We generally think that the observations we make are able to justifysome expectations or predictions about observations we have not yetmade, as well as general claims that go beyond the observed. Forexample, the observation that bread of a certain appearance has thusfar been nourishing seems to justify the expectation that the nextsimilar piece of bread I eat will also be nourishing, as well as theclaim that bread of this sort is generally nourishing. Such inferencesfrom the observed to the unobserved, or to general laws, are known as“inductive inferences”.

The original source of what has become known as the “problem ofinduction” is in Book 1, part iii, section 6 ofA Treatiseof Human Nature by David Hume, published in 1739 (Hume 1739). In1748, Hume gave a shorter version of the argument in Section iv ofAn enquiry concerning human understanding (Hume 1748).Throughout this article we will give references to theTreatise as “T”, and theEnquiry as“E”.

Hume asks on what grounds we come to our beliefs about the unobservedon the basis of inductive inferences. He presents an argument in theform of a dilemma which appears to rule out the possibility of anyreasoning from the premises to the conclusion of an inductiveinference. There are, he says, two possible types of arguments,“demonstrative” and “probable”, but neitherwill serve. A demonstrative argument produces the wrong kind ofconclusion, and a probable argument would be circular. Therefore, forHume, the problem remains of how to explain why we form anyconclusions that go beyond the past instances of which we have hadexperience (T. 1.3.6.10). Hume stresses that he is not disputing thatwe do draw such inferences. The challenge, as he sees it, is tounderstand the “foundation” of the inference—the“logic” or “process of argument” that it isbased upon (E. 4.2.21). The problem of meeting this challenge, whileevading Hume’s argument against the possibility of doing so, hasbecome known as “the problem of induction”.

Hume’s argument is one of the most famous in philosophy. Anumber of philosophers have attempted solutions to the problem, but asignificant number have embraced his conclusion that it is insoluble.There is also a wide spectrum of opinion on the significance of theproblem. Some have argued that Hume’s argument does notestablish any far-reaching skeptical conclusion, either because it wasnever intended to, or because the argument is in some waymisformulated. Yet many have regarded it as one of the most profoundphilosophical challenges imaginable since it seems to call intoquestion the justification of one of the most fundamental ways inwhich we form knowledge. Bertrand Russell, for example, expressed theview that if Hume’s problem cannot be solved, “there is nointellectual difference between sanity and insanity” (Russell1946: 699).

In this article, we will first examine Hume’s own argument,provide a reconstruction of it, and then survey different responses tothe problem which it poses.

• 1. Hume’s Problem
• 2. Reconstruction
• 3. Tackling the First Horn of Hume’s Dilemma
  • 3.1 Synthetica priori
  • 3.2 The Nomological-Explanatory solution
  • 3.3 Bayesian solution
  • 3.4 Partial solutions
  • 3.5 The combinatorial approach
• 4. Tackling the Second Horn of Hume’s Dilemma
  • 4.1 Inductive Justifications of Induction
  • 4.2 No Rules
• 5. Alternative Conceptions of Justification
  • 5.1 Postulates and Hinges
  • 5.2 Ordinary Language Dissolution
  • 5.3 Pragmatic vindication of induction
  • 5.4 Formal Learning Theory
  • 5.5 Meta-induction
• 6. Living with Inductive Skepticism
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Hume’s Problem

Hume introduces the problem of induction as part of an analysis of thenotions of cause and effect. Hume worked with a picture, widespread inthe early modern period, in which the mind was populated with mentalentities called “ideas”. Hume thought that ultimately allour ideas could be traced back to the “impressions” ofsense experience. In the simplest case, an idea enters the mind bybeing “copied” from the corresponding impression (T.1.1.1.7/4). More complex ideas are then created by the combination ofsimple ideas (E. 2.5/19). Hume took there to be a number of relationsbetween ideas, including the relation of causation (E. 3.2). (For moreon Hume’s philosophy in general, see Morris & Brown2014).

For Hume, the relation of causation is the only relation by means ofwhich “we can go beyond the evidence of our memory andsenses” (E. 4.1.4, T. 1.3.2.3/74). Suppose we have an objectpresent to our senses: say gunpowder. We may then infer to an effectof that object: say, the explosion. The causal relation links our pastand present experience to our expectations about the future (E.4.1.4/26).

Hume argues that we cannot make a causal inference by purelyapriori means (E. 4.1.7). Rather, he claims, it is based onexperience, and specifically experience of constant conjunction. Weinfer that the gunpowder will explode on the basis of past experienceof an association between gunpowder and explosions.

Hume wants to know more about the basis for this kind of inference. Ifsuch an inference is made by a “chain of reasoning” (E.4.2.16), he says, he would like to know what that reasoning is. Ingeneral, he claims that the inferences depend on a transition of theform:

I have found that such an object has always been attended withsuch an effect, and I foresee, that other objects, which are, inappearance, similar, will be attended with similar effects. (E.4.2.16)

In theTreatise, Hume says that

if Reason determin’d us, it would proceed upon that principlethat instances, of which we have had no experience, must resemblethose, of which we have had experience, and that the course of naturecontinues always uniformly the same. (T. 1.3.6.4)

For convenience, we will refer to this claim of similarity orresemblance between observed and unobserved regularities as the“Uniformity Principle (UP)”. Sometimes it is also calledthe “Resemblance Principle”, or the “Principle ofUniformity of Nature”.

Hume then presents his famous argument to the conclusion that therecan be no reasoning behind this principle. The argument takes the formof a dilemma. Hume makes a distinction between relations of ideas andmatters of fact. Relations of ideas include geometric, algebraic andarithmetic propositions, “and, in short, every affirmation,which is either intuitively or demonstratively certain”.“Matters of fact”, on the other hand are empiricalpropositions which can readily be conceived to be other than they are.Hume says that

All reasonings may be divided into two kinds, namely, demonstrativereasoning, or that concerning relations of ideas, and moral reasoning,or that concerning matter of fact and existence. (E. 4.2.18)

Hume considers the possibility of each of these types of reasoning inturn, and in each case argues that it is impossible for it to supplyan argument for the Uniformity Principle.

First, Hume argues that the reasoning cannot be demonstrative, becausedemonstrative reasoning only establishes conclusions which cannot beconceived to be false. And, he says,

it implies no contradiction that the course of nature may change, andthat an object seemingly like those which we have experienced, may beattended with different or contrary effects. (E. 4.2.18)

It is possible, he says, to clearly and distinctly conceive of asituation where the unobserved case does not follow the regularity sofar observed (E. 4.2.18, T. 1.3.6.5/89).

Second, Hume argues that the reasoning also cannot be “such asregard matter of fact and real existence”. He also calls this“probable” reasoning. All such reasoning, he claims,“proceed upon the supposition, that the future will beconformable to the past”, in other words on the UniformityPrinciple (E. 4.2.19).

Therefore, if the chain of reasoning is based on an argument of thiskind it will again be relying on this supposition, “and takingthat for granted, which is the very point in question”. (E.4.2.19, see also T. 1.3.6.7/90). The second type of reasoning thenfails to provide a chain of reasoning which is not circular.

In the Treatise version, Hume concludes

Thus, not only our reason fails us in the discovery of theultimate connexion of causes and effects, but even afterexperience has inform’d us of theirconstantconjunction, ’tis impossible for us to satisfy ourselves byour reason, why we shou’d extend that experience beyond thoseparticular instances, which have fallen under our observation. (T.1.3.6.11/91–2)

The conclusion then is that our tendency to project past regularitiesinto the future is not underpinned by reason. The problem of inductionis to find a way to avoid this conclusion, despite Hume’sargument.

After presenting the problem, Hume does present his own“solution” to the doubts he has raised (E. 5, T.1.3.7–16). This consists of an explanation of what the inductiveinferences are driven by, if not reason. In theTreatise Humeraises the problem of induction in an explicitly contrastive way. Heasks whether the transition involved in the inference is produced

by means of the understanding or imagination; whether we aredetermin’d by reason to make the transition, or by a certainassociation and relation of perceptions? (T. 1.3.6.4)

And he goes on to summarize the conclusion by saying

When the mind, therefore, passes from the idea or impression of oneobject to the idea or belief of another, it is not determin’d byreason, but by certain principles, which associate together the ideasof these objects, and unite them in the imagination. (T. 1.3.6.12)

Thus, it is the imagination which is taken to be responsible forunderpinning the inductive inference, rather than reason.

