§2. LATER REFLECTIONS

36. Our knowledge of any subject never goes beyond collecting observations and forming some half-conscious expectations, until we find ourselves confronted with some experience contrary to those expectations. That at once rouses us to consciousness: we turn over our recollections of observed facts; we endeavour so to rearrange them, to view them in such new perspective that the unexpected experience shall no longer appear surprising. This is what we call explaining it, which always consists in supposing that the surprising facts that we have observed are only one part of a larger system of facts, of which the other part has not come within the field of our experience, which larger system, taken in its entirety, would present a certain character of reasonableness, that inclines us to accept the surmise as true, or likely. For example, let a person entering a large room for the first time, see upon a wall projecting from behind a large map that has been pinned up there, three-quarters of an admirably executed copy in fresco of one of Rafael's most familiar cartoons. In this instance the explanation flashes so naturally upon the mind and is so fully accepted, that the spectator quite forgets how surprising those facts are which alone are presented to his view; namely, that so exquisite a reproduction of one of Rafael's grandest compositions should omit one-quarter of it. He guesses that that quarter is there, though hidden by the map; and six months later he will, maybe, be ready to swear that he saw the whole. This will be a case under a logico-psychical law of great importance, to which we may find occasion to revert soon, that a fully accepted, simple, and interesting inference tends to obliterate all recognition of the uninteresting and complex premisses from which it was derived. The brighter the observer's intelligence (unless some circumstance has raised a doubt), the more confident he will soon be that he saw the entire composition. Yet, in fact, the idea of the whole's being on that wall will be merely evolved from his Ichheit: it will be a surmise, conjecture, or guess.

37. We may be aided by previous knowledge in forming our hypotheses. In that case they will not be pure guesses but will be compounds of deductions from general rules we already know, applied to the facts under observation, for one ingredient, and pure guess for the other ingredient. Thus, suppose the surprising facts which puzzle us are the actions of a certain man on a certain occasion; and our conjecture relates to the state of belief that caused such conduct. If we have no previous knowledge of the man, any one state of belief that would account for his conduct might be as good a guess as any other; but if we know that he is particularly inclined, or particularly disinclined, to extravagant beliefs or to any other special kind of belief, we still have to guess; only we shall select our guess from a smaller number of possible hypotheses.

38. In the evolution of science, guessing plays the same part that variations in reproduction take in the evolution of biological forms, according to the Darwinian theory. For just as, according to that theory, the whole tremendous gulf, or ocean rather, between the moner and man has been spanned by a succession of infinitesimal fortuitous variations at birth, so the whole noble organism of science has been built up out of propositions which were originally simple guesses. For my part I refuse to believe that either the one or the other were fortuitous; and indeed I gravely doubt whether there be any tenable meaning in calling them so. As to the biological variations, I will spare the reader my reasons for not believing them fortuitous. For it would only lead us away from our subject. But as to the first guesses out of which science has been developed, I will say a word or two. It is well within bounds to reckon that there are a billion (i.e., a million million) hypotheses that a fantastic being might guess would account for any given phenomenon. For this phenomenon would certainly be more or less connected in the mind of such a being with a million other phenomena (for he would not be restricted to contemporaneous events) and it might be supposed that the special determination of each was connected with the special determinations of each of the others in order to produce the observed phenomenon. I will not carry out this idea further: it suffices to show that according to the doctrine of chances it would be practically impossible for any being, by pure chance, to guess the cause of any phenomenon.

39. There are, indeed, puzzles, and one might well say mysteries, connected with the mental operation of guessing; -- yes; more than one. There can, I think, be no reasonable doubt that man's mind, having been developed under the influence of the laws of nature, for that reason naturally thinks somewhat after nature's pattern. This vague explanation is but a surmise; but there is no room to believe that it was merely by luck that Galileo and other masters of science reached the true theories after so few wrong guesses as they did. This power of divining the truths of physics, -- for such it is, although it is somewhat imperfect, -- is certainly an aid to the instinct for obtaining food, an instinct whose wonders throughout the animal kingdom are exceeded only by that of producing and rearing offspring.

