Lowell Lecture 7 of 1903 (MS 473-4)

transcribed and edited by Gary Fuhrman, July 2018, from the manuscript copies made available by the SPIN project (link above). In a few places punctuation has been added for clarity. Part of this lecture was previously published in CP 7.110-30.

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themselves. I had no time, I very much regret to say, to speak of continuity, which is the principal mathematical conception and the most in need of explanation while its logical importance is far greater than is everything else put together that I could bring into the first six lectures. But there was no possibility of crowding it into this course. In the sixth lecture I was only able to make a few detached remarks concerning statistical deductions. This leaves us but two hours in which to treat of the most important kinds of reasoning, Induction and Abduction. Deduction is of small account beside these but unfortunately these can only be understood in the least in the light of deduction. Without that one could not even comprehend what inductive and abductive reasoning are. All that I have said, therefore, is merely a very inadequate preparation for these last two lectures; and the latter of these will have to be curtailed in a most injurious fashion. Logic, let me tell you, while it is a subject requiring continual stress of mind, is by no means a dry and uninteresting topic. But it becomes so, as any science would, when it has to be compressed to a very injurious and almost fatal degree. Where it differs from other sciences is that fragments of them can be presented very well, while logic must be given in a certain degree of fulness or else it will not be presented at all.

The course of events by which any new subject gets added to our knowledge is most clearly marked in the case of an addition to our scientific knowledge.

In the first place we are already in a previous state of knowledge. Logic has quite nothing to say concerning the primum cognitum. In consequence of this we are in a state of expectation concerning a coming phenomenon,— being that expectation active or passive. If the phenomenon, when it comes, fulfills that expectation, it strengthens the habits of thinking on which that expectation is based, but teaches us nothing new. But if it involves any surprise, as it mostly does, our habits of thinking are deranged, whether little or much. We then feel the need of a new idea which shall serve to bind the surprising phenomenon to our preëxisting experience. One usual phrase is that we want the surprising fact explained. With this end in view we are led to frame a hypothesis, and the process of reasoning by which we come to set up a hypothesis is the kind of reasoning that I call Abduction. Now this hypothesis is a purely ideal state of things, and upon the basis of a purely ideal state of things as a premiss, we can only reason deductively. In fact, deduction always relates to a purely ideal state of things, in this sense, that if the premiss of deduction is known for anything more than that, its being more has nothing to do with the course of the deduction. The Deductions which we base upon the hypothesis which has resulted from Abduction produce conditional predictions concerning our future experience. That is to say, we infer by Deduction that if the hypothesis be true, any future phenomena of certain descriptions must present such and such characters. We now institute a course of quasi-experimentation in order to bring these predictions to the test, and thus to form our final estimate of the value of the hypothesis, and this whole proceeding I term Induction. I speak of quasi-experimentation because the term experiment is, according to the usage of scientific men, restricted to the operation of bringing about certain conditions. The noting of the results of experiments or of anything else to which our attention is directed in advance of our noting it, is called Observation. But by quasi-experimentation I mean the entire operation either of producing or of searching out a state of things to which the conditional predictions deduced from the hypothesis shall be applicable and of noting how far the prediction is fulfilled. I ought to say that the prediction may not be a necessary one. That is it may not be a positive prediction about every individual case, but may be a statistical prediction which simply excludes certain ratios of frequency of the truth [of] a certain species of facts among cases in which a genus of facts are true.

You thus have a preliminary notion of what the three kinds of reasoning are. I will remark however that there is a class of arguments, not a very large or important class, which combines the characters of the three. These are arguments from Analogy.

You will see from what I have said that Deduction is decidedly the least important of the three. Deduction is merely a link by which the result of Abduction,— that is to say the proposed explanatory hypothesis,— is put into a form in which Induction can be applied to it; and it only consists in making thought distinct as to what the Supposition that the Abduction suggests really supposes. While it is a relatively insignificant step in the inquiry, however, Deduction becomes of surpassing importance in logic; and the reason of this is that Logic or rather Critic, which is that branch of logic that evaluates arguments, is itself Deductive.

In order to put this remark into a clear light it is requisite to view the three Classes of Argument from another standpoint.

