1. Introduction

We seek an axiomatic understanding of specific problems of induction. Informally, induction is taken to mean the process of using empirical evidence to validate general claims, and, for our purposes, it is critical to differentiate between two types of epistemic skepticism about induction.

One may doubt it is possible to know whether nature abides by any law.Footnote ¹ Any empirical regularity may be a temporary fluke. Hence, patterns can suggest, but not prove, the existence of universal laws. So, one may ascribe nonvanishing odds to the idea that nature does not follow any law, no matter how numerous and consistent the data may grow to be. We refer to this disposition as Humean skepticism, with the caveat that we do not claim to provide a complete representation of Hume’s (and other authors’) actual statements.

In addition, even if it is taken for granted that nature abides by a law, one may be skeptical that such a law can be inferred with arbitrarily high precision, even when the data grow without bounds. Let us say that in each period either 0 or 1 must occur and that 1 has been observed every period, over a long time, say t periods. The data are consistent with the law “nature produces only 1” and with the law “nature produces 1 until period t and 0 afterward,” among (infinitely) many other laws. So, one may maintain a nonvanishing doubt that empirical evidence can validate a specific law, even under the assumption that the data follow one. We refer to this form of skepticism as Goodman’s skepticism, with the same caveat as above.

We consider a probabilistic framework in which an agent, named Bob, is endowed with a coherent view of the world (i.e., a finitely additive probability measure) over paths (i.e., infinite binary sequences). As the data unfold, Bob updates his view of the world through Bayes’s rule.Footnote ² No restrictions are placed on which paths may be produced. So, no relationship between past and future is, a priori, required (apart from the idea that either 0 or 1 occurs each period). Bob is not dogmatic about induction either. He believes that the data may or may not follow eternal laws.

Under the lenses of this formal framework, we formalize Hume’s and Goodman’s skepticisms by introducing two novel axioms for subjective probabilities. These axioms refer to Bob’s belief as the data unfold and become arbitrarily numerous. Bob’s view of the world is inductive in the sense of Hume, as we define it, if, under data compatible with laws, Bob expects to become almost convinced that nature indeed follows laws. This axiom rules out Humean skepticism about induction. Bob’s view of the world is inductive in the sense of Goodman if he expects to successfully identify nature’s law up to a vanishing degree of error, conditional on nature abiding by one. This axiom rules out Goodman’s skepticism about induction.

A natural starting line of inquiry is the extent of the connection between the two problems of induction. We start by asking whether Goodman’s skepticism implies Hume’s skepticism and the converse implication. Neither is true. Some coherent views exhibit Goodman’s skepticism but not Hume’s skepticism, and, conversely, some coherent views exhibit Hume’s skepticism but not Goodman’s skepticism. Thus, these two types of skepticism are not logically nested.

Of particular interest are the coherent views that express skepticism in the sense of Goodman but not in the sense of Hume. If, say, confronted with the question of whether the data are generated by a Turing machine, such views of the world express conviction that with enough data it is possible to make this determination with near certainty. In spite of this remarkable confidence on the capacity of Bayes’s rule to address this apparently insurmountable inference problem, the same view of the world, if confronted with the (arguably simpler) question of which Turing machine generates the data, assuming that one does, remains skeptical that this determination can be made with arbitrarily high precision.

The celebrated theorems of Lévy (Reference Lévy1937), Doob (Reference Doob1949), and Blackwell and Dubins (Reference Blackwell and Dubins1962) make clear that under σ-additivity a Bayesian must believe that his opinion about a given hypothesis will converge to the truth. In particular, σ-additivity excludes both Hume’s and Goodman’s skepticism, and therefore it implies a form of “Bayesian orgulity” (Belot Reference Belot2013). Different results were obtained by Juhl and Kelly (Reference Juhl and Kelly1994), Kelly (Reference Kelly1996), and Elga (Reference Elga2016), among others, who have shown that there exist non-σ-additive coherent views of the world that allow for Humean skepticism. Hence, in the absence of σ-additivity, epistemic skepticism is allowed.

