No rules of induction: two responses to Hume's problem

1 Introduction

Every person constantly justifies predictions based on observations that they have experienced. Lived experience allows humans the potent and surprisingly versatile ability to infer which unobserved incidences may occur based on extrapolating regularities from observed incidences and formulating general laws about nature. From a finite number of previously observed instances, we can infer that certain items are nourishing (food), that the sun will rise again, and that antibiotic medicine will cure a bacterial infection. This description of inferences in the mundane is a form of inductive inference, whereby we use observed instances to predict the behaviour of unobserved instances. The more instances showing a regularity in the world, the higher our confidence in knowing the behaviour of the unobserved instances. The process of inductive inference has implications for our last few centuries’ scientific enterprises that have been so undeniably successful. The scientific method arguably must use some form of inductive inference as we use observations to predict future and unobserved events. Inductive inference is essential to mentally order the world in daily life and systematically order and understand the world through the acquisition of scientific knowledge. Inductive inference being rationally justified would give us a solid foundation to gain valid knowledge about the universe we inhabit.
The 18th-century Scottish philosopher David Hume was the first to examine the rationality of the ubiquitous inductive inference in his book A Treatise of Human Nature. Hereafter referred to as the Treatise. Whilst first published in 1739-1740; we shall use the 1967 republished version. In the Treatise, Hume gave the first account of what would become to be known as the “problem of induction”. A second more concise account is given in his later work published in 1748 titled An Enquiry Concerning Human Understanding, henceforth referred to as the Enquiry. We shall be using the 1998 republication of the Enquiry. Both texts will be used and referred to throughout this thesis. The exact formulation of his argument will be explicitly made in chapter 2 of this thesis, but simply put, Hume presents a two-pronged argument that concludes that there are no rational grounds for the Uniformity Principle (UP).
The UP is Hume’s premise that nature is homogeneous, both spatially and temporally, i.e., the same type of events will occur in the same way irrespective of their location in space and time, “the future will be conformable to the past” (Hume, 1998/1748, E 4.2.30). To illustrate, instances of consumed bread have been nourishing in the past, and we believe that bread will continue to nourish us in the future. Hume argues that it is logically impossible to argue for the UP deductively. A deductive argument must show that a violation of the UP would lead to a contradiction. Hume states that was the negation of the UP deductively false; the mind could not distinctly conceive the contradiction (Hume, 1998/1748, E 4.1.21). We can, however, distinctly conceive of the sun failing to rise; the violation of the UP remains conceivable. Hence an argument of this type cannot be made. Additionally, arguing for the UP inductively also fails as it will lead to vicious circular reasoning. This is because attempting to justify the UP by its past truth and validating its future truth is using the UP as a premise for which it is the conclusion. Hume himself argues that the UP cannot be justified based on deductive or inductive argument and that we are not rationally justified in making inductive inferences. I will elaborate on Hume’s problem in chapter 2.
Therefore, the beliefs we hold based on inductive inference are not based on solid rational grounds as all inductive inferences use the UP, which either deductive or inductive arguments cannot justify. Hume shows that both arguments fail to validly support inductive inferences (Hume, 1998/1748, E 4.2.32). Due to the far-reaching implications of Hume’s problem for both mundane interaction with the world and the reliability of scientific knowledge, this argument has been one of the most famous in philosophy for nearly three centuries. There have been many responses to Hume attempting to save a rational basis for induction, with varying degrees of success and no lasting consensus. There have also been numerous philosophers, such as Hans Reichenbach and Karl Popper, that the strength has so convinced of Hume’s argument that they have attempted different methods to live with Humean scepticism. I hope not to live with this scepticism and will examine two relatively recent and similar theories by Samir Okasha and John D. Norton responding to Hume’s problem.

The solutions of Okasha and Norton, whilst different, share the similarity that they attack the Humean description of inductive inferences. They similarly attack those philosophers such as Wesley Salmon and Peter Lipton (Okasha 2001, 310). They believe that although Hume’s description of inductive inference is incorrect, there is still a strong prima facie case for inductive scepticism. Okasha and Norton do not believe in general rules of induction, for instance, simple enumerative induction. We shall term these general rule theories of induction universal inductive schemas throughout this paper. Okasha and Norton believe each inductive case is local and unique; any attempt to formulate a general theory of the process of inductive scepticism is incorrect and will suffer from Humean scepticism. Due to the lack of belief in general induction rules, Okasha and Norton propose solutions to Hume’s problem that have been grouped and termed “no-rules” solutions (Henderson 2020, sec. 4.2).
Okasha and Norton are not the first to criticise Hume’s argument regarding the existence and function of the UP. Goodman proposes his famous “new riddle of induction” first published in 1955 and for which we shall use its fourth edition from 1983 (Goodman 1983, 59–83). In a later chapter, we will return to the “new riddle of induction” in some detail. Goodman’s argument demonstrates that not all inductive inferences are legitimate. Humans are adept at generally not inferring nonsensical regularities, such as all the chickens a person has come across are hens. Therefore she believes that all chickens are female. Hume’s argument is made for inferences that are legitimate to draw, but as Goodman points out, he does not address the conditions for inferring a regularity legitimately. The upshot of this argument is the implication that there is perhaps no UP, as the future is only conformable to the past in some respects and not others. It is this implication that there is something wrong with the UP that decades later provides some inspiration and support to Okasha and Norton.
This thesis aims to evaluate the success of two no-rules solutions to Hume’s problem of induction, to find to what degree these solutions are the same, and to see what changes need to be made to make them one solution. The main chapters of this thesis begin with a description of Hume’s problem of induction and the beginnings of a no-rules approach as instigated by Goodman. After we have formally examined Hume’s problem in chapter 2, we will discuss the solution argued for by Samir Okasha and then the solution of John D. Norton. In each chapter on these two philosophers, we will reproduce their arguments and examine the success of these arguments against Hume’s problem. We will then investigate any criticisms they may have garnered from their peers. In the final chapter of this thesis, we will attempt to integrate Okasha and Norton’s theories to see where they differ and if they fundamentally do.