A Deduction is a process of thought by which it is rendered evident that a certain conclusion must be true; and this is so whether this relates to individual single cases or whether it is a statistical proposition concerning a ratio of frequency in indefinitely many single instances. The manner in which the conclusion becomes evident is this. The premisses are stated in general terms. Now the mode of being of a general is that of governing individual cases. Its full expression requires therefore the presentation of something corresponding to those cases,— a quasi-diagram of them. The deduction makes that diagram and when it is made it recognizes that this diagram is and always will be a representation of a state of things describable in a different general statement,— and this latter general statement constitutes the conclusion. To show that this really is a correct description of Deduction, it will suffice to consider the general structure of Euclid's demonstrations,— Euclid's being more formally correct in statement than modern mathematicians think is needful. He invariably begins with a general statement of what he intends to prove. This is called the proposition; in Greek πρότασις, that is the pretension, or preliminary statement. In Aristotle it means a premiss. He then restates the condition of his proposition in such a form as to assign a selective or proper name to every geometrical point, that is every individual of the set concerning which the predication is made. This at the same time and ipso facto describes any diagram of any state of things to which the proposition is applicable. This part ends by stating in terms of those selectives exactly what his assertion makes him responsible for. This part is called the ἔκθεσις, or exposition. Next comes his κατασκευή, or contrivance; which is an ingenious experiment that he performs upon the diagram, by adding lines to it drawn according to an exact precept, or his supposing one part to be moved and superposed in a particular way upon another part. He now shows that it is implied in what is already known that this experiment must give a certain result in every case. This operation is his demonstration ἀπόδειξις, or from-showing. This proof ends by repeating the very statement with which the ἔκθεσις closed, to which he affixes the words, “which is what had to be shown,” ὅπερ ἔδει δεῖξαι. It is, you perceive, simply a showing that the thing is so by actual experience in the imagination. As has often been remarked, by Locke, for example, by Kant, by Stuart Mill, and as is admitted by all logicians the conclusion simply redescribes the state of things which alone is represented in the premisses.

Induction concludes more than would conceivably be true in every case in which what is observed would be true. It cannot, therefore, absolutely guarantee the truth of its conclusion. What can it do? Shall it show that the conclusion would be true in a large proportion of the embodiments of its general condition in the course of experience? This would be nothing but a necessary deduction of a statistical kind. It has to conclude that a hypothesis is true because certain predictions based upon it have been verified. Under what modification is it warranted in asserting that? Without exhausting the inexhaustible future it certainly cannot be justified in asserting that absolutely. It can, however, assert this, subject to such modification as further quasi-experimentation may make.

The distinction between probable deduction and probable induction may be illustrated as follows. Suppose a bag of coffee to be presented to me. I put my hand in the bag and draw out a handful of beans, and on examination of this handful, I find that one bean in ten is a female berry and the rest male berries. If I thence conclude that supposing my handful is a fair representative of the whole bagfull, about one tenth of all the beans in the bag when they come to be drawn out will be found to be female berries, while about nine tenths will be male berries, that will be an Induction. I mean that it either is so or else [??] similar experimentation will correct this ratio. But this having been fully established by further inductions conducted with every precaution, if somebody else draws out a handful of berries, I can say to him, about one tenth of your handful will be found to be female berries, or if not, further handfuls will bring your general average to this. You observe the difference between the two saving clauses. Probable deduction positively states the approximate ratio, and says that if the given sample does not conform to it, other similar deductions will. Induction on the other hand mentions a ratio and says that if it be not quite right, further inductions will make it right. Both inferences virtually assert that there is such a statistical uniformity that samples of experience will in the long run represent the whole long run of experience. In regard to any particular statistical ratio this is not necessarily the case, because there may not be any definite ratio in the long run.

Thus you see that although induction does not necessarily lead to the truth in any particular case and does not even produce a result that is necessarily true in the long run, yet it necessarily in the long run does produce the true result.

It must not be supposed that it does this by the exhaustion or approximate exhaustion of the multitude to which it refers. For it essentially refers to the whole course of experience of which all experience that ever can have taken place must be an infinitesimal proportion. But still the long run of experience must have some general character and to that character sufficient instances must conform. For example, experience may for awhile appear to give a certain value to a ratio, but afterward this ratio may fluctuate so as not to tend to any limiting value; but if this is the case, sufficient instances will bring this general character of experience to light and define it in amplitude and period. Further experience may show variation in respect to the amplitudes and periods so that they tend to no limit, but if so, it can only be by new fluctuations that a further multiplication of instances will bring to light and define; and so on ad infinitum.