Our results reveal a complex relationship between skepticism and subjective probability. There are non-σ-additive coherent views that rule out Humean skepticism and others yet that rule out Goodman’s skepticism. Thus, the spectrum of coherent views is rich enough to allow, at the same time, both orgulity and skepticism about induction. In particular, in the absence of σ-additivity, orgulity and skepticism are allowed. Orgulity is not an exclusive property of σ-additivity and may hold with or without it. This is a difficulty for a clear-cut theory of induction that seeks the root causes of orgulity and skepticism about induction.

We show that while a lack of σ-additivity does not assure Hume’s skepticism and Goodman’s skepticism, it always assures skepticism in at least one of these two ways. This is demonstrated by the structure theorem for coherent views of the world. It shows that a coherent view is inductive in the sense of Hume and in the sense of Goodman if and only if it is σ-additive. Thus, σ-additivity is the definitive condition that assumes away both Hume’s and Goodman’s skepticism about induction. It is not necessary to rule out either type of skepticism, but it is required to rule out both types simultaneously.

The interpretation of the structure theorem requires considerable care. The equivalence between induction and σ-additivity may suggest that the problems of how to conceptually justify either induction or σ-additivity are, in fact, one and the same problem and that σ-additivity is the root and only cause of the conviction in the ultimate success of induction. This reading of the structure theorem may prove incomplete. Consider an alternative approach in which the focus is not on using data to ultimately (i.e., in the limit as data grow) uncover eternal laws of nature but on making predictions within a practical (i.e., bounded) future. Consider the case in which a long sequence of 1s has been observed. One may wonder whether “nature produces only 1s.” One may also wonder whether “nature will produce only 1s for the next 1,000 periods.” Our last result concerns the latter case, in which Bob remains agnostic about the validity of universal claims but asks whether regularities in the past can be used to make sharp predictions about a bounded future. This result shows that any coherent view of the world, no matter how it is formed, must be confident that multiple repetitions of Bayes’s rule transform pattern data into a near infallible guide to a bounded future.

Moreover, after enough data there must be high confidence on limited, but correct, inductive inferences. This holds even if, a priori, no assumption is made on the relationship between past and future in the sense that the data may unfold according to any path, including those without patterns. It also holds even if Bob is a skeptic in regard to the use of data to ultimately validate specific or general laws. Eventually there must be high confidence that the past is a limited, but successful, guide to the future. This conclusion follows from conditional probability alone and holds for any coherent view of the world. Thus, some confidence in inductive inference follows from coherence.

The article speaks to the recent debate on “Bayesian orgulity,” originated with Belot (Reference Belot2013, Reference Belot2017). Central to Belot’s thesis is the argument that the convergence results of Levy, Doob, and Blackwell and Dubins are proof that Bayesianism implies epistemic arrogance. The debate has spurred different views. Huttegger (Reference Huttegger2015) argued that the issue of convergence to the truth should be put in the context of a long but finite horizon. Weatherson (Reference Weatherson2015) revisited Belot’s argument from the perspective of Bayesian imprecise probability. The work closest to this article is Elga (Reference Elga2016), who showed the existence of non-σ-additive subjective probabilities expressing epistemic humility.

This article is also connected to the work of Kelly (Reference Kelly1996), who formalized the connection between inductive inference and finitely additive probabilities, to the work of Gilboa and Samuelson (Reference Gilboa and Samuelson2012), who analyzed how subjectivity can enhance inductive inference, and to Al-Najjar, Pomatto, and Sandroni (Reference Al-Najjar, Pomatto and Sandroni2014), who study how different dispositions toward induction can affect incentive problems.

2. Basic Concepts and Results

2.2. Induction and the Separation Theorem

We now formalize specific forms of induction.

