2. Hume’s problem

Hume asked for the reasoning behind the ready inferences that we typically make from past observed behavior to expectations about the future. For example, we draw inferences such as:

Hume seeks a “chain of reasoning,” which provides the foundation for carrying out such an inference. This reasoning, he says, would have to be based on a principle, often called the “principle of uniformity”: “that instances, of which we have had no experience, must resemble those, of which we have had experience, and that the course of nature continues always uniformly the same (Hume Reference Hume1739, 1.3.6.4).”

Hume’s argument against the possibility of providing the foundation for inductive inferences goes as follows. Hume claims that there are two types of possible arguments: “demonstrative” arguments and “probable” arguments. Demonstrative arguments establish conclusions which cannot be conceived to be false, whereas probable arguments concern “fact and existence.” A natural way to understand the distinction in modern terms is along the lines given by Samir Okasha: “A demonstrative argument, for Hume, has a priori premises and is deductively valid, while a probable argument has one or more empirical premises (Okasha Reference Okasha2001, 313).”

Hume then presents a dilemma. First, the principle of uniformity cannot be supported by a demonstrative argument, because there is no contradiction in supposing that the course of nature might change. And secondly, it cannot be supported by probable reasoning, because this would be circular. All probable arguments “proceed upon the supposition that the future will be conformable to the past (Hume Reference Hume1748, 4.2.19),” and supposing this would be to “take for granted, which is the very point in question (Hume Reference Hume1748, 4.2.19; see also Hume Reference Hume1739, 1.3.6.7/90).” Since neither demonstrative nor probable arguments will serve, the preliminary conclusion PC is that:

One can then add the Humean justification condition HJC:

From PC and HJC, one can draw the conclusion that the inductive inference is not justified.Footnote ¹

3. What needs to be justified?

Hume clearly intended to raise a problem arising for a broad class of inferences. How should we characterize this class? One of the simplest suggestions is that the inferences results from application of a rule such as “simple enumerative induction”:

This may be generalized in a probabilistic setting to a rule such as what Reichenbach calls “the principle of induction,” or “straight rule” (Reichenbach Reference Reichenbach1938):

More sophisticated inductive methods can also be defined (Schurz Reference Schurz2019, section 5.9).

Following Schurz, we can define “object-induction” (abbreviated as OI)” to mean “all methods of induction that are applied at the level of events – the object-level” (Schurz Reference Schurz2019, 83). The general Humean problem of induction may be then seen as the problem of justifying the class of all inferences resulting from the application of object-induction methods. This allows for a reformulation of what the solution to Hume’s problem requires. Rather than looking for specific “chains of reasoning” undergirding the inferences themselves, we can look to properties of the methods which make those methods justified. Inferences then acquire justification from the methods via a principle such as:

4. Justifying a Method

What then does it mean to justify a method? According to the most straightforward way of reformulating Hume’s problem, we could adopt the criterion that the method giving rise to the inductive inferences is justified if it is reliable. To be reliable, the method must mostly reach true conclusions, not only in the past, but also in the future. But then Hume’s argument can be reprised: on the one hand, the reliability of a method cannot be established by a demonstrative argument, since it is quite conceivable that a method which has hitherto proved reliable stops being so tomorrow. On the other hand, we cannot establish the reliability by an argument from the empirical premise that the method has hitherto been highly successful because this would involve extrapolating from the past successes of the method to its probable future successes, which itself requires some kind of principle of uniformity. Thus such an argument would be circular.Footnote ²

Among authors who take Hume’s argument to be decisive against the possibility of showing that the inductive method is reliable, another strategy has been to argue that there are weaker conditions which can justify a method. Reichenbach argued, for example, that a sufficient condition for a method to be justified is that that method is a necessary condition of success:

“Success” here is defined as meeting the aim of induction. Reichenbach gives the following example as an analogy:

Thus, for Reichenbach an alternative formulation of the statement that “M is a necessary condition for success” is “if any method will work (succeed), it will be M.” This allows that there may be cases where no method will work, including M.Footnote ³

Schurz proposes a justification criterion SJC that he sees as having its roots in Reichenbach’s approach (Schurz Reference Schurz2019). The justification of a method is to be based, not on its reliability, but on its optimality—on its being the best available alternative. We are to be justified in deploying a method if it is the best one we’ve got, even if we have no grounds for confidence that it will actually deliver. The justification criterion for the method becomes:

More precisely, the notion of optimality is defined in relation to a class of “prediction games.” A prediction game G=((e),Π) is a pair consisting of a stream of events (e)=(e₁,e₂ …) and a set of prediction methods Π (called “players”). Then, “a method M* is called optimal in relation to a class of prediction games iff for all streams of events (e), M* is at least as good as M for all other $M \in {\rm{\Pi }}$ (Schurz Reference Schurz2019, 95).”