41. All the above, be it understood is sober truth, sedulously freed from all exaggeration and colour. If any reader should incline to deem the narrative apocryphal, it will certainly not be the psycholist, equally versed in the theory of his science and skilled in the application of it; for to him the incidents will present no extraordinary features. I suppose almost everybody has had similar experiences. But however frequently such facts may be encountered, there is certainly something a little mysterious in them; they demand explanation. That explanation must itself be conjectural, and must remain so until exact investigation has tested its sufficiency; and unless some new school of psychology should make its appearance, I do not believe that scientific testing of the theory is likely to be performed in our time.

42. I am going to point out a vera causa -- a known agency which tends to produce effects like the facts to be explained. But whether it would, under the circumstances described, be sufficient to produce the somewhat surprising facts, or whether it was aided by some other agency that has not suggested itself to my mind, I will not presume to opine.

43. My surmise is that at the bottom of the little mystery is buried a principle often enough asserted but never, I believe, supported by scientific observation, until Professor Joseph Jastrow and I carried through, at the Johns Hopkins University, a certain series of experiments. These experiments were mainly designed for quite another purpose, namely, in order to test Fechner's hypothesis of the "Differenzschwelle," which in no wise concerns us now. I proceed to describe, in outline, the essentials of the experiments. Of the two persons engaged in them, the one acted as experimenter and recorder, while the other, who could neither see nor hear the former, was the "subject" or victim of the experimentation. The latter said, "Ready." Thereupon an automatic arrangement, namely, by exposing a card from a well shuffled pack, indicated to the experimenter what pressure he was to bring to bear upon the finger of the subject, who carefully observed the degree of his feeling of pressure. When he was satisfied, perhaps after from five to twenty seconds, he said "Change." Thereupon by an exceedingly delicate contrivance (to avoid any sudden change or shock), the experimenter, according to an automatic operation of chance, either increased or diminished the pressure by less than one per cent of itself. The subject observed the new feeling of pressure, and again said "Change," whereupon the first pressure was brought back. These experiments were interspersed (by the automatic chance arrangement which was intended, of course, to exclude, as far as possible, mental action on the part of the experimenter), by others in which the changes of pressure were somewhat more considerable. The subject having observed the three states of feeling of pressure (of which the first and last were equal), first pronounced one or another of the four numerals, Naught, One, Two, Three. "Three" would mean that he was sure, or almost sure, of being able to say whether the middle pressure was greater or less than the other two. "Two" would mean that he was by no means sure, yet inclined to think he could tell. "One" would mean that he did not think he really perceived any difference; yet suspected that he perhaps might. "Naught" would mean that he was sure he could not perceive the slightest variation of pressure. Having thus indicated the degree of his confidence, he was obliged to say whether the middle pressure was greater or less than the others. In case his confidence was zero, this declaration would be (to his own consciousness) a purely random one, though he would avoid any particular regularity in his declarations, or any great preponderance of either "greater" or "lesser." Of course he never received the slightest intimation of whether he was right or wrong.

44. When our course of experiments had been carried on two hours daily (with such precautions against fatigue as the imperfect psychology of twenty-five years ago prescribed), and for about a month it was found that of the answers supposed to be given at random, which were a good half of the whole number and must, I think (I have not before me the record, which is given in Vol. III of the Memoirs of the U. S. National Academy of Sciences), have approached a thousand in number, about three out of every five were correct. That is to say, among all those cases in which the subject, after carefully searching his consciousness, felt quite sure he had experienced no variation of the sense of pressure, though a change and reverse change had really been made; and had accordingly said, quite at random, as he thought, that the middle pressure was greater or less than the first and last, what he so said agreed with the real fact half as often again as it disagreed. A reader inexpert in dealing with probabilities may think that so small a preponderance of true answers might have come about by chance. But in truth it is among the most certain things that we know that this was not so. So much is demonstrated truth, quite unquestionable. But if you go on to ask me upon what principle I would explain the fact that a person who, after the closest scrutiny of his consciousness, had pronounced that there was no trace of perceptible difference between two sensations of pressure, should in the very next breath have correctly said which of them was the greater, in three cases out of every five, my confidence largely evaporates. I can, indeed, mention a cause which undoubtedly exists and which must have acted toward producing that indubitable fact; but I cannot say whether that cause would or would not have been sufficient, by itself, for that result.