But the conformity of the particular instance to the general character of experience results from the fact that the mode of being of the general consists in its government of individual cases.

When we come to study the third mode of reasoning we shall have to formulate the relation of the three kinds of reasoning in general terms. But for the present it will not be needful, nor practicable.

CP 7.110 begins here

110. Suppose we define Inductive reasoning as that reasoning whose conclusion is justified not by there being any necessity of its being true or approximately true but by its being the result of a method which if steadily persisted in must bring the reasoner to the truth of the matter or must cause his conclusion in its changes to converge to the truth as its limit. Adopting this definition, I find that there are three orders of induction of very different degrees of cogency although they are all three indispensable.

111. The first order of induction, which I will call Rudimentary Induction, or the Pooh-pooh argument, proceeds from the premiss that the reasoner has no evidence of the existence of any fact of a given description and concludes that there never was, is not, and never will be any such thing. The justification of this is that it goes by such light as we have, and that truth is bound eventually to come to light; and therefore if this mode of reasoning temporarily leads us away from the truth, yet steadily pursued, it will lead to the truth at last. This is certainly very weak justification; and were it possible to dispense with this method of reasoning, I would certainly not recommend it. But the strong point of it is that it is indispensable. It goes upon the roughest kind of information, upon merely negative information; but that is the only information we can have concerning the great majority of subjects.

112. I find myself introduced to a man without any previous warning. Now if I knew that he had married his grandmother and had subsequently buried her alive, I might decline his acquaintance; but since I have never heard the slightest suspicion of his doing such a thing, and I have no time to investigate idle surmises, I presume he never did anything of the sort. I know a great many men, however, whose whole stock of reasoning seems to consist in this argument, which they continue to use where there is positive evidence and where this argument consequently loses all force. If you ask such a man whether he believes in the liquefaction of the blood of St. Januarius, he will say no. Why not? Well, nothing of that kind ever came within the range of my experience. But it did come within the range of Sir Humphrey Davy's experience, who was granted every facility for the thorough investigation of it. His careful report simply confirms the usual allegations with more circumstantial details. You are not justified in pooh-poohing such observations; and that the fact is contrary to the apparent ordinary course of nature is no argument whatever. You are bound to believe it, until you can bring some positive reason for disbelieving it.

113. In short this rudimentary kind of induction is justified where there is no other way of reasoning; but it is of all sound arguments the very weakest and must disappear as soon as any positive evidence is forthcoming.

114. The second order of induction consists in the argument from the fulfillment of predictions. After a hypothesis has been suggested to us by the agreement between its consequences and observed fact, there are two different lines that our further studies of it may pursue. In the first place, we may look through the known facts and scrutinize them carefully to see how far they agree with the hypothesis and how far they call for modifications of it. That is a very proper and needful inquiry. But it is Abduction, not Induction, and proves nothing but the ingenuity with which the hypothesis has been adapted to the facts of the case. To take this for Induction, as a great proportion of students do, is one of the greatest errors of reasoning that can be made. It is the post hoc ergo propter hoc fallacy, if so understood. But if understood to be a process antecedent to the application of induction, not intended to test the hypothesis, but intended to aid in perfecting that hypothesis and making it more definite, this proceeding is an essential part of a well-conducted inquiry.

115. The other line which our studies of the relation of the hypothesis to experience may pursue, consists in directing our attention, not primarily to the facts, but primarily to the hypothesis, and in studying out what effect that hypothesis, if embraced, must have in modifying our expectations in regard to future experience. Thereupon we make experiments, or quasi-experiments, in order to find out how far these new conditional expectations are going to be fulfilled. In so far as they greatly modify our former expectations of experience and in so far as we find them, nevertheless, to be fulfilled, we accord to the hypothesis a due weight in determining all our future conduct and thought. It is true that the observed conformity of the facts to the requirements of the hypothesis may have been fortuitous. But if so, we have only to persist in this same method of research and we shall gradually be brought around to the truth. This gradual process of rectification is in great contrast to what takes place with rudimentary induction where the correction comes with a bang. The strength of any argument of the Second Order depends upon how much the confirmation of the prediction runs counter to what our expectation would have been without the hypothesis. It is entirely a question of how much; and yet there is no measurable quantity. For when such measure is possible the argument assumes quite another complexion, and becomes an induction of the Third Order. Inductions of the second order are of two varieties, that are logically quite distinct.

