The justification of induction

Paper read in Cambridge Moral Science Club on the 25th of May 1939.

2 When we speak of inductive inference we mean roughly this: from the fact that known members of a class possess a certain property we conclude that unknown or future members of the class will exhibit the same property. This conclusion can apply either to all the members of the class or to a definite proportion of them; in the former case the induction may be called a Universal Generalization, in the second case a Statistical Generalization. For the sake of simplicity I shall deal in the following mainly with inductions of the former kind. This limitation, however, will be of no relevance for the general inductive problem.

3 It is a very importantan obvious fact that so called ”inductive inference” plays a very important roleoriginal: roll in science as well as in everyday life. Most of our actions are based on induction, i.e. we assume that certain regularities which hitherto have been reliable, can be safely relied on even in the future. Although experience sometimes shows that observed regularities were only apparent or subject to limitations which we had not taken into consideration, there obviously remains a multitude of occasions on which we feel no hesitation in trusting our inductions, or rather: on which it would be seem almost absurd to question their validity. But what is the criterion for that an induction can be trusted, how can we justify the inferences we draw as to what will or what is likely to happen? If there were no justification for this, science as well as practical life would seem to be based on merely haphazard guessing and the fact that we have been so fortunate in trusting our inductions as we actually have must be considered immensely strange if dued only to ”chance”. The demand for a proof that inductive inference if not with certainty necessarily so at least with a considerable degree of probability will be true constitutes what may be called the inductive problem.

4 I shall begin with quoting a passage from one a distinguish|2|ed present-day Britishoriginal: Brittish philosopher, which I think is significant for a fairly wide-spread attitude towards the present state of the inductive problem. He says:

5 original: ”Induction as a habit exists among animals and savages, and is a very frequent source of error. The problem of extracting an acceptable principle to replace this animal habit remains for the moment unsolved. I am convinced that induction must have validity of some kind in some degree, but until the problem of showing how or why it can be valid remains is solved, the rational man will doubt whether his food will nourish him, and whether the sun will rise tomorroworiginal: to-morrow.

6 So far Mr Russell.

7 The ideal way of attacking the inductive problem would obviously be to analyse the idea of ”justification” and consequently to show to what extent we are so to say ”justified” in asking for a justification of induction and to what extent we are not. This way, however, presents some difficulties, owing to a most important peculiarity in the inductive problem. What I wanted to show, namely,is Namely that the problem of finding a justification for inductive inference is in a sense – and as far as I can see just in the sense which commonly is thought of in connection with the inductive problem – a so to say pseudo-problem, a confusion of pictures which we associate with different termsoriginal: termes. This result is often expressed in some such way as: there is no justification of induction, which is utterly misleading because it so easily gives rise to the above-mentioned idea that uniformities in the course of nature are dued to chance, or to the opinion, expressed in the above quotation, that in order to be rational we must doubt our inductions. All this is nonsense, but as far as I can judge it is extremely difficult to separate such ideas from the true meaning of what I wanted to show if one only considers the matter quite abstractly, and therefore I shall not attempt here to give anoriginal: a purely analysis of the meaning of justification, but proceed in a different way.

8 We know from chemistry that the melting pointoriginal: meltingpoint of phosphorus is 44°C. It is obvious that this result has been obtained in what we call an inductive way, i.e. we have melted different pieces of phosphorus and found that they all melted at the same temperature. From these|3|observations we generalize that all pieces of phosphorus will melt at 44°C or, as we say it, that the melting pointoriginal: meltingpoint of phosphorus is 44°C. If we later on found a piece of phosphorus which melted at a different temperature, should we then say that our generalization was false? Obviously we could do so, and as far as I can see that is at least under certain circumstances what we actually should do, but it is important to note that even another attitude is possible and actually has been suggested just in connection with this example. We could namely maintain that the examined substance was no piece of phosphorus at all. This because the melting pointoriginal: meltingpoint – 44°C – which originally was an empirically observed property of substance already called pieces of phosphorus, simultaneously with the establishment of the generalization was made a defining property of phosphorus. Therefore a substance which has all the other defining properties of phosphorus but melts at a different temperature is not phosphorus at all but some other substance. The induction which led to the establishment of the generalization in this case is of a kind sometimes called ”rational induction” and it is obvious that inductions of this type play an important roleoriginal: roll in the formation of scientific concepts. We begin by a set of singular, empirical, i.e. synthetical propositions, which suggests a generalization, but to the generalization itself we give analytical validity. After the establishment of the generalization the originally synthetical singular propositions obviously become analytical also. But this transition from synthetical to analytical is often made almost unconsciously and therefore see very seldom it is explicitely stated, and seems to me therefore to offer a fairly good explanation for some cases were where we believe in an inductive method leading original: leding to absolute certainty. Many inductions obtain their absolute certainty from their being analytical statements, and thus the problem of justifying their assumed truth does not occur at all.

