Studies in Logical Theory Part 8

But it is a.n.a.logy which Welton considers "the chief source from which new hypotheses are drawn." In the second tendency mentioned under enumerative induction, that is, the tendency to a.n.a.lysis of content or intension, we are naturally led to a.n.a.logy, for in our search for the characteristic feature which determines cla.s.sification among the concrete particulars our first step will be an inference by a.n.a.logy. In a.n.a.logy attention is turned from the number of observed instances to their character, and, because particulars have some feature in common, they are supposed to be the same in still other respects. While the best we can reach in a.n.a.logy is probability, the arguments may be such as to result in a high degree of certainty. The form of the argument is valuable in so far as we are able to distinguish between essential and nonessential characteristics on which to base our a.n.a.logy. What is essential and what nonessential depends upon the particular end we have in view.

In addition to enumerative induction, which Welton has mentioned, it is to be noted that there are a number of other processes which are very similar to it in that a number of particulars appear to furnish a basis for a general principle or method. Such instances are common in induction, in instruction, and in methods of proof.

If one is to be instructed in some new kind of labor, he is supposed to acquire a grasp of the method after having been shown in a few instances how this particular work is to be done; and, if he performs the manipulations himself, so much the better. It is not asked why the experience of a few cases should be of any a.s.sistance, for it seems self-evident that an experienced man, a man who has acquired the skill, or knack, of doing things, should deal better with all other cases of similar nature.

There is something very similar in inductive proofs, as they are called.

The inductive proof is common in algebra. Suppose we are concerned in proving the law of expansion of the binomial theorem. We show by actual calculation that, if the law holds good for the _n_th power, it is true for the _n_+first power. That is, if it holds for any power, it holds for the next also. But we can easily show that it does hold for, say, the second power. Then it must be true for the third, and hence for the fourth, and so on. Whether this law, though discovered by inductive processes, depends on deduction for the conclusiveness of its proof, as Jevons holds;[74] whether, as Erdmann[75] contends, the proof is thoroughly deductive; or whether Wundt[76] is right in maintaining that it is based on an exact a.n.a.logy, while the fundamental axioms of mathematics are inductive, it is clear that in such proofs a few instances are employed to give the learner a start in the right direction. Something suggests itself, and is found true in this case, in the next, and again in the next, and so on. It may be questioned whether there is usually a very clear notion of what is involved in the "so on."

To many it appears to mark the point where, after having been taken a few steps, the learner is carried on by the acquired momentum somewhat after the fas.h.i.+on of one of Newton's laws of motion. Whether the few successive steps are an integral part of the proof or merely serve as ill.u.s.tration, they are very generally resorted to. In fact, they are often employed where there is no attempt to introduce a general term such as _n_, or _k_, or _l_, but the few individual instances are deemed quite sufficient. Such, for instance, is the custom in arithmetical processes. We call attention to these facts in order to show that successive cases are utilized in the course of explanation as an aid in establis.h.i.+ng the generality of a law.

In geometry we find a cla.s.s of proofs in which the successive steps seem to have great significance. A common proof of the area of the circle will serve as a fair example. A regular polygon is circ.u.mscribed about the circle. Then as the number of its sides are increased its area will approach that of the circle, as its perimeter approaches the circ.u.mference of the circle. The area of the circle is thus inferred to be [pi]_R_^2, since the area of the polygon is always (1/2)_R_ perimeter, and in case of the circle the circ.u.mference = 2[pi]_R_.

Here again we get under such headway by means of the polygon that we arrive at the circle with but little difficulty. Had we attempted the transition at once, say, from a circ.u.mscribed square, we should doubtless have experienced some uncertainty and might have recoiled from what would seem a rash attempt; but as the number of the sides of our polygon approach infinity--that mysterious realm where many paradoxical things become possible--the transition becomes so easy that our polygon is often said to have truly become a circle.

Similarly, some statements of the infinitesimal calculus rest on the a.s.sumption that slight degrees of difference may be neglected. Though the more modern theory of limits has largely displaced this att.i.tude in calculus and has also changed the method of proof in such geometrical problems as the area of the circle, the underlying motive seems to have been to make transitions easy, and thus to make possible a continued application of some particular method or way of dealing with things.

