plurality, so the logician, as I conceive, must assume the truth of the Laws of Identity and Difference, and occupy himself in developing the variety of forms of reasoning in which their truth may be manifested.

[27] *Principles of Psychology*, Second Ed., vol. ii. p. 86.

[28] *Pure Logic, or the Logic of Quality apart from Quantity*, 1864, pp. 10, 16, 22, 29, 36, &c.

Again, I need hardly dwell upon the question whether logic treats of language, notions, or things. As reasonably might we debate whether a mathematician treats of symbols, quantities, or things. A mathematician certainly does treat of symbols, but only as the instruments whereby to facilitate his reasoning concerning quantities; and as the axioms and rules of mathematical science must be verified in concrete objects in order that the calculations founded upon them may have any validity or utility, it follows that the ultimate objects of mathematical science are the things themselves. In like manner I conceive that the logician treats of language so far as it is essential for the embodiment and exhibition of thought. Even if reasoning can take place in the inner consciousness of man without the use of any signs, which is doubtful, at any rate it cannot become the subject of discussion until by some system of material signs it is manifested to other persons. The logician then uses words and symbols as instruments of reasoning, and leaves the nature and peculiarities of language to the grammarian. But signs again must correspond to the thoughts and things expressed, in order that they shall serve their intended purpose. We may therefore say that logic treats ultimately of thoughts and things, and immediately of the signs which stand for them. Signs, thoughts, and exterior objects may be regarded as parallel and analogous series of phenomena, and to treat any one of the three series is equivalent to treating either of the other series.

*The Process of Inference.*

The fundamental action of our reasoning faculties consists in inferring or carrying to a new instance of a phenomenon whatever we have previously known of its like, analogue, equivalent or equal. Sameness or identity presents itself in all degrees, and is known under various names; but the great rule of inference embraces all degrees, and affirms that *so far as there exists sameness, identity or likeness, what is true of one thing will be true of the other*. The great difficulty doubtless consists in ascertaining that there does exist a sufficient degree of likeness or sameness to warrant an intended inference; and it will be our main task to investigate the conditions under which reasoning is valid. In this place I wish to point out that there is something common to all acts of inference, however different their apparent forms. The one same rule lends itself to the most diverse applications.

The simplest possible case of inference, perhaps, occurs in the use of a *pattern*, *example*, or, as it is commonly called, a *sample*. To prove the exact similarity of two portions of commodity, we need not bring one portion beside the other. It is sufficient that we take a sample which exactly represents the texture, appearance, and general nature of one portion, and according as this sample agrees or not with the other, so will the two portions of commodity agree or differ. Whatever is true as regards the colour, texture, density, material of the sample will be true of the goods themselves. In such cases likeness of quality is the condition of inference.

Exactly the same mode of reasoning holds true of magnitude and figure. To compare the sizes of two objects, we need not lay them beside each other. A staff, string, or other kind of measure may be employed to represent the length of one object, and according as it agrees or not with the other, so must the two objects agree or differ. In this case the proxy or sample represents length; but the fact that lengths can be added and multiplied renders it unnecessary that the proxy should always be as large as the object. Any standard of convenient size, such as a common foot-rule, may be made the medium of comparison. The height of a church in one town may be carried to that in another, and objects existing immovably at opposite sides of the earth may be vicariously measured against each other. We obviously employ the axiom that whatever is true of a thing as regards its length, is true of its equal.

To every other simple phenomenon in nature the same principle of substitution is applicable. We may compare weights, densities, degrees of hardness, and degrees of all other qualities, in like manner. To ascertain whether two sounds are in unison we need not compare them directly, but a third sound may be the go-between. If a tuning-fork is in unison with the middle C of York Minster organ, and we afterwards find it to be in unison with the same note of the organ in Westminster Abbey, then it follows that the two organs are tuned in unison. The rule of inference now is, that what is true of the tuning-fork as regards the tone or pitch of its sound, is true of any sound in unison with it.

