An Introduction to Symbolic Logic PDF

# An Introduction to Symbolic Logic ## Chapter I ## The Study of Forms ### 1. The Importance of Form All knowledge, all sciences and arts have their beginning in the recognition that ordinary, familiar things may take on different forms. Our earliest experience of nature brings this fact to our...

# An Introduction to Symbolic Logic ## Chapter I ## The Study of Forms ### 1. The Importance of Form All knowledge, all sciences and arts have their beginning in the recognition that ordinary, familiar things may take on different forms. Our earliest experience of nature brings this fact to our notice; we see water freezing into a translucent block, or the snow which fell from heaven changing to water before our very eyes. A little reflection makes us aware of further changes. Where do snow and rain come from? From the vaporous masses we call clouds, the white mists that float in the air, and as the snow or rain descends those clouds dissolve. They have turned to water, or to white flakes. Where did the clouds come from? They are made by some process of transformation from the waters of the earth. Different forms of the same thing may be so widely diverse in appearance that it is hard to think of them as essentially the same substance. All science tries to reduce the diversity of things in the world to mere differences of appearance, and treats as many things as possible as variants of the same stuff. When Benjamin Franklin found out that lightning is one form of electricity, he made a scientific discovery, which proved to be but a step in a very great science, for an amazing number of things can be reduced to this same fundamental "something," this protean substance called "electricity": the force that holds the firmament together, the crackle of a cat's fur, the heat in a flat-iron, the flickering Northern Aurora. Electricity is one of the essential things in the world that can take on a vast variety of forms. Its wide mutability makes nature interesting, and its ultimate oneness makes science possible. But if we could not appreciate its different forms as different forms of one thing, then we would have no way of relating them to each other at all; we would not know why sliding our feet over a rug makes the brass door-knob give off a spark at our touch, or why the lightning has a preference for metal crosses on church-steeples, nor see any connection whatever between these two causal facts. We have two kinds of knowledge, which may be called respectively knowledge of things, and knowledge about them. The former is that direct intimacy which our senses give us, the look and smell and feel of a thing--the sort of knowledge a baby has of its own bed, its mother's breast, its usual view of ceiling and window and wall. It knows these objects just as it knows hunger or toothache; it has what Bertrand Russell has called "knowledge by acquaintance" of certain things, a most direct and sensuous knowledge. Yet the baby cannot be said to know anything about beds, food, houses, or toothaches. To know anything about an object is to know how it is related to its surroundings, how it is made up, how it functions, etc., in short, to know what sort of thing it is. To have knowledge about it we must know more than the direct sensuous quality of "this stuff"; we must know what particular shape the stuff is taking in the case of this thing. A child may know the taste and feeling of a scrambled egg, without knowing that it is an egg which has been scrambled; in that case, he will not associate it with a boiled egg or an omelette. He does not know that these are different forms of the same thing. Neither does he know that in its raw, fresh, natural state the egg is a pristine form of chicken, and by more distant relationship an incipient chicken-dinner. The only way we can do business at all with a rapidly changing, shifting, surprising world is to discover the most general laws of its transformations. The very word "transformation" tells us what we are dealing with: changes of form. The growth of science is the most striking demonstration of the importance of forms as distinguished from matter, substance, stuff, or whatever we call that which remains "the same" when the form of a thing changes. Whenever we may truly claim to have a science, we have found some principle by which different things are related to each other as just so many forms of one substrate, or material, and everything that can be treated as a new variation belongs to that science. ### 2. Logical Form Not all the sciences deal with material things. Philology, for instance, deals with the relations among words, and although the production of a word in speech or writing always has a material aspect-involving paper and ink, or movement of speech organs-this aspect is not what interests the philologist. When the word undergoes changes of form, as for instance the word "pater" to "père" and "father" and "padre," this cannot be taken to mean that it changes its shape, as an egg changes shape when it is scrambled. The meaning of "form" is stretched beyond its common connotation of shape. Likewise, in the science or art of musical composition we speak of a rondo form, sonata form, hymn form, and no one thinks of a material shape. Musical form is not material; it is