But granted that this is all true, what has it to do with the origin of the hypothesis? It seems likely that the hypothesis may be suggested by a few successive instances; but are these to be cla.s.sed with the successive steps in proof to which we have referred? In the first place, we attempt to prove our hypothesis because we are not sure it is true; we are not satisfied that there are no other tenable hypotheses. But if we do test it, is not such test enough? It depends upon how thorough a grasp we have of the situation; but, in general, each test case adds to its probability. The value of tests lies in the fact that they strengthen and tend to confirm our hypothesis by checking the force of alternatives. One instance is not sufficient because there are other possible incipient hypotheses, or more properly tendencies, and the enumeration serves to bring one of these tendencies into prominence in that it diminishes other vague and perhaps subconscious tendencies and strengthens the one which suddenly appears as the mysterious product of genius.

The question might arise why the mere repet.i.tion of conflicting tendencies would lead to a predominance of one of them. Why would they not all remain in conflict and continue to check any positive result? It is probably because there never is any absolute equilibrium. The successive instances tend to intensify and bring into prominence some tendency which is already taking a lead, so to speak. And it may be said further in this connection that only as seen from the outside, only as a mechanical view is taken, does there appear to be an excluding of definitely made out alternatives.

In explanation of the part played by a.n.a.logy in the origin of hypotheses, Welton points out that a mere number of instances do not take us very far, and that there must be some "_specification_ of the instances as well as numbering of them," and goes on to show that the argument by enumerative induction pa.s.ses readily into one from a.n.a.logy, as soon as attention is turned from the number of the observed instances to their character. It is not necessary, however, to pa.s.s to a.n.a.logy through enumerative induction. "When the instances presented to observation offer immediately the characteristic marks on which we base the inference to the connection of S and P, we can proceed at once to an inference from a.n.a.logy, without any preliminary enumeration of the instances."[77]

Welton, and logicians generally, regard a.n.a.logy as an inference on the basis of partial ident.i.ty. Because of certain common features we are led to infer a still greater likeness.

Both enumerative induction and a.n.a.logy are explicable in terms of habit.

We saw in our examination of enumerative induction that a form of reaction gains strength through a series of successful applications.

a.n.a.logy marks the presence of an identical element together with the tendency to extend this "partial ident.i.ty" (as it is commonly called) still farther. In other words, in a.n.a.logy it is suggested that a type of reaction which is the same in certain respects may be made similar in a greater degree. In enumerative induction we lay stress on the number of instances in which the habit is applied. In a.n.a.logy we emphasize the content side and take note of the partial ident.i.ty. In fact, the relation between enumerative induction and a.n.a.logy is of the same sort as that existing between a.s.sociation by contiguity and a.s.sociation by similarity. In a.s.sociation by contiguity we think of the things a.s.sociated as merely standing in certain temporal or spatial relations, and disregard the fact that they were elements in a larger experience.

In case of a.s.sociation by similarity we regard the like feature in the things a.s.sociated as a basis for further correction.

In conversion of propositions we try to reverse the direction of the reaction, so to speak, and thereby to free the habit, to get a mode of response so generalized as to act with a minimum cue. For instance, we can deal with A in a way called B, or, in other words, in the same way that we did with other things called B. If we say, "Man is an animal,"

then to a certain extent the term "animal" signifies the way in which we regard "man." But the question arises whether we can regard all animals as we do man. Evidently not, for the reaction which is fitting in case of animals would be only partially applicable to man. With the animals that are also men we have the beginning of a habit which, if unchecked, would lead to a similar reaction toward all animals, _i. e._, we would say: "All animals are men." Man may be said to be the richer concept, in that only a part of the reaction which determines an object to be a man is required to designate it as an animal. On the other hand, if we start with animal, then (except in case of the animals which are men) there is lacking the subject-matter which would permit the fuller concept to be applied. By supplying the conditions under which animal=man we get a reversible habit. The equation of technical science has just this character. It represents the maximum freeing or abstraction of a predicate _qua_ predicate, and thereby multiplies the possible applications of it to subjects of future judgments, and lessens the amount of shearing away of irrelevancies and of re-adaptation necessary when so used in any particular case.