Definition 1. A coherent view of the world P is inductive in the sense of Hume if for every path $\omega \in A$,

(1)$\begin{array}{}& P\left(A|{\omega }^t\right)\to 1\text{ as }t\to \infty .\end{array}$

From sufficient data with a pattern, Bob ultimately concludes, with probability approaching certainty, that nature must follow some law. A view of the world that violates (1) is such that the probability of the set A of law-like paths remains bounded away from 1, regardless of the number of realizations. Any such worldview captures what we refer to as Humean skepticism: Bob maintains a nonvanishing doubt that perhaps nature does not work through eternal laws, no matter how consistent and numerous the data he observes.

At each point in time, the observed finite history ω^t is consistent with a path following a pattern as well as with a path that does not follow a pattern. So, by Bayes’s rule, even a view of the world that is inductive in the sense of Hume will always attach nonzero probability to the event that nature does not abide by laws (under assumption 2). What distinguishes between skepticism and inductivity in the sense of Hume is whether Bob’s doubt on the regularity of nature vanishes as the number of observations that exhibit a pattern goes to infinity.

Definition 2. A coherent view of the world P is inductive in the sense of Goodman if for every path $\omega \in A$,

(2)$\begin{array}{}& P\left(\left\{\omega \right\}|A\cap {\omega }^t\right)\to 1\text{ as }t\to \infty .\end{array}$

If it is granted that nature abides by some law and sufficient data with a pattern are observed, Bob infers nature’s true law with increasing precision and ultimately concludes it is eternal. A view of the world that violates (2) captures what we refer to as Goodman’s skepticism: even assuming that an underlying law exists and that extensive evidence is available, Bob remains skeptical he will ever be able to perfectly single out the data-generating law with arbitrarily high confidence.

As in the case of induction in the sense of Hume, at no point in time is Bob’s inference solved perfectly. He will always attach nonzero odds to multiple paths. However, a view of the world that is inductive in the sense of Goodman is confident Bob must ultimately assign probability close to 1 to the law generating the data.

The distinction we make here need not be seen as the formal counterpart of the classic and the new riddle of induction (see Goodman [Reference Goodman1955] and Stalker [Reference Stalker1994], for a discussion), and the above terminology is used mostly as a mnemonic device. Fundamentally, we ask two direct inference questions: Within the present probabilistic framework can one tell, from sufficient data and with arbitrary precision, (1) whether nature must abide by a law and (2) if so, which law?

We now examine the logical connection between these two questions.

The Separation Theorem. There exist coherent views of the world that are inductive in the sense of Hume but not in the sense of Goodman and views that are inductive in the sense of Goodman but not in the sense of Hume.

The Separation Theorem shows that Hume’s skepticism and Goodman’s skepticism are not logically nested. One does not imply the other. In the appendix we provide simple examples of views satisfying only one of the properties. Given the separation theorem, it is meaningful to consider those coherent views of the world that express both types of faith in induction.

Definition 3. A coherent view of the world P is inductive if it is inductive in the sense of Hume and is inductive in the sense of Goodman.

Under an inductive view of the world, skepticism about induction vanishes. Bob interprets evidence consistent with a pattern as a sign of the existence of an underlying law of nature and expects further evidence to allow him to single out the correct law with virtually exact precision. So, inductive views express great confidence in the power of empirical evidence to predict the future. This can be expressed as follows:

Definition 4. A coherent view of the world P is confident that enough pattern data transform the past into a near infallible guide to the future if for every path $\omega \in A$,

(3)$\begin{array}{}& P\left(\left\{\omega \right\}|{\omega }^t\right)\to 1\text{ as }t\to \infty .\end{array}$

So, conditional on sufficient long pattern data ω^t, the future is forecasted with an arbitrarily high degree of certainty.

Remark 2. A coherent view of the world P is inductive if and only if it is confident that enough pattern data transform the past into a near infallible guide to the future.

So, partial induction is the necessary and sufficient condition for confidence that sufficient pattern data are a near perfect guide to the future. Remark 2 delivers an initial characterization of induction that will prove useful.