In theEnquiry, Hume suggests that the step taken by themind,

which is not supported by any argument, or process of theunderstanding … must be induced by some other principle ofequal weight and authority. (E. 5.1.2)

That principle is “custom” or “habit”. Theidea is that if one has seen similar objects or events constantlyconjoined, then the mind is inclined to expect a similar regularity tohold in the future. The tendency or “propensity” to drawsuch inferences, is the effect of custom:

… having found, in many instances, that any two kinds ofobjects, flame and heat, snow and cold, have always been conjoinedtogether; if flame or snow be presented anew to the senses, the mindis carried by custom to expect heat or cold, and tobelieve,that such a quality does exist and will discover itself upon a nearerapproach. This belief is the necessary result of placing the mindin such circumstances. It is an operation of the soul, when we are sosituated, as unavoidable as to feel the passion of love, when wereceive benefits; or hatred, when we meet with injuries. All theseoperations are a species of natural instincts, which no reasoning orprocess of the thought and understanding is able, either to produce,or to prevent. (E. 5.1.8)

Hume argues that the fact that these inferences do follow the courseof nature is a kind of “pre-established harmony” (E.5.2.21). It is a kind of natural instinct, which may in fact be moreeffective in making us successful in the world, than if we relied onreason to make these inferences.

2. Reconstruction

Hume’s argument has been presented and formulated in manydifferent versions. There is also an ongoing lively discussion overthe historical interpretation of what Hume himself intended by theargument. It is therefore difficult to provide an unequivocal anduncontroversial reconstruction of Hume’s argument. Nonetheless,for the purposes of organizing the different responses to Hume’sproblem that will be discussed in this article, the followingreconstruction will serve as a useful starting point.

Hume’s argument concerns specific inductive inferences suchas:

Let us call this “inferenceI”. Inferences whichfall under this type of schema are now often referred to as cases of“simple enumerative induction”.

Hume’s own example is:

Hume’s argument then proceeds as follows (premises are labeledas P, and subconclusions and conclusions as C):

There have been different interpretations of what Hume means by“demonstrative” and “probable” arguments.Sometimes “demonstrative” is equated with“deductive”, and probable with “inductive”(e.g., Salmon 1966). Then the first horn of Hume’s dilemma wouldeliminate the possibility of a deductive argument, and the secondwould eliminate the possibility of an inductive argument. However,under this interpretation,premise P3 would not hold, because it is possible for the conclusion of adeductive argument to be a non-necessary proposition. PremiseP3 could be modified to say that a demonstrative (deductive) argumentestablishes a conclusion that cannot be false if the premises aretrue. But then it becomes possible that the supposition that thefuture resembles the past, which is not a necessary proposition, couldbe established by a deductive argument from some premises, though notfroma priori premises (in contradiction to conclusionC1).

Another common reading is to equate “demonstrative” with“deductively valid witha priori premises”, and“probable” with “having an empirical premise”(e.g., Okasha 2001). This may be closer to the mark, if one thinks, asHume seems to have done, that premises which can be knownapriori cannot be false, and hence are necessary. If the inferenceis deductively valid, then the conclusion of the inference fromapriori premises must also be necessary. What the first horn ofthe dilemma then rules out is the possibility of a deductively validargument witha priori premises, and the second horn rulesout any argument (deductive or non-deductive), which relies on anempirical premise.

However, recent commentators have argued that in the historicalcontext that Hume was situated in, the distinction he draws betweendemonstrative and probable arguments has little to do with whether ornot the argument has a deductive form (Owen 1999; Garrett 2002). Inaddition, the class of inferences that establish conclusions whosenegation is a contradiction may include not just deductively validinferences froma priori premises, but any inferences thatcan be drawn usinga priori reasoning (that is, reasoningwhere the transition from premises to the conclusion makes no appealto what we learn from observations). It looks as though Hume doesintend the argument of the first horn to rule out anyapriori reasoning, since he says that a change in the course ofnature cannot be ruled out “by any demonstrative argument orabstract reasoninga priori” (E. 5.2.18). On thisunderstanding,a priori arguments would be ruled out by thefirst horn of Hume’s dilemma, and empirical arguments by thesecond horn. This is the interpretation that I will adopt for thepurposes of this article.

In Hume’s argument, the UP plays a central role. As we will seeinsection 4.2, various authors have been doubtful about this principle. Versions ofHume’s argument have also been formulated which do not makereference to the UP. Rather they directly address the question of whatarguments can be given in support of the transition from the premisesto the conclusion of the specific inductive inferenceI. Whatarguments could lead us, for example, to infer that the next piece ofbread will nourish from the observations of nourishing bread made sofar? For the first horn of the argument, Hume’s argument can bedirectly applied. A demonstrative argument establishes a conclusionwhose negation is a contradiction. The negation of the conclusion ofthe inductive inference is not a contradiction. It is not acontradiction that the next piece of bread is not nourishing.Therefore, there is no demonstrative argument for the conclusion ofthe inductive inference. In the second horn of the argument, theproblem Hume raises is a circularity. Even if Hume is wrong that allinductive inferences depend on the UP, there may still be acircularity problem, but as we shall see insection 4.1, the exact nature of the circularity needs to be carefully considered.But the main point at present is that the Humean argument is oftenformulated without invoking the UP.

Since Hume’s argument is a dilemma, there are two main ways toresist it. The first is to tackle the first horn and to argue thatthere is after all a demonstrative argument –here taken to meanan argument based ona priori reasoning—that canjustify the inductive inference. The second is to tackle the secondhorn and to argue that there is after all a probable (or empirical)argument that can justify the inductive inference. We discuss thedifferent variants of these two approaches in sections3 and4.

There are also those who dispute the consequences of the dilemma. Forexample, some scholars have denied that Hume should be read asinvoking a premise suchpremise P8 at all. The reason, they claim, is that he was not aiming for anexplicitly normative conclusion about justification such asC5. Hume certainly is seeking a “chain of reasoning” from thepremises of the inductive inference to the conclusion, and he thinksthat an argument for the UP is necessary to complete the chain.However, one could think that there is no further premise regardingjustification, and so the conclusion of his argument is simplyC4: there is no chain of reasoning from the premises to the conclusion ofan inductive inference. Hume could then be, as Don Garrett and DavidOwen have argued, advancing a “thesis in cognitivepsychology”, rather than making a normative claim aboutjustification (Owen 1999; Garrett 2002). The thesis is about thenature of the cognitive process underlying the inference. According toGarrett, the main upshot of Hume’s argument is that there can beno reasoning process that establishes the UP. For Owen, the message isthat the inference is not drawn through a chain of ideas connected bymediating links, as would be characteristic of the faculty ofreason.

There are also interpreters who have argued that Hume is merely tryingto exclude a specific kind of justification of induction, based on aconception of reason predominant among rationalists of his time,rather than a justification in general (Beauchamp & Rosenberg1981; Baier 2009). In particular, it has been claimed that it is“an attempt to refute the rationalist belief that at least someinductive arguments are demonstrative” (Beauchamp &Rosenberg 1981: xviii). Under this interpretation,premise P8 should be modified to read something like:

• If there is no chain of reasoning based on demonstrative argumentsfrom the premises to the conclusion of inferenceI, theninferenceI is not justified.

Such interpretations do however struggle with the fact thatHume’s argument is explicitly a two-pronged attack, whichconcerns not just demonstrative arguments, but also probablearguments.

The question of how expansive a normative conclusion to attribute toHume is a complex one. It depends in part on the interpretation ofHume’s own solution to his problem. As we saw insection 1, Hume attributes the basis of inductive inference to principles of theimagination in the Treatise, and in the Enquiry to“custom”, “habit”, conceived as a kind ofnatural instinct. The question is then whether this alternativeprovides any kind of justification for the inference, even if not onebased on reason. On the face of it, it looks as though Hume issuggesting that inductive inferences proceed on an entirely arationalbasis. He clearly does not think that they do not succeed in producinggood outcomes. In fact, Hume even suggests that this operation of themind may even be less “liable to error and mistake” thanif it were entrusted to “the fallacious deductions of ourreason, which is slow in its operations” (E. 5.2.22). It is alsonot clear that he sees the workings of the imagination as completelydevoid of rationality. For one thing, Hume talks about the imaginationas governed byprinciples. Later in theTreatise, heeven gives “rules” and “logic” forcharacterizing what should count as a good causal inference (T.1.3.15). He also clearly sees it as possible to distinguish betweenbetter forms of such “reasoning”, as he continues to callit. Thus, there may be grounds to argue that Hume was not trying toargue that inductive inferences have no rational foundationwhatsoever, but merely that they do not have the specific type ofrational foundation which is rooted in the faculty of Reason.

All this indicates that there is room for debate over the intendedscope of Hume’s own conclusion. And thus there is also room fordebate over exactly what form a premise (such aspremise P8) that connects the rest of his argument to a normative conclusionshould take. No matter who is right about this however, the factremains that Hume has throughout history been predominantly read aspresenting an argument for inductive skepticism.