2 Hume’s problem

Before discussing either of our two responses to Hume’s scepticism, we must first examine Hume’s position, as we must know in some detail the problem that Okasha and Norton are attempting to solve. To do this, we will reconstruct Hume’s argument from the Treatise and the Enquiry and then examine which part of Hume’s argument it is that forms the no-rules critique. To do this, we will look at an earlier criticism of Hume’s problem from Nelson Goodman as it sows the seeds of the no-rules approach. Goodman’s “new riddle of induction” (Goodman 1983, 59–83) will serve as a convenient springboard to discuss these solutions in the proceeding chapters.

2.1 Reconstructing the argument

The argument that Hume originally sets out concerns inferences of the type called “simple enumerative induction” (Henderson 2020, sec. 2.0), this is Hume’s universal inductive schema. These are inductive inferences of the type ‘all observed instances of A have been B’; the next instance of A will be B. It is important to note that according to Hume, this inductive rule presupposes the aforementioned UP for it to remain an eternally true universal inductive schema. I remind the reader here that the UP is the Uniformity Principle and states that the future will resemble the past (Hume, 1998/1748, E 4.2.19). In other words, as the past, present and future are related and presumed fundamentally the same as one another, it is possible to make predictions based on the past and, in general, spot regularities within nature. It seems key to our reality that we exist in a universe friendly to inductive inference. Hume’s argument does presuppose the UP based on the effectiveness of inductive inference. Nevertheless, due to the success of our empirical science Hume sees no other option but to identify it as the underlying mechanism that allows universal inductive schemas to function, which he hopes he could justify.
From Hume’s chosen underlying assumption, the UP, which is the underlying assumption that validates his universal inductive schema, he then asserts that two types of argument can be made. These Hume terms as demonstrative and probable (Hume, 1998/1748, E 4.2.30). The 1^st horn of the argument, i.e., the demonstrative argument, we shall treat similarly to Salmon as a deductive argument (Salmon 2017, 8). The 2^nd horn of the argument, i.e., the probable argument, we will again treat it in the same way as Salmon, as an inductive argument (Salmon 2017, 16). We must note that there is some controversy and an ongoing debate as to what exactly Hume meant by the terms demonstrative and probable and whether he did take them to mean deductive and inductive arguments, respectively (Henderson 2020, sec. 2.0). For the sake of simplicity and brevity, we follow Salmon’s example.
Hume’s 1^st horn is extremely short in the Treatise, only a short paragraph, and essentially states that as we can conceive a case in nature where the next instance of A is not B, rules out a deductive argument (Hume, 1967/1739-1740, T 1.3.6.5). In other words, the negation of the UP is not a contradiction; therefore there cannot be a deductive argument for the UP (Henderson 2020, sec. 2.0). For a deductive argument to be valid, it must possess a contradiction if it is negated. The negation of the UP is not a contradiction; for example, it is humanely possible for us to imagine the sun not rising, even after many instances of it rising. The sun not rising is unlike the negation of a triangle possessing three vertices; this creates a contradiction. We cannot claim to be able to imagine a triangle with vertices not equal to three. Its conception is impossible.
In the 2^nd horn Hume argues that any inductive argument for the UP based on experience (observed instances of the UP being correct), must presuppose the UP. This is the argument that the justification of the UP is its exemplary record of success in empirical science. The inductive argument, however, is not valid as the argument’s conclusion, i.e., the UP, is being used to support the validity of the UP. This is a circular argument and we are not justified in making it (Hume, 1998/1748, E 4.2.32). Therefore, due to vicious circularity, there is not a valid inductive argument for the UP. Whilst Hume does believe the UP to be fundamentally correct; he cannot justify it here via valid inductive argument. Hume’s argument shows that his chosen inductive inference cannot be rationally made and is therefore not justified based on his premises. Hume argues that we do not make inductive inferences with reason but by association and habit (Hume, 1967/1739-1740, T 1.3.6.16).
Many authors believe that Hume’s argument, whilst made for a specific type of universal inductive schema, i.e., simple enumerative induction the argument still applies to all universal inductive schema. Therefore the argument applies to any inductive inference. Wesley Salmon (Salmon 1965, 268–69) and Peter Lipton (Lipton 2003, 11) in particular argue that no matter our inductive process, Humean scepticism holds. Our proposed solutions by Okasha and Norton attack this claim, as they both believe that the root of inductive scepticism comes from the universality and formality of universal inductive schemas. Both Okasha and Norton argue that we do not use such universal schemas in practice. Moving away from arguments against Hume of this type might secure us from Humean scepticism.
As we have seen, many philosophers believe that the actual inductive process humans follow is unimportant, as Hume’s argument will always hold. We shall see in the following subsection another methodology used by Goodman that highlights the importance of the inductive process by which inductive inference actually proceeds. Finally, we shall examine why Goodman’s argument shows there may not be a general law of inductive inference as is, in essence, is asserted by Salmon and Lipton and provide the foundation for no-rules solutions.