3. Orgulity and σ-Additive Coherent Views

This section examines inductive properties of σ-additive coherent views. These results are known and adapted to our framework. We refer to known results as “propositions” and to novel ones as “theorems.”

The proof of this result can be found in Kelly (1996). Under σ-additivity, after multiple observations consistent with a pattern, Bob infers nature’s underlying law with arbitrary accuracy and concludes with almost certainty that nature cannot follow a different law. However, σ-additivity entails even stronger forms of faith in induction.

Definition 5. A coherent view of the world P is completely inductive in the sense of Hume if

$P\left(A|{\omega }^t\right)\to 1\text{ as }t\to \infty ,\text{ for every path }\omega \in A,$

and

(4)$\begin{array}{}& P\left(A^c|{\omega }^t\right)\to 1\text{ as }t\to \infty ,\text{ for }P\text{-almost every path }\omega \text{ in }{A}^c\mathrm{.}\end{array}$

A coherent view of the world that is completely inductive in the sense of Hume expresses full confidence that, with sufficient data, laws and nonlaws can be distinguished empirically and with near certainty. So, complete induction in the sense of Hume is an expression of confidence that a remarkably difficult inference problem can be resolved with arbitrarily high precision.

This result has led Belot (Reference Belot2013) to speak of “Bayesian orgulity.” The basic inference problem is difficult. Yet, σ-additive coherent views are confident that finite, but long enough, data suffice to determine with arbitrarily high precision whether nature is governed by a law.

In addition,

Combining propositions 1 and 2 yields:

Under σ-additivity, Bob must express the following viewpoint on induction:

So, under σ-additivity, Bob believes that Bayes’s rule resolves these essential problems of induction. With sufficient data, nature’s law is eventually uncovered. A false inference of laws, when nature follows none, is unlikely. The intuition behind these results is as follows: First, let us assume, for simplicity, that nature abides either by the law “always 1” or by a law “1 until period t and 0 thereafter,” for some $t>0$. No sequence of 1s, either large or small, suffices to infer nature’s law conclusively, but there is a crucial difference between a short and a long sequence. Ex ante, the odds of the law “always 1” are fixed and strictly positive. The odds of the law “1 until some period $t\ge m$ and 0 thereafter” are arbitrarily small if m is sufficiently high. It is here that the assumption of σ-additivity is used. Under σ-additivity, such tail events must be unlikely. It now follows, by Bayes’s rule, that conditional on a sufficiently long sequence of 1s, the likelihood of the law “always 1” eventually dominates the likelihood of any competing standing theories. Thus, under σ-additivity, Bob cannot express Goodman’s skepticism.

The intuition regarding Hume’s skepticism is related but not identical. Assume, for simplicity, that nature either abides by the law “always 1” or does not abide by any law. Once again, no sequence of 1s, either large or small, suffices for conclusive inference. For any sequence of 1s, no matter how long, there are still many nonlaws that are consistent with it. However, the set of nonlaws that are consistent with consecutive 1s until period t shrinks monotonically to the empty set as t goes to infinity. This follows because no nonlaw is consistent with an infinite sequence of 1s. So, under σ-additivity, the ex ante odds of the set of standing nonlaws (i.e., those consistent with data of consecutive 1s until period t) goes to zero as t goes to infinity. Hence, by Bayes’s rule, conditional on a sufficiently long sequence of 1s, the relative likelihood of the law “always 1” is much higher than the competing and still standing nonlaws. Thus, under σ-additivity, Bob cannot express Hume’s skepticism. Finally, the intuition regarding property (4) is also similar. The set of laws consistent with nonpattern data of length t shrinks monotonically to the empty set as t goes to infinity (because no law is consistent with an infinite sequence of nonpattern data). Thus, under σ-additivity, it is unlikely that laws are consistent with long nonpattern data. Hence, property (4) holds and so does complete induction in the sense of Hume.