45. Everybody knows how self-consciousness makes one awkward and may even quite paralyze the mind. Nobody can have failed to remark that mental performances that are gone through with lightly are apt to be more adroit than those in which every little detail is studied while the action is proceeding, nor how a great effort -- say to write a particularly witty letter -- or even to recall a word or name that has slipped one's memory may spoil one's success. Perhaps it is because in trying very hard we are thinking about our effort instead of about the problem in hand. At any rate my own experience is that self-consciousness, and especially conscious effort, are apt to carry me to the verge of idiocy and that those things that I have done spontaneously were the best done. Now, in the experiments I have described, the so-called "subject," the victim of the experimentation, would not seldom sit in the darkened and silent room, straining with all his might for two or three minutes, to detect the slightest difference between two pressures. Finding himself unable to do so he would utter his "zero" that this inability might be recorded. Thereupon all straining ceased; for all it then remained for him to do was mention at random which one of the pressures he would mark as the heavier -- and here his perfect unconsciousness greatly increased his power of discrimination -- a discrimination below the surface of consciousness, and not recognized as a real judgment, yet in very truth a genuine discrimination, as the statistical results showed. The circumstances of my talking with the waiters on the boat were almost identical. While I was going through the row, chatting a little with each, I held myself in as passive and receptive a state as I could. When I had gone through the row I made a great effort to detect in my consciousness some symptoms of the thief, and this effort, I suppose, prevented my success. But then finding I could detect nothing I said to myself, "Well, anyway, I must fasten on someone, though it be but a random choice," and instantly I knew which of the men it was. . . .

46. I could tell many other true tales of successful guessings; but I have mentioned here two principles which I have been led to conjecture furnish at least a partial explanation of the mystery that overhangs this singular guessing instinct. I infer in the first place that man divines something of the secret principles of the universe because his mind has developed as a part of the universe and under the influence of these same secret principles; and secondly, that we often derive from observation strong intimations of truth, without being able to specify what were the circumstances we had observed which conveyed those intimations.

47. It is a chapter of the art of inquiry.

48. Our faculty of guessing corresponds to a bird's musical and aeronautic powers; that is, it is to us, as those are to them, the loftiest of our merely instinctive powers. I suppose that if one were sure of being able to discriminate between the intimations of this instinct and the self-flatteries of personal desire, one would always trust to the former. For I should not rate high either the wisdom or the courage of a fledgling bird, if, when the proper time had come, the little agnostic should hesitate long to take his leap from the nest on account of doubts about the theory of aerodynamics.

680. It is idle to say that the doctrine of chances would account for man's ultimately guessing right. For if there were only a limited number n of hypotheses that man could form, so that 1/n would be the chance of the first hypothesis being right, still it would be a remarkable fact that man only could form n hypotheses, including in the number the hypothesis that future experimentation would confirm. Why should man's n hypotheses include the right one? The doctrine of chances could never account for that until it was in possession of statistics of the hypotheses that are inconceivable by man. But even that is not the real state of things. It is hard to say how many hypotheses a physicist could conceive to account for a phenomenon in his laboratory. He might suppose that the conjunctions of the planets had something to do with it, or some relation between the phases of variability of the stars in {ö} Centauri, or the fact of the Dowager empress having blown her nose 1 day 2 hours 34 minutes and 56 seconds after an inhabitant of Mars had died. The truth is that very few hypotheses will appear to the physicist to be reasonable; and the one true hypothesis is usually of this small number. Why is that? It may be answered, very truly, that experience has taught us that astrology, correspondences, magic, and many hypotheses formerly considered reasonable are to be put aside. Yes, but if primitive man had not had, at the very outset, some decided tendency toward preferring truthful hypotheses, no length of time, -- absolutely none, -- would have been sufficient to educate him even to the state of mind of Aristotle in his Book of Physical Auscultations, ridiculous as all that now seems to us. No, it is absolutely necessary to admit some original connection between human ideas, and the events that the future was destined to unfold.