_Formation and test of the hypothesis._--The formation of the hypothesis is commonly regarded as essentially different from the process of testing, which it subsequently undergoes. We are said to observe facts, invent hypotheses, and _then_ test them. The hypothesis is not required for our preliminary observations; and some writers, regarding the hypothesis as a formulation which requires a difficult and elaborate test, decline to admit as hypotheses those more simple suppositions, which are readily confirmed or rejected. A very good ill.u.s.tration of this point of view is met with in Wundt's discussion of the hypothesis, by an examination of which we hope to show that such distinctions are rather artificial than real.

The subject-matter of science, says Wundt,[78] is const.i.tuted by that which is actually given and that which is actually to be expected. The whole content is not limited to this, however, for these facts must be supplemented by certain presuppositions, which are not given in a factual sense. Such presuppositions are called hypotheses and are justified by our fundamental demand for unity. However valuable the hypothesis may be when rightly used, there is constant danger of illegitimately extending it by additions that spring from mere inclinations of fancy. Furthermore, the hypothesis in this proper scientific sense must be carefully distinguished from the various inaccurate uses, which are prevalent. For instance, hypotheses must not be confused with expectations of fact. As cases in point Wundt mentions Galileo's suppositions that small vibrations of the pendulum are isochronous, and that the s.p.a.ce traversed by a falling body is proportional to the square of the time it has been falling. It is true that such antic.i.p.ations play an important part in science, but so long as they relate to the facts themselves or to their connections, and can be confirmed or rejected any moment through observation, they should not be cla.s.sed with those added presuppositions which are used to co-ordinate facts. Hence not all suppositions are hypotheses. On the other hand, not every hypothesis can be actually experienced. For example, one employs in physics the hypothesis of electric fluid, but does not expect actually to meet with it. In many cases, however, the hypothesis becomes proved as an experienced fact. Such was the course of the Copernican theory, which was at first only a hypothesis, but was transformed into fact through the evidence afforded by subsequent astronomical observation.

Wundt defines a theory as a hypothesis taken together with the facts for whose elucidation it was invented. In thus establis.h.i.+ng a connection between the facts which the hypothesis merely suggested, the theory furnishes at the same time partly the foundation (_Begrundung_) and partly the confirmation (_Bestatigung_) of the hypothesis.[79] These aspects, Wundt insists, must be sharply distinguished. Every hypothesis must have its _Begrundung_, but there can be _Bestatigung_ only in so far as the hypothesis contains elements which are accessible to actual processes of verification. In most cases verification is attainable in only certain elements of the hypothesis. For example, Newton was obliged to limit himself to one instance in the verification of his theory of gravitation, viz., the movements of the moon. The other heavenly bodies afforded nothing better than a foundation in that the supposition that gravity decreases as the square of the distance increases enabled him to deduce the movements of the planets. The main object of his theory, however, lay in the deduction of these movements and not in the proof of universal gravity. With the Darwinian theory, on the contrary, the main interest is in seeking its verification through examination of actual cases of development. Thus, while the Newtonian and the greater part of the other physical theories lead to a deduction of the facts from the hypotheses, which can be verified only in individual instances, the Darwinian theory is concerned in evolving as far as possible the hypothesis out of the facts.

Let us look more closely at Wundt's position. We will ask, first, whether the distinction between hypotheses and expectations is as p.r.o.nounced as he maintains; and, second, whether the relation between _Begrundung_ and _Bestatigung_ may not be closer than Wundt would have us believe.

As examples of the hypothesis Wundt mentions the Copernican hypothesis, Newton's hypothesis of gravitation, and the predictions of the astronomers which led to the discovery of Neptune. As examples of mere expectations we are referred to Galileo's experiments with falling bodies and pendulums. In case of Newton's hypothesis there was the a.s.sumption of a general law, which was verified after much labor and delay. The heliocentric hypothesis of Copernicus, which was invented for the purpose of bringing system and unity into the movements of the planets, has also been fairly well substantiated. In the discovery of Neptune we have, apparently, not the proof of a general law or the discovery of further peculiarities of previously known data, but rather the discovery of a new object or agent by means of its observed effects.

