THE MOST REASONABE RESPONSE TO THE HUMEAN PROBLEM OF INDUCTION ##

Draft in english of an article published in portuguese.

 

 

THE MOST REASONABLE RESPONSE TO THE HUMEAN PROBLEM OF INDUCTION

 

It would be impossible to truly say that the universe is chaotic, for if the universe were genuinely chaotic there could be no language with which to say so. Language depends on things and qualities that have sufficient persistence in time to be identified by words, and that very persistence is a form of uniformity.

Jenny Teichman

 

I would like to discuss here the apparently insoluble problem of the justification of induction introduced by David Hume. To do so, I want to begin by reconstructing Hume’s famous critique of the possibility of grounding our inductive inferences. Then I will outline, in very general terms, a solution to the problem that seems to me the only truly viable one.

 

1. The Humean Argument

I begin by briefly reconstructing Hume’s argument. He presented the problem through a critique of causal necessity, but in the reconstruction that follows I detach Hume’s anti‑inductivist argument from that critique, in order to make clearer what concerns us. According to Hume, our inductive inferences—that is, those that amplify our knowledge by moving from the observed to the unobserved—require metaphysical principles of regularity or uniformity of nature that guarantee them. Although induction may proceed not only from past to future but also from future to past, or from one spatial region to another, whether in the present or not, for the sake of simplicity I will restrict myself here to the first case, whose principle of uniformity can be stated as:

PF: the future will resemble the past.

If this principle is true, it will guarantee inductive inferences from the past to the future. Consider the following very simple example of justifying an inductive argument by introducing PF as the first premise:

1.     The future will resemble the past (PF).

2.     The sun has always risen every day.

3.     Hence: the sun will rise tomorrow.

This seems at first sight a natural way to justify the inference that if the sun has always risen every day, it will also rise tomorrow—an inference that could also be extended in the form of the generalization “The sun will always rise.” Here the problem of induction begins to take shape. It starts with the observation that the first premise of the argument, the formulation of PF, is not a relation of ideas (Hume) or a truth of reason (Leibniz), characterized by the contradictory or inconsistent nature of its negation; that is, it is not an analytic proposition (Kant). It is perfectly conceivable, Hume writes, that the future might become very different from the past—for example, that trees bloom in winter and that snow burns like fire. Even so (although Hume did not present it this way), it seems that we can acquire the conviction that the future will resemble the past based on our experience of past futures, which were similar to their own pasts. Here is the inference that seems to justify PF:

1.     Past futures have always resembled their own pasts.

2.     Hence: the future will resemble the past.

The whole problem is that this is an inductive inference. In other words: to justify induction we appeal to PF, the principle that the future will resemble the past, and to justify PF we again appeal to induction. The attempted justification of induction thus proves circular, since it depends on a principle that itself ends up depending on induction in order to be established.

   Hume’s well-known conclusion is that no rational justification for induction is possible; therefore, there is no rational justification either for the expectations created by empirical scientific laws or even for our everyday common‑sense expectations, since both clearly rely on induction. It is true that we possess a very strong disposition to believe in our inductive inferences. But for Hume this disposition has no rational basis; it is due only to our psychological constitution. We are instinctively disposed to acquire certain habits that produce inductive expectations, and once formed, these habits make us act much like moths, which are naturally disposed to fly toward light. This is an extremely skeptical conclusion, and it is not surprising that only a few philosophers have followed Hume on this point. Most think that something must be wrong somewhere.

 

2. Proposal of an analytic-conceptual solution

The strategy that I believe to be able to solve the Humean problem of induction in its very foundations, admits a priori inductive principles; however, they are not something with the strength of synthetic a priori principles. Rather, they are analytic–conceptual principles, in the sense that they are said to be true by virtue of what they mean — more precisely, by virtue of the combination of the meanings of their semantic constituents — such that their denial would involve contradiction or incoherence with our common rules of reasoning. My strategy also preserves something of a common‑sense solution, insofar as it turns much of Hume’s argument into a pseudo‑problem. I want first to present my general thesis and then show how it would apply to a chosen inductive principle.