2.2 Seeds of a no-rules solution

Nelson Goodman, in his famous book Fact, fiction, and forecast, claims to dissolve the ‘old’ problem of induction. Goodman then proposes a reformulation in his “new riddle of induction”. This claim of dissolution and the method by which Goodman believes he achieves it act as a foundation for Okasha and Norton. Goodman exposes the tension between the universality of a universal inductive schema and its failure to make exclusively sensible, natural and valid inferences.
Goodman draws a parallel between the validity of deductive inference and the validity of inductive inference. He states that to justify a deductive conclusion, one only needs to see if it satisfies all our highly developed and valid rules of deductive inference, those rules for a deductive argument that have existed since the time of the ancient Greeks (Goodman 1983, 62–63). The comparison comes when Goodman argues that our deductive rules for inference yield acceptable conclusions, and rules that do not are discarded (Goodman 1983, 63). Goodman calls this process virtuously circular (Goodman 1983, 64) as reflective and uses accepted conclusions to produce the most justified deductive rules. Goodman argues that in fact the problem of justifying induction is not something over and above the problem of describing or defining valid inductive rules (Goodman 1983, 64–65).
This questioning of the inductive process gives spark to our later developing no-rules camp of Okasha and Norton. Our actual specific inductive practices are crucial to resolving Hume’s problem. This contrasts Salmon and Lipton, who believe that any method of inductive inference will suffer from Humean scepticism. To clearly show why Hume’s old problem can be dissolved, Goodman shows that the particular inductive method that Hume uses, simple enumerative induction is not a good description of a universal inductive schema. Hume’s inductive method says nothing about which inductive inferences are valid and which regularities we are legitimate in drawing from instances of observation. Goodman uses an example in which he invents a predicate “grue”, whereby something is grue if it is green before a time t, and blue after t. If we observe some emeralds before t then in Hume’s schema, we can equally infer that the emeralds are grue or green (Goodman 1983, 74). Goodman shows that the inductive rules Hume used virtually exclude nearly no inferences. To have genuinely confirmed inferences, we can only make predictions under lawlike hypotheses. Distinguishing between lawlike and accidental hypotheses such as green and grue (Goodman 1983, 77) is once again the problem of finding some valid inductive rules under which to make inferences, something that Goodman doubts can be done (Goodman 1983, 83).
Goodman believes in his “new riddle of induction” that the inability to make the distinction between lawlike and accidental hypotheses will extend to any inductive rules (Goodman 1983, 83). Goodman thus explicitly questions the existence of a general UP (Goodman 1983, 61) as it is at least as likely to produce accidental hypotheses as lawlike ones when drawing inferences, as shown in Goodman’s “grue” objection. Many philosophers accept that Hume’s description was insufficient but that the force of his argument is unaffected. Goodman has planted the seeds of doubt in the generality of the UP. The generality and universality of universal inductive schemas as the root of scepticism and an incorrect description of our inductive process will be built upon by Okasha and Norton as we shall see in the next two chapters.

3 Okasha’s solution

The first of our two theories proposed by Samir Okasha seeks to undermine Hume’s problem on its terms using the aforementioned no-rules approach. Okasha accuses Hume of objecting to inductive inference because it is not deductive and can therefore not achieve certainty in the same manner as deductive arguments. Okasha believes this is impossible and unnecessary (Okasha 2005a, 193) and aims to show us why. Furthermore, Okasha aims to show us how we conduct inductive inference is crucial and that we do not follow a universal inductive schema such as the UP as Hume believed. Additionally, Okasha will present that not following a universal inductive schema is rational, justified and fits the actual practice of empiricism. In this chapter, we will discuss Okasha’s argument against Humean scepticism, to what degree it presents a singular, coherent position in the no-rules camp, and we will look at the success of Okasha’s argument by looking at the criticisms levied at it since its publication.

3.1 Inductive practice matters

The crux of Okasha’s claim is that the description Hume gives of our inductive practices is incorrect and that a correct description of our inductive practice is required (Okasha 2001, 308). To support this, Okasha cites and uses Goodman’s “new riddle of induction” that is described in some detail in chapter 2.3.
Okasha and Goodman claim that Hume’s sceptical argument is not universally applicable to all theories of inductive inference and that the specifics of inductive practice matter. Okasha believes Hume’s argument to be valid but unsound as Hume has not found any inductive rule that human beings, in fact, use. As Goodman showed, the UP justified simple enumerative induction does not discriminate between law-like and accidental hypotheses. Therefore, Hume has not provided any universal inductive schema on which to assess the reasonableness of predictions from experience (Okasha 2001, 314). Okasha states that the UP can be thought of as an underlying empirical assumption about reality that must commit us to order the world in a certain way (Okasha 2001, 314). The underlying assumption is vital because Okasha believes that Hume’s argument needs to be of a specific form for Humean scepticism to apply to an inductive practice. The form is as follows: Hume identifies an order to reality, the UP, Hume claims that arguments from experience are only justifiable if there is a reason to make the empirical assumption¹, that justifies the universal inductive schema. However, only an argument from experience is available, and this is circular (Okasha 2001, 314). Many philosophers, such as Salmon and Lipton, believe that once a valid inductive schema that we universally use is identified, Humean scepticism will remain in place, irrefutable. Okasha does seem to agree with this position, but only when assuming we make a commitment in our inductive inferences to a particular universal inductive schema. If we do not use universal inductive schemas to infer inductively, we may be able to reject Humean scepticism.
From this exposé, Okasha shows that what is required to move from inductive fallibilism, the claim that inductive inferences possess some uncertainty, to scepticism is a universal inductive schema. A universal inductive schema shows we order reality in some way that Hume’s argument can attack. What if there are no rules of inductive inference? What if humans do not order the world in a way consistent with one or more universal inductive schemas? Then we would only be subject to boring old fallibilism; this is Okasha’s solution to Humean scepticism. In the next section, we shall see his argument for why there are no rules of inference.

3.2 The no-rules rejection of Humean scepticism

In this section, we shall see an elaboration of the argument for why the no-rules argument allows us to reject inductive scepticism. Then, why we do not use universal inductive schemas, and what does Okasha think our inductive practices could be like if we do not use such schemas. This is followed by Okasha’s justification for why he thinks our inductive practices are like this, and finally, some criticisms levied at Okasha.

3.2.1 No Humean scepticism

We recall the earlier formulation of Hume’s argument essentially being that our beliefs about unobserved instances are conditional on the UP, for which Hume justifiably required a reason for us to believe it true. Of course, this subjects us to a vicious circularity and leads to inductive scepticism if we do form beliefs based on some commitment to order in reality. For any such order, it is easy to imagine data that would rationally demand us to change our empirical assumption, i.e. the UP (Okasha 2001, 321). Okasha argues that empiricists should only take empirical assumptions and the universal inductive schemas they attempt to justify provisionally. They should not hold onto them beyond, what Hume would agree, was rational (Okasha 2001, 321). This clearly shows that universal inductive schemas are subject to inductive scepticism as they rely on one or more empirical assumptions, such as the UP, which by Hume’s argument, we can never justify. However, humans frequently change their beliefs in ways that defy definition in terms of universal inductive schemas. At worst, and indeed at best, inductive fallibilism rules the day. Next, we shall examine the evidence for why humans behave in a no-rules manner when making inductive inferences.