4. Orgulity and General Coherent Views

As shown, σ-additive coherent views rule out skepticism about induction. We now consider Bob’s conclusions about the ultimate fate of multiple repetitions of Bayes’s rule for general, not necessarily σ-additive, coherent views of the world. We start with an important result, a corollary of Elga (Reference Elga2016) (related results can also be found in Juhl and Kelly [Reference Juhl and Kelly1994] and Kelly [Reference Kelly1996]):Footnote ⁵

Proposition 3. Let $\epsilon >0$. There exists a coherent view of the world P such that

$P\left(A|{\omega }^t\right)\le \epsilon \text{ for every }t\text{ and every }\omega \in A.$

The view P displays a complete failure of induction in the sense of Hume. Under P, no evidence can overturn Bob’s initial pessimistic belief on the existence of laws. Hence, σ-additivity suffices to rule out Hume’s skepticism about induction, and this condition cannot be completely disposed of. Elga (Reference Elga2016) shows that not all coherent views are inductive in the sense of Hume. But, the separation theorem shows that some non-σ-additive coherent views are inductive in the sense of Hume. Moreover, there are also coherent views that are not σ-additive but nevertheless are inductive in the sense of Goodman. Lack of σ-additivity does not assure skepticism in the sense of Hume and does not assure skepticism in the sense of Goodman either. Other strong forms of induction can also be obtained without σ-additivity.

Insomuch as confidence about Humean induction must be granted under σ-additivity, the same confidence must also be granted without σ-additivity, for some coherent views of the world. An example of such a view can be found in the appendix.

Consider, for instance, the case in which A is the set of computable paths. The Complete Humean Induction Theorem shows that some, but not all, coherent views of the world express the belief that even a fundamental problem such as whether nature can be reduced to a Turing machine can be solved (up to a vanishing error) empirically, even in the absence of σ-additivity. In this sense, Bayesian orgulity is not restricted to σ-additivity. It extends to other coherent views of the world as well.

5. The Axiomatization of Induction

The Separation and the Complete Humean Induction theorems present a difficulty for the development of a crisp theory of inductive inference. The difficulty is that confidence on solving induction problems is a product of not only well-understood conditions, such as σ-additivity, but also properties coherent views might have, which are less understood and intuitively less clear. The Complete Humean Induction Theorem is particularly challenging because it shows that confidence on empirical solutions to strong forms of inference problems can be obtained under conditions other than σ-additivity. However, let $\overline{\Sigma }$ be the smallest algebra that contains all finite histories and all singletons {ω} for $\omega \in A$. This is the smallest algebra that allows the expression of property (3), which is equivalent to a view P being inductive. The key point of this algebra is as follows: it is possible to obtain property (1) and also property (2) without σ-additivity. It is even possible to combine properties (1) and (4) without σ-additivity (and, hence, produce complete Humean induction). However, on $\overline{\Sigma }$, it is not possible for property (3) without σ-additivity. This makes σ-additivity not only sufficient but necessary for partial induction (and, hence, for complete induction as well). Thus,

The Structure Theorem is a full characterization result that delivers an axiomatic understanding of induction. The key result is the demonstration that while lack of σ-additivity does not assure skepticism in the sense of Goodman and it does not ensure skepticism in the sense of Hume either, it always assures skepticism in at least one of these two senses. So, on $\overline{\Sigma }$, any result that holds without σ-additivity holds under some skepticism over induction. Conversely, results that require σ-additivity, also require induction.

The collection $\overline{\Sigma }$ is smaller than the σ-algebras commonly used in probability theory. While σ-algebras are mathematically convenient under σ-additivity, they do not play a particular role under finite additivity. What makes $\overline{\Sigma }$ appealing in the context of induction is that $\overline{\Sigma }$ is the simplest (i.e., the smallest) algebra that allows distinguishing between inductive and noninductive views of the world. Small algebras such as $\overline{\Sigma }$ have an additional advantage. Because $\overline{\Sigma }$ is countable, finitely additive measures can be defined on P using only elementary mathematics and without invoking the (uncountable) Axiom of Choice.