161. Reasoning is of three types, Deduction, Induction, and Abduction.Ý3 In deduction, or necessary reasoning, we set out from a hypothetical state of things which we define in certain abstracted respects. Among the characters to which we pay no attention in this mode of argument is whether or not the hypothesis of our premisses conforms more or less to the state of things in the outward world. We consider this hypothetical state of things and are led to conclude that, however it may be with the universe in other respects, wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be true invariably. Our inference is valid if and only if there really is such a relation between the state of things supposed in the premisses and the state of things stated in the conclusion. Whether this really be so or not is a question of reality, and has nothing at all to do with how we may be inclined to think. If a given person is unable to see the connection, the argument is none the less valid, provided that relation of real facts really subsists. If the entire human race were unable to see the connection, the argument would be none the less sound, although it would not be humanly clear. Let us see precisely how we assure ourselves of the reality of the connection. Here, as everywhere throughout logic, the study of relatives has been of the greatest service. The simple syllogisms, which are alone considered by the old inexact logicians, are such very rudimentary forms that it is practically impossible to discern in them the essential features of deductive inference until our attention has been called to these features in higher forms of deduction.

162. All necessary reasoning without exception is diagrammatic. That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. Otherwise, although our conclusions may be correct, they will not be the particular conclusions at which we are aiming. But the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations. Let me say that it would make a grand life-study to give an account of this operation of planning a mathematical demonstration. Sundry sporadic maxims are afloat among mathematicians, and several meritorious books have been written upon the subject, but nothing broad and masterly. With the modern reformed mathematics and with my own and other logical results as a basis, such a theory of the plan of demonstration is no longer a superhuman task.

163. Having thus determined the plan of the reasoning, we proceed to the reasoning itself, and this I have ascertained can be reduced to three kinds of steps. The first consists in copulating separate propositions into one compound proposition. The second consists in omitting something from a proposition without possibility of introducing error. The third consists in inserting something into a proposition without introducing error.

164. You can see precisely what these elementary steps of inference are in Baldwin's Dictionary under Symbolic Logic. As a specimen of what they are like you may take this:

A is a bay horse,

Therefore, A is a horse.

If one asks oneself how one knows that this is certain, one is likely to reply that one imagines a bay horse and on contemplating the image one sees that it is a horse. But that only applies to the single image. How large a horse did this image represent? Would it be the same with a horse of very different size? How old was the horse represented to be; was his tail docked? Would it be so if he had the blind-staggers, and if so are you sure it would be so whatever of the numerous diseases of the horse afflicted him? We are perfectly certain that none of these circumstances could affect the question in the least. It is easy enough to formulate reasons by the dozen; but the difficulty is that they are one and all far less evident than the original inference. I do not see that the logician can do better than to say that he perceives that when a copulative proposition is given, such as "A is a horse and A has a bay color" any member of the copulation may be omitted without changing the proposition from true to false. In a psychological sense I am willing to take the word of the psychologist if he says that such a general truth cannot be perceived. But what better can we do in logic?

165. Somebody may answer that the copulative proposition contains the conjunction "and" or something equivalent, and that the very meaning of this "and" is that the entire copulation is true if and only if each of the members is singly true; so that it is involved in the very meaning of the copulative proposition that any member may be dropped.

To this I assent with all my heart. But after all, what does it amount to? It is another way of saying that what we call the meaning of a proposition embraces every obvious necessary deduction from it. Considered as the beginning of an analysis of what the meaning of the word "meaning" is, it is a valuable remark. But I ask how it helps us to understand our passing from an accepted judgment A to another judgment C of which we not only feel equally confident but in point of fact are equally sure, barring a possible blunder which could be corrected as soon as attention was called to it, barring another equivalent blunder?

To this the advocate of the explanation by the conception of "meaning" may reply: that is meant which is intended or purposed; that a judgment is a voluntary act, and our intention is not to employ the form of the judgment A, except to the interpretation of images to which judgments, corresponding in form to C, can be applied.

166. Perhaps it may reconcile the psychologist to the admission of perceptual judgments involving generality to be told that they are perceptual judgments concerning our own purposes. I certainly think that the certainty of pure mathematics and of all necessary reasoning is due to the circumstance that it relates to objects which are the creations of our own minds, and that mathematical knowledge is to be classed along with knowledge of our own purposes. When we meet with a surprising result in pure mathematics, as we so often do, because a loose reasoning had led us to suppose it impossible, this is essentially the same sort of phenomenon as when in pursuing a purpose we are led to do something that we are quite surprised to find ourselves doing, as being contrary, or apparently contrary, to some weaker purpose.