116. The weaker of these is where the predictions that are fulfilled are merely of the continuance in future experience of the same phenomena which originally suggested and recommended the hypothesis, expectations directly involved in holding the hypothesis. Even such confirmation may have considerable weight. This, for example, is the way in which the undulatory theory of light stood before Maxwell. The phenomena of interference suggested undulations, which measures of the velocity of light in different media confirmed; and the phenomena of polarization suggested transverse vibrations. All the direct expectations involved in the hypothesis were confirmed, except that there no phenomena due to longitudinal vibrations were found. But all physicists felt that it was a weakness of the theory that no unexpected predictions occurred. The rotation of the plane of polarization was an outstanding fact not accounted for.

117. The other variety of the argument from the fulfillment of predictions is where truths ascertained subsequently to the provisional adoption of the hypothesis or, at least, not at all seen to have any bearing upon it, lead to new predictions being based upon the hypothesis of an entirely different kind from those originally contemplated and these new predictions are equally found to be verified.

118. Thus Maxwell, noticing that the velocity of light had the same value as a certain fundamental constant relating to electricity, was led to the hypothesis that light was an electromagnetic oscillation. This explained the magnetic rotation of the plane of polarization, and predicted the Hertzian waves. Not only that, but it further led to the prediction of the mechanical pressure of light, which had not at first been contemplated.

119. The second order of induction only infers that a theory is very much like the truth, because we are so far from ever being authorized to conclude that a theory is the very truth itself, that we can never so much as understand what that means. Light is electro-magnetic vibrations; that is to say, it [is] something very like that. In order to say that it is precisely that, we should have to know precisely what we mean by electro-magnetic vibrations. Now we never can know precisely what we mean by any description whatever.

120. The third order of induction, which may be called Statistical Induction, differs entirely from the other two in that it assigns a definite value to a quantity. It draws a sample of a class, finds a numerical expression for a predesignate character of that sample and extends this evaluation, under proper qualification, to the entire class, by the aid of the doctrine of chances. The doctrine of chances is, in itself, purely deductive. It draws necessary conclusions only. The third order of induction takes advantage of the information thus deduced to render induction exact.

121. This family of inductions has three different kinds quite distinct logically. Beginning with the lowest and least certain, we have cases in which a class of individuals recur in endless succession and we do not know in advance whether the occurrences are entirely independent of one another or not. But we have some reason to suppose that they would be independent and perhaps that they have some given ratio of frequency. Then what has to be done is to apply all sorts of consequences of independence and see whether the statistics support the assumption. For instance, the value of the ratio of the circumference of a circle to its diameter, a number usually called π has been calculated in the decimal notation, to over seven hundred figures. Now as there is not the slightest reason to suppose that any law expressible in a finite time connects the value of π with the decimal notation or with any whole number, we may presume that the recurrences of any figure, say 5, in that succession are independent of one another and that there is simply a probability of 1/10 that any figure will be a 5.

122. In order to illustrate this mode of induction, I have made a few observations on the calculated number. There ought to be, in 350 successive figures, about 35 fives. The odds are about 2 to 1 that there will be 30-39 [and] 3 to 1 that there will be 29-41. Now I find in the first 350 figures 33 fives, and in the second 350, 28 fives, which is not particularly unlikely under the supposition of a chance distribution. During the process of counting these 5's, it occurred to me that as the expression of a rational fraction in decimals takes the form of a circulating decimal in which the figures recur with perfect regularity, so in the expression of a quantity like π, it was naturally to be expected that the 5's, or any other figure, should recur with some approach to regularity. In order to find out whether anything of this kind was discernible I counted the fives in 70 successive sets of 10 successive figures each. Now were there no regularity at all in the recurrence of the 5's, there ought among these 70 sets of ten numbers each to be 27 that contained just one five each; and the odds against there being more than 32 of the seventy sets that contain just one five each is about 5 to 1. Now it turns out upon examination that there are 33 of the sets of ten figures which contain just one 5. It thus seems as if my surmise were right that the figures will be a little more regularly distributed than they would be if they were entirely independent of one another. But there is not much certainty about it. This will serve to illustrate what this kind of induction is like, in which the question to be decided is how far a given succession of occurrences are independent of one another and if they are not independent what the nature of the law of their succession is.