9 Now it has been maintained that if we take a corresponding attitude in the case of all natural laws, the inductive problem would altogether disappear. It is easy to see, however that this would not be|4| the case. – Suppose that we regarded all natural laws as analytical propositions. Then we could coordinate to at least a great multitude of them general synthetical propositions after the following pattern: We have first of all the analytical proposition, say, that all pieces of phosphorus melt at 44°C. But in addition we get a proposition of the type: if a substance has all the defining properties of phosphorus except perhaps the melting pointoriginal: meltingpoint which is still unexamined, then it is very likely that this substance really is phosphorus. The possibility of adding this proposition to the first one is actually of the greatest importance. Because if it were not possible to coordinate propositions of this kind to our inductions, regarded as analytical statements, propositions of this kind, which generally speaking assert that we regard the applicability of an analytic statement as highly, as we say, probable as soon as some defining properties are known to be present, this would imply that it were impossible to make any successful predictions in science. But as a matter of fact we are constantly making predictions and at the same time assuming that we are justified in doing so. But for this assumption the view that natural laws are analytical propositions gives no justification. Thus even within this view the inductive problem reoccurs.

10 Before we entirely leave this view, however, it may be useful to consider shortly another case where a similar attitude is possible. Let us take the well-known example with the billiard ballsoriginal: billiard-balls, the stroke of the one ball being the cause of the second ball’s movement, this implying that whenever one ball strikes another the second one will move. Suppose it happened that although a ball was struck by another it did not move. When we investigate the circumstances we find, say, that the ball was fixed on the table and could not move at all. Obviously nobody would say in this case that our causal law had been refuted, we should simply say that the cause did not bring about its effect because on account of a counteracting cause, namely the ball being fixed on the table. All this may sound most trivial, but deserves in fact great interest, becau|5|se it indicates that our original formulation of the law was in a sense incomplete. What we really ought to have said was: always when one ball strikes another this will move, provided that certain additional conditions are fulfilled, e.g. that there are no counteracting causes. Only if we knew for certain that these additional conditions are fulfilled we should speak of a falsification of the general law if the supposed cause did not bring about its supposed effect. But what are the criteria that really all additional circumstances are enumerated, or that no counteracting cause can be present? Obviously there are no so to speak given criteria for this at all, and even if we in many cases do fix these criteria arbitrarily, e.g. in all the cases where we wish to bring about an experimentum crucis, there are obviously other cases in which we do not wishoriginal: whish to decide in advance whether really all relevant circumstances are taken into consideration or not. And here again an attitude similar to the one in the example with the melting pontoriginal: meltingpoint of phosphorus is not only theoretically possible but, as far as I can see, in fact often resorted to when we wish to justify well-established inductions, namely the following: we make the truth, or better the working of the law the criterion that the circumstances necessary for its application are present. We define in other words the presence of a counteracting cause by the law’s not-working. – Something of this kind occurs e.g. in physics when we speak of isolated systems.