In each of these instances we admit that the hypothesis was not readily suggested or easily and directly tested.

If we turn to Galileo's pendulum and falling bodies, it is clear first of all that he did not have in mind the discovery of some object, as was the case in the discovery of Neptune. Did he, then, either contribute to the proof of a general law or discover further characteristics of things already known in a more general way? Wundt tells us that Galileo only determined a little more exactly what he already knew, and that he did this with but little labor or delay.

What, then, is the real difference between hypothesis and expectation?

If we compare Galileo's determination of the law of falling bodies with Newton's test of his hypothesis of gravitation, we see that both expectation and hypothesis were founded on observation and took the form of mathematical formulae. Each tended to confirm the general law expressed in its formula, though there was, of course, much difference in the time and labor required. If we compare the Copernican hypothesis with Galileo's supposition concerning the pendulum, we find again that they agree in regard to general purpose and method, and differ in the difficulty of verification. If the experiment with the pendulum only subst.i.tuted exactness for inexactness, did the Copernican theory do anything different in _kind_? It is true that the more exact statement of the swing of the pendulum was expressed in quant.i.tative form, but quant.i.tative statement is no criterion of either the presence or the absence of the hypothesis.

Again, we may compare the pendulum with Kepler's laws. What was Kepler's hypothesis, that the square of the periodic times of the several planets are proportional to the cubes of their mean distances from the sun, except a more exact formulation of facts which were already known in a more general way? Wundt's position seems to be this: whenever a supposition or suggestion can be tested readily, it should not be cla.s.sed as a hypothesis. This would make the distinction one of degree rather than kind, and it does not appear how much labor we must expend, or how long our supposition must evade our efforts to test it, before it can win the t.i.tle of hypothesis.

In the second place, we have seen that Wundt draws a sharp line between _Begrundung_ and _Bestatigung_. It is doubtless true that every hypothesis requires a certain justification, for unless other facts can be found which agree with deductions made in accordance with it, its only support would be the data from which it is drawn. Such support as this would be obtained through a process too clearly circular to be seriously entertained. The distinction which Wundt draws between _Begrundung_ and _Bestatigung_ is evidently due to the presence of the experimental element in the latter. For descriptive purposes this distinction is useful, but is misleading if it is understood to mean that there is mere experience in one case and mere inference in the other. The difference is rather due to the relative parts played by inference and by accepted experience in each. In _Begrundung_ the inferential feature is the more prominent, while in _Bestatigung_ the main emphasis is on the experiential aspect. It must not be supposed, however, that either of these aspects can be wholly absent. It is difficult to understand how any hypothesis can be entertained at all unless it meets in some measure the demand with reference to which it was invented, viz., a unification of conflicts in experience. And, _in so far_, it is confirmed. The motive which casts doubt upon its adequacy is the same that leads to its re-forming as a hypothesis, as a mental concept.

The difficulties in Wundt's position are thus due to a failure to take account of the reconstructive nature of the judgment. The predicate, supposition, or hypothesis, whatever we may choose to call it, is formed because of the check of a former habit. The judgment is an ideal application of a new habit, and its test is the attempt to act in accordance with this ideal reconstruction. It must not be thought, however, that our supposition is first fully developed and then tried and accepted or rejected without modification. On the contrary, its growth is the result of successive minor tests and corresponding minor modifications in its form. Formation and test are merely convenient distinctions in a larger process in which forming, testing, and _re_-forming go on together. The activity of experimental verification is not only a testing, a confirming or weakening of the validity of a hypothesis, but it is equally well an evolution of the _meaning_ of the hypothesis through bringing it into closer relations with specific data not previously included in defining its import. _Per contra_, a purely reflective and deductive consideration which develops the idea as hypothesis, _in so far_ as it introduces the determinateness of previously accepted facts within the scope, comprehension, or intension of the idea, is in so far forth, a verification.