123. In the second variety of statistical induction, we are supposed to know whether the occurrences are independent or not, and if not, exactly how they are connected, and the inquiry is limited to ascertaining what the ratio of frequency is, after the effects of the law of succession have been eliminated. As a very simple example, I will take the following. The dice that are sold in the toy shops as apparatus for games that are sold are usually excessively irregular. It is no great fault, but rather enhances the christmas gaiety. Suppose, however, some old frump with an insatiable appetite for statistics to get hold of a die of that sort, and he will spend his Christmas in throwing it and recording the throws in order to find out the relative frequency with which the different faces turn up. He assumes that the different throws are independent of one another and that the ten thousand or so which he makes will give the same relative frequencies of the different faces as would be found among any similar large number of throws until the die gets worn down. At least he can safely assume that this will be the case as long as the die is thrown out of the same box by the same person in the same fashion.

124. This second variety is the usual and typical case of statistical induction. But it occasionally happens that we can sample a finite collection of objects by such a method that in the long run any one object of the collection would be taken as often as every other and any one succession as often as any other. This may [be] termed a random selection. It is obviously possible only in the case of an enumerable collection. When this sort of induction is possible it far surpasses every other in certainty and may closely approach that of demonstration itself.

125. I have now passed in review all the modes of pure induction with which I am acquainted. Induction may, of course, be strengthened or weakened by the addition of other modes of argument leading to the same conclusion or to a contrary conclusion. It may also be strengthened or weakened by arguments which do not directly affect the conclusion of the induction but which increase or diminish the strength of its procedure. There are in particular four kinds of uniformities which may greatly affect an induction.

126. In the first place the members of a class may present a greater or less general resemblance as regards certain kinds of characters. Birds for example are, generally speaking, much more alike than are fishes or mammals; and that will strengthen any induction about birds. Orchids, on the other hand, are extraordinarily various.

127. In the second place a character may have a greater or less tendency to be present or absent throughout the whole of certain kinds of groups. Thus, coloration often differs within one species, while the number of the principal bones of the skeleton, and almost all characters which are developed early in individual life and which persist to maturity are common to all the members of large classes.

128. In the third place, a certain set of characters may be more or less intimately connected, so as probably to be present or absent together in certain kinds of objects. Thus, we generally associate insistency upon minute forms with narrowness of mind, cleanliness with godliness, and so on.

129. In the fourth place, an object may have more or less tendency to possess the whole of certain sets of characters when it possesses any of them. Thus, one meets one man whose views whatever they may be are extreme, while the opinions of another form a strange mosaic.

130. From the knowledge of a uniformity of any one of these four classes or from the knowledge of the lack of such uniformity it may be deductively inferred that a given induction is either stronger or weaker than it otherwise would be.

CP ends here.

I will now say a few words concerning the Rationale of Induction.

I leave to one side the Psychological question of what causes us to make Inductions, which the German logicians are never able to dismiss from their minds.