11 This attitude towards inductions differs from the former therein that whereas we in the former case started from singular synthetical propositions and ascended byoriginal: be a sort of quasi-induction to general analytical propositions, we in the last-mentioned case only operate with synthetical propositions, which, however, are regarded as incomplete, as fragments so to say of certain ”ideal” propositions, whose truth never can be contested because we per definitionem have excluded any falsification of them. But although this again gives a sort of justifications of particular inductions it is easily seen that the inductive problem as a whole cannot be eliminated in this way. Let us namely ass|6|ume, that we wished to regard all natural laws as incomplete propositions in the sense described above. Then we could be perfectly sure that all our inductions are true, i.e. that we need not fear a falsification of any of them. But now we must not forget, that when we apply these natural laws we apply them as incomplete, and from this again follows that if we were not did not regard, not only the law itself as true, but also the fragmentary part of it which we actually use as sufficient for a reliable prediction, we could again not rely on inductions for any practical purposes. But as a matter of fact we use inductions for making predictions and in most cases we rely on our predictions or regard them as highly probable. But what justifies us in relying upon our predictions, what can assure us that because it hitherto has been possible to predict events successfully it will be so even in the future? This is the inductive problem which again appears.

12 I have dealt with these views to some length because they seem to me to offer a very good explanation for a great many puzzling problems connected with so called inductive inference, and although they are not able to solve or to eliminate the inductive problem as a whole they leave it in a much less so to say excitingoriginal: exiting state than it was before. But I do not pretend to have made the point clear at all. When we state the view in terms of oversimplified examples as here has been the case the whole thing has a tendency to look so utterly naiv and everybody is likely to say: ”yes, surely it is so, but this is not interesting”. – Perhaps Ioriginal: a could say it better in this way: When Hume says that causal laws are never necessary connections, because we can always, as in the case of the billiard ballsoriginal: billiard-balls, imagineoriginal: immagine things to be otherwise without having a contradiction, he certainly is essentially right. But if we actually e.g. in the case of billiard ballsoriginal: billiard-balls try to imagineoriginal: immagine that the assumed causal law does not hold, we shall soon find first of all that it is not at all clear what this really comes to, and secondly that if we give it a definite meaning this would be something very so to say ”upsetting”. This is because the induction we consider is not an isolated uniformity, but bound up with an immense bulk of knowledge, e.g., the laws of mechanics and a multitude of other experiences of a very general kind. These connections again introduce an element of necessity, first of all in the form that propositions inductions are logically dependent upon each other, and secondly in the form that among these propositions some only apparently assert anything about matters of fact but play actually the roleoriginal: roll of definitions what, above all is important in this connection, it is usually not at all settled which propositions are to be regarded as definitions and which are not.tillagt av utgivaren All these facts considered together throw a peculiar light upon the inductive problem and give a partial explanation why the idea of a justification of induction is so natural and so dear to philosophers We shall, however, proceed furtheroriginal: futher and examine if there is no other way of securing the truth of inductions which at the same time would completely solve the inductive problem. Such a solution has in fact several times been proposed by philosophers in the form of what has been called an inductive logic.

13 The main idea underlying the trials to establish an inductive logic could be described as follows. We wish to establish the law, say, that A implies S for all values of x. – In the following I omitoriginal: omitt the variables. – We examine for this purpose a multitude of cases having A and S in common and beside these naturally a good many other properties which must not remain constantly present from one case to another. So we get instances|7|

14 As long as there remain other characteristics beside A and S common to all the instances the possibility is present that some of these other characteristics and not A entails S. As we cannot assume that we have complete knowledge of all the characteristics present in a given case, we can never know for certain that the examined cases do not have other characteristics beside A and S in common. But by multiplying the instances and not findingoriginal: finging any other characteristics than A and S common to all of them we make it, as we say, more and more probable that A and S are the only common characteristics. Let us, however, for a moment assume that we were sure that A and S are the only characteristics common to all the instances. Then we have established what is sometimes called a perfect analogy between A and S. Does it follow from this that the law A entails S really is true, i.e. that whenever in the future A will be present S will be present also? Obviously this does not follow from the mere fact that we have established a perfect analogy between A and S, unless we establish a special postulate to the effect that a perfect analogy is sufficient for the proof the a law of the form A entails S. Such a postulate or a postulate of a similar type occurs in most systems of an inductive logic; let us here call it the postulate of Uniformity of Nature.