If the view which we have maintained is correct, the hypothesis is not to be limited to those elaborate formulations of the scientist which he seeks to confirm by crucial tests. The hypothesis of the investigator differs from the comparatively rough conjecture of the plain man only in its greater precision. Indeed, as we have attempted to show, the hypothesis is not a method which we may employ or not as we choose; on the contrary, as predicate of the judgment it is present in a more or less explicit form if we judge at all. Whether the time and labor required for its confirmation or rejection is a matter of a lifetime or a moment, its nature remains the same. Its function is identical with that of the predicate. In short, the hypothesis is the predicate so brought to consciousness and defined that those features which are not noticed in the ordinary judgment are brought into prominence. We then recognize the hypothesis to be what in fact the predicate always is, viz., a method of organization and control.

VIII

IMAGE AND IDEA IN LOGIC

The logic of sense-impressions and of ideas as copies of sense-impressions has had its day. It engaged in a conflict with dogmatism, and scored a decisive victory. It overthrew the dynasty of prescribed formulae and innate ideas, of ideas derived ready-made from custom and social usage, ancient enough to be lost in the remote obscurity of divine sources; and enthroned in their place ideas derived from, and representative of, the sense-experiences of a very real and present world. It marked a reaction from dogma back to the original meaning of dogma, back to the seeming, the appearance, of things. So thoroughly did Bacon and Hobbes, Locke and Hume, to mention only these four, do their work, that many of the problems growing out of the conflict itself, to say nothing of the scholastic traditions that were combated, have come to have merely a historical rather than a logical interest. Logic no longer concerns itself very eagerly with the content or sensuous qualities of ideas, with their derivation from sense-impressions, or with questions as to the relation of copy to original, of representative to that which is presented. It is concerned rather with the constructive operations of thought, with meaning, reference to reality, inference--with intellectual processes. Perhaps in no respect is this s.h.i.+fting of logical standpoint indicated more clearly than in the unregretful way with which the old logical interest in the sense-qualities of ideas is now made over to psychology. States of consciousness as such, we are told, are the proper study of psychology; whereas logic concerns itself with the relation of thought to its object. True, these states of consciousness include thought-states, as well as sense-impressions; ideas and concepts, as well as feelings and fancies; and the business of psychology is to observe, compare and cla.s.sify, describe and chronicle, these states and whatever else is carried along in the stream of consciousness. But logic is concerned, not with these states of consciousness _per se_, least of all with the flotsam and jetsam of the stream, but with its reference to reality; not with the true, but with truth; not even with what consciousness does, but with how consciousness is to outdo itself, transcend itself, in a rational and universal whole. Even an empirical logic has to arrange somehow the way to get from one sense-impression to another.

In drawing this distinction between logic and psychology--a distinction which virtually amounts to a separation--two things are overlooked: first, that the distinction itself is a logical distinction, and may properly const.i.tute a problem falling under the province of logical inquiry and theory; and, second, that the rather arbitrary and official setting apart of psychology to look after the task of studying states of consciousness does not carry with it the guarantee that psychology will confine itself exclusively to that task. This last point in particular must be my excuse for discussing the question of image and idea from the psychological rather than from the logical standpoint. The logic of ideas derived from sense-impressions has had its day. But even the very leavings of the past may have been gathered up and reconstructed by psychology in such a way as to antic.i.p.ate some of the newer developments of logical theory and meet some of its difficulties. One can hardly hope to justify in advance a discussion based on such a sheer possibility.

Let us begin, rather, by noting down from the standpoint of logic some of the distinctions between image and idea, and the estimate of the logical function and value of mental imagery, and see in what direction they take us and whether they suggest a resort to an a.n.a.lysis from the standpoint of psychology.

Proceeding from the standpoint of logic to inquire into the logical function of mental imagery and into the distinction between image and idea, we shall come upon two opposed but characteristic answers. If the inquiry be directed to a member of the empirical school of logic, he would be bound to answer in the affirmative, so far as the question regarding the function of mental imagery is concerned. He would be likely to say, if he were loyal to the traditions of his school, that mental imagery is the counterpart of sense-perception, and is thus the representative of the data with which empirical logic is concerned.