But if it is supposed that any such considerations afford any logical justification of primary logical principles I must say that, on the contrary, at the very best they beg the question by assuming premisses far less certain than the conclusion to be established.

§3. INDUCTIVE REASONING

167. A generation and a half of evolutionary fashions in philosophy has not sufficed entirely to extinguish the fire of admiration for John Stuart Mill -- that very strong but Philistine philosopher whose inconsistencies fitted him so well to be the leader of a popular school -- and consequently there will still be those who propose to explain the general principles of formal logic, which are now fully shown to be mathematical principles, by means of induction. Anybody who holds to that view today may be assumed to have a very loose notion of induction; so that all he really means is that the general principles in question are derived from images of the imagination by a process which is, roughly speaking, analogous to induction. Understanding him in that way, I heartily agree with him. But he must not expect me in 1903 to have anything more than a historical admiration for conceptions of induction which shed a brilliant light upon the subject in 1843. Induction is so manifestly inadequate to account for the certainty of these principles that it would be a waste of time to discuss such a theory.

168. However, it is now time for me to pass to the consideration of Inductive Reasoning. When I say that by inductive reasoning I mean a course of experimental investigation, I do not understand experiment in the narrow sense of an operation by which one varies the conditions of a phenomenon almost as one pleases. We often hear students of sciences, which are not in this narrow sense experimental, lamenting that in their departments they are debarred from this aid. No doubt there is much justice in this lament; and yet those persons are by no means debarred from pursuing the same logical method precisely, although not with the same freedom and facility. An experiment, says Stöckhardt, in his excellent School of Chemistry, is a question put to nature. Like any interrogatory, it is based on a supposition. If that supposition be correct, a certain sensible result is to be expected under certain circumstances which can be created, or at any rate are to be met with. The question is, Will this be the result? If Nature replies "No!" the experimenter has gained an important piece of knowledge. If Nature says "Yes," the experimenter's ideas remain just as they were, only somewhat more deeply engrained. If Nature says "Yes" to the first twenty questions, although they were so devised as to render that answer as surprising as possible, the experimenter will be confident that he is on the right track, since 2 to the 20th power exceeds a million.

169. Laplace was of the opinion that the affirmative experiments impart a definite probability to the theory; and that doctrine is taught in most books on probability to this day, although it leads to the most ridiculous results, and is inherently self-contradictory. It rests on a very confused notion of what probability is. Probability applies to the question whether a specified kind of event will occur when certain predetermined conditions are fulfilled; and it is the ratio of the number of times in the long run in which that specified result would follow upon the fulfillment of those conditions to the total number of times in which those conditions were fulfilled in the course of experience. It essentially refers to a course of experience, or at least of real events; because mere possibilities are not capable of being counted. You can, for example, ask what the probability is that a given kind of object will be red, provided you define red sufficiently. It is simply the ratio of the number of objects of that kind that are red to the total number of objects of that kind. But to ask in the abstract what the probability is that a shade of color will be red is nonsense, because shades of color are not individuals capable of being counted. You can ask what the probability is that the next chemical element to be discovered will have an atomic weight exceeding a hundred. But you cannot ask what the probability is that the law of universal attraction should be that of the inverse square until you can attach some meaning to statistics of the characters of possible universes. When Leibniz said that this world is the best that was possible, he may have had some glimmer of meaning, but when Quételet says that if a phenomenon has been observed on m occasions, the probability that it will occur on the (m + 1)th occasion is (m+1)/(m+2), he is talking downright nonsense. Mr. F.Y. Edgeworth asserts that of all theories that are started one half are correct. That is not nonsense, but it is ridiculously false. For of theories that have enough to recommend them to be seriously discussed, there are more than two on the average to each general phenomenon to be explained. Poincaré, on the other hand, seems to think that all theories are wrong, and that it is only a question of how wrong they are.