orderliness, but not shape. So we must recognize a wider sense of the word than the geometric sense of physical shape. And indeed, we admit this wider sense in ordinary parlance; we speak of "formality" in social intercourse, of "good form" in athletics, of "formalism" in literature, music, or dancing. Certainly we cannot refer to the shape of a dinner or a poem or a dance. In this wider sense, anything may be said to have form that follows a pattern of any sort, exhibits order, internal connection; our many synonyms for "form" indicate how wide the range of this notion really is. We speak of physical, grammatical, social forms; of psychological types; norms of conduct, of beauty, of intelligence; fashions in clothing, speech, behaviour; new designs of automobiles or motor boats; architectural plans, or the plans for a festival; pattern, standard, mode, and many other words signify essentially the same thing in specialized usage or subtle variations of meaning. But all these words refer to "form" in that most general sense in which we are going to use it here. It is this most general sense which we always give to the word in logic. Therefore I shall call it "logical form," to distinguish it from "form" in any more restricted sense, particularly its usual connotation of physical shape. ### 3. Structure "Logical form" is a highly general notion, and like all generalizations, it covers a large number of particular ideas. But if these various particular ideas may all be called by one general name, they must have something-some general feature-in common. But what is common to, say, the form of declensions and conjugations in a language, and the topography of a continent? Both are "forms" in the logical sense. What bridge can be thrown across the gap between such widely diverse notions as the shape of the continent, and the orderly sequence of word-endings in the language? Is there any justification for applying the same name to such different matters? The bridge that connects all the various meanings of form-from geometric form to the form of ritual or etiquette-is the notion of structure. The logical form of a thing is the way that thing is constructed, the way it is put together. Any thing that has a definite form is constructed in a definite way. This does not mean, of course, that it has been deliberately put together by somebody; forms may be preconceived, or they may be natural, or accidental. The famous "Old Man of the Mountain" in New Hampshire is an accidental rock-formation, put together like a human profile; a piece juts out just a little, as eyebrows do, then there is a gap like the hollow of the eyes, a large projection, exactly as a nose projects, a straight piece, a fissure beneath it, as the mouth is beneath the lip, another straight piece ending in a projection with a great overhang that resembles the projection of a strong chin and jaw. The outline of the rock is made up like a human profile; the powers of nature-glacier, water, and frost-have accidentally constructed the "Old Man of the Mountain" out of granite; the cliff, which has nothing else in common with human flesh, yet resembles a man by its form. Nature is full of most elaborate constructs, from the crude structure of strata in a geological fault to the infinitesimal dynamic pattern of protons and electrons in an atom. One must not make the mistake of associating "structure" always with something put together out of parts that were previously separate. A snowflake is a detailed construct of very recognizable individual parts, but these have not been "put together"; they crystallized out of one homogeneous drop of water. They were never separate, and there has been no process of combination. Yet the flake has a structure (in fact, an interesting structure) and is therefore called a natural construct. Now let us consider a form which is not geometric, i.e., which is not a shape: the form, for instance, of the ordinary musical scale, in the so-called "major mode," do, re, mi, fa, sol, la, si, high do. There are eight notes in this scale, counting the customary repetition of do. But to have a major scale we must have more than the eight notes; we must have them put together in just this particular way. The major scale is constructed by letting these notes follow each other in just this order. Supposing we put them together differently, as for instance; do, mi, re, fa, si-la sol, do (high). We have then constructed a perfectly singable melody, but it is not a major scale. It has the notes but not the form of the scale. We may group them yet differently, and construct another tune; sol-mi-do (high), si, la-fa, re-do. Here we have the same notes again, but the musical form is not the same; this is a "skippy" tune in waltz-time. The two melodies have every note in common; that is to say, exactly the same ingredients have gone into both. Were these ingredients material things, we should say the two constructs were made of exactly the same substance, and differed only in form. Logically we may say this even though the "substance" is immaterial, being a collection of individual sounds, for order and arrangement among sounds is just as much "logical form" as the arrangement of parts in a physical thing. If we compare these tunes with each other and with the conventional "major scale," we see