Mental imagery, he would continue, is a representative in a literal sense, a copy, a reflection, of what comes to us through the avenues of sensation. True, it is not the perfect twin of sense-experience; else we could not tell them apart; indeed, there are times when the copy becomes so much like the original that we are deceived by it, as in dreams or in hallucinations. Ordinarily, however, we are able to distinguish one from the other. Two criteria are usually present; (1) imagery is fainter, more fleeting, than the corresponding sense-experience; and (2), save in the case of accurate memory-images, it is subject to a more or less arbitrary rearrangement of its parts, as when, for example, we make over the images of scenes we have actually experienced, to furnish forth the setting of some remote historical event.

Barring, or controlling and rectifying, its tendencies toward both arbitrary and constructive variations from the original, mental imagery is on the same level as sense-experience, and serves the same logical purpose. That is to say, it contributes to the data which const.i.tute the foundations of empirical logic. It furnishes materials for the operations of observing, comparing, abstracting and generalizing.

Mental imagery helps to piece out the fragments that may be presented to sense-experience. It supplies the entire anatomy when only a single bone, say, is actually given. Yet, however useful as a servant of truth, it has to be carefully watched, lest its spontaneous tendency to vary the actual order and coexistence of data lead the investigator astray.

The copy it presents is, after all, a temporary makes.h.i.+ft, until it can be shown to correspond point for point to the now absent reality. Mental imagery furnishes one with an ill.u.s.trated edition of the book of nature, but the ill.u.s.trations await the confirmation of comparison with the originals.

Mental imagery functions logically when it extends the area of data beyond the range of the immediate sense-perceptions of any given time, and thus makes possible a more comprehensive application of the empirical methods of observation, comparison, abstraction, and generalization. It functions logically when it acts as a feeder of logical machinery, though it is not indispensable to this machinery and does not modify its principles. The logical mill could grind up in the same way the pure grain of sense-perceptions, unmixed with mental images, but it would have to grind more slowly for lack of material. In other words, empirical logic could carry on its operations of observing, comparing, abstracting, and generalizing, solely on the basis of objects or data present to the senses, and with no extension of this basis in terms of imagery, or copies of objects not immediately present; but it would take more time for it to apply and carry through its operations.

The logical machinery is the same in each case. The materials fed and the product issuing are the same in each case. Imagery simply fulfils the function of providing a more copious grist.

The empiricist's answer to our question regarding the logical function of mental imagery leaves that function in an uncertain and parlous state. Imagery lacks the security of sense-perception on the one hand, and it has no part in the operation of thought on the other. It is a sort of hod-carrier, whose function it is to convey the raw materials of sense-perception to a more exalted position where someone else does all the work. I suppose this could be called a functional interpretation of a logical element. The question, then, would be whether an element so functioning is in any sense logical. As an element lying outside of the thought-process it owes no responsibility to logic; it is not amenable to its regulations. Thought simply finds it expedient to operate with an agent over which it has no intrinsic control. The case might be allowed to rest here. Yet were this extra-logical element of imagery to abandon thought, all conscious thinking as opposed to sense-perception would cease. A false alarm, perhaps. Imagery may be so const.i.tuted that it is inseparably subordinated to thought and can never abandon it. Thought may simply exude imagery. But imagery somehow has to represent sense-perception, also. It can hardly be a secretion of thought and a copy of sense-perceptions at one and the same time, unless the empiricist is willing to turn absolute idealist! Before taking such a desperate plunge as this, it might be desirable to see whether there is any other recourse.

There is another and a very different answer to the question regarding the logical function of mental imagery. To distinguish this answer from that of the a.s.sociationist or empiricist, I will call it the answer of the conceptualist. I am not at all positive that this label would stick even to those to whom it might be applied with considerable justification. The terms "rationalistic" and "transcendental" might be preferred in opposition to the term "empirical." And we have the term "apperceptionist" in opposition to the term "a.s.sociationist." If the term "conceptualist" is admissible, it should be brought down to date, perhaps, by making it "neo-conceptualist." The present difficulties regarding terminology would be eased considerably if we only had a convenient set of derivatives made from the word "meaning." Since we have not, I will use derivatives made from the word "concept" to denote views opposite to those held by the empirical school.