6. Pragmatism, Induction, and de Finetti

So far, we have focused on induction in the sense of the empirical validation of eternal laws of nature. There are, however, other perspectives on induction, such as the one in which Bob is concerned with making accurate predictions about the practical future, rather than uncovering universal laws of nature or even questioning their existence.Footnote ⁶

If a law or theory makes predictions that are accurate within some finite horizon, then the theory predicts as if it were correct. Thus, the argument goes, data need not uncover the actual data-generating process. Nor do data need to reveal whether a law exists. It only needs to allow for accurate predictions for the practical future. To fix ideas, we refer to this perspective as pragmatism, with no claim that our narrow use of this terminology comprehends most associations with this word.

We now revisit the different problems on induction but from a more pragmatic perspective. In doing so we take a shortcut in the conceptual development. We define pragmatic inductive views as requiring that enough pattern data lead to a near infallible guide to a bounded future, instead of first making a distinction between induction in the sense of Hume and Goodman and then obtaining accurate predictions as a result of both conditions as we did in remark 2.

Definition 7. A coherent view of the world P is pragmatically inductive if, for every path $\omega \in A$ and every natural number k,

(5)$\begin{array}{}& P\left({\omega }^{t+k}|{\omega }^t\right)\to 1\text{ as }t\to \infty .\end{array}$

So, with enough pattern data, Bob is convinced that the next outcomes can be predicted with near certainty. This follows, in Bob’s belief, even if nature abides by no laws or if it abides by a law that cannot be inferred from the data. The only claim is that after enough pattern data nature behaves as if it abides by a (data-inferred) law for a bounded but arbitrarily long future.

We now turn to the concept of complete induction in the sense of Hume, from the pragmatic perspective. Let $U$ be the set of unions of finite histories. So, a set $U\in U$ is a union of finite histories such as ω^t, where $\omega \in \Omega$ and t is a natural number. Any arbitrarily complex set $E\subseteq \Omega$ can be approximated in terms of finite histories by choosing a set $U\in U$ such that $E\subseteq U$.Footnote ⁷

Definition 8. A coherent view of the world P is pragmatically completely inductive in the sense of Hume if for any set $U\in U$ such that $A\subseteq U$,

$P\left(U|{\omega }^t\right)\to 1\text{ as }t\to \infty \text{ on every }\omega \text{ in }A,$

and for any set $V\in U$ such that $A^c\subseteq V$,

$P\left(V|{\omega }^t\right)\to 1\text{ as }t\to \infty \text{ on }P\text{-almost every }\omega \text{ in }{A}^c.$

Given the requirement for any set in $U$ that contains A or A^c, there is, in particular, the same requirement for sets arbitrarily close to A or A^c. Sufficient pattern data lead to near certainty of finite histories associated with laws, and sufficient nonpattern data lead to near certainty of finite histories associated with nonlaws. Combining the two definition yields

So, in particular, enough pattern data lead to a near infallible guide to a bounded future, and enough nonpattern data lead to near certainty of future finite histories associated with nonlaws.

Unlike the previous results, the Pragmatic Induction Theorem holds for all coherent views of the world. No matter how coherent beliefs are formed, they must express confidence that mechanical repetitions of Bayes’s rule transform sufficiently numerous pattern data into a near infallible guide to a bounded future. In the case of nonpattern data then, provided that the data are sufficiently long, there must be confidence, approaching certainty, of an observable future associated with nonlaws. This holds without any other assumption such as σ-additivity. Therefore, any coherent view of the world contains a seed of orgulity.

