Appendix to chapter V
SOLVING HUME’S PROBLEM OF INDUCTION
It would be impossible to say truly that the universe is a chaos, since if the universe were genuinely chaotic there could not be a language to tell it. A language depends on things and qualities having enough persistence in time to be identified by words and this same persistence is a form of uniformity.
J. Teichman & C. C. Evans
Here I will first reconstruct in the clearest possible way the essentials of Hume’s skeptical argument against the possibility of induction (Hume 1987 Book I, III; 2004 sec. IV, V, VII), separating it from its amalgamated analysis of causality. My aim in this is to find an argumentative model that allows me to outline what seems to me the only adequate way to react to the Humean argument in order to re-establish the credibility of inductive reasoning.
1. Formulating a Humean argument
According to Hume, our inductive inferences require metaphysical principles of the uniformity of nature to support them. Although induction can move not only from the past to the future, but also from the future to the past and from one spatial region to another, for the sake of simplicity I will limit myself here to the first case. A Humean principle of uniformity from the past to the future can be stated as:
PF: the future will resemble the past.
If this principle is true, it ensures the truth of inductive inferences from the past to the future. Consider the following very simple example of an inductive argument justifying the (implicit) introduction of PF as a first premise:
1. The future will resemble the past. (PF)
2. The Sun has always risen in the past.
3. Hence, the Sun will rise tomorrow.
This seems at first glance a natural way to justify the inference according to which if the Sun rose each day in the past then it will also rise tomorrow, an inference which could be extended as a generalization, ‘The Sun will always rise in the future.’ We make these inferences because we unconsciously believe that the future will be like the past.
It is at this point that the problem of induction begins to delineate itself. It starts with the observation that the first premise of the argument – a formulation of the principle of uniformity of nature from the past to the future – is not a truth of reason characterized by the inconsistency of its negation, one could say, it is not an analytic thought-content. According to Hume, it is perfectly imaginable that the future could be very different from the past, for instance, that in the future trees could bloom in the depths of winter and snow taste like salt and burn like fire (1748, IV).
We can still try to ground our certainty that the future will resemble the past on the past permanence of uniformities that once belonged to the future, that is, on past futures. This is the inference that at first glance seems to justify PF:
1. Already past futures were always similar to their own pasts.
2. Hence the future of the present will also resemble its own past.
The problem with this inference is that it is also inductive. That is, in order to justify this induction we need to use PF, the principle that the future will resemble the past; but PF itself is the issue. Thus, when we try to justify PF, we need to appeal once more to induction, which will require PF again... Consequently, the above justification is circular.
From similar considerations, Hume concluded that induction cannot be rationally justified. The consequences are devastating: there is no rational justification either for expectations created by the laws of empirical science, or for our own expectations of everyday life, since both are grounded on induction. We have no reason to believe that the floor will not sink under us when we take our next step.
It is true that we are always willing to believe in our inductive inferences. But for Hume, this disposition is only due to our psychological constitution. We are by nature inclined to acquire habits of having inductive expectations. Once we form these expectations, they force us to obey them like moths flying towards bright lights without any warrant. This is an extremely skeptical conclusion, and it is not without reason that only a few philosophers accepted Hume’s conclusion. Most think that something must be wrong somewhere.
There have been many interesting attempts to solve or dissolve Hume’s problem; all of them in some way unsatisfactory. I believe my approach, although only sketched out, has the virtue of being on the right track. I want to first present a general argument and then show how it could influence PF.
2. The basic idea
My fundamental idea has a mildly Kantian flavor, but without its indigestible synthetic a priori. We can sum it up in the view that any idea of a world (nature, reality) that we are able to have must be intrinsically open to induction. I see this as a conceptual truth in the same way as, say, the truth of our view that any imaginary external world must in principle be accessible to perceptual experience.