The conceptualist could be depended upon to answer our question in the negative. Logical functions begin where the image leaves off. They begin with the _idea_, with meaning. The conceptualist distinguishes sharply between the image as a psychical existence and the idea, or concept, as logical meaning. On the one hand, you have the "image," not only as a mere psychical existence, but a mocking existence at that, fleeting, inconstant, s.h.i.+fting, never perhaps twice alike; yet, mind you, an _existence_, a _fact_--that must be admitted. On the other hand, you have the "idea," with "a fixed content or logical meaning,"[80] which is referred by an act of judgment to a reality beyond the act.[81]

The "idea," the logical meaning, begins where the "image" leaves off.

Does this mean that the "idea" is wholly independent of the "image"? Yes and no. The "idea" is independent of that which is ordinarily regarded as the special characteristic of an "image," namely, its quality, its sense-content. That is to say, the "idea" is independent of any particular "image," any special embodiment of sense-content. Any image will do. As Mr. Bosanquet remarks in comparing the psychical images that pa.s.s through our minds to a store of signal flags:

Not only is it indifferent whether your signal flag of today is the same bit of cloth that you hoisted yesterday, but also, no one knows or cares whether it is clean or dirty, thick or thin, frayed or smooth, as long as it is distinctly legible as an element of the signal code. Part of its content, of its attributes and relations, is a fixed index which carries a distinct reference; all the rest is nothing to us, and, except in a moment of idle curiosity, we are unaware that it exists.[82]

On the other hand, the "idea" could not operate as an idea, could not be in consciousness, save as it involves some imagery, however old, dirty, thin, and frayed. Take the statement, "The angles of a triangle are equal to two right angles." If the statement means anything to a given individual, if it conveys an idea, it must necessarily involve some form of imagery, some qualitative or conscious content. But so far as the _meaning_ is concerned, it is a matter of complete indifference as to _what_ qualities are involved. These qualities may be in terms of visual, auditory, tactual, kinaesthetic, or verbal imagery. The individual may visualize a blackboard drawing of a triangle with its sides produced, or he may imagine himself to be generating a triangle while revolving through an angle of 180. Any imagery anyone pleases may be employed, so long as there goes with it somehow the _idea_ of the relation of equality between the angles of a triangle and two right angles. But the conceptualist does not stop here. The act of judgment comes in to affirm that the "idea" is no mere idea, but is a quality of the real. "The act [of judgment] attaches the floating adjective [the idea, the logical meaning] to the nature of the world, and, at the same time, tells one it was there already."[83] The "idea," the logical meaning, begins where the "image" leaves off. Yet, somehow, the "idea"

could not begin, unless there were an "image" to leave off.

An "image" is not an "idea," says the conceptualist. An "idea" is not an "image." (1) An "image" is not an "idea," because an "image" is a particular, individual fragment of consciousness. It is so bound up with its own existence that it cannot reach out to the existence of an "idea," or to anything beyond itself. Chemically speaking, it is an _avalent_ atom of consciousness, if such a thing is thinkable. Mr.

Bosanquet raises the question:

Are there at all ideas which are not symbolic?... The answer is that _(a)_ in judgment itself the idea can be distinguished _qua_ particular in time or psychical fact, and _so far_ is not symbolic; and _(b)_ in all those human experiences from which we draw our conjectures as to the animal intelligence, when in languor or in ignorance image succeeds image without conscious judgment, we feel what it is to have ideas as facts and not as symbols.[84]

(2) An "idea" is not an "image," because an idea _is_ meaning, which consists in a part of the content of the image, cut off, and considered apart from the _existence_ of the content or sign itself.[85] This meaning, this fragment of psychical existence, lays down all claim to existence on its own account, that it may refer through an act of judgment to a reality beyond itself and beyond the act also. An "image"

is not an "idea" and an "idea" is not an "image," because an "image"