The skilful employment of this substitutive process enables us to make measurements beyond the powers of our senses. No one can count the vibrations, for instance, of an organ-pipe. But we can construct an instrument called the *siren*, so that, while producing a sound of any pitch, it shall register the number of vibrations constituting the sound. Adjusting the sound of the siren in unison with an organ-pipe, we measure indirectly the number of vibrations belonging to a sound of that pitch. To measure a sound of the same pitch is as good as to measure the sound itself.

Sir David Brewster, in a somewhat similar manner, succeeded in measuring the refractive indices of irregular fragments of transparent minerals. It was a troublesome, and sometimes impracticable work to grind the minerals into prisms, so that the power of refracting light could be directly observed; but he fell upon the ingenious device of compounding a liquid possessing the same refractive power as the transparent fragment under examination. The moment when this equality was attained could be known by the fragments ceasing to reflect or refract light when immersed in the liquid, so that they became almost invisible in it. The refractive power of the liquid being then measured gave that of the solid. A more beautiful instance of representative measurement, depending immediately upon the principle of inference, could not be found.[29]

[29] Brewster, *Treatise on New Philosophical Instruments*, p. 273. Concerning this method see also Whewell, *Philosophy of the Inductive Sciences*, vol. ii. p. 355; Tomlinson, *Philosophical Magazine*, Fourth Series, vol. xl. p. 328; Tyndall, in Youmans’ *Modern Culture*, p. 16.

Throughout the various logical processes which we are about to consider--Deduction, Induction, Generalisation, Analogy, Classification, Quantitative Reasoning--we shall find the one same principle operating in a more or less disguised form.

*Deduction and Induction.*

The processes of inference always depend on the one same principle of substitution; but they may nevertheless be distinguished according as the results are inductive or deductive. As generally stated, deduction consists in passing from more general to less general truths; induction is the contrary process from less to more general truths. We may however describe the difference in another manner. In deduction we are engaged in developing the consequences of a law. We learn the meaning, contents, results or inferences, which attach to any given proposition. Induction is the exactly inverse process. Given certain results or consequences, we are required to discover the general law from which they flow.

In a certain sense all knowledge is inductive. We can only learn the laws and relations of things in nature by observing those things. But the knowledge gained from the senses is knowledge only of particular facts, and we require some process of reasoning by which we may collect out of the facts the laws obeyed by them. Experience gives us the materials of knowledge: induction digests those materials, and yields us general knowledge. When we possess such knowledge, in the form of general propositions and natural laws, we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment. In its ultimate foundation, then, all knowledge is inductive--in the sense that it is derived by a certain inductive reasoning from the facts of experience.

It is nevertheless true,--and this is a point to which insufficient attention has been paid, that all reasoning is founded on the principles of deduction. I call in question the existence of any method of reasoning which can be carried on without a knowledge of deductive processes. I shall endeavour to show that *induction is really the inverse process of deduction*. There is no mode of ascertaining the laws which are obeyed in certain phenomena, unless we have the power of determining what results would follow from a given law. Just as the process of division necessitates a prior knowledge of multiplication, or the integral calculus rests upon the observation and remembrance of the results of the differential calculus, so induction requires a prior knowledge of deduction. An inverse process is the undoing of the direct process. A person who enters a maze must either trust to chance to lead him out again, or he must carefully notice the road by which he entered. The facts furnished to us by experience are a maze of particular results; we might by chance observe in them the fulfilment of a law, but this is scarcely possible, unless we thoroughly learn the effects which would attach to any particular law.

Accordingly, the importance of deductive reasoning is doubly supreme. Even when we gain the results of induction they would be of no use unless we could deductively apply them. But before we can gain them at all we must understand deduction, since it is the inversion of deduction which constitutes induction. Our first task in this work, then, must be to trace out fully the nature of identity in all its forms of occurrence. Having given any series of propositions we must be prepared to develop deductively the whole meaning embodied in them, and the whole of the consequences which flow from them.