Before explaining it in more detail, I need to say that my view is so near to self-evidence that it would be strange if no one had thought of it earlier, as the citation at the start of this appendix proves. More technically, Keith Campbell followed a similar clue in developing a short argument to show the inevitability of applying inductive procedures in any world circumstances (1974: 80-83). As he noted, in order to experience a world cognitively – as an objectively structured reality – we must continually apply empirical concepts, which, in turn, if we are to postulate, learn from and use them, require a re-identification of the designata of their applications as identical. However, he thinks this is only possible if there is a degree of uniformity in the world that is sufficient to allow re-identification. Indeed, if the world were to lose all the regularities implicitly referred to, no concept would be re-applicable and the experience of a world would be impossible.
Coming back to my general idea, and understanding the concept of world minimally as any set of empirical entities compatible with each other, this idea can be unpacked as follows. First, I consider it an indisputable truism that an external world can only be said to exist if it is at least conceivable. However, we cannot conceive of any external world without any degree of uniformity or regularity. Now, since we can only experience what we are able to conceive, it follows that we cannot experience any world completely devoid of regularity. This brings us to the point where it seems reasonable to think that the existence of regularity is all that is necessary for at least some inductive procedure to be applicable. However, if this is the case, then it is impossible for us to conceive of any world of experience that is not open to induction. Consequently, it is a conceptual truth that if a world is given to us, then some inductive procedure should be applicable to it.
There is a predictable objection to this idea: why should we assume that we cannot conceive the existence of a chaotic world – a world devoid of regularities and therefore closed to induction? In my view, the widespread belief in this possibility has been a deplorable mistake, and I am afraid that David Hume was chiefly responsible for this. His error was to choose causal regularity as the focus of his discussion, strengthening it with interesting selected examples like those of trees blooming in winter and snow burning like fire. This was misleading, and in what follows, I hope to explain why.
Causal regularity is what I would call a form of diachronic regularity, that is, one in which a given kind of phenomenon is regularly followed by another kind. We expect the ‘becoming’ (werden) of our world to include regular successions.
However, induction applies not only to diachronic regularities, but also to something that Hume, with his fixation on causality, did not consider, namely, synchronic regularities. Synchronic regularities are what we could also call structures: states of affairs that endure over time in the constitution of anything we can conceive of. The world has not only a ‘becoming’ (werden), but also a ‘remaining’ (bleiben), with its multiple patterns of permanence. This remaining must also be reached inductively.
We can make this last view clear by conceiving of a world without any diachronic regularity, also excluding causal regularities. This world would be devoid of change, static, frozen. It still seems that we could properly call it a world, since even a frozen world must have regularities to be conceivable; it must have a structure full of synchronic regularities. However, insofar this frozen world is constituted by synchronic regularities, it must be open to induction: we could foresee that its structural regularities would endure for some time – the period of its existence – and this already allows a strong degree of inductive reasoning.
Considerations like this expose the real weakness in Hume’s argument. By concentrating on diachronic patterns and thinking of them as if they were the only regularities that could be inductively treated, it becomes much easier to suppose the possibility of the existence of a world to which induction does not apply or cannot be applicable, while the world still continues to exist.
To clarify these points, try to imagine a world lacking both synchronic and diachronic regularities. Something close to this can be grasped if we imagine a world made up of irregular, temporary, random repetitions of a single point of light or sound. However, even if the light or sound occurs irregularly, it will have to be repeated at intervals (as long as the world lasts), which demonstrates that it still displays the regularity of randomly intermittent repetition open to recognition. But what if this world didn’t have even random repetitions? A momentary flash of light… Then it would not be able to be fixed by experience and consequently to be said to exist. The illusion that it could after all be experienced arises from the fact that we already understand points of light or sounds based on previous experiences.
My conclusion is that a world absolutely deprived of both species of regularity is as such inconceivable, hence inaccessible to experience – a non-world. We cannot conceive of any set of empirical elements without assigning it some kind of static or dynamic structure. But if that’s the case, if a world without regularities is unthinkable, whereas the existence of regularities is all we need for some kind of inductive inference to be applicable, then it is impossible that there is for us a world closed to induction. And since the concept of a world is nothing but the concept of a world for us, there is no world at all that is closed to induction.