I leave to the other side the metaphysical question of what makes an induction hold true. Only, I will here pause for an instant to drop a tear upon the tomb of John Stuart Mill. Here was a young man brought up in the strictest sect of the Ockhamists, of whom his father was the Grand Master in his day. That which was dearest to the heart of the Ockhamist was to deny the reality of anything of the nature of Law in nature. If we go back to Ockham himself and ask ourselves what excuse he had for such a contention, the answer is that it was because Duns Scotus, his master, evoked too many of these general beings without stopping to prove their reality. That is true. The Scotist saw laws or generals on every hand, laws of laws, laws of laws of laws, and left experience a little too much out of account. And yet Scotus never did leave individuals out of account. Only he was not an experimental philosopher. He was, on the contrary, a monk, sworn to separate himself from the things of this world. Now the things of this world and the Papacy, which at that day appeared to the logician as a wholly inconsistent meddling of general ideas with existences, came into conflict, and Ockham stood up manfully for experience and for positive fact. Very well; Ockhamism became a great tradition. The tradition was that generals are mere words, and have no reality. James Mill, the father of John Stuart Mill brought out all the truth there was in that view in a masterly work,— a great solitary performance in the history of philosophy,— his Analysis of the Human Mind. John Stuart Mill was his son. Imbued, as were Bain and Grote and others, with this doctrine,— this sacred truth as it seemed to them and as it really was, in so far as it was opposed to certain ideas, yet extravagantly pushed and denying what it had no justification in denying. It was on the whole a Denying doctrine. From the sacred Denial John Sturart Mill drew his mother's milk. To him loyalty to Truth and loyalty to Ockhamism were inseparable. But should it happen by any impossible conjuncture that they were opposed there can be no doubt that it would be Truth to which Stuart Mill would be faithful to even to the stake. It was a most dreary world,— the universe as the Ockhamists conceived it. It utterly sickened Stuart Mill. But that, as he conceived it, was the only universe there was,— the only universe that was so much as imaginable. Anything of the nature of a Law was, according to that creed, nothing but a word,— a nothing. Yet Stuart Mill having to ask himself what makes Inductions hold good, replied the Uniformity of Nature. The word law was concealed for his apprehension behind this word Uniformity. But after all, what is the difference between saying that there is a Uniformity and that there is a Law? It is simply that phenomena being conceived as having already occurred, any agreement in the ideas they excite in the mind, is a Uniformity while a Law is something which has some sort of being in advance of the occurrences and the occurrences all conform to it. It is plain that this Uniformity of Nature was for Stuart Mill a great Law, although it never occurred to him that in holding to it he was really cutting loose from his traditional Nominalism.

I have long ago fully expressed myself about Mill's Logic. I am not a Nominalist. I am a scholastic Realist of an extreme type. I not only believe that Laws really are, but it seems to me evident that Laws really live, and that they are the only things that do live, the only things whose being is complete. Accordingly, I have no disposition to deny Mill's Uniformity of Nature simply because it is a law. Nevertheless, I do deny it. It seems to me plain enough that Nature is not absolutely uniform in any positive sense. Variety, on the contrary, is the leading characteristic of Nature; and the uniformity of it is only just barely sufficient to make things go pretty regularly.

But this is metaphysics; and it is neither the psychological question nor the metaphysical question concerning induction,— interesting as each of them certainly is,— that concerns us now. It is the logical question how can we be justified in drawing an induction? “Justified,” I say. That is, how far does it conduce to our purpose. What is our purpose? It is to form such habits of thinking that events will not surprise us. So the question is how does the making of an induction conduce to the formation of such habits of thinking that the events when they come shall merely realize our anticipations, and prevent anticipations that are not realized?

My own answer to this question has already been made clear. Namely, the answer is that induction is such a way of inference that if one persists in it one must necessarily be led to the truth, at last. It is true that this condition is most imperfectly[?] fulfilled in the Pooh-pooh argument. For here the unexpected, when it comes, comes with a bang. But then, on the other hand, until the fatal day arrives, this argument causes us to anticipate just what does happen and prevents us from anticipating a thousand things that do not happen. I engage a stateroom; I purchase a letter of credit for fifty thousand dollars, and I start off determined to have a good time. On the way down the bay, my wife says to me ‘Aren't you afraid the house may be struck by lightning while we are gone?’ Pooh-pooh! ‘But aren't you afraid there will be a war and Boston will be bombarded?’ Pooh-pooh! ‘But aren't you afraid that when we are in the heart of Hungary or somewhere you will get the Asiatic plague, and I shall be left unable to speak the language?’ Pooh-pooh! On the morning of the fourth day out there is a terrific explosion and I find myself floating about in the middle of the Atlantic with my letter of credit safe in my breast pocket. I say to myself, my pooh- pooh argument broke down that time sure enough, but after all it made my mind easy about a number of possibilities that did not occur, and even about this one for three days. So I had better be content with my lot. This little parable is intended to illustrate how even the pooh-pooh argument, the weakest of all sound inductions, does satisfy the essential condition of saving me from surprises both positive and negative; that is from the happening of things not anticipated and the non-occurrence of imaginary disasters.

The argument from the fulfillment of predictions does still better. It may deceive one sometimes; but if I am constantly reasoning in this way, I shall get an early warning of the danger of my theory's breaking down and can prepare myself for it. The argument from sampling simply evaluates a ratio. The value may be wrong. But that I know already, and the inductive process itself is continually correcting its own evaluation and goes on making it approximate indefinitely to the truth, which is all the process ever promised to do.