*Symbolic Expression of Logical Inference.*

In developing the results of the Principle of Inference we require to use an appropriate language of signs. It would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary. Special examples of reasoning, too, may seem to be more readily apprehended than general symbolic forms. But it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear, brief, and appropriate system of symbols. Not only is such a language convenient, but it is almost essential to the expression of those general truths which are the very soul of science. To apprehend the truth of special cases of inference does not constitute logic; we must apprehend them as cases of more general truths. The object of all science is the separation of what is common and general from what is accidental and different. In a system of logic, if anywhere, we should esteem this generality, and strive to exhibit clearly what is similar in very diverse cases. Hence the great value of *general symbols* by which we can represent the form of a reasoning process, disentangled from any consideration of the special subject to which it is applied.

The signs required in logic are of a very simple kind. As sameness or difference must exist between two things or notions, we need signs to indicate the things or notions compared, and other signs to denote the relations between them. We need, then, (1) symbols for terms, (2) a symbol for sameness, (3) a symbol for difference, and (4) one or two symbols to take the place of conjunctions.

Ordinary nouns substantive, such as *Iron*, *Metal*, *Electricity*, *Undulation*, might serve as terms, but, for the reasons explained above, it is better to adopt blank letters, devoid of special signification, such as A, B, C, &c. Each letter must be understood to represent a noun, and, so far as the conditions of the argument allow, *any noun*. Just as in Algebra, *x*, *y*, *z*, *p*, *q*, &c. are used for *any quantities*, undetermined or unknown, except when the special conditions of the problem are taken into account, so will our letters stand for undetermined or unknown things.

These letter-terms will be used indifferently for nouns substantive and adjective. Between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view. But grammatical usage sanctions the conversion of adjectives into substantives, and *vice versâ*; we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved. Here, as throughout this work, I shall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical science, this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject.

Every noun or term denotes an object, and usually implies the possession by that object of certain qualities or circumstances common to all the objects denoted. There are certain terms, however, which imply the absence of qualities or circumstances attaching to other objects. It will be convenient to employ a special mode of indicating these *negative terms*, as they are called. If the general name A denotes an object or class of objects possessing certain defined qualities, then the term Not A will denote any object which does not possess the whole of those qualities; in short, Not A is the sign for anything which differs from A in regard to any one or more of the assigned qualities. If A denote “transparent object,” Not A will denote “not transparent object.” Brevity and facility of expression are of no slight importance in a system of notation, and it will therefore be desirable to substitute for the negative term Not A a briefer symbol. De Morgan represented negative terms by small Roman letters, or sometimes by small italic letters;[30] as the latter seem to be highly convenient, I shall use *a*, *b*, *c*, ... *p*, *q*, &c., as the negative terms corresponding to A, B, C, ... P, Q, &c. Thus if A means “fluid,” *a* will mean “not fluid.”

[30] *Formal Logic*, p. 38.

*Expression of Identity and Difference.*

To denote the relation of sameness or identity I unhesitatingly adopt the sign =, so long used by mathematicians to denote equality. This symbol was originally appropriated by Robert Recorde in his *Whetstone of Wit*, to avoid the tedious repetition of the words “is equal to;” and he chose a pair of parallel lines, because no two things can be more equal.[31] The meaning of the sign has however been gradually extended beyond that of equality of quantities; mathematicians have themselves used it to indicate equivalence of operations. The force of analogy has been so great that writers in most other branches of science have employed the same sign. The philologist uses it to indicate the equivalence of meaning of words: chemists adopt it to signify identity in kind and equality in weight of the elements which form two different compounds. Not a few logicians, for instance Lambert, Drobitsch, George Bentham,[32] Boole,[33] have employed it as the copula of propositions. De Morgan declined to use it for this purpose, but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred;[34] and Herbert Spencer has applied it in a like manner.[35]

[31] Hallam’s *Literature of Europe*, First Ed., vol. ii. p. 444.