   The general thesis has a faint Kantian flavor, though without an indigestible seasoning of the synthetic a priori. It is the idea that it is part of our very capacity to conceive a world of any kind — indeed part of the concept of experiencing any world whatsoever — that the world to which the concepts constitutive of it apply must be open to induction. I want to argue that this is a conceptual truth (analytic, since obtained a priori).

   Defining a possible world as a world like ours, though also conceivable as in greater or lesser degree different from ours, the argument can be formulated more carefully as follows:

   A world can possibly exist only if it is at least conceivable. But one cannot conceive a world with no degree whatsoever of uniformity or regularity. Now, since one can only experience what one can conceive, one cannot experience any world completely devoid of uniformity or regularity. And since the existence of regularity or uniformity is sufficient for some inductive procedure to be applicable, it follows that no conceivable or experienceable world can be closed to induction. It is, therefore, a conceptual truth that if a world is given to us, then some inductive procedure must be applicable to it. (A world that is in principle non‑experienceable or inconceivable, cannot qualify as such.)

   The objection to this thesis is predictable: what authorizes anyone to suppose that a chaotic world cannot exist — a world devoid of any regularity and therefore closed to induction? After all, the hypothesis of a world inaccessible to induction has traditionally been accepted.

   Nonetheless, the widespread belief in this possibility has, in my view, been a major mistake, one whose responsibility lies with David Hume himself. This mistake was introduced at the very beginning by the fact that Hume mixed the problem of induction with the problem of causation. He chose causal regularity as the focus of his discussion of induction and selected his examples accordingly, which was from begin misleading. What follows is meant to justify this claim.

   Causal regularity is what I would call a diachronic regularity, that is, one in which we think that a given thing is regularly followed by another thing different from the first, often in a way deeply entrenched in our belief system. Note also that not all diachronic regularities are causal; some are merely sequencial. After night there always comes a new day, but night does not cause day. Still, this is a diachronic regularity. Diachronic regularities constitute what might be called the becoming of the world.

   Yet it is a fact that we can conceive a world without becoming, without diachronic regularities, including causal regularity. This would be the case of a world without change: a static, frozen world. Even so, it seems it could still be correctly called a world. After all, worlds without diachronic regularities are conceivable and even in principle knowable, although things like the induction of causal laws would not apply to them. (I am abstracting here from any causal interaction between a frozen world and the epistemic subject.) The problem with Hume’s argumentative focus restricted to causal inductive inference — which is diachronic — is that it diverts our attention from the fact that an empirical world is equally constituted by synchronic regularities, which, just like diachronic regularities, can only be known through inductive procedures.

   But what, after all, are these synchronic regularities? We can define them as the relations simultaneously holding among things differently located in space, insofar as these relations endure through time. In contrast to becoming, they are responsible for what we could call the permanence of the world. This is the case of the relations that exist among the faces of a crystal, to take a distinctive example. It is induction that must justify the persistence of synchronic relations, leading us to believe, for instance, that the crystal will remain recognizable as having the same form when observed again in the future. The domain of synchronic regularities is extremely broad, since not only any object but any complex property and any recognizable state of affairs possesses constitutive relations among its parts — relations that must endure as long as the object, property, or state of affairs exists. The most interesting form of synchronic regularity is what could be called a structure. Synchronic regularities are often structures in the sense of possessing an internal identity that persists over time. A Gothic cathedral, with its pointed arches, large stained‑glass windows, ornaments, and statuary, can serve as an illustrative example of synchronic structurality. But a pile of books, although constituting a synchronic regularity while it remains the same, does not have what we more properly call a structure.

   A frozen world is a world only because, although it lacks diachronic regularities, it still possesses synchronic regularities, and it is expected to possess some structure. Moreover, induction is certainly applicable to this structure, since it is always applicable to synchronic regularities in predicting their persistence in the future. If the frozen world has the exagonal structure of a snow crystal, this structure must endure, and after a while, one could inductively predict that in the following moment, this structure will remain. What makes the frozen world knowable are its synchronic regularities.