170. Induction consists in starting from a theory, deducing from it predictions of phenomena, and observing those phenomena in order to see how nearly they agree with the theory. The justification for believing that an experiential theory which has been subjected to a number of experimental tests will be in the near future sustained about as well by further such tests as it has hitherto been, is that by steadily pursuing that method we must in the long run find out how the matter really stands. The reason that we must do so is that our theory, if it be admissible even as a theory, simply consists in supposing that such experiments will in the long run have results of a certain character. But I must not be understood as meaning that experience can be exhausted, or that any approach to exhaustion can be made. What I mean is that if there be a series of objects, say crosses and circles, this series having a beginning but no end, then whatever may be the arrangement or want of arrangement of these crosses and circles in the entire endless series must be discoverable to an indefinite degree of approximation by examining a sufficient finite number of successive ones beginning at the beginning of the series. This is a theorem capable of strict demonstration. The principle of the demonstration is that whatever has no end can have no mode of being other than that of a law, and therefore whatever general character it may have must be describable, but the only way of describing an endless series is by stating explicitly or implicitly the law of the succession of one term upon another. But every such term has a finite ordinal place from the beginning and therefore, if it presents any regularity for all finite successions from the beginning, it presents the same regularity throughout. Thus the validity of induction depends upon the necessary relation between the general and the singular. It is precisely this which is the support of Pragmatism.

§4. INSTINCT AND ABDUCTION

171. Concerning the validity of Abductive inference, there is little to be said, although that little is pertinent to the problem we have in hand.

Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis.

Deduction proves that something must be; Induction shows that something actually is operative; Abduction merely suggests that something may be.

Its only justification is that from its suggestion deduction can draw a prediction which can be tested by induction, and that, if we are ever to learn anything or to understand phenomena at all, it must be by abduction that this is to be brought about.

No reason whatsoever can be given for it, as far as I can discover; and it needs no reason, since it merely offers suggestions.

172. A man must be downright crazy to deny that science has made many true discoveries. But every single item of scientific theory which stands established today has been due to Abduction.

But how is it that all this truth has ever been lit up by a process in which there is no compulsiveness nor tendency toward compulsiveness? Is it by chance? Consider the multitude of theories that might have been suggested. A physicist comes across some new phenomenon in his laboratory. How does he know but the conjunctions of the planets have something to do with it or that it is not perhaps because the dowager empress of China has at that same time a year ago chanced to pronounce some word of mystical power or some invisible jinnee may be present. Think of what trillions of trillions of hypotheses might be made of which one only is true; and yet after two or three or at the very most a dozen guesses, the physicist hits pretty nearly on the correct hypothesis. By chance he would not have been likely to do so in the whole time that has elapsed since the earth was solidified. You may tell me that astrological and magical hypotheses were resorted to at first and that it is only by degrees that we have learned certain general laws of nature in consequence of which the physicist seeks for the explanation of his phenomenon within the four walls of his laboratory. But when you look at the matter more narrowly, the matter is not to be accounted for in any considerable measure in that way. Take a broad view of the matter. Man has not been engaged upon scientific problems for over twenty thousand years or so. But put it at ten times that if you like. But that is not a hundred thousandth part of the time that he might have been expected to have been searching for his first scientific theory.

You may produce this or that excellent psychological account of the matter. But let me tell you that all the psychology in the world will leave the logical problem just where it was. I might occupy hours in developing that point. I must pass it by.

You may say that evolution accounts for the thing. I don't doubt it is evolution. But as for explaining evolution by chance, there has not been time enough.

173. However man may have acquired his faculty of divining the ways of Nature, it has certainly not been by a self-controlled and critical logic. Even now he cannot give any exact reason for his best guesses. It appears to me that the clearest statement we can make of the logical situation -- the freest from all questionable admixture -- is to say that man has a certain Insight, not strong enough to be oftener right than wrong, but strong enough not to be overwhelmingly more often wrong than right, into the Thirdnesses, the general elements, of Nature. An Insight, I call it, because it is to be referred to the same general class of operations to which Perceptive Judgments belong. This Faculty is at the same time of the general nature of Instinct, resembling the instincts of the animals in its so far surpassing the general powers of our reason and for its directing us as if we were in possession of facts that are entirely beyond the reach of our senses. It resembles instinct too in its small liability to error; for though it goes wrong oftener than right, yet the relative frequency with which it is right is on the whole the most wonderful thing in our constitution.