There are a number of approaches which effectively, if not explicitly,take issue withpremise P8 and argue that providing a chain of reasoning from the premises tothe conclusion is not a necessary condition for justification of aninductive inference. According to this type of approach, one may admitthat Hume has shown that inductive inferences are not justified in thesense that we have reasons to think their conclusions true, but stillthink that weaker kinds of justification of induction are possible (section 5). Finally, there are some philosophers who do accept the skepticalconclusionC5 and attempt to accommodate it. For example, there have been attemptsto argue that inductive inference is not as central to scientificinquiry as is often thought (section 6).

3. Tackling the First Horn of Hume’s Dilemma

The first horn of Hume’s argument, as formulated above, is aimedat establishing that there is no demonstrative argument for the UP.There are several ways people have attempted to show that the firsthorn does not definitively preclude a demonstrative orapriori argument for inductive inferences. One possible escaperoute from the first horn is to denypremise P3, which amounts to admitting the possibility of syntheticapriori propositions (section 3.1). Another possibility is to attempt to provide ana prioriargument that the conclusion of the inference is probable, though notcertain. The first horn of Hume’s dilemma implies that therecannot be a demonstrative argument to the conclusion of an inductiveinference because it is possible to conceive of the negation of theconclusion. For instance, it is quite possible to imagine that thenext piece of bread I eat will poison me rather than nourish me.However, this does not rule out the possibility of a demonstrativeargument that establishes only that the bread is highly likely tonourish, not that it definitely will. One might then also challengepremise P8, by saying that it is not necessary for justification of an inductiveinference to have a chain of reasoning from its premises to itsconclusion. Rather it would suffice if we had an argument from thepremises to the claim that the conclusion is probable or likely. Thenana priori justification of the inductive inference wouldhave been provided. There have been attempts to provideapriori justifications for inductive inference based on Inferenceto the Best Explanation (section 3.2). There are also attempts to find ana priori solution basedon probabilistic formulations of inductive inference, though many nowthink that a purelya priori argument cannot be found becausethere are empirical assumptions involved (sections3.3 -3.5).

3.1 Synthetica priori

As we have seen insection 1, Hume takes demonstrative arguments to have conclusions which are“relations of ideas”, whereas “probable” or“moral” arguments have conclusions which are“matters of fact”. Hume’s distinction between“relations of ideas” and “matters of fact”anticipates the distinction drawn by Kant between“analytic” and “synthetic” propositions (Kant1781). A classic example of an analytic proposition is“Bachelors are unmarried men”, and a synthetic propositionis “My bike tyre is flat”. For Hume, demonstrativearguments, which are based ona priori reasoning, canestablish only relations of ideas, or analytic propositions. Theassociation between a prioricity and analyticity underpinspremise P3, which states that a demonstrative argument establishes a conclusionwhose negation is a contradiction.

One possible response to Hume’s problem is to denypremise P3, by allowing the possibility thata priori reasoning couldgive rise to synthetic propositions. Kant famously argued in responseto Hume that such synthetica priori knowledge is possible(Kant 1781, 1783). He does this by a kind of reversal of theempiricist programme espoused by Hume. Whereas Hume tried tounderstand how the concept of a causal or necessary connection couldbe based on experience, Kant argued instead that experience only comesabout through the concepts or “categories” of theunderstanding. On his view, one can gaina priori knowledgeof these concepts, including the concept of causation, by atranscendental argument concerning the necessary preconditions ofexperience. A more detailed account of Kant’s response to Humecan be found in de Pierris and Friedman 2013.

3.2 The Nomological-Explanatory solution

The “Nomological-explanatory” solution, which has been putforward by Armstrong, BonJour and Foster (Armstrong 1983; BonJour1998; Foster 2004) appeals to the principle of Inference to the BestExplanation (IBE). According to IBE, we should infer that thehypothesis which provides the best explanation of the evidence isprobably true. Proponents of the Nomological-Explanatory approach takeInference to the Best Explanation to be a mode of inference which isdistinct from the type of “extrapolative” inductiveinference that Hume was trying to justify. They also regard it as atype of inference which although non-deductive, is justifiedapriori. For example, Armstrong says “To infer to the bestexplanation is part of what it is to be rational. If that is notrational, what is?” (Armstrong 1983: 59).

Thea priori justification is taken to proceed in two steps.First, it is argued that we should recognize that certain observedregularities require an explanation in terms of some underlying law.For example, if a coin persistently lands heads on repeated tosses,then it becomes increasingly implausible that this occurred justbecause of “chance”. Rather, we should infer to the betterexplanation that the coin has a certain bias. Saying that the coinlands heads not only for the observed cases, but also for theunobserved cases, does not provide an explanation of the observedregularity. Thus, mere Humean constant conjunction is not sufficient.What is needed for an explanation is a “non-Humean,metaphysically robust conception of objective regularity”(BonJour 1998), which is thought of as involving actual naturalnecessity (Armstrong 1983; Foster 2004).

Once it has been established that there must be some metaphysicallyrobust explanation of the observed regularity, the second step is toargue that out of all possible metaphysically robust explanations, the“straight” inductive explanation is the best one, wherethe straight explanation extrapolates the observed frequency to thewider population. For example, given that a coin has some objectivechance of landing heads, the best explanation of the fact that \(m/n\)heads have been so far observed, is that the objective chance of thecoin landing heads is \(m/n\). And this objective chance determineswhat happens not only in observed cases but also in unobservedcases.

The Nomological-Explanatory solution relies on taking IBE as arational,a priori form of inference which is distinct frominductive inferences like inferenceI. However, one mightalternatively view inductive inferences as a special case of IBE(Harman 1968), or take IBE to be merely an alternative way ofcharacterizing inductive inference (Henderson 2014). If either ofthese views is right, IBE does not have the necessary independencefrom inductive inference to provide a non-circular justification ofit.

One may also object to the Nomological-Explanatory approach on thegrounds that regularities do not necessarily require an explanation interms of necessary connections or robust metaphysical laws. Theviability of the approach also depends on the tenability of anon-Humean conception of laws. There have been several seriousattempts to develop such an account (Armstrong 1983; Tooley 1977;Dretske 1977), but also much criticism (see J. Carroll 2016).

Another critical objection is that the Nomological-Explanatorysolution simply begs the question, even if it is taken to belegitimate to make use of IBE in the justification of induction. Inthe first step of the argument we infer to a law or regularity whichextends beyond the spatio-temporal region in which observations havebeen thus far made, in order to predict what will happen in thefuture. But why could a law that only applies to the observedspatio-temporal region not be an equally good explanation? The mainreply seems to be that we can seea priori that laws withtemporal or spatial restrictions would be less good explanations.Foster argues that the reason is that this would introduce moremysteries:

For it seems to me that a law whose scope is restricted to someparticular period is more mysterious, inherently more puzzling, thanone which is temporally universal. (Foster 2004)

3.3 Bayesian solution

Another way in which one can try to construct ana prioriargument that the premises of an inductive inference make itsconclusion probable, is to make use of the formalism of probabilitytheory itself. At the time Hume wrote, probabilities were used toanalyze games of chance. And in general, they were used to address theproblem of what we would expect to see, given that a certain cause wasknown to be operative. This is the so-called problem of “directinference”. However, the problem of induction concerns the“inverse” problem of determining the cause or generalhypothesis, given particular observations.

One of the first and most important methods for tackling the“inverse” problem using probabilities was developed byThomas Bayes. Bayes’s essay containing the main results waspublished after his death in 1764 (Bayes 1764). However, it ispossible that the work was done significantly earlier and was in factwritten in direct response to the publication of Hume’s Enquiryin 1748 (see Zabell 1989: 290–93, for discussion of what isknown about the history).

We will illustrate the Bayesian method using the problem of drawingballs from an urn. Suppose that we have an urn which contains whiteand black balls in an unknown proportion. We draw a sample of ballsfrom the urn by removing a ball, noting its color, and then putting itback before drawing again.

Consider first the problem of direct inference. Given the proportionof white balls in the urn, what is the probability of various outcomesfor a sample of observations of a given size? Suppose the proportionof white balls in the urn is \(\theta = 0.6\). The probability ofdrawing one white ball in a sample of one is then \(p(W; \theta = 0.6)= 0.6\). We can also compute the probability for other outcomes, suchas drawing two white balls in a sample of two, using the rules of theprobability calculus (see section 1 of Hájek 2011). Generally,the probability that \(n_w\) white balls are drawn in a sample of sizeN, is given by the binomial distribution:

This is a specific example of a “sampling distribution”,\(p(E\mid H)\), which gives the probability of certain evidenceE in a sample, on the assumption that a certain hypothesisH is true. Calculation of the sampling distribution can ingeneral be donea priori, given the rules of the probabilitycalculus.

However, the problem of induction is the inverse problem. We want toinfer not what the sample will be like, with a known hypothesis,rather we want to infer a hypothesis about the general situation orpopulation, based on the observation of a limited sample. Theprobabilities of the candidate hypotheses can then be used to informpredictions about further observations. In the case of the urn, forexample, we want to know what the observation of a particular samplefrequency of white balls, \(\frac{n_w}{N}\), tells us about\(\theta\), the proportion of white balls in the urn.