3.2.2 Evidence for no-rules

Okasha outlines what qualities a rule for inductive inference would need to possess if it does exist. He argues that it “would be a rule for forming new beliefs based on evidence, where the evidence does not entail the belief” (Okasha 2001, 315). Okasha states that philosophers have failed to identify any inductive rules that are universally valid in the same manner as our rules for deduction (Van Fraassen 1989, 279). However, there is a persistent belief that such rules must exist. Okasha, supported by Van Fraassen, argues that there is no rational basis for believing that such universal inductive schemas must exist if no one has discovered them (Okasha 2001, 316).
Okasha argues that the universal inductive schema model for inductive inference does not account for a very relevant, powerful, and widely used model for belief change, the Bayesian model. Bayesianism argues that the correct method for modifying our beliefs based on experience by updating our prior probability function is via Bayesian conditionalisation. This is where our prior belief/hypothesis probability distribution is updated based on some experience using Bayes’ rule (Joyce 2021, sec. 1.0). This form of updating belief is not a rule of inductive inference as conditionalisation does not give us the optimised belief based solely on the data but also based on our previous belief(s) (Okasha 2001, 316). Bayesianism makes no demands that the world is ordered in a particular way as a universal inductive schema does. For a universal inductive schema to function, i.e., be truth-conducive, a person must believe that the universal inductive schema of the inference is correct. Bayesianism is based upon what we believe and is therefore not strictly a rule of inductive inference. The Bayesian process can be encapsulated in the phrase “belief guides action” (Lin 2022, secs. 1,6).
Bayesianism can be justified by using a Dutch Book argument (DBA) which exemplifies degrees of belief in a betting scenario. A DBA will show how Bayesianism can be used without requiring anything like the UP to justify it. The DBA will show that it is irrational to violate probability calculus when holding beliefs. In this scenario, we imagine a simple betting situation in which a person holds the belief (perhaps provided by a shrewd bookmaker) that a particular horse (\(\mathit{A}\)) will win a race \(75\%\) of the time and that any other horse will win (\(\neg\mathit{A}\)) \(30\%\) of the time. The sum of these belief percentages is \(105\%\). Naturally, this is impossible. Our person is willing to bet \(\pounds75\) on horse \(\mathit{A}\) as there is a \(75\%\) chance of winning \(\pounds25\) and a \(25\%\) chance of losing \(\pounds75\), netting zero. The person is also willing to bet \(\pounds30\) on our \(30\%\) chance of any other horse (\(\neg\mathit{A}\)) since there is a \(30\%\) chance to win \(\pounds70\) and a \(70\%\) chance to lose \(\pounds30\), also netting zero. However, if we sum these bets, as shown in table 1 our poor gambler would always suffer a net loss. We see that following a Bayesian system in which we do not violate probabilism, our agent would make a rational and coherent betting decision without ordering our world beyond accepting probability calculus. Probabilism is founded on mathematics, the most deductive and ironclad of our knowledge systems, so this seems an excellent place to rest the weight of our beliefs that drive our actions.

A criticism that can be levied at Bayesianism is that it can be wildly useless at producing an optimal belief if an initial prior belief is incorrect. However, the possibility of error, in other words, fallibilism, does not imply scepticism in and of itself (Okasha 2001, 316). We can also not choose some prior belief, even if we make efforts that it is as minimal as possible, so we are justifiably and always forced into taking an initial position.
Bayesianism is used throughout many modern scientific disciplines, so whilst it is not necessarily the psychological mechanism by which humans constantly form opinions, it is a hugely successful and widely adopted method. Bayesianism is a concrete example of a no inductive rule method being a reasonable substitute for the Humean picture of forming a belief from the evidence that does not entail the belief. Bayesianism reveals that the dichotomy of there either being inductive rules or those beliefs formed from experience being random, as false (Okasha 2001, 317). The Bayesian example plus the complete lack of any description of our supposed universally valid inductive rules over the near three centuries since the publication of the Treatise form the cornerstone of Okasha’s argument against the existence of justifiable universal inductive schemas.
Further evidence for no-rules comes from being clear about using the concept of a universal inductive schema. Okasha phrases this in terms of inductive rules, in which rule is to be understood as universal schema. Okasha states that occasionally using an inductive “rule” does not mean that we are following this rule (Okasha 2001, 317–18), following a rule some of the time is hardly universal. The inductive rule is an inductive method we are licensed to use in a particular inference. Arguing that we use a particular universal inductive schema on occasion, such as the UP, stretches the definition of a universal inductive schema beyond its status as a law for inductive inference being meaningful anymore (Okasha 2001, 318). To use a universal inductive schema, sometimes, when one could state some background information licenses, it is to remove the universal aspect.
The argument against universality from background information leads into a related argument against universal inductive schemas presented by Okasha that hinges on the role that background beliefs have in our responses to experience (Okasha 2001, 318). He states that a no universal inductive schema approach like Bayesianism easily handles this as the whole process begins with the realistic scenario that we will have some prior beliefs (Okasha 2001, 318). Universal inductive schemas have a more difficult time. It is straightforward to imagine that for any inductive rule, we could imagine one or more background beliefs that would lead us to discard that rule for at least a specific inference (Okasha 2001, 318), if not more generally. To illustrate this, we turn to a simple example, say a patient is visiting a particularly careless doctor. The patient presents a host of symptoms that fit perfectly a particular medical condition associated with pregnancy. The doctor has used the hypothetico-deductive schema for inductive inference, in short, “to entail the evidence is the mark of its truth” (Norton 2003, 653). The doctor has assumed the patient is pregnant and can explain the patient’s symptoms (it entails the evidence, the symptoms). The doctor tells the patient this information and prescribes the patient’s treatment. At this point, the patient points out to the doctor that they are a male. The doctor somehow missed a piece of crucial background information that would not have licensed this particular use of the hypothetico-deductive schema. Once again, we have seen the supposed universality of an inductive method essentially cause a failure to produce a rational inference, as in the universal schema form, such methods are not sensitive to local and licensing facts.
Next, we shall try and encapsulate what Okasha’s no-rules inductive practice is actually like, albeit a little vague. However, this is perhaps to be expected when there are no rules to describe.