even in this rudimentary and trivial case how very diverse may be the appearance, the character, the value, of things which are merely different forms of the same given material. This fact should always be borne in mind; often it is hard to believe that a philosopher or a scientist has any chance whatever of reducing widely different things to the same category, because they look and "feel" so incommensurable, but he has found a principle by which he can describe them as two forms of one substance, and we are amazed to see how precisely and usefully he then relates them. Logic is full of such surprises; that is why it sharpens and broadens one's outlook, scientific or metaphysical, upon the whole world. ### 4. Form and Content So far, we have dealt entirely with the different forms that may be exhibited by the same material, and which may make it look like an essentially different thing in each case. The "material" may not be physical at all; the words "matter" and "substance" are not very fortunate, because they only reinforce our natural prejudice in favour of imagining form as shape, and whatever has form, as stuff. Logicians usually avoid this connotation by calling that medium wherein a form is expressed, its content. We may say, then, that we have so far considered how one and the same content may appear in several forms. But equally important is the fact that one and the same form may be exemplified by different contents. Different things may take the same form. Consider the example given above (page 25) of structure determining form-the great stone face called the "Old Man of the Mountain." There it was pointed out that the cliff had nothing in common with a human face, except the arrangement of its parts, but this arrangement gave it the form of a man's profile. A form which is usually expressed in flesh and blood is here articulated in stone; the extraordinary content arouses our wonder and perhaps our superstitious imagination. A very much homelier example of different contents for the same form, one which confronts us repeatedly in practical life, derives from the fact that two suits of the same pattern may be cut of different cloths. In this day of standardization, any number of suits, in any number of materials, exhibit exactly the same form, and if we chose to sew up the paper pattern from which they were cut, instead of leaving it spread out flat, we would have yet another suit made of paper. Likewise, one may think of a bread-pudding, a fruit-jelly, a blanc-mange, etc., all cooled in the same mould; they will, then, all have the same form, but differ in material, or content. There is nothing unfamiliar about the notion that diverse things may follow the same pattern. Everybody takes it for granted, just as everybody grants the fact that one substance may assume various forms. The form of a suit or a pudding is, of course, a geometric form, a shape; so is the form of a face, whether it is moulded of flesh or of stone. But non-material structures may also have various contents. Suppose we return to our former example of a non-material construct, the C-major scale, and imagine it transposed one half-tone higher, so that it reads: C#, D#, E#, F#, G#, A#, B#, high C#. Not a single note in the C#-major scale figures in the scale of C-natural.* Yet the two have exactly the same form, which is commonly called "the major scale." It is a peculiar fact that in music forms are easier to recognize than contents; most people can tell whether a given succession of tones is a major scale or not-they will mark any deviation from the form-but very few can tell whether the given major scale is C, C#, or any other particular key. A normal ear will apprehend the form, but only persons blessed with so-called "absolute pitch" can identify the content. Furthermore, two different contents for the same form may vary so widely that they belong to entirely different departments of human experience. Suits of cloth or paper are, after all, equally tangible, physical contents for a geometric form. Tones in a major scale are all equally auditory contents. However we vary our material, as from C to C#, D to D#, etc., our scale is still a musical form, and its content is some sort of sound. But why is the standard arrangement of these tones called a "scale"? "Scale" means "ladder." The fact is that ordinary, common sense sees a similarity of form between the order of successive tones, each new tone being a little higher than its predecessor, and the successive rungs of a ladder, each a little higher than the one before it. The word "scale" or "ladder" is transferred from one to the other. In this way, what was once the name of a certain kind of object has become the name of a certain form. Any series whose separate parts are arranged so that each is either higher or lower than any other part, is a "scale." Thus we speak of "going up in the social scale," or call a certain series of successively "higher" spiritual experiences "the ladder of faith," without any danger of being misunderstood, and being thought to refer to a series of tones, or a wooden contraption with steps or rungs. Everybody admits the propriety of our usage, by analogy; and analogy is nothing but the recognition of a common form in different things. ### 5. The Value of Analogy Whenever we draw a