[32] *Outline of a New System of Logic*, London, 1827, pp. 133, &c.

[33] *An Investigation of the Laws of Thought*, pp. 27, &c.

[34] *Formal Logic*, pp. 82, 106. In his later work, *The Syllabus of a New System of Logic*, he discontinued the use of the sign.

[35] *Principles of Psychology*, Second Ed., vol. ii. pp. 54, 55.

Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience; but I hold that the common use of this sign = in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importance--one indeed which it is a principal purpose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved. Accordingly De Morgan’s refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion.

I conceive that the sign = as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus “6,720 pounds = 3 tons” is evidently an equation of quantities. The formula - × - = + expresses the equivalence of operations. “Exogens = Dicotyledons” is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants.

We have great need in logic of a distinct sign for the copula, because the little verb *is* (or *are*), hitherto used both in logic and ordinary discourse, is thoroughly ambiguous. It sometimes denotes identity, as in “St. Paul’s is the *chef-d’œuvre* of Sir Christopher Wren;” but it more commonly indicates inclusion of class within class, or partial identity, as in “Bishops are members of the House of Lords.” This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on.

When with this sign of equality we join two nouns or logical terms, as in

Hydrogen = The least dense element,

we signify that the object or group of objects denoted by one term is identical with that denoted by the other, in everything except the names. The general formula

A = B

must be taken to mean that A and B are symbols for the same object or group of objects. This identity may sometimes arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature; as when we say

Gravitating matter = Matter possessing inertia, Exogenous plants = Dicotyledonous plants, Plagihedral quartz crystals = Quartz crystals causing the plane of polarisation of light to rotate.

We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature; but at present we are considering only the mode of expression which may be the same in either case.

Sometimes, but much less frequently, we require a symbol to indicate difference or the absence of complete sameness. For this purpose we may generalise in like manner the symbol ~, which was introduced by Wallis to signify difference between quantities. The general formula

B ~ C

denotes that B and C are the names of two objects or groups which are not identical with each other. Thus we may say

Acrogens ~ Flowering plants. Snowdon ~ The highest mountain in Great Britain.

I shall also occasionally use the sign ᔕ to signify in the most general manner the existence of any relation between the two terms connected by it. Thus ᔕ might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A ᔕ B I mean, then, any two objects of thought related to each other in any conceivable manner.

*General Formula of Logical Inference.*

The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The *Substitution of Similars* is a phrase which seems aptly to express the capacity of mutual replacement existing in any two objects which are like or equivalent to a sufficient degree. It is matter for further investigation to ascertain when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form

A = B.

Now if we take the letter C to denote any third conceivable object, and use the sign ᔕ in its stated meaning of *indefinite relation*, then the general formula of all inference may be thus exhibited:--

From A = B ᔕ C we may infer A ᔕ C

or, in words--*In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing.* The identity between A and B allows us indifferently to place A where B was, or B where A was; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign ᔕ, we may say that if *C is the weight of B*, then *C is also the weight of A*. Similarly

If C is the father of B, C is the father of A; If C is a fragment of B, C is a fragment of A; If C is a quality of B, C is a quality of A; If C is a species of B, C is a species of A; If C is the equal of B, C is the equal of A;

and so on *ad infinitum*.

We may also endow with special meanings the letter-terms A, B, and C, and the process of inference will never be false. Thus let the sign ᔕ mean “is height of,” and let

A = Snowdon, B = Highest mountain in England or Wales, C = 3,590 feet;

then it obviously follows since “3,590 feet is the height of Snowdon,” and “Snowdon = the highest mountain in England or Wales,” that, “3,590 feet is the height of the highest mountain in England or Wales.”