That, then, is my answer to the question, and it seems to me entirely satisfactory. But many other answers have been proposed by different writers. For it has always seemed to logicians to be a great how the knowledge of one thing should ever justify a man in any assertion about another thing quite independent of the first. I will give you some examples of the answers that have been proposed to this question.

The abbé Gratry, who was an original thinker of a far higher order than many of those who called him a fool, after showing in a most interesting way that mathematical differentiation is of the nature of inductive thought as integration is of deductive thought,— an eternal truth, and by no means an obvious one, with which the name of Gratry must be forever associated,— is led to believe that it is an act of Divine illumination whenever we form a true Induction. Now I am loath to believe that such perfectly single-minded, sincere, serious and sacred search as that of Gratry is in any given case utterly and irretrievably erroneous. There is some truth in what he says. Only it does not apply to Induction, which is merely the process of appraising a theory already proposed to which Gratry's devout thought should be understood to apply, but to Abduction, which alone brings forth all new conceptions.

Dismissing that, let us consider the answer to our question that is given by DeMorgan and others, and which virtually has the support of Laplace, although he did not formulate it in general terms. This answer is that we are justified in drawing an induction because the conclusion of an induction is rendered probable by the premisses, according to the doctrine of chances. This I deny, and maintain that the doctrine of chances, rightly applied, does not give any determinate probability to an inductive conclusion unless we make an assumption which really renders the reasoning deductive.

In order to show what sort of reasoning it is by which the Laplacians support their dogma, I will show you how they would calculate the probabilities in an example. In order to avoid too advanced mathematics and too great intricacies, I am compelled to take for my example a problem where the Laplacian method leads to much better results than it does in those problems to which Laplace applies it. I must ask you to make allowance for this.

The problem is this. Suppose there comes to the port of New York a large cargo of many thousand boxes of toys, each containing ten toys, all different in any one box, but possibly recurring in different boxes. Now the custom-house inspector proceeds in this way. He selects a box by a method calculated in the long run to bring out one box as often as another. He shakes up the box well and then putting his hand into it grabs the first toy his hand touches, which he examines as to its being dutiable or not, after which he then goes on in the same way with another box. Now suppose that proceeding in this way the Inspector has found the first five toys to be dutiable, the question is what is the probability that the sixth toy is so.

I will first give my answer to this problem, secondly I will give Laplace's presumable treatment of it, and thirdly I will go on to show you that the latter is absurd. My answer is that we are tolerably safe in inferring that the great majority of toys in the boxes are dutiable. Because in reasoning that the proportion of dutiable toys in such a sample is roughly equal to that of the toys in the boxes we shall be proceeding according to a rule which would generally lead us as near to the truth as any rule could and which moreover if persisted in, in this very case, would ultimately bring us indefinitely near to the truth. And if it be true that pretty nearly all the toys are dutiable, then in almost all similar cases the sixth toy would be found dutiable. But we can know nothing more than this without further facts to go upon. It may be, for example, that such boxes invariably contain three toys of a kind not dutiable and that the fact that all the first five drawn were dutiable is simply an extraordinary coincidence. How this may be we do not know, as long as we do not know how the contents [of] such boxes are usually made up,— even in the long run, to say nothing of this particular cargo.

The Laplacians reason, however, as follows. They say that there might be in each box either 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 or no dutiable toys. These they say are simple possible events. They lay great stress on events being simple, although they never tell us what they mean by it and appear to call any events simple which they are not led to view as compounded. They further say that these happening are equally possible, “également possible” is the phrase which Laplace introduces at the beginning of his book, without telling us what scintilla of meaning can be attached to it. Certainly anybody talking French would say they were également possible,— that is, one as truly possible as another. But également, in that sense, is not an expression of quantity. But Laplace always proceeds as if events which are également possible and which are also simple events, were equally probable, which if equally probable means equally frequent in the long run, the only sense that it can usefully bear, is a monstrously unwarrantable assumption. These eleven proportions, then, being equally possible, he calculates in how many different equally probable ways the first five toys would be dutiable in each of these ways. This is shown in the table.

After that first error, the rest of the calculation is unexceptionable except for certain simplifications which I make, in order to avoid the integral calculus. All the ways in which the observed event could have been brought about are reckoned up, and the probabilities of the causes after you once suppose[dly] know their antecedent probabilities etc. are clearly proportionate to the ways the known event could have come about. The probability of each proportion is then to be multiplied by the probability in case of that proportion of the sixth toy being dutiable, in order to find the probability of its becoming dutiable in that way; and these being summed the total is the probability sought.