The concerns one may have about the orgulity of Bayesians may not go away, at least completely, by abandoning σ-additivity. The Pragmatic Induction Theorem relies on multiple repetitions of Bayes’s rule alone; hence, it holds with or without σ-additivity. However, the extent to which this remaining form of orgulity is a difficulty for the Bayesian paradigm is a question beyond the scope of this article. According to one viewpoint, the cases of successful inference that follow from the repetition of Bayes’s rule can be seen as a desideratum that provides support to the Bayesian approach. According to a different viewpoint, the Pragmatic Induction Theorem can be seen as an expression of excessive confidence of the same paradigm. This article does not resolve this fundamental tension, but it helps to make precise the conditions under which orgulity holds.

While the Pragmatic Induction Theorem relies only on coherence and Bayes’s rule, it is embedded in a standpoint that can be traced back to de Finetti. The key conceptual point advanced by de Finetti is that the Bayesian perspective on inference effectively solves the problem of induction. As de Finetti (Reference de Finetti1970, 201) stated:Footnote ⁸ “In the philosophical arena, the problem of induction, its meaning, use and justification, has given rise to endless controversy, which, in the absence of an appropriate probabilistic framework, has inevitably been fruitless, leaving the major issues unresolved. It seems to me that the question was correctly formulated by Hume. … In our formulation, the problem of induction is, in fact, no longer a problem: we have, in effect, solved it without mentioning it explicitly. Everything reduces to the notion of conditional probability.”

In this sense, the Pragmatic Induction Theorem can be seen as formalization of de Finetti’s viewpoint on induction. However, to the best our knowledge, de Finetti never made a distinction between the two basic inference problems (i.e., does nature abides by laws, and if so which one?) and never examined these problems in a formal model. While pragmatism is the additional element necessary for the formalization of this viewpoint, there is a yet more basic contribution. De Finetti mostly wrote about induction in the context, as in de Finetti (Reference de Finetti1969), of exchangeable beliefs (i.e., beliefs such that the order in which different outcomes occur over time is irrelevant). Exchangeability not only rules out elementary laws such as “1 until period t and 0 afterward,” it is also a critical assumption on the data and, hence, an assumption on how past and future must relate to each other. In contrast, in the Pragmatic Induction Theorem, confidence on limited, but successful, inductive inference about the future holds without assumptions on how the past and the future must relate to each other. The conclusions about the future depend on the data, but there is no restriction on the data-generating process itself.

7. Extensions

This article dealt with some inductive inference problems but left others unexamined. Perhaps the most basic limitation in this article is that the data-generating processes are deterministic. A natural extension could go as follows: the Blackwell-Dubins theorem extends proposition 1 to stochastic data-generating processes. Let us say that there are countably many (possibly stochastic) data-generating processes P ₁, P ₂, P ₃, …, and Bob’s belief (a prior over {P ₁, P ₂, P ₃, …}) assigns, ex ante, strictly positive probability to each of them. If all probabilities are σ-additive, then Bob’s predictions eventually will become indistinguishable from the data-generating process, no matter which one.

In spite of the power of the Blackwell-Dubins theorem, new difficulties arise in the case of stochastic data-generating processes. For example, if two processes are identical in all but the first period, then it may be impossible to empirically determine which process runs the data. This determination is not relevant for predicting the future after period 1 (see Lehrer and Smorodinsky [Reference Lehrer and Smorodinsky1996] and Acemoglu, Cherzonukov, and Yildiz [Reference Acemoglu, Cherzonukov and Yildiz2016] on this problem). Other difficulties may prove currently intractable. The Blackwell-Dubins theorem relies heavily on σ-additivity. For general coherent views, there are some conceptual advances, and some analytical methods for Bayesian learning were developed in Pomatto, Al-Najjar, and Sandroni (Reference Pomatto, Al-Najjar and Sandroni2014). With some effort, these techniques can be applied to prove a version of the Pragmatic Induction Theorem for stochastic data-generating processes. The Complete Humean Induction and the Separation theorems are existence results and so still hold when the set of data-generating processes is expanded. The main hurdle is the Structure Theorem. For a counterpart of that result, one must find an algebra on which induction is equivalent to σ-additivity when the data-generating processes can be stochastic. This is a (very) difficult problem.