The idea of the Bayesian approach is to assign probabilities not onlyto the events which constitute evidence, but also to hypotheses. Onestarts with a “prior probability” distribution over therelevant hypotheses \(p(H)\). On learning some evidenceE,the Bayesian updates the prior \(p(H)\) to the conditional probability\(p(H\mid E)\). This update rule is called the “rule ofconditionalisation”. The conditional probability \(p(H\mid E)\)is known as the “posterior probability”, and is calculatedusing Bayes’ rule:

Here the sampling distribution can be taken to be a conditionalprobability \(p(E\mid H)\), which is known as the“likelihood” of the hypothesisH on evidenceE.

One can then go on to compute the predictive distribution for as yetunobserved data \(E'\), given observationsE. The predictivedistribution in a Bayesian approach is given by

where the sum becomes an integral in cases whereH is acontinuous variable.

For the urn example, we can compute the posterior probability\(p(\theta\mid n_w)\) using Bayes’ rule, and the likelihoodgiven by the binomial distribution above. In order to do so, we alsoneed to assign a prior probability distribution to the parameter\(\theta\). One natural choice, which was made early on by Bayeshimself and by Laplace, is to put a uniform prior over the parameter\(\theta\). Bayes’ own rationale for this choice was that thenif you work out the probability of each value for the number of whitesin the sample based only on the prior, before any data is observed,all those probabilities are equal. Laplace had a differentjustification, based on the Principle of Indifference. This principlestates that if you don’t have any reason to favor one hypothesisover another, you should assign them all equal probabilities.

With the choice of uniform prior, the posterior probability andpredictive distribution can be calculated. It turns out that theprobability that the next ball will be white, given that \(n_w\) ofN draws were white, is given by

This is Laplace’s famous “rule of succession”(1814). Suppose on the basis of observing 90 white balls out of 100,we calculate by the rule of succession that the probability of thenext ball being white is \(91/102=0.89\). It is quite conceivable thatthe next ball might be black. Even in the case, where all 100 ballshave been white, so that the probability of the next ball being whiteis 0.99, there is still a small probability that the next ball is notwhite. What the probabilistic reasoning supplies then is not anargument to the conclusion that the next ball will be a certain color,but an argument to the conclusion that certain future observations areverylikely given what has been observed in the past.

Overall, the Bayes-Laplace argument in the urn case provides anexample of how probabilistic reasoning can take us from evidence aboutobservations in the past to a prediction for how likely certain futureobservations are. The question is what kind of solution, if any, thistype of calculation provides to the problem of induction. At firstsight, since it is just a mathematical calculation, it looks as thoughit does indeed provide ana priori argument from the premisesof an inductive inference to the proposition that a certain conclusionis probable.

However, in order to establish this definitively, one would need toargue that all the components and assumptions of the argument area priori and this requires further examination of at leastthree important issues.

First, the Bayes-Laplace argument relies on the rules of theprobability calculus. What is the status of these rules? Doesfollowing them amount toa priori reasoning? The answer tothis depends in part on how probability itself is interpreted. Broadlyspeaking, there are prominent interpretations of probability accordingto which the rules plausibly havea priori status and couldform the basis of a demonstrative argument. These include theclassical interpretation originally developed by Laplace (1814), thelogical interpretation (Keynes (1921), Johnson (1921), Jeffreys(1939), Carnap (1950), Cox (1946, 1961), and the subjectivistinterpretation of Ramsey (1926), Savage (1954), and de Finetti (1964).Attempts to argue for a probabilistica priori solution tothe problem of induction have been primarily associated with theseinterpretations.

Secondly, in the case of the urn, the Bayes-Laplace argument is basedon a particular probabilistic model—the binomial model. Thisinvolves the assumption that there is a parameter describing anunknown proportion \(\theta\) of balls in the urn, and that the dataamounts to independent draws from a distribution over that parameter.What is the basis of these assumptions? Do they generalize to othercases beyond the actual urn case—i.e., can we see observationsin general as analogous to draws from an “Urn of Nature”?There has been a persistent worry that these types of assumptions,while reasonable when applied to the case of drawing balls from anurn, will not hold for other cases of inductive inference. Thus, theprobabilistic solution to the problem of induction might be ofrelatively limited scope. At the least, there are some assumptionsgoing into the choice of model here that need to be made explicit.Arguably the choice of model introduces empirical assumptions, whichwould mean that the probabilistic solution is not ana priorione.

Thirdly, the Bayes-Laplace argument relies on a particular choice ofprior probability distribution. What is the status of this assignment,and can it be based ona priori principles? Historically, theBayes-Laplace choice of a uniform prior, as well as the whole conceptof classical probability, relied on the Principle of Indifference.This principle has been regarded by many as ana prioriprinciple. However, it has also been subjected to much criticism onthe grounds that it can give rise to inconsistent probabilityassignments (Bertrand 1888; Borel 1909; Keynes 1921). Suchinconsistencies are produced by there being more than one way to carveup the space of alternatives, and different choices give rise toconflicting probability assignments. One attempt to rescue thePrinciple of Indifference has been to appeal to explanationism, andargue that the principle should be applied only to the carving of thespace at “the most explanatorily basic level”, where thislevel is identified according to ana priori notion ofexplanatory priority (Huemer 2009).

The quest for ana priori argument for the assignment of theprior has been largely abandoned. For many, the subjectivistfoundations developed by Ramsey, de Finetti and Savage provide a moresatisfactory basis for understanding probability. From this point ofview, it is a mistake to try to introduce any furtherapriori constraints on the probabilities beyond those dictated bythe probability rules themselves. Rather the assignment of priors mayreflect personal opinions or background knowledge, and no prior isa priori an unreasonable choice.

So far, we have considered probabilistic arguments which placeprobabilities over hypotheses in a hypothesis space as well asobservations. There is also a tradition of attempts to determine whatprobability distributions we should have, given certain observations,from the starting point of a joint probability distribution over allthe observable variables. One may then postulate axioms directly onthis distribution over observables, and examine the consequences forthe predictive distribution. Much of the development of inductivelogic, including the influential programme by Carnap, proceeded inthis manner (Carnap 1950, 1952).

This approach helps to clarify the role of the assumptions behindprobabilistic models. One assumption that one can make about theobservations is that they are “exchangeable”. This meansthat the joint distribution of the random variables is invariant underpermutations. Informally, this means that the order of theobservations does not affect the probability. For instance, in the urncase, this would mean that drawing first a white ball and then a blackball is just as probable as first drawing a black and then a white. DeFinetti proved a general representation theorem that if the jointprobability distribution of an infinite sequence of random variablesis assumed to be exchangeable, then it can be written as a mixture ofdistribution functions from each of which the data behave as if theyare independent random draws (de Finetti 1964). In the case of the urnexample, the theorem shows that it isas if the data areindependent random draws from a binomial distribution over a parameter\(\theta\), which itself has a prior probability distribution.

The assumption of exchangeability may be seen as a naturalformalization of Hume’s assumption that the past resembles thefuture. This is intuitive because assuming exchangeability meansthinking that the order of observations, both past and future, doesnot matter to the probability assignments.

However, the development of the programme of inductive logic revealedthat many generalizations are possible. For example, Johnson proposedto assume an axiom he called the “sufficientnesspostulate”. This states that outcomes can be of a number ofdifferent types, and that the conditional probability that the nextoutcome is of typei depends only on the number of previoustrials and the number of previous outcomes of typei (Johnson1932). Assuming the sufficientness postulate for three or more typesgives rise to a general predictive distribution corresponding toCarnap’s “continuum of inductive methods” (Carnap1952). This predictive distribution takes the form:

for some positive numberk. This reduces to Laplace’srule of succession when \(t=2\) and \(k=1\).

Generalizations of the notion of exchangeability, such as“partial exchangeability” and “Markovexchangeability”, have been explored, and these may be thoughtof as forms of symmetry assumption (Zabell 1988; Skyrms 2012). As lessrestrictive axioms on the probabilities for observables are assumed,the result is that there is no longer a unique result for theprobability of a prediction, but rather a whole class of possibleprobabilities, mapped out by a generalized rule of succession such asthe above. Therefore, in this tradition, as in the Bayes-Laplaceapproach, we have moved away from producing an argument which producesa unique a priori probabilistic answer to Hume’s problem.

One might think then that the assignment of the prior, or the relevantcorresponding postulates on the observable probability distribution,is precisely where empirical assumptions enter into inductiveinferences. The probabilistic calculations are empirical arguments,rather thana priori ones. If this is correct, then theprobabilistic framework has not in the end provided anapriori solution to the problem of induction, but it has ratherallowed us to clarify what could be meant by Hume’s claim thatinductive inferences rely on the Uniformity Principle.