174. One little remark and I will drop this topic. If you ask an investigator why he does not try this or that wild theory, he will say, "It does not seem reasonable." It is curious that we seldom use this word where the strict logic of our procedure is clearly seen. We do [not] say that a mathematical error is not reasonable. We call that opinion reasonable whose only support is instinct. . . .

§10. KINDS OF REASONING

65. There are in science three fundamentally different kinds of reasoning, Deduction (called by Aristotle {synagögé} or {anagögé}), Induction (Aristotle's and Plato's {epagögé}) and Retroduction (Aristotle's {apagögé}, but misunderstood because of corrupt text, and as misunderstood usually translated abduction). Besides these three, Analogy (Aristotle's {paradeigma}) combines the characters of Induction and Retroduction.

66. Deduction is that mode of reasoning which examines the state of things asserted in the premisses, forms a diagram of that state of things, perceives in the parts of that diagram relations not explicitly mentioned in the premisses, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth. For example, let the premiss be that there are four marked points upon a line which has neither extremity nor furcation. Then, by means of a diagram,

[to be provided ...]

we may conclude that there are two pairs of points such that in passing along the line in any way from one to the other point of either pair, one point of the second pair will be passed an odd number of times and the other point an even (or zero) number of times. This is deduction.

67. Induction is that mode of reasoning which adopts a conclusion as approximate, because it results from a method of inference which must generally lead to the truth in the long run. For example, a ship enters port laden with coffee. I go aboard and sample the coffee. Perhaps I do not examine over a hundred beans, but they have been taken from the middle, top, and bottom of bags in every part of the hold. I conclude by induction that the whole cargo has approximately the same value per bean as the hundred beans of my sample. All that induction can do is to ascertain the value of a ratio.

68. Retroduction is the provisional adoption of a hypothesis, because every possible consequence of it is capable of experimental verification, so that the persevering application of the same method may be expected to reveal its disagreement with facts, if it does so disagree. For example, all the operations of chemistry fail to decompose hydrogen, lithium, glucinum, boron, carbon, nitrogen, oxygen, fluorine, sodium, . . . gold, mercury, thallium, lead, bismuth, thorium, and uranium. We provisionally suppose these bodies to be simple; for if not, similar experimentation will detect their compound nature, if it can be detected at all. That I term retroduction.

69. Analogy is the inference that a not very large collection of objects which agree in various respects may very likely agree in another respect. For instance, the earth and Mars agree in so many respects that it seems not unlikely they may agree in being inhabited.

70. The methods of reasoning of science have been studied in various ways and with results which disagree in important particulars. The followers of Laplace treat the subject from the point of view of the theory of probabilities. After corrections due to Boole and others, that method yields substantially the results stated above. Whewell described the reasoning just as it appeared to a man deeply conversant with several branches of science as only a genuine researcher can know them, and adding to that knowledge a full acquaintance with the history of science. These results, as might be expected, are of the highest value, although there are important distinctions and reasons which he overlooked. John Stuart Mill endeavored to explain the reasonings of science by the nominalistic metaphysics of his father. The superficial perspicuity of that kind of metaphysics rendered his logic extremely popular with those who think, but do not think profoundly; who know something of science, but more from the outside than the inside, and who for one reason or another delight in the simplest theories even if they fail to cover the facts.

71. Mill denies that there was any reasoning in Kepler's procedure. He says it is merely a description of the facts. He seems to imagine that Kepler had all the places of Mars in space given him by Tycho's observations; and that all he did was to generalize and so obtain a general expression for them. Even had that been all, it would certainly have been inference. Had Mill had even so much practical acquaintance with astronomy as to have practised discussions of the motions of double stars, he would have seen that. But so to characterize Kepler's work is to betray total ignorance of it. Mill certainly never read the De Motu [Motibus] Stellae Martis, which is not easy reading. The reason it is not easy is that it calls for the most vigorous exercise of all the powers of reasoning from beginning to end.