   Let us now try to imagine a world without synchronic or diachronic regularities, without permanence or becoming. At first sight, this minimalist world seems illustrable if we think of it as consisting of irregular repetitions of a single luminous point or a single sound. However, even if the luminous point or the sound occurs irregularly, it must repeat at least once (as long as the world lasts), which already demonstrates at least the diachronic regularity of repetition; hence, induction applies to such minimalist worlds as long as they endure. But what about a world absolutely devoid of both kinds of regularity, without permanence and without becoming — is it conceivable? The answer is clear: a world without any regularity cannot be genuinely conceivable, and therefore cannot be accessible to experience. We cannot think of any set of compatible empirical elements without giving it some permanence or becoming. But if this is so — if a world without regularities is inconceivable — and since the existence of regularities is all we need for some inductive inference to be applicable, then it is impossible for there to exist a world that is not open to induction. Wherever there is a world, there must be some regularity; and wherever there is some regularity, some inductive access will be logically possible. To conceive a world to which induction does not apply would amount to conceiving a world without any regularity, which contradicts our very concept of a world.

To summarize: by concentrating on causal relations, Hume leads us to ignore that the world is also constituted by synchronic regularities, which in turn leads us to believe that we can conceive the existence of a world whose becoming is devoid of diachronic regularities — that is, an entirely chaotic world and therefore inaccessible to inductive inference. When we properly take into account both kinds of regularity to which induction applies, we realize that an entirely chaotic world, without any regularity, is inconceivable and therefore impossible, since any possible world is made of its regularities and is thus intrinsically open to induction.

   Searching in the literatur, I discovered that I am not the only person to have noted that any conceivable world must be open to induction. According to Keith Campbell, in order for us to cognitively experience a world — an objectively structured reality — we must always be reapplying empirical concepts, which, in turn, in order to be fixed, learned, and used, require a reidentification of their designations as being identical; but this is only possible if there is a degree of uniformity in the world sufficient to allow reidentification. Indeed, if the world could completely lose its regularities (not only diachronic ones, but also synchronic ones), then no concept could be reapplied, the experience of the world would cease, and it would, for us, cease to exist. Such considerations only corroborate what has already been suggested.

   Another reason why people tend to admit the possibility of a world whose degree of irregularity would make induction impossible is the neglect of the fact that induction has a self‑adjusting nature; that is, the application of the inductive procedure must always be calibratable in accordance with the nature of that to which it is applied. The requirement of inductive basis — repeated and varied inductive experimentation — can theoretically always be increased, the more improbable the expected uniformity is; consequently, even a world with minimal uniformity would still end up making inductive success possible, since it would require a maximized inductive search. In other words: it is enough that there be some uniformity for some requirement of inductive basis to ideally allow us to find it.

   To illustrate, imagine that in a nearby possible world a team of zoologists is searching for wild camels in the Gobi Desert in the final of the last century. This desert is immense, covering northern China and all of Mongolia. Moreover, these shy camels were are extremely difficult to find. The zoologists would have to visually inspect, with binoculars, a vast expanse of desert, climbing and descending dunes until they eventually find, with some luck, this almost mythical animal. Here the pressure of inductive calibration must be very much increased.

   The general considerations made so far suggest the following interwoven set of conceptual inferences: effective cognitive‑conceptual experience of the world ↔ applicability of empirical concepts ↔ applicability of inductive procedures ↔ existence of a world intrinsically possessing regularities.

   These concepts are internally related in the sense that they can be intrinsically inferred from one another. By this means, contrary to what Hume believed, when adequately formulated the principles of uniformity should reveal themselves as analytic‑conceptual truths understood as truths applicable to any possible world and which cannot be denied without contradiction. Thus, my goal now is to establish at least one formulation of these principles in a way that makes it sound like the analytic‑conceptual truths they must be.

 

3.     UNIFORMITY OF NATURE IN A NEW FORM

To show how the thesis just presented could be applied to the reformulation of the principles of uniformity or induction, I would like to reconsider PF in some detail. Could it be transformed into an analytic‑conceptual truth? As I have already noted, I understand an analytic‑conceptual proposition as one whose truth depends only on the combination of its semantic constituents. This truth is characterized by not being ampliative of our knowledge (as opposed to synthetic propositions), possessing as its criterion of identification the feature that its negation is contradictory, incoherent, or impossible to conceive.