3.2.3 Okasha’s no-rules inductive practice

Beyond a conviction that we are not rule-governed when performing belief change based on experience, Okasha does not provide a complete picture of our actual inductive practices. However, he is clear that a concept from a liberal version of Bayesianism may be helpful. Okasha uses a permissive conception of rationality (Van Fraassen 1989, 171–73). Rationality tells us what we can believe, not what we must believe (Okasha 2001, 317). Permissive rationality allows radical changes to belief and opinion where necessary; for example, Bayesian conditionalisation cannot tell a scientist or the reader when a new hypothesis is needed, only how to modify their beliefs given a starting probability distribution. Van Fraassen and Okasha believe that we are allowed complete rational freedom in choosing new hypotheses ((Okasha 2001, 321); (Van Fraassen 1989, 172)).
So Okasha seems to believe that whilst it is improbable that we modify our beliefs using universal inductive schemas, we also possess complete freedom to form new hypotheses from finite data. We are therefore not beholden to a rule-governed process. Okasha believes that in daily life and empirical scientific practice, we do not follow universal inductive schemas; we may choose to apply a method of inductive inference in a certain situation. Our background information and licensing facts govern this. Having beliefs and updating them depending on experience is a more realistic picture of how we make inferences than the universal inductive schema model.
The local situation of a particular instance of inference is fundamental to understanding our actual inductive practice and rejecting the Humean scepticism that any universal inductive schema suffers. It is imperative to note again that something cannot be a universal inductive schema if we only occasionally follow it. It is correct that, as good empiricists, we should only use specific inductive methods when background information licenses us.
We saw in 3.2 the role that background information plays in making valid inductive inferences; now, we shall examine an implication of our license to use background information. Okasha does seem to believe that justifying a piece of background information may require an inductive inference from a different piece of background information. That piece of information may require justification from yet another piece of background information (Okasha 2005b, 251–53). This layering of licensing information would result in a regress of pieces of background information justifying other pieces of information to infinity (Okasha 2005b, 253). Okasha does not believe this is much of a problem as it is coherent with empiricism. Furthermore, we are not committed to assumptions about the world as we are licensed to change our background information based on new information becoming available (Okasha 2005b, 252). An infinite regress is therefore not as problematic as circularity as it reflects the process of empiricism that any attempt at solving Hume’s problem is trying to justify. Finally, for this chapter, we shall look at a criticism that Okasha’s solution has garnered.

3.3 Criticism

This subsection will analyse the most common criticism of Okasha’s solution and see to what degree they are valid and sound. We start with a critique by Marc Lange in his paper Okasha on Inductive Scepticism. Lange’s criticism centres on the problem of background beliefs (or priors in the example case of Bayesianism) (Lange 2002, 227). Lange argues against Okasha that any initial state of belief strong enough to support an inductive inference embodies some information; some inductive leap is made. Lange states that no prior belief that is capable of this is justified due to Humean scepticism, as it is an inductive inference (Lange 2002, 227). Lange has shifted the Humean scepticism to the priors. In Okasha’s 2001 paper, this is refuted in the case of Bayesianism as there is no alternative to having some prior probability distribution as having an information-free prior is simply impossible (Okasha 2001, 323). Lange, however, argues that what Okasha has done is to beg the question against the inductive sceptic (Lange 2002, 228). Okasha forcefully responds to this criticism in reply to Lange, reiterating that as there cannot logically be a viable principle of indifference, i.e., an information-free or theoretically barren state, Lange’s argument is not realistic or fair (Okasha 2003, 420).
Okasha’s argument against Lange’s criticism is compelling and in line with the rest of his thinking as we must examine actual inductive practice and not some hypothetical inductive schema or impossible knowledge state. Lange’s use of an impossible situation, the theoretically barren context situation, to criticise Okasha’s no-rules theory fails as an attack. Lange demands something from a no-rules theory that is not fair nor based on reality to demand. It is something only solvable by a universal inductive schema. However, it is not an advantage to universal inductive schemas that they can dissolve an impossible problem and may even highlight how unlike our actual inductive practice such schemas are.

3.4 Concluding remarks about Okasha’s solution

Okasha has made a compelling argument that we do not follow a general rule of induction such as the UP to form our beliefs, showing that in practice, we are only subject to inference fallibilism. This claim is reassuring and comfortable with our beliefs about epistemic uncertainties. We concur with Okasha that he has successfully rebuffed Lange’s criticism concerning shifting Humean scepticism onto our prior beliefs or background information.
In the next chapter, we shall examine our second no-rules solution proposed by John D. Norton. Norton’s solution will receive the same analysis treatment as has been presented for Okasha in this chapter. Once we have a full view of Okasha’s and Norton’s respective theories, we shall see in chapter 5 to what degree we can integrate these solutions into a single no-rules solution.

4 Norton’s solution

In this chapter, we will examine our second solution to Hume’s problem of induction a “material theory of induction”² proposed by John D. Norton. This solution shows that all induction is local and depends upon solid facts. Norton believes that such a solution does not suffer from Humean scepticism in the same manner as accounts based upon the existence of universal inductive schemas. We will first characterise Norton’s argument against Hume and other inductive rule schemas. Then we shall present Norton’s solution and analyse the strength of the most common criticisms against Norton’s theories. Once chapter 4 is concluded, we will have all the information required to assess the possible integration of our two no-rules solutions to Hume’s problem.