diagram, say the ground-plan of a house, or a street-plan to show the location of its site, or a map, or an isographic chart, or a "curve" representing the fluctuations of the stock-market, we are drawing a "logical picture" of something. A "logical picture" differs from an ordinary picture in that it need not look the least bit like its object. Its relation to the object is not that of a copy, but of analogy. We do not try to make an architect's drawing look as much as possible like the house; that is, even if the floor is to be brown, the floor-plan is not considered any better for being drawn in brown; and if the house is to be large, the plan need not convey an impression of vastness. All that the plan must do is to copy exactly the proportions of length and width, the arrangement of rooms, halls and stairs, doors and windows. The narrow dash that represents a window is not intended to look like one; it resembles the object for which it stands only by its location in the plan, which must be analogous to the location of the window in the room. The dissimilarity in appearance between a "logical picture" and what it represents is even more marked in the case of a graph. Supposing a graph in the newspaper conveys to you the growth, acceleration, climax and decline of an epidemic. The graph is spatial, its form is a shape, but the series of events does not have shape in a literal sense. The graph is a picture of events only in a logical sense; its constituents, which are little squares on paper, are arranged in the same proportions to each other as the constituents of the epidemic, which are cases of illness. If the epidemic has lasted twenty days, the graph will show twenty vertical columns, and if the third day of the epidemic brought sixty cases of illness, the third column of the graph will show black squares up to sixty. Most of us have no difficulty in seeing an order and configuration of events graphically; yet the only form which the graph and the events have in common is a logical form. They have an analogous structure, though their contents are more incongruous than cabbages and kings. It is only by analogy that one thing can represent another which does not resemble it. By analogy, a map can "mean" a certain place; and obviously it cannot "mean" any place which it does not fit, i.e. which has not a contour analogous to the map. If two things have the same logical form, one of them may represent the other, and not otherwise. Six mice may represent six horses, but even a fairy godmother would have had a hard time making six horses out of five, or seven, mice. Likewise, seven lean cows may mean seven poor years, and seven fat cows, seven years of plenty; but had the cows been all alike they would have lost their significance, for the analogy would have been broken. A rosary of beads may represent the number and order of prayers to be repeated, for the beads can be moved one after another, as the successive prayers are accomplished; if the beads were strung so they could not move, the band might be a necklace or a bracelet, but it could not be a rosary, for it would not represent what the shifting of the prayer-beads is supposed to stand for. Perhaps the most elaborate structure ever invented for purely representative purposes is the syntactical structure of language. Its content, in itself, is trivial; it is a system of various little sounds, not beautiful and arresting sounds, like the content of musical structures, but rather ridiculous little squeaks and hums and groans. Yet in their arrangement and organization, these noises have such a developed pattern that they constitute a great system, any fragment of which has its logical form, its so-called "grammatical structure." What this structure can represent, is the order and connection of ideas in our minds. Our ideas are not mere fleeting images without definite relations to each other; whenever we are really thinking, not merely dozing in a haze of passive impressionism, our ideas exhibit sequence, arrangement, connection, a definite pattern. Some ideas belong together more intimately than others; some lead to others; some arise out of others, etc. It is this pattern which the elaborate pattern of language reflects. Separate words usually (though not always) stand for separate impressions or ideas, and such words are put together to make sentences, which express completed, organic thoughts, or propositions. Because the ideas we want to represent are of this complex sort, language cannot be a mere collection of words, such as a spelling-book might offer. Language must have a more articulated logical form. As soon as we are old enough to apprehend this form (not to comprehend it, for that is another matter), we learn to speak; our concerted ideas are reflected in the concerted patterns of sound that we utter. There are many ways of combining the elementary notions in our minds, and the commonest, most general of these ways are reflected in the laws of language, which we call syntax. Syntax is simply the logical form of our language, which copies as closely as possible the logical form of our thought. To understand language is to appreciate the analogy between the syntactical construct and the complex of ideas, letting the former function as a representative, or "logical picture," of the