   The first question that arises is whether PF, stating that the future will be similar to the past, can satisfy this usual characterization of analyticity. Hume thought not. As we have seen, he considers that we can conceive of snow beginning to burn like fire and trees beginning to blossom in winter… But these examples of Hume are as suggestive as they are illusory. For since a multitude of other regularities, especially synchronic ones, would continue to remain, these examples are far from making the future so dissimilar from the past as to invalidate inductive procedures. Nevertheless, it is still clearly conceivable that if some unpredictable cosmic cataclysm were to profoundly modify the future, making it different from the past, this shows us that PF is conceivably deniable and therefore non‑analytic. However, we can redo PF. Consider the following reformulation:

PF\*: The future must have some similarity with its past.

Unlike PF, PF* can clearly be understood as expressing an analytic‑conceptual truth. After all, PF* seems to satisfy the characterization of analyticity presented above. Here is how this can be shown: understanding the notion of future in terms of successive sets of regularities constituting the world to be given at times later than the present, and clarifying the concept of past in terms of successive sets of regularities at times earlier than the present, we can say the following: it belongs to the concept of future that it is the future of its own past. It cannot be the future of any other past. But if a future had nothing to do with its past, we could not even recognize it as being the future of its own past, for it could then be the future of some other past. In other words: the future F of the actual world m can only be the future of m, that is, Fm, which can only be the future of the past of m, that is, Pm; it cannot be the future of the many other possible worlds m1, m2, m3… which had as pasts the sequences Pm1, Pm2, Pm3… There must, therefore, be something that identifies Fm as being the future of Pm. Now, that something can only be some margin of similarity. That is: the notion of future must be in some way conceptually linked to the notion of its past as being to some extent, in some manner, similar to it — at least to the extent and in the manner that allow the temporal association of Fm with Pm. This is why PF* satisfies our characterization of analyticity: to deny it is to render the words “future” and “past” meaningless by making it impossible to relate them in the way conventionally done; if I deny PF, then it seems clear that the future no longer needs to be distinguished from its past as being the future of its past. Moreover, PF does not seem to enlarge our knowledge. PF* satisfies the criterion for identifying analytic propositions, for we are not capable of coherently denying it; we are not capable of thinking that the future has no similarity whatsoever with its past without inconsistency.

   Indeed, it seems that, in an attempt to reject PF, whenever we try to conceive a dissimilarity so great between future and past that it invalidates all inductive procedures, we fail to conceive any objective structure and even any possible world. This point can be easily illustrated through examples. Imagine, in an attempt to conceive a future completely different from its past, a “complete transformation of the world” such as that narrated in the biblical text of Revelation. It is difficult to imagine more drastic alterations than those described there. After all, it is the narration of the very end of the world as we know it! But it is a mistake to think that the destruction of our world described in Revelation would imply a negation of PF, since the idea of a “complete transformation” is not understood here in a literal sense. If we examine the text more closely, we will see that the great majority of things with which we are familiar — that is, the basic synchronic regularities and even most diachronic regularities — remain unchanged after the transformation, although they have been bizarrely combined, as in the biblical passage describing the locusts sent by the fifth angel:

The appearance of these locusts was like horses prepared for battle. On their heads they wore something like crowns with golden reflections. Their faces were like human faces. Their hair was like women’s hair, and their teeth were like lions’ teeth. Their chests seemed to be covered in iron, and the sound of their wings was like the noise of chariots with many horses rushing into battle. They had tails like scorpions, with stingers, and the power to afflict human beings for five months.

 

Now, nothing in this account puts PF* into question. In fact, a careful examination of the example shows that it does not even challenge a loose understanding of PF! For although these biblical locusts appear deliriously strange to us, they are composed of combinations of parts with which we are already very familiar — such as hair, women, men, teeth, scorpions, stingers — which internally and externally include a vast sum of regularities, of structural associations (such as those that form locusts, those that form crowns, those that form human faces…) and sequential associations (such as the causal relation between crowns and their golden reflections, between the scorpion’s sting and the effects of its venom on human beings for five months, between the beating of wings and the noises they produce…), all of which remain preserved and inductively accessible despite the alterations. Indeed, if this were not so, the Apocalypse would not be comprehensible, thinkable, conceivable, or capable of linguistic description — and what is none of these is also impossible to experience. The account illustrates the idea already mentioned: that the future world must, at least insofar as it remains sufficiently close to the present, continue sufficiently similar to its past so that it can be conceived as the future of that same past; that is, it must remain sufficiently similar to its past to warrant the application of inductive procedures in recognizing its continuity as a world.