Summarizing the argument: By focusing on causal relationships, Hume invited us to ignore the fact that the world consists of not only diachronic, but also synchronic regularities. If we overlook this point, we are prone to believe that we are able to conceive of a world inaccessible to inductive inference. If, by contrast, we take into account both general types of regularity to which induction is applicable, we realize that a world that is entirely unpredictable, chaotic, devoid of any regularity, is impossible, because any possible world is conceivable and any conceivable world must contain regularities, which makes it intrinsically open to some form of induction.
One could insist on thinking that at least a partially chaotic world could be given, with a minimum of structure or uniformity, so that it would be insufficient for the application of our inductive procedures. However, I think this is a theoretical impossibility, for induction has a self-adjusting nature, that is, the application of its principles must always be calibrated to match with the degree of uniformity given in its field of application. The requirement of an inductive basis, of repeated and varied inductive attempts, can always be further extended, the greater the improbability of the expected uniformity. Consequently, even a system with a minimum of uniformity requiring a maximum of inductive search would always end up enabling the success of induction.
These general considerations suggest a variety of internal conceptual inferences, such as the following:
Conceivable cognitive-conceptual experience of a world ↔ applicability of inductive procedures ↔ existence of regularities in the world ↔ existence of a world ↔ conceivable cognitive-conceptual experience of a world…
These phenomena are internally related to each other in order to derive each other extensionally, so that their existence already implies these relations. But this means, contrary to what Hume believed, that when properly understood the principles of uniformity should be analytic-conceptual truths, that is, truths of reason applicable in any possible world.
3. Reformulating PF
To show how I would use the just offered proposal to reformulate the principles of uniformity or induction, I will reconsider in some detail PF, the principle that the future will be like the past. If my suggestion is correct, then it must be possible to turn this principle into an analytic-conceptual truth constituting our only possibilities of conceiving and experiencing the world. – I understand an analytic-conceptual thought-content to be simply one whose truth depends only on the combination of its semantic constituents; its truth isn’t ampliative of our knowledge, in opposition to synthetic propositions, and is such that its denial implies a contradiction or inconsistency (cf. Ch. V, sec. 12).
To show how the aforementioned suggestion could be applied to reformulating the principles of uniformity or induction it is necessary to reformulate PF. If my general thesis is correct, then it must be possible to turn this principle into an analytic-conceptual truth, constituting a way of conceiving and experiencing the world. Here is a first attempt to reformulate PF in a clearly analytical form:
PF*: The future must have some resemblance to its past.
Unlike PF, PF* can easily be accepted as expressing an analytic-conceptual truth, for PF* can be clearly seen as satisfying the above characterization of analyticity. Certainly, it belongs to the concept of the future to be the future of its own past. It cannot be the future of another past belonging to some alien world. If a future had nothing to do with its past, we could not even recognize it as being the future of its own past, because it could be the future of any other past... In still clearer words: the future of our actual world w, as Fw, can only be the future of the past of w, that is, Pw. It cannot be the future of infinitely many possible worlds, w1, w2, w3... that have as their past respectively Pw1, Pw2, Pw3... It is necessary, therefore, that there is something that identifies Fw as being the future of Pw, and this something can only be some degree of resemblance.
Against this proposal, we can try to illustrate by means of examples the possibility of complete changes of the world, only to see that we will always be unsuccessful. Suppose, in an attempt to imagine a future totally different from its past, a ‘complete transformation of the world’ as foretold in the Book of Revelations. It is hard to imagine changes more drastic than those described in the Apocalypse, since it intends to describe the end of the world as we know it. Here is the biblical passage describing the locusts sent by the fifth angel:
In appearance the locusts were like horses equipped for battle. And on their heads were what looked like golden crowns; their faces were like human faces and their hair like women’s hair; they had teeth like lions’ teeth and they wore breastplates like iron; the sound of their wings was like the noise of horses and chariots rushing to battle; they had tails like scorpions with stings in them, and in their stings lay their power to plague mankind for five months.