latter. Bertrand Russell has given an excellent account of the sort of "form" that belongs to language, and by virtue of which we understand it to mean what it does. I quote the passage chiefly because it shows clearly the distinction of form and content in a sentence, and the relation of that form to structure, or arrangement of parts. >"In every proposition and every inference there is, besides the particular subject-matter concerned, a certain form, a way in which the constituents of the proposition or inference are put together. If I say, 'Socrates is mortal,' 'Jones is angry,' 'The sun is hot,' there is something in common in these three cases, something indicated by the word 'is.' What is in common is the form of the proposition, not an actual constituent. If I say a number of things about Socrates-that he was an Athenian, that he married Xantippe, that he drank the hemlock-there is a common constituent, namely Socrates, in all the propositions I enunciate, but they have diverse forms. If, on the other hand, I take any one of these propositions and replace its constituents, one at a time, by other constituents, the form remains constant, but no constituent remains. Take (say) the series of propositions, 'Socrates drank the hemlock,' 'Coleridge drank the hemlock,' 'Coleridge drank opium,' 'Coleridge ate opium.' The form remains unchanged throughout this series, but all the constituents are altered. Thus form is not another constituent, but is the way the constituents are put together.... We might understand all the separate words of a sentence without understanding the sentence: if a sentence is long and complicated, this is apt to happen. In such a case we have knowledge of the constituents, but not of the form. We may also have knowledge of the form without having knowledge of the constituents. If I say, 'Rorarius drank the hemlock,' those among you who have never heard of Rorarius (supposing there are any) will understand the form, without having knowledge of all the constituents. In order to understand a sentence, it is necessary to have knowledge both of the constituents and of the particular instance of the form.... Thus some kind of knowledge of logical forms, though with most people it is not explicit, is involved in all understanding of discourse. It is the business of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit and pure."* The great value of analogy is that by it, and it alone, we are led to seeing a single "logical form" in things which may be entirely discrepant as to content. The power of recognizing similar forms in widely various exemplifications, i.e. the power of discovering analogies, is logical intuition. Some people have it by nature; others must develop it (and I believe all normal minds can develop it), and certainly all may sharpen the precision of their understanding, by a systematic study of the principles of structure. ### 6. Abstraction The consideration of a form, which several analogous things may have in common, apart from any contents, or "concrete integuments," is called abstraction. If we speak of the major scale apart from any particular key, we are treating it as an abstracted form. If we note what is common to a couple of days, a pair of gloves, a brace of partridges, and a set of twins, we are abstracting a form which each of these items exhibits, namely its numerosity, two. If we speak simply of a couple, without reference to any content, or simply of "two-ness" or "two," we are treating of this form in abstracto. Or again, if we consider the order in which hours of a day follow each other-always one after another, never two at once following the same predecessor -and then regard the order of inches on a ruler, or rungs on a ladder, or the succession of volumes of the Encyclopaedia Britannica, or the sequence of Presidents of the United States, we see at once that there is a common form in all these progressions. They are all analogous, all different contents for a pattern which is a section of the ordinal number series: first, second, third, etc. It is easy to see that it is but a short step from the recognition of analogies, or *"Logic as the Essence of Philosophy" in Our Knowledge of the External World, London, 1914.* Most people shy at the very word "abstraction." It suggests to them the incomprehensible, misleading, difficult, the great intellectual void of empty words. But as a matter of fact, abstract thinking is the quickest and most powerful kind of thinking, as even an elementary study of symbolic logic tends to show. The reason people are afraid of abstraction is simply that they do not know how to handle it. They have not learned to make correct abstractions, and therefore become lost among the empty forms, or worse yet, among the mere words for such forms, which they call "empty words" with an air of disgust. It is not the fault of abstraction that few people can really think abstractly, any more than it is the fault of mathematics that not many people are good mathematicians. There is nothing in our educational curriculum that would teach anyone to deal in abstracted forms. The only notable abstraction we ever meet is that empty form of arithmetic which is called algebra; and this is taught to us in such a way that most of us can pass a fairly hard examination in algebraic technique