One result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole. To alter is to make a difference; but if in replacing a part I make no difference, there is no alteration of the whole. Many inferences which have been very imperfectly included in logical formulas at once follow. I remember the late Prof. De Morgan remarking that all Aristotle’s logic could not prove that “Because a horse is an animal, the head of a horse is the head of an animal.” I conceive that this amounts merely to replacing in the complete notion *head of a horse*, the term “horse,” by its equivalent *some animal* or *an animal*. Similarly, since

The Lord Chancellor = The Speaker of the House of Lords,

it follows that

The death of the Lord Chancellor = The death of the Speaker of the House of Lords;

and any event, circumstance or thing, which stands in a certain relation to the one will stand in like relation to the other. Milton reasons in this way when he says, in his Areopagitica, “Who kills a man, kills a reasonable creature, God’s image.” If we may suppose him to mean

God’s image = man = some reasonable creature,

it follows that “The killer of a man is the killer of some reasonable creature,” and also “The killer of God’s image.”

This replacement of equivalents may be repeated over and over again to any extent. Thus if *person* is identical in meaning with *individual*, it follows that

Meeting of persons = meeting of individuals;

and if *assemblage* = *meeting*, we may make a new replacement and show that

Meeting of persons = assemblage of individuals.

We may in fact found upon this principle of substitution a most general axiom in the following terms[36]:--

[36] *Pure Logic, or the Logic of Quality*, p. 14.

*Same parts samely related make same wholes.*

If, for instance, exactly similar bricks and other materials be used to build two houses, and they be similarly placed in each house, the two houses must be similar. There are millions of cells in a human body, but if each cell of one person were represented by an exactly similar cell similarly placed in another body, the two persons would be undistinguishable, and would be only *numerically* different. It is upon this principle, as we shall see, that all accurate processes of measurement depend. If for a weight in a scale of a balance we substitute another weight, and the equilibrium remains entirely unchanged, then the weights must be exactly equal. The general test of equality is substitution. Objects are equally bright when on replacing one by the other the eye perceives no difference. Objects are equal in dimensions when tested by the same gauge they fit in the same manner. Generally speaking, two objects are alike so far as when substituted one for another no alteration is produced, and *vice versâ* when alike no alteration is produced by the substitution.

*The Propagating Power of Similarity.*

The relation of similarity in all its degrees is reciprocal. So far as things are alike, either may be substituted for the other; and this may perhaps be considered the very meaning of the relation. But it is well worth notice that there is in similarity a peculiar power of extending itself among all the things which are similar. To render a number of things similar to each other we need only render them similar to one standard object. Each coin struck from a pair of dies not only resembles the matrix or original pattern from which the dies were struck, but resembles every other coin manufactured from the same original pattern. Among a million such coins there are not less than 499,999,500,000 *pairs of coins* resembling each other. Similars to the same are similars to all. It is one great advantage of printing that all copies of a document struck from the same type are necessarily identical each with each, and whatever is true of one copy will be true of every copy. Similarly, if fifty rows of pipes in an organ be tuned in perfect unison with one row, usually the Principal, they must be in unison with each other. Similarity can also reproduce or propagate itself *ad infinitum*: for if a number of tuning-forks be adjusted in perfect unison with one standard fork, all instruments tuned to any one fork will agree with any instrument tuned to any other fork. Standard measures of length, capacity, weight, or any other measurable quality, are propagated in the same manner. So far as copies of the original standard, or copies of copies, or copies again of those copies, are accurately executed, they must all agree each with every other.

It is the capability of mutual substitution which gives such great value to the modern methods of mechanical construction, according to which all the parts of a machine are exact facsimiles of a fixed pattern. The rifles used in the British army are constructed on the American interchangeable system, so that any part of any rifle can be substituted for the same part of another. A bullet fitting one rifle will fit all others of the same bore. Sir J. Whitworth has extended the same system to the screws and screw-bolts used in connecting together the parts of machines, by establishing a series of standard screws.