   But what should we say of a future immensely later than the present? Could it not be totally different from the past? It seems that it could. If we interpreted PF* as referring not to the future as a whole from the present onward, but to a very remotely distant future, detached from all those that preceded it, then it seems clear that PF* could be falsified, for it is not inconceivable that a continuous sequence of small alterations in regularities could, over a very long period of time, give rise to completely different regularities. But this is not the sense in which I intended PF*, for when I presented it, it was already implicit that it concerned the continuation of its own past, including at least the future that immediately follows the present.

   This last consideration reminds us of another conceptual truth already observable in the relation considered by PF*. The closer we approach the point of junction between future and past — that is, the present — the greater the similarity between them must be, with future and past becoming identical at their limit, which is the present. This point can be approached when we recall Aristotle’s analysis of the concept of change as presupposing the permanence of something that remains identical and that continuously gains or loses. The suggestion is that all change presupposes some basis of permanence, that is, some synchronic regularity, which not only allows inductive inference but requires it in order to be known.

   But that is not all. There is a relevant observation that still needs to be made, now concerning the measure of the permanence presupposed. While change occurs, the measure of permanence as a whole must be inversely proportional to the period in which the change occurs. This means that if we are given a sequence of changes that are part of a more complete change, then, compared to the whole, the changes that are part of the sequence presuppose more permanence than the more complete change.

   The principle I have just presented may at first seem somewhat obscure, but it can be well illustrated through an example: consider the changes resulting from heating a piece of wax starting at T0. First we have the change from solid to liquid at T1. With greater heating we have the change from liquid wax to carbon ash at T2. If this ash is heated to many millions of degrees Celsius, we finally have the dissolution of the carbon atoms and the formation of a plasma of subatomic particles at T3. Here is a scheme showing how changes typically presuppose greater permanence the more partial and brief they are:

 

 PHYSICAL         COURSE OF TIME:

 ENTITY:

 

                             T0                  T1                  T2                   T3

 Solid wax              X                

 Liquid wax           X                   X

 Carbon ashes        X                   X                   X

 Subatomic            X                   X                   X                   X

 Particles

 (plasma)

Notice how synchronic regularities are lost over time. From T0 to T1, what is presupposed as permanent is the wax and its atomic constituents — carbon, oxygen, and hydrogen atoms — as well as subatomic constituents. From T1 to T2, only the carbon atoms and their subatomic constituents remain presupposed as permanent. Finally, from T1 to T3, all that remains are certain subatomic constituents. The change here is gradual in its loss of regularities.

   The model of change suggested above is not restricted to cases such as the chemical structure of a compound. It applies to physical, biological, psychological, social, economic alterations — in short, to any empirical domain we can conceive. Consider, for example, the Industrial Revolution. It began in the eighteenth century with the introduction of weaving machines, division of labor, and a small rural exodus. The social and economic structure of England remained practically the same at first. In the course of the nineteenth century, however, the changes deepened. Ironworks appeared, steam locomotives, a railway network, a large rural exodus… English society ceased to be the same, although many uniformities remained. There are successive losses and gains of regularities here. But the alterations can only be identified on the basis of permanence.

   What I am suggesting is that this model of change applies to everything for the simple reason that it constitutes part of the metaphysical structure of reality, as it is conceptually conceivable. It is constitutive of the very structure of the world of real or conceivable experience that changes occurring over a shorter period typically presuppose more permanence than the more complete changes of which they are part: natura non facit saltus. Even the abrupt change of an electron’s orbit only occurs under a structure of permanence, which in this case is the atom.

   The metaphysical model of change just suggested has consequences for our epistemological understanding of induction. The future closest to us must, by necessity, if taken as a sufficiently proximate whole, be more similar to its past than more distant futures (which, as we have already noted, may become even unrecognizably different from the present). This should already have been clear when we examined the example of the heated wax: if T0 is the present, T1 retains more similarities with T0 than T2, and T2 retains more similarities with T0 and T1 than the more distant T3.