3.4 Partial solutions

Some think that although the problem of induction is not solved, thereis in some sense a partial solution, which has been called a“logical solution”. Howson, for example, argues that“Inductive reasoning is justified to the extent that it issound, given appropriate premises” (Howson 2000: 239, hisemphasis). According to this view, there is no getting away from anempirical premise for inductive inferences, but we might still thinkof Bayesian conditioning as functioning like a kind of logic or“consistency constraint” which “generatespredictions from the assumptions and observations together”(Romeijn 2004: 360). Once we have an empirical assumption,instantiated in the prior probability, and the observations, Bayesianconditioning tells us what the resulting predictive probabilitydistribution should be.

The idea of a partial solution also arises in the context of thelearning theory that grounds contemporary machine learning. Machinelearning is a field in computer science concerned with algorithms thatlearn from experience. Examples are algorithms which can be trained torecognise or classify patterns in data. Learning theory concernsitself with finding mathematical theorems which guarantee theperformance of algorithms which are in practical use. In this domain,there is a well-known finding that learning algorithms are onlyeffective if they have ‘inductive bias’ — that is, ifthey make some a priori assumptions about the domain they are employedupon (Mitchell 1997).

The idea is also given formal expression in the so-called‘No-Free-Lunch theorems’ (Wolpert 1992, 1996, 1997). Thesecan be interpreted as versions of the argument in Hume’s firstfork since they establish that there can be no contradiction in thealgorithm not performing well, since there area prioripossible situations in which it does not (Sterkenburg andGrünwald 2021:9992). Given Hume’s premiseP3, this rules out a demonstrative argument for its good performance.

PremiseP3 can perhaps be challenged on the grounds thata priorijustifications can also be given for contingent propositions. Eventhough an inductive inference can fail in some possible situations, itcould still be reasonable to form an expectation of reliability if wespread our credence equally over all the possibilities and have reasonto think (or at least no reason to doubt) that the cases whereinductive inference is unreliable require a ‘very specificarrangement of things’ and thus form a small fraction of thetotal space of possibilities (White 2015). The No-Free-Lunch theoremsmake difficulties for this approach since they show that if we put auniform distribution over all logically possible sequences of futureevents, any learning algorithm is expected to have a generalisationerror of 1/2, and hence to do no better than guessing at random(Schurz 2021b).

The No-Free-Lunch theorems may be seen as fundamental limitations onjustifying learning algorithms when these algorithms are seen as‘purely data-driven’ — that is as mappings from possibledata to conclusions. However, learning algorithms may also beconceived as functions not only of input data, but also of aparticular model (Sterkenburg and Grünwald 2021). For example,the Bayesian ‘algorithm’ gives a universal recipe fortaking a particular model and prior and updating on the data. A numberof theorems in learning theory provide general guarantees for theperformance of such recipes. For instance, there are theorems whichguarantee convergence of the Bayesian algorithm (Ghosal, Ghosh and vander Vaart 2000, Ghosal, Lember and van der Vaart 2008). In eachinstantiation, this convergence is relative to a particular specificprior. Thus, although the considerations first raised by Hume, andlater instantiated in the No-Free-Lunch theorems, preclude anyuniversal model-independent justification for learning algorithms, itdoes not rule out partial justifications in the form of such general apriori ‘model-relative’ learning guarantees (Sterkenburgand Grünwald 2021).

3.5 The combinatorial approach

An alternative attempt to use probabilistic reasoning to produce ana priori justification for inductive inferences is theso-called “combinatorial” solution. This was first putforward by Donald C. Williams (1947) and later developed by DavidStove (1986).

Like the Bayes-Laplace argument, the solution relies heavily on theidea that straightforwarda priori calculations can be donein a “direct inference” from population to sample. As wehave seen, given a certain population frequency, the probability ofgetting different frequencies in a sample can be calculatedstraightforwardly based on the rules of the probability calculus. TheBayes-Laplace argument relied on inverting the probabilitydistribution using Bayes’ rule to get from the samplingdistribution to the posterior distribution. Williams instead proposesthat the inverse inference may be based on a certain logicalsyllogism: the proportional (or statistical) syllogism.

The proportional, or statistical syllogism, is the following:

1. Of all the things that areM, \(m/n\) areP.
2. a is anM

Therefore,a isP, with probability \(m/n\).

For example, if 90% of rabbits in a population are white and weobserve a rabbita, then the proportional syllogism says thatwe infer thata is white with a probability of 90%. Williamsargues that the proportional syllogism is a non-deductive logicalsyllogism, which effectively interpolates between the syllogism forentailment

1. AllMs areP
2. a is anM

Therefore,a isP.

And the syllogism for contradiction

1. NoM isP
2. a isM

Therefore,a is notP.

This syllogism can be combined with an observation about the behaviorof increasingly large samples. From calculations of the samplingdistribution, it can be shown that as the sample size increases, theprobability that the sample frequency is in a range which closelyapproximates the population frequency also increases. In fact,Bernoulli’s law of large numbers states that the probabilitythat the sample frequency approximates the population frequency tendsto one as the sample size goes to infinity. Williams argues that suchresults support a “general over-all premise, common to allinductions, that samples ‘match’ their populations”(Williams 1947: 78).

We can then apply the proportional syllogism to samples from apopulation, to get the following argument:

1. Most samples match their population
2. S is a sample.

Therefore,S matches its population, with highprobability.

This is an instance of the proportional syllogism, and it uses thegeneral result about samples matching populations as the first majorpremise.

The next step is to argue that if we observe that the sample containsa proportion of \(m/n\)Fs, then we can conclude that sincethis sample with high probability matches its population, thepopulation, with high probability, has a population frequency thatapproximates the sample frequency \(m/n\). Both Williams and Stoveclaim that this amounts to a logicala priori solution to theproblem of induction.

A number of authors have expressed the view that the Williams-Stoveargument is only valid if the sampleS is drawn randomly fromthe population of possible samples—i.e., that any sample is aslikely to be drawn as any other (Brown 1987; Will 1948; Giaquinto1987). Sometimes this is presented as an objection to the applicationof the proportional syllogism. The claim is that the proportionalsyllogism is only valid ifa is drawn randomly from thepopulation ofMs. However, the response has been that thereis no need to know that the sample is randomly drawn in order to applythe syllogism (Maher 1996; Campbell 2001; Campbell & Franklin2004). Certainly if you have reason to think that your samplingprocedure is more likely to draw certain individuals thanothers—for example, if you know that you are in a certainlocation where there are more of a certain type—then you shouldnot apply the proportional syllogism. But if you have no such reasons,the defenders claim, it is quite rational to apply it. Certainly it isalways possible that you draw an unrepresentative sample—meaningone of the few samples in which the sample frequency does not matchthe population frequency—but this is why the conclusion is onlyprobable and not certain.

The more problematic step in the argument is the final step, whichtakes us from the claim that samples match their populations with highprobability to the claim that having seen a particular samplefrequency, the population from which the sample is drawn has frequencyclose to the sample frequency with high probability. The problem hereis a subtle shift in what is meant by “high probability”,which has formed the basis of a common misreading ofBernouilli’s theorem. Hacking (1975: 156–59) puts thepoint in the following terms. Bernouilli’s theorem licenses theclaim that much more often than not, a small interval around thesample frequency will include the true population frequency. In otherwords, it is highly probable in the sense of “usuallyright” to say that the sample matches its population. But thisdoes not imply that the proposition that a small interval around thesample will contain the true population frequency is highly probablein the sense of “credible on each occasion of use”. Thiswould mean that for any given sample, it is highly credible that thesample matches its population. It is quite compatible with the claimthat it is “usually right” that the sample matches itspopulation to say that there are some samples which do not match theirpopulations at all. Thus one cannot conclude from Bernouilli’stheorem that for any given sample frequency, we should assign highprobability to the proposition that a small interval around the samplefrequency will contain the true population frequency. But this isexactly the slide that Williams makes in the final step of hisargument. Maher (1996) argues in a similar fashion that the last stepof the Williams-Stove argument is fallacious. In fact, if one wants todraw conclusions about the probability of the population frequencygiven the sample frequency, the proper way to do so is by using theBayesian method described in the previous section. But, as we theresaw, this requires the assignment of prior probabilities, and thisexplains why many people have thought that the combinatorial solutionsomehow illicitly presupposed an assumption like the principle ofindifference. The Williams-Stove argument does not in fact give us analternative way of inverting the probabilities which somehow bypassesall the issues that Bayesians have faced.

4. Tackling the Second Horn of Hume’s Dilemma

So far we have considered ways in which the first horn of Hume’sdilemma might be tackled. But it is of course also possible to take onthe second horn instead.

One may argue that a probable argument would not, despite what Humesays, be circular in a problematic way (we consider responses of thiskind insection 4.1). Or, one might attempt to argue that probable arguments are notcircular at all (section 4.2).