without even knowing that algebra is, indeed, the abstracted form of arithmetical calculations. No wonder, then, that we feel unfamiliar with pure forms! No wonder that some philosophers and almost all laymen believe abstraction to be vicious and intrinsically false. Without logical insight and training, they cannot go very far before falling into confusion, and then they blame the abstract nature of the ideas they are trying to handle for their own inability to handle them. Yet these same people are not afraid that a problem in, say, distances and horse-powers will become "purely verbal" if they apply algebra to its solution; that is because they know the manipulation of their algebra, and have learned to choose such forms as actually can be applied. There is nothing abstruse, esoteric or "unreal" about abstract thinking. As Lord Russell remarks, everybody has "some kind of knowledge of logical forms"; it only needs to be made explicit, conscious, and familiar. And this is what the study of logic is supposed to do. We all deal with pure forms in a practical, intuitive way. Whenever we draw the ground-plan of a house, we not only see the analogy between the plan and the prospective edifice, but we intend to convey the mere form of the house without any indication or thought of the material to be used in building it. When we buy a paper pattern for a dress, we intend to use the form, without mentally committing ourselves to follow the suggestion on the envelope that it be made in blue chiffon or flowered voile. When we compose a tune, we recognize it whether it be sung, played, or whistled, and even if it is transposed into a higher or lower key. It is the form that interests us, not the medium wherein this form is expressed. ### 7. Concepts This process of attending only to the form of a thing or a situation, and conveying this "abstracted" form, which we carry out unconsciously as part of our common sense, becomes increasingly important when we pass from mere common sense to scientific thinking. Such abstracted forms are our scientific concepts. And because there are an astounding number of analogies in nature, we can form concepts which apply to a very great range of events. In fact, a few powerful concepts can systematize, or perhaps revolutionize, a whole field of observation, experiment, and hypothesis, called "a science." Consider, for instance, how many motions follow the general pattern called "oscillation." The swing of a pendulum, the swaying of a skyscraper, the vibration of a violin-string over which the bow is passing, the chatter of our teeth on a cold day-all these are examples of the type-form called "oscillation." Now, if we were to define this type-form, we would omit all reference to skyscrapers and fiddle-strings and teeth, and describe it, probably, as "rhythmic motion to and fro," or in some such terms that would connote only the sort of motion we are talking about and not the sort of thing that moves. Probably, each of us has learned the meaning of oscillation through a different medium; but whether we gathered our first idea of it from the shaking of Grandpa's palsied hands or from the quiver of a tuning-fork-or from the vibration of a parked automobile with the motor running-however our mental pictures may differ from each other, they have one thing in common: they are all derived from some rhythmic motion to and fro. The things exemplifying this type of motion are not necessarily alike in other respects; the swaying skyscraper and the vibrating violin-string are certainly not alike in appearance, origin, or purpose. But their motions have the common property of going rhythmically to and fro. This property is the logical form of their motions, and so we may call all these motions diverse instances of the same form. When we consider the common form of various things, or various events, and call it by a name that does not suggest any particular thing or event, or commit us to any mental picture for instance, when we consider this common form of various movements, and call it by a name such as "oscillation"-we are consciously, deliberately abstracting the form from all things which have it. Such an abstracted form is called a concept. From our concrete experiences we form the concept of oscillation. The fact that so many things in nature exemplify the same forms makes it possible for us to collect our enormously variegated experiences of nature under relatively few concepts. If this were not the case, we could have no science. If there were not fundamental concepts such as oscillation, gravitation, radiation, etc., exemplified in nature over and over again, we could have no formulae of physics and discover no laws of nature. Scientists proceed by abstracting more and more fundamental forms (often seeing similarities among the abstracted forms, or concepts, themselves, and thus gathering several concepts into one); and by finding more and more things that fall under certain concepts, i.e. that exhibit certain general forms. ### 8. Interpretation The latter of these two procedures, finding applications for concepts, is called interpretation of an abstract form. It is a process of looking about for kinds of things to which a certain form belongs. If, for instance, we would interpret the abstract concept of "rotation," we would