15 What is the logical nature of this postulate? It is obviously a general synthetical proposition. But how then do we know that it is true? If we try to base it on induction we move in a vicious circle. And if we do not base it on induction it must be true a priori. Without entering the old question, whether synthetic judgements a priori are possible or not, it ought not to be difficult to see that the assumption of such judgements could not help us to solve the inductive problem. If we only say that whenever A is present S must also be present because it follows from a synthetic proposition a priori, we obviously do not assure us of anything at all, unless we can give reasons why the a pri|8|ori proposition must be true. If we give these reasons by pointing to other, still more general propositions from which it follows we only remove the problem a step further away, but do not solve it. An escape from this infinite regress was tried by Kant, who based the necessity of certain synthetic truths a priori as e.g. the Law of Causation, which roughly correponds to what we have called the postulate of Uniformity of Nature, upon the principle that if these truths were necessary in order to make experience possible. But even this does not help us to solve the inductive problem. Because first of all: the truths which we deem necessary in order to make experience possible may be formulated in such a way general way that it is never possible to draw any conclusions from them as to concrete happenings. In this case, which Kant apparently had in mind, the synthetic judgements a priori do not help us to solve the inductive problem for any single case, and consequently does not help us at all. This was by the way pointed out by the followers of Kant who hardly himself realized the full amount of this truth. Or secondly we could formulate these principles in such a way, that it were possible at least in certain exceptional cases to draw conclusions from them as to matters of fact concrete fact. This would in our example amount to the same as to saying, that under certain peculiar circumstances S must always be present when A is present, because if this were not the case, we had violated a general rule, which is necessary in order to make experience possible. But in this case if so, then we are of course as justified to ask: how do we know that experience – in the sense of future experience – really is possible in this case? as we are justified to ask how do we know that whenever A will be present S will be present also? because in this situation these two questions come to the same thing. And to ask this is nothing but to ask for a solution of the inductive problem. This way of dealing with the idea of an inductive logic and with Kant’s famous attempt to solve the problem of Hume may again seem too oversimplified and naiv. But I suspect that the difficulty in a way consists just in seeing that we can simplify matters here to this enormous extent. It may e.g. be objected that the important difference between Kant and Hume lies therein, that whereas Hume maintained that causal connections are necessary connections, Kant showed that even if we cannot be certain, say, that A is the cause of S we can be sure that there must exist some causeoriginal: cuase for S, i.e. some character or combination of characters having the power always to produce S, and that this is a great step forward in comparison with the state in which Hume left the problem. This sounds very plausible. But to maintain merely that there must be some cause for S every phenomenon, is much the same as to maintain e.g. that for every set of points in a system of coordinates there exists a function such that the values of the coordinates of the points correspond to ordered pairs of the values of satisfying the function, – which by the way is an analytical statement. And if we again try to specify what we mean by saying that every phenomenon has a cause in such a way that it gives some real assistance in our search for causes then we introduce assumptions which themselves are inductive.

16 Finally I shall make some comments on what may be considered most important of all in connection with the inductive problem, namely the so called inductive probability. It is very commonly said often stated, both|9| in philosophical text-books and in ordinary language that all inductive inference is merely probable, that we can know anything about the future only with probability, although in some cases, e.g. where we have employed scientific methods, with a higher degree of probability than in other cases again with a mediocre or small probability. I feel certain that we in many cases can assign a definite meaning to propositions about the probability of inductions and hypotheses, that we e.g. can explain what we mean by saying that a simple hypothesis is more probably than a complex one or that the verification of an unexpected or strange consequence of a hypothesis increases it probability more than the verification of even a great number of more ”trivial” consequences. But although this may be the case it is important to note that although this may be so, probability-considerations do not give us any better justification for inductive inference than e.g. the above-mentioned idea of an inductive logic does. This I wanted to point out by means of some very brief and very general deliberations as to the nature of the probability-concept.