4.1 Inductive Justifications of Induction

One way to tackle the second horn of Hume’s dilemma is to rejectpremise P6, which rules out circular arguments. Some have argued that certainkinds of circular arguments would provide an acceptable justificationfor the inductive inference. Since the justification would then itselfbe an inductive one, this approach is often referred to as an“inductive justification of induction”.

First we should examine how exactly the Humean circularity supposedlyarises. Take the simple case of enumerative inductive inference thatfollows the following pattern (X):

Hume claims that such arguments presuppose the Uniformity Principle(UP). According to premisesP7 andP8, this supposition also needs to be supported by an argument in orderthat the inductive inference be justified. A natural idea is that wecan argue for the Uniformity Principle on the grounds that “itworks”. We know that it works, because past instances ofarguments which relied upon it were found to be successful. This alonehowever is not sufficient unless we have reason to think that sucharguments will also be successful in the future. That claim mustitself be supported by an inductive argument (S):

But this argument itself depends on the UP, which is the verysupposition which we were trying to justify.

As we have seen insection 2, some reject Hume’s claim that all inductive inferencespresuppose the UP. However, the argument that basing the justificationof the inductive inference on a probable argument would result incircularity need not rely on this claim. The circularity concern canbe framed more generally. If argumentS relies onsomething which is already presupposed in inferenceX, then argumentS cannot be used to justifyinferenceX. The question though is what precisely thesomething is.

Some authors have argued that in factS does not rely on anypremise or even presupposition that would require us to already knowthe conclusion ofX.S is then not a “premisecircular” argument. Rather, they claim, it is“rule-circular”—it relies on a rule of inference inorder to reach the conclusion that that very rule is reliable. Supposewe adopt the ruleR which says that when it is observed thatmostFs areGs, we should infer that mostFs areGs. Then inferenceX relies on ruleR. We want to show that ruleR is reliable. We couldappeal to the fact thatR worked in the past, and so, by aninductive argument, it will also work in the future. Call thisargumentS*:

Since this argument itself uses ruleR, using it to establishthatR is reliable is rule-circular.

Some authors have then argued that although premise-circularity isvicious, rule-circularity is not (Cleve 1984; Papineau 1992). Onereason for thinking rule-circularity is not vicious would be if it isnot necessary to know or even justifiably believe that ruleRis reliable in order to move to a justified conclusion using the rule.This is a claim made by externalists about justification (Cleve 1984).They say that as long asR isin fact reliable, onecan form a justified belief in the conclusion of an argument relyingonR, as long as one has justified belief in thepremises.

If one is not persuaded by the externalist claim, one might attempt toargue that rule circularity is benign in a different fashion. Forexample, the requirement that a rule be shown to be reliable withoutany rule-circularity might appear unreasonable when the rule is of avery fundamental nature. As Lange puts it:

It might be suggested that although a circular argument is ordinarilyunable to justify its conclusion, a circular argument is acceptable inthe case of justifying a fundamental form of reasoning. After all,there is nowhere more basic to turn, so all that we can reasonablydemand of a fundamental form of reasoning is that it endorse itself.(Lange 2011: 56)

Proponents of this point of view point out that even deductiveinference cannot be justified deductively. Consider LewisCarroll’s dialogue between Achilles and the Tortoise (Carroll1895). Achilles is arguing with a Tortoise who refuses to performmodus ponens. The Tortoise accepts the premise thatp, and the premise thatp impliesq but hewill not acceptq. How can Achilles convince him? He managesto persuade him to accept another premise, namely “ifpandp impliesq, thenq”. But theTortoise is still not prepared to infer toq. Achilles goeson adding more premises of the same kind, but to no avail. It appearsthen thatmodus ponens cannot be justified to someone who isnot already prepared to use that rule.

It might seem odd if premise circularity were vicious, and rulecircularity were not, given that there appears to be an easyinterchange between rules and premises. After all, a rule can always,as in the Lewis Carroll story, be added as a premise to the argument.But what the Carroll story also appears to indicate is that there isindeed a fundamental difference between being prepared to accept apremise stating a rule (the Tortoise is happy to do this), and beingprepared to use that rule (this is what the Tortoise refuses todo).

Suppose that we grant that an inductive argument such asS(orS*) can support an inductive inferenceX withoutvicious circularity. Still, a possible objection is that the argumentsimply does not provide a full justification ofX. After all,less sane inference rules such as counterinduction can supportthemselves in a similar fashion. The counterinductive rule is CI:

Consider then the following argument CI*:

This argument therefore establishes the reliability of CI in arule-circular fashion (see Salmon 1963).

ArgumentS can be used to support inferenceX, butonly for someone who is already prepared to infer inductively by usingS. It cannot convince a skeptic who is not prepared to relyupon that rule in the first place. One might think then that theargument is simply not achieving very much.

The response to these concerns is that, as Papineau puts it, theargument is “notsupposed to do very much”(Papineau 1992: 18). The fact that a counterinductivist counterpart ofthe argument exists is true, but irrelevant. It is conceded that theargument cannot persuade either a counterinductivist, or a skeptic.Nonetheless, proponents of the inductive justification maintain thatthere is still some added value in showing that inductive inferencesare reliable, even when we already accept that there is nothingproblematic about them. The inductive justification of inductionprovides a kind of important consistency check on our existingbeliefs.

4.2 No Rules

It is possible to go even further in an attempt to dismantle theHumean circularity. Maybe inductive inferences do not even have a rulein common. What if every inductive inference is essentially unique?This can be seen as rejecting Hume’s premiseP5. Okasha, for example, argues that Hume’s circularity problem canbe evaded if there are “no rules” behind induction (Okasha2005a,b). Norton puts forward the similar idea that all inductiveinferences are material, and have nothing formal in common (Norton2003, 2010, 2021).

Proponents of such views have attacked Hume’s claim that thereis a UP on which all inductive inferences are based. There have longbeen complaints about the vagueness of the Uniformity Principle(Salmon 1953). The future only resembles the past in some respects,but not others. Suppose that on all my birthdays so far, I have beenunder 40 years old. This does not give me a reason to expect that Iwill be under 40 years old on my next birthday. There seems then to bea major lacuna in Hume’s account. He might have explained ordescribed how we draw an inductive inference, on the assumption thatit is one wecan draw. But he leaves untouched the questionof how we distinguish between cases where we extrapolate a regularitylegitimately, regarding it as a law, and cases where we do not.

Nelson Goodman is often seen as having made this point in aparticularly vivid form with his “new riddle of induction”(Goodman 1955: 59–83). Suppose we define a predicate“grue” in the following way. An object is“grue” when it is green if observed before timetand blue otherwise. Goodman considers a thought experiment in which weobserve a bunch of green emeralds before timet. We coulddescribe our results by saying all the observed emeralds are green.Using a simple enumerative inductive schema, we could infer from theresult that all observed emeralds are green, that all emeralds aregreen. But equally, we could describe the same results by saying thatall observed emeralds are grue. Then using the same schema, we couldinfer from the result that all observed emeralds are grue, that allemeralds are grue. In the first case, we expect an emerald observedafter timet to be green, whereas in the second, we expect itto be blue. Thus the two predictions are incompatible. Goodman claimsthat what Hume omitted to do was to give any explanation for why weproject predicates like “green”, but not predicates like“grue”. This is the “new riddle”, which isoften taken to be a further problem of induction that Hume did notaddress.

One moral that could be taken from Goodman is that there is not onegeneral Uniformity Principle that all probable arguments rely upon(Sober 1988; Norton 2003; Okasha 2001, 2005a,b, Jackson 2019). Rathereach inductive inference presupposes some more specific empiricalpresupposition. A particular inductive inference depends on somespecific way in which the future resembles the past. It can then bejustified by another inductive inference which depends on some quitedifferent empirical claim. This will in turn need to bejustified—by yet another inductive inference. The nature ofHume’s problem in the second horn is thus transformed. There isno circularity. Rather there is a regress of inductive justifications,each relying on their own empirical presuppositions (Sober 1988;Norton 2003; Okasha 2001, 2005a,b).

One way to put this point is to say that Hume’s argument restson a quantifier shift fallacy (Sober 1988; Okasha 2005a). Hume saysthat there exists a general presupposition for all inductiveinferences, whereas he should have said that for each inductiveinference, there is some presupposition. Different inductiveinferences then rest on different empirical presuppositions, and theproblem of circularity is evaded.

What will then be the consequence of supposing that Hume’sproblem should indeed have been a regress, rather than a circularity?Here different opinions are possible. On the one hand, one might thinkthat a regress still leads to a skeptical conclusion (Schurz and Thorn2020). So although the exact form in which Hume stated his problem wasnot correct, the conclusion is not substantially different (Sober1988). Another possibility is that the transformation mitigates oreven removes the skeptical problem. For example, Norton argues thatthe upshot is a dissolution of the problem of induction, since theregress of justifications benignly terminates (Norton 2003). AndOkasha more mildly suggests that even if the regress is infinite,“Perhaps infinite regresses are less bad than vicious circlesafter all” (Okasha 2005b: 253).