At first view these changes are formidable. Nonetheless, there is nothing in this report that puts PF* at risk. In fact, closer reflection on the example demonstrates that even PF isn’t seriously challenged. Although these biblical locusts are indeed very strange creatures, they are described as combinations of things already very familiar to us. These things are horses, women, hairs, men, heads, teeth, scorpion tails with stings, faces of persons, etc. Both internally and externally, they include a vast quantity of synchronic regularities, of permanent structural associations, together with familiar diachronic associations, like the causal relationship between the noise produced and the movement of wings or the sting of the scorpion and the effects of its poison on humans…
In fact, were it not for these uniformities, the Revelation of St. John would not be conceivable, understandable and able to be the subject of any linguistic description. The future, at least in proportion to its greater proximity to the present, must maintain sufficient similarity to its past to allow an application of inductive procedures to recognize the continuity of the same world.
Now one could object that maybe it is possible that at some time in a remote future we could find a dissimilarity so great between the future and our past that it invalidates any of our reasonably applicable inductive procedures – a remote future that would be radically different from its past. Indeed, it seems conceivable that a continuous sequence of small changes could in the course of a very long period of time lead to something, if not completely different, at least extremely different. I think that this would not destroy PF*, because its formulation is too weak, requiring only that some similarity must remain. However, it also seems that this weakness of PF*, even if not destroying its analytic-conceptual character, exposes PF* to disproportionate poverty as a way to assure the force of our inductive forecasts.
However, precisely this weakness of PF* indicates a way to improve it. It leads us to see that the closer we get to the point of junction between the future and the past, the greater must be the similarity between future and past, becoming both identical at their limit, which is the present. We can approximate this issue by remembering the Aristotelian analysis of change as always assuming the permanence of something that remains identical in a continuous way, without gains or losses (Aristotle: Physics, 200b, 33-35); in other words, the intuitive idea is that every change must occur upon some basis of permanence.
This leads us to create another variant of PF, namely, the principle according to which in a process of change the amount of permanence must be inversely proportional to the period of time in which the change occurs. In other words: if there is a sequence of changes that are parts of a more complete change, the changes that belong to a shorter sequence typically presuppose a greater number of permanent structural (and sequential) associations than the sequence constitutive of the more complete change.
This principle can be illustrated with many examples. Consider a simple one: the changes resulting from heating a piece of wax. The change from the solid state to the liquid state assumes the permanence of the same wax-like material. However, the next change, from liquid wax to carbon ash, assumes only the permanence of carbon atoms. If then the heat is much more intense, carbon will lose its atomic configuration, giving place to a super-heated plasma of subatomic particles. We have here four periods of time in a row: regarding the shortest period of time from t1 to t2, we assume that we will be left with (i) the same wax, made up of (ii) the same carbon atoms, which in turn are composed of (iii) their same subatomic constituents. In the longer period of time from t1 to t3 we assume the identity of only (ii) and (iii): carbon atoms and subatomic particles. And in the still longer period of time from t1 to t4 the only things that remain the same are (iii): subatomic constituents.
Note that this model is not restricted to changes in the physical material world! As Leibniz saw: Natura non facit saltus. The same examples repeat in every domain that one can imagine, chemical, biological, psychological, social, economic, historical… with the same patterns: the closer the future is to its junction with its past, the more structural identities must be in some way assumed. For example: the process of industrialization. The Industrial Revolution was a period of social and economic changes from an agrarian society to an industrialized society, which suffered an upheaval in the mid-19th century. As a whole, after its second period it included the refinement of the steam engine, invention of the internal combustion engine, harnessing of electricity, construction of infrastructure such as railways… and, socially, the exodus of families from rural areas to the large cities where factories were constructed… However, when we choose to consider a short period in this process, for instance, at the end of the 18th century, what changes might be only the invention of a primitive piston engine and a minor exodus from the countryside, all other characteristics of the society remaining essentially the same.
We conclude that it belongs to the very structure of the world of experience that changes taking place in a shorter period of time tend to presuppose more permanence than the most comprehensive changes in which they occur. Consequently, the future closer to its present should in some way be inevitably more similar to its past in more aspects than more distant futures would be (which, as already noted, may become nearly unrecognizably distinct) until the point of junction between future and past (the present), when no difference is conceivable.