*Anticipations of the Principle of Substitution.*

In such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained. The germ at least of every doctrine will be found in earlier writers, and novelty must arise chiefly in the mode of harmonising and developing ideas. When I first employed the process and name of *substitution* in logic,[37] I was led to do so from analogy with the familiar mathematical process of substituting for a symbol its value as given in an equation. In writing my first logical essay I had a most imperfect conception of the importance and generality of the process, and I described, as if they were of equal importance, a number of other laws which now seem to be but particular cases of the one general rule of substitution.

[37] *Pure Logic*, pp. 18, 19.

My second essay, “The Substitution of Similars,” was written shortly after I had become aware of the great simplification which may be effected by a proper application of the principle of substitution. I was not then acquainted with the fact that the German logician Beneke had employed the principle of substitution, and had used the word itself in forming a theory of the syllogism. My imperfect acquaintance with the German language had prevented me from acquiring a complete knowledge of Beneke’s views; but there is no doubt that Professor Lindsay is right in saying that he, and probably other logicians, were in some degree familiar with the principle.[38] Even Aristotle’s dictum may be regarded as an imperfect statement of the principle of substitution; and, as I have pointed out, we have only to modify that dictum in accordance with the quantification of the predicate in order to arrive at the complete process of substitution.[39] The Port-Royal logicians appear to have entertained nearly equivalent views, for they considered that all moods of the syllogism might be reduced under one general principle.[40] Of two premises they regard one as the *containing proposition* (propositio continens), and the other as the *applicative proposition*. The latter proposition must always be affirmative, and represents that by which a substitution is made; the former may or may not be negative, and is that in which a substitution is effected. They also show that this method will embrace certain cases of complex reasoning which had no place in the Aristotelian syllogism. Their views probably constitute the greatest improvement in logical doctrine made up to that time since the days of Aristotle. But a true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow restrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part.

[38] Ueberweg’s *System of Logic*, transl. by Lindsay, pp. 442–446, 571, 572. The anticipations of the principle of substitution to be found in the works of Leibnitz, Reusch, and perhaps other German logicians, will be noticed in the preface to this second edition.

[39] *Substitution of Similars* (1869), p. 9.

[40] *Port-Royal Logic*, transl. by Spencer Baynes, pp. 212–219. Part III. chap. x. and xi.

*The Logic of Relatives.*

There is a difficult and important branch of logic which may be called the Logic of Relatives. If I argue, for instance, that because Daniel Bernoulli was the son of John, and John the brother of James, therefore Daniel was the nephew of James, it is not possible to prove this conclusion by any simple logical process. We require at any rate to assume that the son of a brother is a nephew. A simple logical relation is that which exists between properties and circumstances of the same object or class. But objects and classes of objects may also be related according to all the properties of time and space. I believe it may be shown, indeed, that where an inference concerning such relations is drawn, a process of substitution is really employed and an identity must exist; but I will not undertake to prove the assertion in this work. The relations of time and space are logical relations of a complicated character demanding much abstract and difficult investigation. The subject has been treated with such great ability by Peirce,[41] De Morgan,[42] Ellis,[43] and Harley, that I will not in the present work attempt any review of their writings, but merely refer the reader to the publications in which they are to be found.

[41] *Description of a Notation for the Logic of Relatives, resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic.* By C. S. Peirce. *Memoirs of the American Academy*, vol. ix. Cambridge, U.S., 1870.

[42] *On the Syllogism No IV., and on the Logic of Relations.* By Augustus De Morgan. *Transactions of the Cambridge Philosophical Society*, vol. x. part ii., 1860.

[43] *Observations on Boole’s Laws of Thought.* By the late R. Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S. *Report of the British Association*, 1870. *Report of Sections*, p. 12. Also, *On Boole’s Laws of Thought*. By the Rev. Robert Harley, F.R.S., *ibid.* p. 14.