Any dissolution of Hume’s circularity does not depend only onarguing that the UP should be replaced by empirical presuppositionswhich are specific to each inductive inference. It is also necessaryto establish that inductive inferences share no commonrules—otherwise there will still be at least somerule-circularity. Okasha suggests that the Bayesian model ofbelief-updating is an illustration how induction can be characterizedin a rule-free way, but this is problematic, since in this model allinductive inferences still share the common rule of Bayesianconditionalisation. Norton’s material theory of inductionpostulates a rule-free characterization of induction, but it is notclear whether it really can avoid any role for general rules(Achinstein 2010, Kelly 2010, Worrall 2010).

5. Alternative Conceptions of Justification

Hume is usually read as delivering a negative verdict on thepossibility of justifying inferenceI, via a premise such asP8, though as we have seen in sectionsection 2, some have questioned whether Hume is best interpreted as drawing aconclusion about justification of inferenceI at all. In thissection we examine approaches which question in different ways whetherpremise P8 really does give a valid necessary condition for justification ofinferenceI and propose various alternative conceptions ofjustification.

5.1 Postulates and Hinges

One approach has been to turn to general reflection on what is evenneeded for justification of an inference in the first place. Forexample, Wittgenstein raised doubts over whether it is even meaningfulto ask for the grounds for inductive inferences.

If anyone said that information about the past could not convince himthat something would happen in the future, I should not understandhim. One might ask him: what do you expect to be told, then? What sortof information do you call a ground for such a belief? … Ifthese are not grounds, then what are grounds?—If you say theseare not grounds, then you must surely be able to state what must bethe case for us to have the right to say that there are grounds forour assumption…. (Wittgenstein 1953: 481)

One might not, for instance, think that there even needs to be a chainof reasoning in which each step or presupposition is supported by anargument. Wittgenstein took it that there are some principles sofundamental that they do not require support from any furtherargument. They are the “hinges” on which enquiryturns.

Out of Wittgenstein’s ideas has developed a general notion of“entitlement”, which is a kind of rational warrant to holdcertain propositions which does not come with the same requirements as“justification”. Entitlement provides epistemic rights tohold a proposition, without responsibilities to base the belief in iton an argument. Crispin Wright (2004) has argued that there arecertain principles, including the Uniformity Principle, that we areentitled in this sense to hold.

Some philosophers have set themselves the task of determining a set orsets of postulates which form a plausible basis for inductiveinferences. Bertrand Russell, for example, argued that five postulateslay at the root of inductive reasoning (Russell 1948). Arthur Burks,on the other hand, proposed that the set of postulates is not unique,but there may be multiple sets of postulates corresponding todifferent inductive methods (Burks 1953, 1955).

The main objection to all these views is that they do not really solvethe problem of induction in a way that adequately secures the pillarson which inductive inference stands. As Salmon puts it,“admission of unjustified and unjustifiable postulates to dealwith the problem is tantamount to making scientific method a matter offaith” (Salmon 1966: 48).

5.2 Ordinary Language Dissolution

Rather than allowing undefended empirical postulates to give normativesupport to an inductive inference, one could instead argue for acompletely different conception of what is involved in justification.Like Wittgenstein, later ordinary language philosophers, notably P.F.Strawson, also questioned what exactly it means to ask for ajustification of inductive inferences (Strawson 1952). This has becomeknown as the “Ordinary language dissolution” of theproblem of induction.

Strawson points out that it could be meaningful to ask for a deductivejustification of inductive inferences. But it is not clear that thisis helpful since this is effectively “a demand that inductionshall be shown to be really a kind of deduction” (Strawson 1952:230). Rather, Strawson says, when we ask about whether a particularinductive inference is justified, we are typically judging whether itconforms to our usual inductive standards. Suppose, he says, someonehas formed the belief by inductive inference that Allf’s areg. Strawson says that if that personis asked for their grounds or reasons for holding that belief,

I think it would be felt to be a satisfactory answer if he replied:“Well, in all my wide and varied experience I’ve comeacross innumerable cases off and never a case offwhich wasn’t a case ofg”. In saying this, he isclearly claiming to haveinductive support,inductive evidence, of a certain kind, for his belief.(Strawson 1952)

That is just because inductive support, as it is usually understood,simply consists of having observed many positive instances in a widevariety of conditions.

In effect, this approach denies that producing a chain of reasoning isa necessary condition for justification. Rather, an inductiveinference is justified if it conforms to the usual standards ofinductive justification. But, is there more to it? Might we not askwhat reason we have to rely on those inductive standards?

It surely makes sense to ask whether a particular inductive inferenceis justified. But the answer to that is fairly straightforward.Sometimes people have enough evidence for their conclusions andsometimes they do not. Does it also make sense to ask about whetherinductive procedures generally are justified? Strawson draws theanalogy between asking whether a particular act is legal. We mayanswer such a question, he says, by referring to the law of theland.

But it makes no sense to inquire in general whether the law of theland, the legal system as a whole, is or is not legal. For to whatlegal standards are we appealing? (Strawson 1952: 257)

According to Strawson,

It is an analytic proposition that it is reasonable to have a degreeof belief in a statement which is proportional to the strength of theevidence in its favour; and it is an analytic proposition, though nota proposition of mathematics, that, other things being equal, theevidence for a generalisation is strong in proportion as the number offavourable instances, and the variety of circumstances in which theyhave been found, is great. So to ask whether it is reasonable to placereliance on inductive procedures is like asking whether it isreasonable to proportion the degree of one’s convictions to thestrength of the evidence. Doing this is what “beingreasonable”means in such a context. (Strawson 1952:256–57)

Thus, according to this point of view, there is no further question toask about whether it is reasonable to rely on inductiveinferences.

The ordinary language philosophers do not explicitly argue againstHume’spremise P8. But effectively what they are doing is offering a whole differentstory about what it would mean to be justified in believing theconclusion of inductive inferences. What is needed is just conformityto inductive standards, and there is no real meaning to asking for anyfurther justification for those.

The main objection to this view is that conformity to the usualstandards is insufficient to provide the needed justification. What weneed to know is whether belief in the conclusion of an inductiveinference is “epistemically reasonable or justified in the sensethat …there is reason to think that it is likely to betrue” (BonJour 1998: 198). The problem Hume has raised iswhether, despite the fact that inductive inferences have tended toproduce true conclusions in the past, we have reason to think theconclusion of an inductive inference we now make is likely to be true.Arguably, establishing that an inductive inference is rational in thesense that it follows inductive standards is not sufficient toestablish that its conclusion is likely to be true. In fact Strawsonallows that there is a question about whether “induction willcontinue to be successful”, which is distinct from the questionof whether induction is rational. This question he does take to hingeon a “contingent, factual matter” (Strawson 1952: 262).But if it is this question that concerned Hume, it is no answer toestablish that induction is rational, unless that claim is understoodto involve or imply that an inductive inference carried out accordingto rational standards is likely to have a true conclusion.

5.3 Pragmatic vindication of induction

Another solution based on an alternative criterion for justificationis the “pragmatic” approach initiated by Reichenbach (1938[2006]). Reichenbach did think Hume’s argument unassailable, butnonetheless he attempted to provide a weaker kind of justification forinduction. In order to emphasize the difference from the kind ofjustification Hume sought, some have given it a different term andrefer to Reichenbach’s solution as a “vindication”,rather than a justification of induction (Feigl 1950; Salmon1963).

Reichenbach argued that it was not necessary for the justification ofinductive inference to show that its conclusion is true. Rather“the proof of the truth of the conclusion is only a sufficientcondition for the justification of induction, not a necessarycondition” (Reichenbach 2006: 348). If it could be shown, hesays, that inductive inference is a necessary condition of success,then even if we do not know that it will succeed, we still have somereason to follow it. Reichenbach makes a comparison to the situationwhere a man is suffering from a disease, and the physician says“I do not know whether an operation will save the man, but ifthere is any remedy, it is an operation” (Reichenbach 1938[2006: 349]). This provides some kind of justification for operatingon the man, even if one does not know that the operation willsucceed.

In order to get a full account, of course, we need to say more aboutwhat is meant for a method to have “success”, or to“work”. Reichenbach thought that this should be defined inrelation to the aim of induction. This aim, he thought, is“to find series of events whose frequency of occurrenceconverges towards a limit” (1938 [2006: 350]).