The performance here is measured in terms of predictive success rates. Each prediction method in the set Π has a certain success rate, which is a cumulative score for how well that method’s predictions have panned out.Footnote ⁴ “M* is at least as good as M” can be defined in a long-run sense—meaning that in the long run as the number of events observed becomes very large, M* always achieves a higher cumulative success rate than M. It can also be defined in a more short-run sense according to which other methods can do better than M* in the short run, but the amount by which they do so converges quite quickly to zero as the number of events observed increases. In either sense of optimality then, what is to justify the method is its performance, relative to other available methods in a relevant class, over a long run of data for all possible data-streams. The justification for performing a particular inductive inference remains parasitic, via Inf-M, on the justification of the method M. Now however, the justification of the method is in terms of its optimality rather than its reliability.

5. Schurz’s two-step justification of induction

In the first step of his argument, Schurz appeals to results from the regret-based learning framework of Cesa-Bianchi and Lugosi that show that there is a meta-inductive strategy that is predictively optimal among all predictive methods that are accessible to an epistemic agent (Cesa-Bianchi and Lugosi Reference Cesa-Bianchi and Lugosi2006; Schurz Reference Schurz2008, Reference Schurz2017). In general terms, “meta-induction” is the method of applying induction at the meta-level to the success rates of all accessible prediction methods. One kind of meta-inductive strategy is simply to imitate the most successful prediction method available. More sophisticated meta-inductive methods will produce a combination of available prediction methods weighted according to their past success. For example, the meta-inductive strategy “wMI” predicts a weighted average of the predictions of the accessible methods, where the weights are “attractivities,” which measure the difference between the method’s own success rate and the success rate of wMI. The main result is that the wMI strategy is long-run optimal in the sense that it converges to the maximum success rate of the accessible prediction methods. Worst-case bounds for short-run performance can also be derived. The optimality results form the basis for an a priori justification for the use of wMI. Namely, the thought is, it is justified to use wMI because it satisfies the justification criterion SJC.

The first stage of Schurz’s argument can be reconstructed as follows:

1. 1. If method M is predictively optimal, method M is justified. (SJC)

2. 2. wMI is predictively optimal. (mathematical result)

3. 3. wMI is justified

The second step of Schurz’s argument is to show how the justification can be transferred from meta-induction to object-induction. Schurz claims that the a priori justification of wMI, given in the first step, together with the contingent fact that inductive methods have so far been much more successful than non-inductive methods, gives rise to an a posteriori justification of induction which is non-circular. “We know by experience that in our world, object-inductive prediction methods have been more successful than noninductive methods so far, whence it is meta-inductively justified to favor object-inductivistic strategies in the future (my emphasis, Schurz Reference Schurz2019, 85).”

The reasoning here is very compressed. In the next section, we explore how exactly to unpack the second stage of Schurz’s argument.

6. Unpacking the argument of the second step

What exactly is meant here by “meta-inductively justified?” The first point to notice is that following object-inductive strategies is not justified according to the criterion SJC, since object-induction has not been shown to be predictively optimal. The result we have is that there are meta-inductive strategies which are predictively optimal. OI and MI are not the same strategy, and in general their predictions may not coincide. The meta-inductivist using wMI does not assign all the weight to the most successful method, but spreads it among the other competing methods. If the competitors are not very successful, this may mean that in practice wMI is fairly well-approximated by the most successful object-method. We may further observe that the most successful object-method up to now has been OI. But wMI remains a different method than OI. It can result in different predictions on the very next application if OI stops working and a non-inductive method does better. In that case, wMI would begin to favor the non-inductive method, and wMI would start to come apart from OI. Thus we have no argument that OI has the same longer-term properties that produce the optimality of wMI. We know by following meta-induction we are en route to doing as well as we can do predictively, but we have no similar guarantees for OI. Thus, we do not have an argument yet that we are justified in favoring object-inductivistic strategies in the future.

The argument for transferring justification to OI must be somewhat more elaborate. In fact, Schurz does not claim that OI is justified in exactly the same way as wMI. For one thing, the argument for OI is a posteriori, since it is based on an empirical premise EP:

Given the empirical premise, the claim is that there is a non-circular argument for OI. The argument is non-circular because it depends only on steps of a priori reasoning once the empirical premise is given.

It looks as though the basis for transferring the justification from wMI to OI is the close approximation between OI and wMI. The methods are expected to produce approximately the same predictions. However, the empirical premise Schurz invokes—namely EP—does not in fact ensure such an approximation. If OI has been more successful than noninductive methods, that only means that OI should be weighted highly in the combination of methods that wMI produces on the basis of the past experience, not necessarily that it should produce a close approximation of wMI.