But I insist that a number that has no practical use (giving “practical” as wide a sense as you will) simply has no meaning. And in particular that the numerical probability of of the sixth toy being dutiable has no sense unless it means that among precisely similar experiments under the same conditions performed with these very boxes in the long run such a proportion would turn out in that way. If, indeed, among the data of the problem, it had been added that an endless series of similar cargoes of boxes of toys were coming in, then the probability need not be a statistical ratio true of these very boxes, so long as it was a statistical ratio true of the general run of those cargoes. But no such thing has been stated; and even if it had been, we cannot find out what that statistical ratio is until we know what the facts are in regard to the habits of making up the contents of such boxes.

For that reason, I object in toto to all calculations founded upon assumptions in regard to “equal possibility”, because such assumptions are only made when we are ignorant of conditions essential to the solution of problems; and a number which is made determinate only by an assumption founded on ignorance can have no practical utility. But I am going to waive for a moment my objection to all such processes of calculation, because by doing so I shall be able to bring the force of the objection into a stronger light. Grant, then, that we do go on some assumption of equal possibility. Still, such an assumption ought to be made on some consistent principle. Even if we are to assume some things to be equally possible I continue to object to assuming that all the different probabilities of an event are equally possible, because that proceeds on no principle that is consistent with itself. For there is no reason in assuming that the different probabilities of one event are equally possible rather than those of any other event. The distinction between a simple and a compound event is not a distinction of fact, nor a distinction having any basis whatever in objective fact. We assume any events we like to be simple without ever dreaming of being called to account for it. Nor is there any principle on which we could distinguish between simple and compound events. What is simple and what compound depends on the wording of the problem; and nothing else has anything to do with the distinction. Now if you assume that the different probabilities of a certain event are equally possible, ipso facto the different probabilities of certain compound and disjunct events will not be equally possible. Such an assumption is therefore entirely arbitrary. There is just as much reason for supposing the probabilities of a disjunct event to be equally possible in which case the probabilities of the first event are determined, and are determined not to be equally possible. In short, if you are going to make any assumption of equal possibility,— which I hold to be altogether unwarranted,— you ought at least to adopt some self-consistent principle in doing so. Now there is only one self-consistent principle on which such assumptions can be made; and that is that what Boole called the different constitutions of the universe are equally possible.

In making up the contents of the box of toys, the first toy put in might be dutiable D or free F. Then the next article, quite independently will be, in either case, dutiable or free. For we may assume that those who make up the box know nothing about our American duties. Thus for those first two toys, these cases may be assumed to be equally possible
DD
DF
FD
FF
So on for the third and the following; and so one shall have 2¹⁰, or 1024, different possible constitutions of the contents of the boxes. If you suppose these to be equally possible, you will make an approach to the only consistent principle, that of making all constitutions of the universe equally possible. You will fall short of doing so, only because you have considered only ten different grades instead of making an infinite number of them. Were that done, while your asumptions would have been utterly without warrant, it would have attained, in the only way possible for any such assumption, the merit of proceeding on a principle that does not cut its own throat.

At this point, I should not wonder if some unwary Laplacian were to say, “There seems to be some reason in what you urge. I think, myself, that it would be a sounder proceeding to assume the different constitutions of the universe to be equally possible. But that is a mere detail, and will alter the numerical result a bit, but that is all.” Well, let us see how much the result will be altered by assuming first the different constitutions of the box to be equally possible, and then the different constitutions of the universe to be equally possible. Here is the table for the equal possibility of the constitutions of the box. It thus turns out that on this hypothesis, which is certainly not so absurd as that of Laplace, that the odds in favor of the sixth toy being dutiable after the first five are found dutiable are only 7 to 3. Common sense revolts against this. But we ought to have made the constitutions of the universe equally possible. That is, we ought to have neglected the fact of the toys being packed in boxes of ten, which really has nothing to do with the matter and have treated them as if they were packed in boxes of indefinitely many toys. I have made the calculation for 25 in a box, and the result is shown in the following table.

We thus find the odds only 5 to 4. If instead of 25 divisions I had made 50 or 100 we should have found it an even chance.

Lowell Lecture 8