4.1 Locality matters

Norton’s argument against Hume comes in several parts. First, Norton states that there has been no success in producing rules for inductive inference on par with the deductive rules (Norton 2003, 647–48). The lack of success should be a warning; Norton does not believe that this will be remedied for any inductive account based on universal schemas (Norton 2003, 648). We shall expand upon this in the solution section of this chapter as Norton provides evidence for his solution by seeing how the broad groupings of inductive rule theories can be rescued from scepticism by introducing licensing facts termed material postulates³ (Norton 2003, 650).
Concerning this, Norton believes that Hume’s argument is only directly applicable with the force that many understand it to have when applied to universal inductive schemas, or as Norton frequently calls them, formal theories of induction (Norton 2003, 667). Norton understands formal theories to be broadly defined by their appeal to a universal template to legitimise an inductive inference, analogous to formal theories of deductive inference. Norton notes that this results in the exact type of tension revealed by Goodman, as discussed in chapter 2. To apply a universal template, Norton argues that we search for the properties for which a universal schema works (Norton 2003, 650). Norton argues that listing these properties amounts to specifying the specific facts that allow the application of an inductive rule to instances that belong to a uniform totality (Norton 2003, 649). Without mentioning particular facts, Norton believes it impossible to even give a possible meaning to the term uniform (Norton 2003, 649). No inductive schema can be applied to the correct instances of uniformity in nature without knowing specific facts, such as an object being made of a single chemical element.
We shall now examine why this appeal to universality makes inductive rules weak to Humean scepticism. Moreover, prioritising function and locality may make a material theory that depends upon particular facts immune to the sceptic. Norton argues that universal inductive rules separate any factual content from a specific induction with the template of the formal scheme used to make the inference (Norton 2003, 666). The pattern is by now familiar, the inductive rule cannot be shown to be justified deductively, and an inductive argument, justifying an inductive rule by the success of the inductive rule, is circular. Norton builds on this pattern; he adds the situation in which we have many successful instances of an inductive rule and then performs a meta-induction by using a different inductive rule on these successful instances (Norton 2003, 667). Meta-induction, he argues, triggers an infinite regress of meta-inductions that get increasingly broad in scope and removed from any inductive practice performed by anyone simply no one behaves in this manner. We are unjustified in doing so (Norton 2003, 667).
In section 4.2.3, we shall examine Norton’s claim that a theory that does not separate the universal schema of the argument from particular material facts does not suffer from a necessarily infinite vicious regress. Whilst discussing the additional claim that, unlike infinite meta-inductions, the equivalent process for a material theory reflects our actual inductive inference behaviour.

4.2 Rejecting scepticism with material induction

First, we shall look at Norton’s evidence for the material theory by seeing how its application to the broad families of universal inductive schemas would allow us to rescue these theories for local use. Then we shall present a description of Norton’s solution, how it functions, and why this allows it to reject Humean scepticism.

4.2.1 Evidence for a material theory

Norton examines three broad families of inductive rule schemas that he takes to include all of the contemporary positions regarding inductive rules. Norton aims to show that these inductive rule schemas are insufficient, unjustified, and suffer from Humean scepticism. Unless they additionally appeal to material postulates (Norton 2003, 652). Norton believes that he can build his argument that all inductions rely on local information in the relevant domain. To do this, he gives many examples to illustrate that successfully functioning inductive inferences must abandon universality (Norton 2003, 652–54). We shall see the examples of universal inductive schemas that Norton shows seem to require material postulates to be at all reasonable in the rest of 4.2.1. To paraphrase Peter Achinstein, Norton not only takes on Hume, Norton takes on everyone (Achinstein 2010, 728).
Norton first attacks the family of inductive rule schemas he terms inductive generalisation. The shared characteristic of this family is that generalisation is confirmed by multiple instances, the most famous of which is simple enumerative induction (Norton 2003, 652). The criticisms are many and familiar but are, in essence, once again Goodman’s objection; this family of schemas equally justifies lawlike hypotheses as it does accidental ones. Norton argues that in this simplest familial case, whether multiple instances confirm a law or generalisation and the strength of the generalisation is entirely dependent on the relevant material postulate (Norton 2003, 653). Norton’s analysis is very much in line with our other theorists, Goodman and Okasha, and it additionally fits our intuitions about how we perceive the correct regularities in nature by selecting rational predicates based on the domain of our inference.
The second family is termed hypothetical induction and is defined by the principle that the ability of a theory to entail the evidence means that it is truth conducive (Norton 2003, 653). Norton notes that this principle is generally not used in isolation due to its indiscriminate nature as it can logically entail many unwanted and unreasonable results. This is shown in the example that A&B is confirmed by A being true. This may seem innocent but if A, then B is supported, without the addition of material postulates to stipulate a legitimate relation between A and B the internal sum of angles of a triangle being \(180\degree\) (A), can support Freudian psychology (B) (Norton 2003, 653). Hence material postulates are already employed here to restrict hypothetical induction to the relevant domain; a Freudian would not consult a geometry textbook to help her understand Freudian psychology. Material postulates are crucial here to the functioning of this family of schemas, outside the most utterly basic and boring cases of inductive inferences. (Norton 2003, 653).
The third and final family of inductive rule schemas is called probabilistic accounts; the most famous and influential in this family is Bayesianism. Degrees of belief represented by magnitudes characterise schemas in this family; new evidence then alters the ordering of these magnitudes according to a definite calculus (Norton 2003, 659). Here Norton argues that using schemas in this family to make inferences about physical systems requires a material postulate that contains the relevant information on the stochastic properties of that system (Norton 2003, 660). For example, our beliefs about what numbers will come up if we roll a six-sided die depends upon the material postulate that we have a fair die with six possible outcomes. Once again, we have been shown by Norton how to legitimise the use of an inductive schema family, we appeal to local and particular facts based on the situation and inference, not a general and universal inductive rule.

4.2.2 Norton’s material inductive practice

Norton frames his material theory of induction in terms of controlling inductive risk. He argues that in any induction, we take a risk, and our conclusion has a chance of not being entailed by the premises. His theory aims to use a strategy that puts the weight of this risk on a firmer footing, i.e., the material postulates, than the failed universal inductive schemas (Norton 2003, 664).
Norton exposes the strategy used in universal inductive schemas to control risk. The first strategy is to gather as much evidence as possible to strengthen the inference (Norton 2003, 664). Secondly, we could expand the number of inductive methods available to use. Norton states that whilst the first strategy is used constantly in empirical science and daily life; the second strategy has mostly been neglected. The reason that Norton gives for this neglect is that we are all, scientists especially, performing material induction. We amass evidence, we use this evidence to reduce inductive risk, and we learn from this evidence. We will expand upon the implications of the regress in this process in 4.2.3 and 4.3. As we have seen in 4.2.1, all of our families of universal schemas ultimately seem to rely on material postulates as licensing facts to function in particular and local domains. Hence the more we know, the better we can infer, but only in the relevant domain of our inquiry (Norton 2003, 664). Norton’s methodology gives rise to something of a light theory of scientific progress; learning more and having more material postulates automatically improves scientists’ inductive powers automatically (Norton 2003, 664). I believe this to be an uncontroversial statement that increasing knowledge (number and reliability of material postulates) is what powers scientific discovery and allows for the increasing bounds of scientific knowledge.
It is a strength of Norton’s theory that it not only provides justifications for itself in terms of controlling inductive risk as used by scientists. In addition, something of a theory of the development of scientific knowledge can also be deduced from it. We transport inductive risk from unreliable universal schema to a relevant and local material postulate (Norton 2003, 665) that we can strengthen by gathering ever-increasing quantities of data until uncertainty and risk are reduced to negligible.