Sterkenburg suggests that one could invoke a stronger empirical premise EP* in order to ensure that the approximation holds:

This is a fairly strong claim about what has actually been empirically shown so far. It requires not just that OI has proved more successful than other methods, but it has proved so much more successful that it gains the lion’s share of the weight of the meta-inductive method.

Now let us see how the argument could be formulated using EP*. Let a “prediction situation s” include the observed past events, a candidate pool of methods, their track records and their actual predictions. Suppose the current situation s* is one in which EP* holds. Then we could use EP* to justify a claim that “application of OI in s* yields approximately the same result as wMI.” We also need to add a premise (“Approx”) to the effect that close approximation of methods allows for transfer of justification from one method to the other.

The argument would then go as followsFootnote ⁵ :

1. 1. wMI is justified (from stage 1)

2. 2. EP*

3. 3. Application of OI in s* yields approximately the same result as wMI (given EP*)

4. 4. If the application of a method ${M_1}$ to a situation s is justified and the application of a method ${M_2}$ to situation s yields approximately the same result as ${M_1}$ , then the application of ${M_2}\;$ to situation s is justified too. (Approx)

Conclusion: Application of OI in s* is justified too.

Sterkenburg allows this conclusion, given the assumption of EP*. However, he points out that it is not equivalent to Schurz’s stated conclusion that “it is meta-inductively justified to favor object-inductivistic strategies in the future” (Schurz Reference Schurz2019, 85). Rather, Sterkenburg says, the argument gives “justification for sticking to strategy OI for now”—in other words, in situation s*. That is, it provides only a justification for following the prediction of the strategy OI “at this point in time (Sterkenburg Reference Sterkenburg2020, 539).” However, he takes it that “this would still be an important result,” since the problem of induction is important insofar as it it concerns whether we should follow scientific induction or its alternatives at the current moment (Sterkenburg Reference Sterkenburg2020, 539).

However, in my view, the argument formulated above fails, and hence even the more limited conclusion cannot be drawn. The problem is that the premise Approx is not true. Suppose ${M_1}$ is the method of painstakingly applying careful scientific reasoning, and ${M_2}$ is the method of making a prediction by flipping a coin. It is possible that the application of ${M_2}$ to a situation yields approximately the same prediction as ${M_1}$ , but we would not say in this case that the application of the coin-flipping method to the situation is itself justified. It is generally thought that one of the functions of justification is as a condition that enforces a certain kind of non-accidentality. It is an anti-luck requirement. Thus, if it simply happens that an otherwise crazy method happens to approximate a good method at a particular point, that in itself provides no reason to regard the crazy method as justified. The approximation is itself an accident, so you just got lucky by following the crazy method.

Is there a way to fix the argument to exclude these kinds of accidental approximation cases? It seems as though justification could transfer from ${M_1}$ to ${M_2}$ , if we have a good reason for expecting ${M_2}$ to approximate ${M_1}$ . Perhaps we know that the working of ${M_2}$ is relevantly similar to ${M_1}$ , or even for example that they share some common way of working. Perhaps, for example, ${M_2}$ is a more coarse-grained way of making predictions, which omits some of the details that ${M_1}$ takes into account, but still achieves decent results. In such a case, it seems that we might allow that the fact that ${M_1}$ closely approximates ${M_2}$ is a reason to allow the justification for ${M_1}$ to transfer to ${M_2}$ . However, the transfer of justification in this way would be one-way. The detailed working of the more fine-grained method would not be justified in any way by its coincidence with the coarse-grained method. The transfer of justification would only occur when you know that the reasons for the success of the first method are likely to be the same for the second method.

This is a condition which does not hold in the case where ${M_1}$ is wMI and ${M_2}$ is OI. The reasons for the success of wMI are its ability to make use of the successes of object-level methods such as OI. But the converse does not hold. The reasons for the success of OI, if it is successful, are something to do with the way that the world is structured, perhaps its uniformity in certain respects. Thus, although we might want to say that wMI could inherit justification from justification for OI, we would not want to put this the other way round. OI is not justified on the basis of an approximation to wMI, because that approximation does not come about because of the way that OI works in relation to wMI.

Furthermore, attempting to justify Approx as an empirical premise by appeal to scientific practice will also not work. Such a justification would be based on arguing that success in scientific practice is based in part on Approx, and then inductively projecting that success to the future. However, this presupposes that the success of approximation practices in the past will extend to the future, and since this is part of “the very point in question,” this reintroduces the Humean circularity concern.

Thus, it seems that a plausible formulation of an argument for transferring justification from wMI to OI is still missing. We do not appear to have an argument which allows any conclusion about justification of the method OI either in general, or its application to the specific case.