Regarding induction, this principle assures that inductive predictions will become more likely the closer the future is to the present. On this basis, we can improve principle PF* as:
PF**: As a rule, the closer the future is to the junction point with its own past, the more it will tend to resemble its past, the two being indistinguishable at the point of junction (the present).
For a correct understanding of PF** we need to add two specifying sub-conditions:
(i) that this principle should be applied to a future that is sufficiently close to its past and not to an indefinitely distant future.
(ii) to safeguard the possibility of anomalous but conceivable cases in which we find shorter sequential periods in which states of affairs of a more distant future are closer to the present than those of an earlier future.
Although I admit that PF** deserves more detailed and precise consideration, it seems to me intuitive that so understood this principle already meets a standard of analyticity.
Moreover, it is the truth of PF** which explains why it is natural for us to think that the more distant the future, the less probable our inductive forecasts will be. This is the very familiar case of weather forecasts: they are presently reliable for two or three days, less so for a week or more... It also explains why our inductive generalizations about the future cannot be applied to a very distant future. When we say, for instance, that induction allows us to infer that the Sun will ‘always’ rise, the word must be placed in quotation marks. On the basis of induction, it makes sense to affirm that the sun will rise tomorrow or even a thousand years from now. But it doesn’t make any sense (and is for astrophysical reasons false) to use the same inductive basis to say that the Sun will rise in seventeen billion years.
Finally, PF** can ensure restricted applications of PF: If the future is sufficiently close to its junction with the past, then the future is unavoidably similar to its past. The problem, of course, is that we need to establish criteria for measuring how close we have to expect that the future will be to its past so that PF will apply. We can guess whether the response does not depend on the background represented by the domain of regularities in which we are considering the change, a domain of regularities to which a whole system of sufficiently well entrenched beliefs applies.
For example: the inductive conclusion that the Sun will rise tomorrow belongs to a domain of regularities implicated by changes predicted by astronomy, which may include a very distant future in which broader changes, such as the death of the Sun, are also predictable based on the previously observed fate of similar stars.
Of course, it is always possible that the Sun will not rise tomorrow! However, this is only conceivable at the price of an immense loss of other well-entrenched beliefs about astronomical regularities and, subsequently, the loss of the current intelligibility of a considerable portion of the physical world around us. Still, what makes us consider as highly likely the future occurrence of regularities such as that the Sun will rise tomorrow? The ultimate answer seems, based on the inevitable assumption of the world we experience, namely, our world as a whole will continue to exist as a system of regularities, at least in the form prescribed by PF**. Taken as a whole this assumption – I am forced to admit – is a real and inevitable gamble! There is nothing preventing our whole world from disappearing in the next few seconds: we could disappear or suddenly wake up on a completely different world. However, once we accept the general assumption that our world as a whole will continue to exist, it looks as though the rest will take place in the form prescribed by PF**: we are inevitably led to admit that certain fields of cohesive regularities are likely to remain. In other words: there is no reason that makes it improbable or probable that the whole world will disappear a moment from now; however, we can find reasons that make it improbable that a dependent portion of the world will disappear in the next moment, while others continue to exist, since this already presupposes that we are assuming the permanence of our world as an inductive basis.
The above outlined argument serves only a single form of induction: from the past to the future. Nevertheless, the attempt to better specify it and to generalize through further development would be worthwhile, since it indicates an open path. This may be of some interest regarding a problem that from any other angle seems to remain disorienting and intangible.
 For example, Hans Reichenbach (1938), D. C. Williams (1947), P. F Strawson (1952), Max Black (1954), Karl Popper (1959). Original as they may be, when faced with the real difficulties, all these attempts are disappointing. (For critical evaluation see W. C. Salmon 1966 and Laurence Bonjour 1997, Ch. 7.)
 For the sake of the argument, I am abstracting here the subject of experience.
 Strangely enough, the idea of a chaotic world to which induction isn’t applicable has been uncritically assumed as possible in the literature on the problem, from P. F. Strawson to Wesley C. Salmon. This seems to exemplify the weight of tradition as a knife with two edges.
 Revelation of St. John 9, 7.