72. What Kepler had given was a large collection of observations of the apparent places of Mars at different times. He also knew that, in a general way, the Ptolemaic theory agrees with the appearances, although there were various difficulties in making it fit exactly. He was furthermore convinced that the hypothesis of Copernicus ought to be accepted. Now this hypothesis, as Copernicus himself understood its first outline, merely modifies the theory of Ptolemy so far as [to] impart to all the bodies of the solar system one common motion, just what is required to annul the mean motion of the sun. It would seem, therefore, at first sight, that it ought not to affect the appearances at all. If Mill had called the work of Copernicus mere description he would not have been so very far from the truth as he was. But Kepler did not understand the matter quite as Copernicus did. Because the sun was so near the centre of the system, and was of vast size (even Kepler knew its diameter must be at least fifteen times that of the earth), Kepler, looking at the matter dynamically, thought it must have something to do with causing the planets to move in their orbits. This retroduction, vague as it was, cost great intellectual labor, and was most important in its bearings upon all Kepler's work. Now Kepler remarked that the lines of apsides of the orbits of Mars and of the earth are not parallel; and he utilized various observations most ingeniously to infer that they probably intersected in the sun. Consequently, it must be supposed that a general description of the motion would be simpler when referred to the sun as a fixed point of reference than when referred to any other point. Thence it followed that the proper times at which to take the observations of Mars for determining its orbit were when it appeared just opposite the sun -- the true sun -- instead of when it was opposite the mean sun, as had been the practice. Carrying out this idea, he obtained a theory of Mars which satisfied the longitudes at all the oppositions observed by Tycho and himself, thirteen in number, to perfection. But unfortunately, it did not satisfy the latitudes at all and was totally irreconcilable with observations of Mars when far from opposition.

73. At each stage of his long investigation, Kepler has a theory which is approximately true, since it approximately satisfies the observations (that is, within 8', which is less than any but Tycho's observations could decisively pronounce an error), and he proceeds to modify this theory, after the most careful and judicious reflection, in such a way as to render it more rational or closer to the observed fact. Thus, having found that the centre of the orbit bisects the eccentricity, he finds in this an indication of the falsity of the theory of the equant and substitutes, for this artificial device, the principle of the equable description of areas. Subsequently, finding that the planet moves faster at ninety degrees from its apsides than it ought to do, the question is whether this is owing to an error in the law of areas or to a compression of the orbit. He ingeniously proves that the latter is the case.

74. Thus, never modifying his theory capriciously, but always with a sound and rational motive for just the modification he makes, it follows that when he finally reaches a modification -- of most striking simplicity and rationality -- which exactly satisfies the observations, it stands upon a totally different logical footing from what it would if it had been struck out at random, or the reader knows not how, and had been found to satisfy the observation. Kepler shows his keen logical sense in detailing the whole process by which he finally arrived at the true orbit. This is the greatest piece of Retroductive reasoning ever performed.

§22. THE UNCERTAINTY OF SCIENTIFIC RESULTS

120. It is a great mistake to suppose that the mind of the active scientist is filled with propositions which, if not proved beyond all reasonable cavil, are at least extremely probable. On the contrary, he entertains hypotheses which are almost wildly incredible, and treats them with respect for the time being. Why does he do this? Simply because any scientific proposition whatever is always liable to be refuted and dropped at short notice. A hypothesis is something which looks as if it might be true and were true, and which is capable of verification or refutation by comparison with facts. The best hypothesis, in the sense of the one most recommending itself to the inquirer, is the one which can be the most readily refuted if it is false. This far outweighs the trifling merit of being likely. For after all, what is a likely hypothesis? It is one which falls in with our preconceived ideas. But these may be wrong. Their errors are just what the scientific man is out gunning for more particularly. But if a hypothesis can quickly and easily be cleared away so as to go toward leaving the field free for the main struggle, this is an immense advantage.

121. Retroduction goes upon the hope that there is sufficient affinity between the reasoner's mind and nature's to render guessing not altogether hopeless, provided each guess is checked by comparison with observation. It is true that agreement does not show the guess is right; but if it is wrong it must ultimately get found out. The effort should therefore be to make each hypothesis, which is practically no more than a question, as near an even bet as possible.