4.2.3 No Humean scepticism

Now we must examine to what degree the material theory of induction defeats Humean scepticism, and we will look at the analogous argument applied to Norton’s theory. The first horn of the argument, the deductive fork, would involve using a material postulate that is a known a priori universal truth. A universal truth immediately fails as it violates the critical point of the locality of the material theory (Norton 2003, 666). Next, we examine the second horn, the inductive fork. In this analogy, we see circularity immediately as we attempt to justify a material postulate with itself. This failure is un-concerning, however, as it does not matter because this is not how we justify the material postulate. It should not concern the empiricist that we cannot and do not justify a material postulate with the material postulate itself. This process is not how science behaves. We now perform the analogous meta-induction that was employed at the end of 4.1. The analogy entails that we now justify our material postulate with another material postulate and that material postulate with yet a different material postulate, and so on infinitely (Norton 2003, 666). Knowledge begets the ability to acquire knowledge. A regress is initiated here, but Norton aims to show that this is not necessarily vicious or infinite.
First, this regress is familiar as it is the natural progress of empirical science as described in 4.2.2. Once again, the material theory represents something of our actual inductive practice. What remains is an open question about how these chains (or branches) of reasoning terminate if they do terminate (Norton 2003, 668). Norton argues that Humean scepticism is uncertain as long as this remains uncertain. He admits that circularity could persist somehow but that it is unlikely as he cannot think of any examples in the history of science that do so. He argues that the chains may end benignly, possibly in what he terms “brute facts of experience” that do not require further justification, such that an infinite regress is avoided (Norton 2003, 668). Norton agrees that we may doubt that such a single brute fact may not exist and may not be substantial enough to license inductive inference. However, possibly many less grand facts may interplay and interact to license our grander inductions (Norton 2003, 668).
Without closure on this open question, Humean scepticism is not inevitable, and any fatal difficulty in a material theory can be questioned due to a reasonable chance of benign termination. In contrast to universal schemas, this is a massive improvement. Changing the focus from universal formal theories to local, relevant domain theories has for now evaded certain scepticism and provided a framework for thinking about how scientists acquire knowledge through inductive inference.

4.3 Criticism

A criticism arrives from an exegetical concern of Thomas Kelly about what type of thing Norton’s material postulate is. Kelly wishes to examine an implied commitment of the material theory which he terms the principle of prior knowledge (Kelly 2010, 760). This principle consists of Norton’s implicit assertion that uses a material postulate to gain knowledge via induction; one must have prior knowledge of a material postulate to license the induction. Kelly’s concern is that if we follow the chain to the bottom to a position E that represents the totality of our knowledge before we gain our first piece of inductive knowledge (Kelly 2010, 760). What does E contain? Kelly argues that logically E can include propositions that we could know a priori. It could also contain empirical knowledge that is gained through direct observation (Kelly 2010, 760) i.e., no inductive leap required such as physical relations (a seen biro touches a table) of some objects. Kelly argues that we must use a material postulate to move beyond E using the material theory. This material postulate must therefore be a proper part of E so that we are justified to use the material theory of induction (Kelly 2010, 761). Kelly’s concern is that there is nothing that can suitably play the role of a material postulate within E, so how do we move beyond our minimal knowledge state? The only rational positions are that a material postulate can either be known a priori deductively or is a piece of innate knowledge, both positions that are seemingly rejected by Norton (Kelly 2010, 761–62). One could respond to this criticism that a knowledge state E does not exist, so Norton’s regress issue remains open. However, Kelly’s concern does remain compelling as it points out a serious challenge to the possibility of benign termination in the material theory.
Another criticism comes from Daniel Steel. This criticism is that Norton, whilst commendable in his attempt to tell a particularistic and pluralistic story of induction, omits something crucial. Steel alleges Norton misses the role played by the standards, goals, and norms of induction as applied in a particular situation (Steel 2005, 189–90). Steel argues that this application of normativity drives the licensed choice of different methods of inductive inference, not solely and wholly the material postulates (Steel 2005, 189–90). A response to this criticism in the vein of Norton would be that the material postulates tell the inductor what norms are relevant and applicable in a particular case using domain-specific knowledge.

4.4 Concluding remarks about Norton’s solution

As chapter 4 closes, we have examined Norton’s theory, the strength of his arguments first against universal inductive schemas and then for the material theory. We have seen that whilst the final status of Humean scepticism in the material theory remains open, the shift from the universal to local, from the formal rule to no rule, has avoided scepticism for now. We have also seen two criticisms of the theory that though serious and well-considered, also do not defeat material induction for now.
In the next and final chapter, we shall attempt to reconcile Okasha and Norton with one another. We shall see what this means for our no-rules theorists’ current strength and success and whether they can better fend off Humean scepticism together rather than alone.

5 Integration

In this final chapter, we will attempt to find to what degree Okasha’s and Norton’s theories and arguments are similar. In what way do they represent the no-rules family of replies to Hume’s problem of induction, and what may need to be changed to unify them as a singular no-rules theory. To do this, we assess these two theories under three categories; critique of universal inductive schemas, a look at their inductive practices and the role of regress in place of circularity.

5.1 Category assessment

5.1.1 Critique of universal inductive schemas

We have seen in chapters 3 and 4 that both Okasha and Norton strongly believe that how inductive inference proceeds is of the utmost importance. First, we will look at the similarity of their criticisms regarding the weakness to scepticism of all universal inductive schemas. Secondly, we will examine the role licensing facts have in Okasha’s and Norton’s theories.
Okasha and Norton ascribe to the belief that the nature of universal inductive schemas makes them immediately vulnerable to Humean scepticism. Okasha frames this by assuming to order the world. We are only licenced to use a universal inductive schema if reality is ordered in such a way to justify that schema. Making a commitment of this magnitude to metaphysical order is beyond what it is possible and justifiable for the empiricist to make, something Okasha believes Hume would agree with (Okasha 2001, 321). Norton agrees with this as universal inductive schemas attempt to apply a universal template for which circularity will always apply (Norton 2003, 650).
Norton is particularly vivacious in his criticism of the attempted universality of inductive schemas. He systematically makes it clear that the only possibility to make these inductive schemas function at all is to add licensing facts (Norton 2003, 649). Okasha agrees, terming it background information, which allows the use of a schema for a particular inference (Okasha 2003, 318). Background information to rationally license particular methods removes the universal aspect of our inductive schemas.
Both philosophers take remarkably similar lines of argumentation regarding understanding Hume’s problem in terms of universal inductive schemas. So it is perhaps unsurprising that in light of these criticisms, they should both arrive at no-rules descriptions of inductive inference, as we shall see in 5.1.2.