think of the rolling of a wheel, the motion of a heavenly body, the spinning of a top, the whirl of a propeller. Wheel-rolling, globe-turning, top-spinning, propeller-whirling, are all interpretations of the form, all different contents for the abstract concept "rotation." In one sense, two exactly similar spinning tops might be taken as two contents for one form, but in order to avoid confusion I shall call them two instances of one content for the same form. By two contents for a form I shall always mean two sorts of thing having the same form, i.e. falling under the same concept. That two instances do so goes without saying. There will be further discussion of this usage when we come to the distinction between "concrete" and "specific" elements. Scientific concepts are forms which are exemplified in some general and important part of reality. The natural sciences all deal with abstract forms, but only with a selected set of them, namely those which will take a special sort of thing for their contents. Physics deals with any forms which may have physical things for their contents. Biology deals with just those forms that apply to living matter. That is to say, the special sciences take cognizance of all those and only those conceptual patterns, or formulae, to which they can give some interpretation relevant to their chosen subject-matter. Interpretation is the reverse of abstraction; the process of abstraction begins with a real thing and derives from it the bare form, or concept, whereas the process of interpretation begins with an empty concept and seeks some real thing which embodies it. In the sciences as in ordinary life, we are interested in forms only in so far as they are the patterns of certain things that concern us. Most of the abstract concepts we employ are handed down to us-in language, like all our simple adjectival and adverbial concepts; or by deliberate training, like our common knowledge of mathematics, mechanics, and so forth. We learn them as they apply to certain things; we learn number by counting things, shapes by fitting objects together, qualities by comparing various articles, rules of conduct by gradually collecting and judging instances of good and evil. The easiest way to teach a formula is to present several instances and point out their common formal properties. But this is not the easiest way to discover new patterns, which no one points out to us. There are, essentially, two ways in which new forms of things are discovered: (1) by abstraction from instances which nature happens to collect for us (the power to recognize a common form in such a chance collection is scientific genius); (2) by interpretation of empty forms we have quite abstractly constructed. The latter way is usually the easier, because if we know a great variety of possible forms, we have at least an idea of what we are looking for. Technical inventions are discoveries of this sort; the inventor rarely, if ever, propounds a new principle, i.e. a new fundamental concept of physics, but cleverly combines the principles he has learned into a new interesting pattern; he may then construct a physical thing, or actual model, of that pattern. His abstract form-the calculation he makes on paper-is a mathematical theorem from purely conceptual principles; his application of it to the realm of physics is an interpretation; and his model is an instance of this interpreted special form (and of the principles it combines). He is concerned essentially about interpretations, and his abstract work is all performed for the sake of finding physically interpretable forms. ### 9. The Field of Logic But if we would hold aloof awhile from any special science and really gain insight into the great storehouse of forms which may be interpretable physically, or psychically, or for any realm of experience whatever, we must consider abstracted patterns as such-the orders in which any things whatever may be arranged, the modes under which anything whatever may present itself to our understanding. This sounds like an utterly impossible and elusive task; to a casual observer it certainly seems as though there must be as many incommensurable forms in the world as there are different kinds and departments of experience. But, happily for our restricted intellects, this is not the case. Many things which look utterly unlike in experience-far more unlike than the motions of skyscrapers and of violin-strings, or tops and planets are really made up in very similar ways, only it requires a good deal of practice to see this. *“Orderliness and system,” said Josiah Royce,* *“are much the same in their most general characters, whether they appear in a Platonic dialogue, or in a modern textbook of botany, or in the commercial conduct of a business firm, or in the arrangement and discipline of an army, or in a legal code, or in a work of art, or even in a dance or in the planning of a dinner. Order is order. System is system. Amidst all the variations of systems and of orders, certain general types and characteristic relations can be traced.”* The tracing of such types and relations among abstracted forms, or concepts, is the business of logic. *“The Principles of Logic” in Windelband and Ruge’s Encyclopaedia of the Philosophical Sciences, vol. i, p. 81 (further volumes did not appear).*