   With respect to induction, this principle guarantees, given a certain frame of reference, that inductive predictions become all the more probable the closer the future to which they refer is. On this basis we can replace the excessively weak principle PF* with:

 

PF**: A sufficiently proximate future must retain some similarity with its sufficiently proximate past such that the closer it is to the point of junction with its own past (the present), the more it must tend to resemble that past, becoming identical to it at the point of junction.

 

For the correct understanding of PF**, ir ia important to consider that the futures must be sufficiently close to their pasts, since it is necessary that something in them contain situations belonging to the past or the continuation of changes arising from the past. After all, we can conceibe a longinquous future where there is nothing or nearly nothing that wouldd be similar to its past. Firthermore, it is important to emphasize the tendential aspect of the process: we must add that nothing prevents the anomaly that part of this sufficiently proximate future may remain the same or even be more diverse from the present than a part of it that is more distant, provided that at its point of junction with the past the future becomes identical to the present. (Consider, for a simple example, the cyclical crises that occur in the economy, despite its long‑term ever‑increasing pattern. For instance: the U.S. GDP of 1930 returned to that of 1920, although overall, from 1900 to 2000, it was increasing…)

   I belive that PF** could be considered in greater detail and even formally. But it seems to me that this principle already clearly satisfies the characterization of analyticity defended here, for it shows that it belongs to the very concept of a future sufficiently close to the present that, taken as a whole, it tends to resemble its past exponentially more, the closer it is to its point of junction with that past, converging toward identity at the very point of junction, which is the present. We may even attempt the Fregean strategy of showing that PF** can be transformed into a tautology, where x may be occupied by any event or by a variety of events, Fa = …what belongs to the sufficiently proximate future, Pa = …what belongs to the sufficiently proximate past, ≈a≈ = tend, in proportion to their proximity to the present, to be identical to one another. Here is how this relation may be presented:

 

PF**: (x) (Fax ≈a≈ Tax)

 

Something like this could be understood as a tautology within a temporal logic, insofar as it seems to be part of what we understand and can define as the flow of events in time; something reminiscent of a logarithmic scale which, indicating the present, runs from past to future within a given frame of reference.

   A consequence of admitting PF** is that it becomes natural to think that the more distant from its point of junction with the past a future period is, the less probable the inductive predictions concerning it will be. This explains why our inductive generalizations about the future are never truly about an indefinitely remote future, as might seem at first glance. When we say, for example, that induction allows us to infer that the Sun will always rise, the word “always” should be placed in quotation marks. It makes sense to affirm, on the inductive basis that the Sun has always risen, that it will rise tomorrow and even a thousand years from now. But it makes no sense at all (and in fact astronomy asserts it to be predictively false) to use the same inductive basis to say that the Sun will rise seventeen billion years from now.

   There is, of course, the case of scientific laws such as those of general relativity or quantum mechanics, which hypothetically apply to the entire universe. These laws generally result from inferences to the best explanation. An inference to the best explanation is inductive because ampliative, resting surreptitiously on an immense volume of prior enumerative inductions. Even so, it is questionable whether PF** would not be valid even for them. The field equations of general relativity apply well to known regions of the universe. Would they truly apply to everything in the universe? Suppose, for example, that multiple universes exist. In that case we would no longer feel so secure about the scope of their application.

   Finally, PF** can guarantee restricted applications of PF, making PF analytic when understood as restricted to the domain of those applications: if the future in question is sufficiently close to its point of junction with the past, then that future will necessarily tend to resemble its past. The problem, naturally, is that we lack criteria for determining how close a future must be to its past for PF to apply to it. We may speculate that the answer depends on assuming a domain of regularities to which the change under consideration belongs — a domain of regularities understood as one to which an entire system of well‑entrenched beliefs applies. Thus, the inductive conclusion that the Sun will rise tomorrow belongs to the domain of regularities implicated in the changes investigated by astronomy, which includes a future distant enough for the broader changes to occur, such as the death of the Sun. It is possible, although highly improbable, that the Sun will not rise tomorrow, as the inductive procedure itself predicts. But this would only be conceivable at the cost of an immense loss of other regularities and, subsequently, of our present intelligibility of a considerable part of what surrounds us.