17 When we speak say that something is probable we can mean, roughly speaking, one of two things. Either we mean something which already involves explicit assumptions about the future, e.g. about the stability of statistical series. As these assumptions are themselves inductive, this meaning of probability will not help us to a justification of induction. Or we make the probability values depend upon something which does not involve considerations assumptions about the future, as e.g. when we say in the case of a symmetrical and homogenous penny that the probability of getting head is 1/2, meaning simply something like: we have no reason to expect the occurrence of head more than the occurrence of tail. (The meaning of this is of course extremely obscure. All I wanted to point to is that in order to give get a justification of induction in terms of probability we must not base probability-judgements upon assumptions which are not themselves inductive. Mr Keynes’ concept of probability-relation for instance is obviously meantoriginal: ment to be a relation not involving any inductive assumptions.) This probability-concept we then connect with ind|10|uction by saying that it determines degrees of rational belief in future events. This expression again will have no bearing upon the justification of induction if it does not imply e.g. in the case of two different degrees of belief in two different inductions that we expect that in making predictions we shall on the whole be more successful with the induction we regard more probable than with the other. As this phrase is somewhat ambiguous I shall illustrate what I mean by an example. Let us assume that we wanted to can produce an in the Bacon’ian sense of the word, a property S either by producing first another property S A, or by producing the two properties A and B, and that we regard the former latteroriginal: later way of producing a A S as giving leading to the desired result with a higher degree if probability than the former. This can mean two things. Either it means that A and B more frequently lead to the desired result than only A, or we it means that as far as our experience goes both ways of producing S always have given the desired result but that we know of analogous cases, A´ producing S´ and B producing S´, where B has been after all shown to be necessary in order always to get the desired result, and that we therefore expect it to be necessary even in this case. But from this, and that is the crucial point, does not follow anything as to whether we actually shall succeed better in the long run in making predictions according to the more probable law than in making predictions according to the less probable. In other words – and we could express it almost paradoxically –: it does not follow in the least that it is ”rational” to act in accordance with our rational beliefs. That this actually is the case is by the way not at all so easily seen as is shown by the famous attempts to justify induction starting from purely a priorioriginal: a priori considerations by means of the Laws of Great Number. That this attempt is a failure is nowadays admitted by almost every philosopher as far as I know, but it deserves in fact greater interest than many other proposed systems ways of justifying inductions. I can, however, not enter into these questions here. – Anyway, we can say that if probability does not tell us anything about howoriginal: hoe to act in order to be successful then it gives no justification of induction. – Here I think a supporter of the theory of inductive probability would make an immediate objection. Of course, he would say, does he not maintain, that we could prove that if we acted in accordance with rational beliefs we must in the long run be more successful than if we acted in any other way; he only maintains that it is a way of acting which most probably will be successful. But this is only to remove the problem and precisely the same questions things could be said about this probability of a second order. To say merely that probability tells us something not about success but about probable success does not help us at all in the inductive problem. This is just the difference between the older school of justifying inductions by means of probability considerations and the more recent one of e.g. Mr Keynes: In the seve eighteenthoriginal: eightteenth century philosophers maintained that we from a priori considerations about probability could deduce propositions about what actually was going to happen in the long run, whereas modern philosophers rightly maintain that we can deduce what is going to happen in the long run only with probability. The difficulty consists in seeing that when we say so we are not giving any more the slightest justification of induction. This difficulty is not so easily overcome and I do not pretend to have made the point quite transparent, so to say, for you.

18 I have tried to show in this paper that if we by demanding a justification of induction demand some sort of proof that the future will be in uniformity with the past or that our beliefs are rational in the sense that if we act according to them we shall ultimately have more success than if we do not act so, then we are crying for the moon. I think that to realize the full amount of this truth, is to see – what I indeed have not explicitely tried to show here – that the problem of finding a justification of induction is no problem at all in the proper sense of the word, that what matters is not that the justification|11| of induction is lacking, but rather: that here is nothing to justify at all. The inductive problem – as so many problems in philosophy – is like a mist, and to solve the problem is merely to make the mist disappear. – What I have said is not meantoriginal: ment to be a proof for this, I have merely tried to to point out a way; which – if followed – ought to contribute to a clarification of the ideas which we are apt to connect with this particular problem.