5.1.2 Inductive practice

Okasha adopts the aforementioned permissive conception of rationality; being rational tells us what we may believe, not what we must believe (Van Fraassen 1989, 171–73). From this, Okasha essentially argues that any inductive method is permitted if an inference does not violate some instance of rationality. This is contingent on the specific background information for a specific inference. Hence, universal inductive schemas are reduced to entirely rational and justifiable inductive methods we can choose as appropriate. Norton’s material theory of induction also embodies this sentiment and describes how background information, i.e., material postulates, are key to a no-rules theory. Both philosophers believe that a more accurate description of human inductive practice is the freedom to choose a local inductive schema that can be utilised for an instance and then rationally discarded.
Both Okasha and Norton believe that they have created a more accurate description of human inductive practice. Okasha believes that from the complete freedom given by local inductive schemas, he has captured one aspect of how inductive inference in daily life and science proceeds. Humans do not attempt to make inferences based on schemas; rather, they look for facts and employ schemas appropriate to a particular inference. Norton’s description is highly similar but goes further. Norton insists that we use material postulates for which we are confident of the veracity to build further inferences and gain more knowledge.
Okasha’s more psychological proposition and Norton’s more epistemic development approach do not exclude one another but realise two possible explanatory levels. First, Okasha attempts to provide an individual psychological account of how the realistic cognitive process of inductive inference proceeds. Okasha does this by presenting cases of inductive inference, such as the Bayesian case. In this example, we reason and infer from the strength of belief instead of a more classical universal inductive schema approach such as simple enumerative induction. To me, this seems more psychologically plausible as inductive schemas are made here into tools for impacting the strength of belief and are used where applicable instead of universally. Second, Norton opts for using his theory to explain how the acquisition of more material postulates, using inductive inference from previously held material postulates, describes scientific progress. This is an excellent area to propose integrating these two distinct levels of explanation in the no-rules camp. Norton’s theory is more developed to explain scientific progress, whereas Okasha’s is more focused on the permissive conception of rationality governing our inductive inferences on a more individual basis.

5.1.3 Role of regress

In Norton’s material theory of induction, a regress is seen when attempting to justify the material postulates. As discussed, this seems in line with how empirical science proceeds. Unlike the circularity that universal inductive schemas are susceptible to, Norton believes it is an open question whether regression is also vicious.
Norton harbours a belief that this regress will not be infinite and may end benignly in some “brute” facts of existence. Norton believes this will most likely be many small material postulates that form a branching structure rather than a chain (Norton 2003, 668). Okasha is not entirely as convinced of a benign termination in a brute fact as Norton. Okasha argues that this belief of Norton’s in finite benign termination may still allow scepticism. Ultimately, all inductive inference would rest on some overarching assumption about the world (Okasha 2005b, 252). Okasha supplements this argument with a point about the conduct and nature of science. He argues that any attempt to provide any underlying justification, such as a fundamental material postulate, would make science and empiricism seem irrational due to the reliance on a blind faith assumption (Okasha 2005b, 252). Okasha believes that justifying background information inductively will also lead to regresses, but he remains optimistic. Okasha argues that regresses, even if infinite, are a less bad result than the vicious circularity that is only available to universal inductive schemas (Okasha 2005b, 253).
There is clearly disagreement about the exact nature of the regresses in the no-rules camp between Okasha and Norton. However, we should stress that both agree that as the result of the regresses is open, we are in a better position to reject scepticism than any theory utilising the universal inductive schema framework. To formalise a no-rules position on the role of regresses, it is sufficient to say that a characteristic of the no-rules framework is that regresses are less immediately problematic than circularity in the argument justifying inductive inferences.

5.2 Concluding remarks about integration

Assessing Okasha’s and Norton’s respective theories, we have found many large similarities, enough to characterise the general features of the no-rules approach to Hume’s problem.
The universality is the crucial cause of weakness to Humean scepticism in universal inductive schemas. Reducing these schemas to local inductive methods licensed by particular background information is rational and a more accurate description of our actual inductive practice. Additionally, it is vital to no-rules philosophers that any justified response to Hume’s problem possess an accurate description of our inductive practice, something they both argue is missing from universal inductive schemas.
The no-rules approach attempts a psychological and developmental approach that attempts to explain our personal use of inductive inference as well as how this applies and relates to the epistemology of science. Using local inductive inference grounded in scientifically acquired postulates fits many descriptions of scientific practice and intuition regarding the field.
Finally, the no-rules approach views regress in the place of circularity, a far more preferable situation as the question remains open as to whether this also suffers scepticism. It additionally fits empiricism as a programme as it means we are not forced to rely on blind faith assumptions about the nature of our knowledge determined from inductive inference.

6 Conclusion

This paper aimed to assess the effectiveness of two no-rules approach theories to Hume’s problem of induction and to see to what degree they could be integrated into a single no-rules theory. I found several strengths in the methodology of both Okasha and Norton. They identify several legitimate concerns about the predominant faith in the existence of a valid universal inductive schema for deductive inference. These concerns include irrational commitments to universality that open the door to Humean scepticism. The role background information plays when using any inductive schema and the unrealistic representation of the actual process of inductive inference that universal inductive schemas provide.
They both present a fruitful approach of replying to Hume with actual human inductive practice in mind instead of assumed generality in Hume’s argument. The no-rules framework holds locality and background information as key to describing our inductive practice. The no-rules approach also provides a theory of scientific development and does seem to avoid the circularity that permanently threatens universal inductive inference. Finally, the no-rules framework views regress as ultimately unsettled to its status regarding Humean scepticism, but this, combined with a good description of inductive inference practice, leaves us in a good position for future research.

References