   Still, what leads us to consider highly probable the future permanence of particular regularities, such as the Sun rising each day? The answer seems to begin with the inevitable assumption of the brute fact that the world exists as a system of regularities, since we can conceive it and experience it. Along with this, it seems that we also assume that this system of regularities that is our world will continue to exist. But are we not once again falling into the Humean abyss? After all, what guarantees that our world will not suddenly cease to exist five seconds from now, along with everything within it, including ourselves?

   The answer is that although we may find reasons that make it improbable that part of our world will disappear in the near future while other parts remain, it makes no sense at all to suppose that the entire world could vanish in an instant. The reason for this is verificationist. According to a reasonable principle of verification, a statement that cannot be verified or refuted in any way cannot have cognitive meaning. Statements such as (i) “My brother died after tomorrow” and (ii) “During this night the entire world doubled in size along with everything belonging to it” may have grammatical meaning — that provided by grammatical rules — but they do not have the much more important and proper cognitive meaning of being capable of saying something about the world, a meaning to be given by verification rules. (Obviously, someone might object here that verificationism is a doctrine refuted by the later development of much of analytic philosophy of language and even by its main defenders, the logical positivists… However, this would be a serious mistake! Semantic verificationism was first proposed by Wittgenstein to the logical positivists in 1929 and, as I tried to show in the first chapter, it was a commonsense proposal, far more flexible and plausible than theirs, and not open to the same objections.)

   Now, in the same way as with statements (i) and (ii), we may suggest that the statement (iii) “The entire world will disappear five seconds from now” is unverifiable. (Note that a statement such as (iv) “Possibly the entire world will disappear five seconds from now” is verifiable, for it will be verified after five seconds, since what is certain is also possible. But statement (iv) is different from assertion (iii), which has no justificatory basis.) We can easily deceive ourselves on this point by imagining ourselves outside the world, perceiving its disappearance. But since we ourselves belong to the world, it is simply impossible to verify such an event, for the epistemic subject capable of verifying it would also disappear, and, no longer existing, would be logically incapable of performing such verification. We conclude, therefore, that the statement above only seems to possess cognitive meaning, but does not truly possess it. But what about its negation? What about the statement (v) “The entire world will not disappear five seconds from now” or (vi) “The entire world will continue to exist five seconds from now”? The negation of a meaningless statement must also be meaningless: the negation of “My brother died after tomorrow,” which is “My brother did not die after tomorrow,” is equally meaningless. Consequently, statements (v) and (vi) must also lack cognitive meaning. However, one might suggest that (v) and (vi) escape this rule, since after the five seconds have elapsed these statements will have been verified. What is certain? I prefer to think that (v) and (vi) truly make no sense as such, for these statements assert the inassertible — that the world will continue to exist — unlike a statement such as (vii) “It is possible that the world will continue to exist five seconds from now,” which can be verified by the fact of the world’s non‑disappearance after the five seconds, since existence implies possibility. That we have the disposition to continue acting as if the world will continue to exist seems to be a purely dispositional matter. Apart from this natural disposition, we are not capable of finding any reason to believe that everything will exist or cease to exist in the near future. This is why this dispositional fact cannot lead us into skepticism: it remains cognitively unredeemable.

   Such considerations do not prevent us from admitting as probable the existence of certain cohesive domains of regularities, and of the particular regularities inferred within these domains as being of probable permanence, following PF**. The consequence of conceiving things in this way is that if we reject the future permanence of a regularity — such as the Sun rising each day — we must reject the future permanence of the entire domain of regularities in which it is included. But since the very probability of the regularity in question is measured on the basis of admitting the permanence of that domain of regularities (a point demonstrable by Bayes’ theorem), it ceases to be rational for us to put it into question.

   The solution I have just sketched is schematic and inconclusive, limited to a single form of induction. Nevertheless, it has the advantage of not approaching the Humean problem equivocally, as seems to me to be the case with all others. It approaches the Humean problem head‑on, in the condition in which it presents itself, without reducing it to something else, which may already be of some help for a problem that, viewed from any other angle, has appeared disorienting and intangible.