Reichenbach applied his strategy to a general form of“statistical induction” in which we observe the relativefrequency \(f_n\) of a particular event inn observations andthen form expectations about the frequency that will arise when moreobservations are made. The “inductive principle” thenstates that if after a certain number of instances, an observedfrequency of \(m/n\) is observed, for any prolongation of the seriesof observations, the frequency will continue to fall within a smallinterval of \(m/n\). Hume’s examples are special cases of thisprinciple, where the observed frequency is 1. For example, inHume’s bread case, suppose bread was observed to nourishn times out ofn (i.e. an observed frequency of100%), then according to the principle of induction, we expect that aswe observe more instances, the frequency of nourishing ones willcontinue to be within a very small interval of 100%. Following thisinductive principle is also sometimes referred to as following the“straight rule”. The problem then is to justify the use ofthis rule.

Reichenbach argued that even if Hume is right to think that we cannotbe justified in thinking for any particular application of the rulethat the conclusion is likely to be true, for the purposes ofpractical action we do not need to establish this. We can insteadregard the inductive rule as resulting in a “posit”, orstatement that we deal with as if it is true. We posit a certainfrequencyf on the basis of our evidence, and this is likemaking a wager or bet that the frequency is in factf. Onestrategy for positing frequencies is to follow the rule ofinduction.

Reichenbach proposes that we can show that the rule of induction meetshis weaker justification condition. This does not require showing thatfollowing the inductive principle will always work. It is possiblethat the world is so disorderly that we cannot construct series withany limits. In that case, neither the inductive principle, nor anyother method will succeed. But, he argues, if there is a limit, byfollowing the inductive principle we will eventually find it. There issome element of a series of observations, beyond which the principleof induction will lead to the true value of the limit. Although theinductive rule may give quite wrong results early in the sequence, asit follows chance fluctuations in the sample frequency, it isguaranteed to eventually approximate the limiting frequency, if such alimit exists. Therefore, the rule of induction is justified as aninstrument of positing because it is a method of which we know that ifit is possible to achieve the aim of inductive inference we shall doso by means of this method (Reichenbach 1949: 475).

One might question whether Reichenbach has achieved his goal ofshowing that following the inductive rule is a necessary condition ofsuccess. In order to show that, one would also need to establish thatno other methods can also achieve the aim. But, as Reichenbach himselfrecognises, many other rules of inference as well as the straight rulemay also converge on the limit (Salmon 1966: 53). In fact, any methodwhich converges asymptotically to the straight rule also does so. Aneasily specified class of such rules are those which add to theinductive rule a function \(c_n\) in which the \(c_n\) converge tozero with increasingn.

Reichenbach makes two suggestions aimed at avoiding this problem. Onthe one hand, he claims, since we have no real way to pick betweenmethods, we might as well just use the inductive rule since it is“easier to handle, owing to its descriptive simplicity”.He also claims that the method which embodies the “smallestrisk” is following the inductive rule (Reichenbach 1938 [2006:355–356]).

There is also the concern that there could be a completely differentkind of rule which converges on the limit. We can consider, forexample, the possibility of a soothsayer or psychic who is able topredict future events reliably. Here Reichenbach argues that inductionis still necessary in such a case, because it has to be used to checkwhether the other method works. It is only by using induction,Reichenbach says, that we could recognise the reliability of thealternative method, by examining its track record.

In assessing this argument, it is helpful to distinguish betweenlevels at which the principle of induction can be applied. FollowingSkyrms (2000), we may distinguish between level 1, where candidatemethods are applied to ordinary events or individuals, and level 2,where they are applied not to individuals or events, but to thearguments on level 1. Let us refer to “object-induction”when the inductive principle is applied at level 1, and“meta-induction” when it is applied at level 2.Reichenbach’s response does not rule out the possibility thatanother method might do better than object-induction at level 1. Itonly shows that the success of that other method may be recognised bya meta-induction at level 2 (Skyrms 2000). Nonetheless,Reichenbach’s thought was later picked up and developed into thesuggestion that a meta-inductivist who applies induction not only atthe object level to observations, but also to the success ofothers’ methods, might by those means be able to do as wellpredictively as the alternative method (Schurz 2008; seesection 5.5 for more discussion of meta-induction).

Reichenbach’s justification is generally taken to be a pragmaticone, since though it does not supply knowledge of a future event, itsupplies a sufficient reason for action (Reichenbach 1949: 481). Onemight question whether a pragmatic argument can really deliver anall-purpose, general justification for following the inductive rule.Surely a pragmatic solution should be sensitive to differences inpay-offs that depend on the circumstances. For example, Reichenbachoffers the following analogue to his pragmatic justification:

We may compare our situation to that of a man who wants to fish in anunexplored part of the sea. There is no one to tell him whether or notthere are fish in this place. Shall he cast his net? Well, if he wantsto fish in that place, I should advise him to cast the net, to takethe chance at least. It is preferable to try even in uncertainty thannot to try and be certain of getting nothing. (Reichenbach 1938 [2006:362–363])

As Lange points out, the argument here “presumes that there isno cost to trying”. In such a situation, “the fishermanhas everything to gain and nothing to lose by casting his net”(Lange 2011: 77). But if there is some significant cost to making theattempt, it may not be so clear that the most rational course ofaction is to cast the net. Similarly, whether or not it would makesense to adopt the policy of making no predictions, rather than thepolicy of following the inductive rule, may depend on what thepractical penalties are for being wrong. A pragmatic solution may notbe capable of offering rationale for following the inductive rulewhich is applicable in all circumstances.

Another question is whether Reichenbach has specified the aim ofinduction too narrowly. Finding series of events whose frequency ofoccurrence converges to a limit ties the vindication to the long-run,while allowing essentially no constraint on what can be posited in theshort-run. Yet it is in the short run that inductive practice actuallyoccurs and where it really needs justification (BonJour 1998: 194;Salmon 1966: 53).

5.4 Formal Learning Theory

Formal learning theory can be regarded as a kind of extension of theReichenbachian programme. It does not offer justifications forinductive inferences in the sense of giving reasons why they should betaken as likely to provide a true conclusion. Rather it offers a“means-ends” epistemology -- it provides reasons forfollowing particular methods based on their optimality in achievingcertain desirable epistemic ends, even if there is no guarantee thatat any given stage of inquiry the results they produce are at allclose to the truth (Schulte 1999).

Formal learning theory is particularly concerned with showing thatmethods are “logically reliable” in the sense that theyarrive at the truth given any sequence of data consistent with ourbackground knowledge (Kelly 1996). However, it goes further than this.As we have just seen, one of the problems for Reichenbach was thatthere are too many rules which converge in the limit to the truefrequency. Which one should we then choose in the short-run? Formallearning theory broadens Reichenbach’s general strategy byconsidering what happens if we have other epistemic goals besideslong-run convergence to the truth. In particular, formal learningtheorists have considered the goal of getting to the truth asefficiently, or quickly, as possible, as well as the goal ofminimising the number of mind-changes, or retractions along the way.It has then been argued that the usual inductive method, which ischaracterised by a preference for simpler hypotheses (Occam’srazor), can be justified since it is the unique method which meets thestandards for getting to the truth in the long run as efficiently aspossible, with a minimum number of retractions (Kelly 2007).

Steel (2010) has proposed that the Principle of Induction (understoodas a rule which makes inductive generalisations along the lines of theStraight Rule) can be given a means-ends justification by showing thatfollowing it is both necessary and sufficient for logical reliability.The proof is an a priori mathematical one, thus it allegedly avoidsthe circularity of Hume’s second horn. However, Steel also doesnot see the approach as an attempt to grasp Hume’s first horn,since the proof is only relative to a certain choice of epistemicends.

As with other results in formal learning theory, this solution is alsoonly valid relative to a given hypothesis space and conception ofpossible sequences of data. For this reason, some have seen it as notaddressing Hume’s problem of giving grounds for a particularinductive inference (Howson 2011). An alternative attitude is that itdoes solve a significant part of Hume’s problem (Steel 2010).There is a similar dispute over formal learning theory’streatment of Goodman’s riddle (Chart 2000, Schulte 2017).

5.5 Meta-induction

Another approach to pursuing a broadly Reichenbachian programme isGerhard Schurz’s strategy based on meta-induction (Schurz 2008,2017, 2019). Schurz draws a distinction between applying inductivemethods at the level of events—so-called“object-level” induction (OI), and applying inductivemethods at the level of competing prediction methods—so-called“meta-induction” (MI). Whereas object-level inductivemethods make predictions based on the events which have been observedto occur, meta-inductive methods make predictions based on aggregatingthe predictions of different available prediction methods according totheir success rates. Here, the success rate of a method is definedaccording to some precise way of scoring success in makingpredictions.

The starting point of the meta-inductive approach is that the aim ofinductive inference is not just, as Reichenbach had it, findinglong-run limiting frequencies, but also predicting successfully inboth the long and short run. Even if Hume has precluded showing thatthe inductive method is reliable in achieving successful prediction,perhaps it can still be shown that it is “predictivelyoptimal”. A method is “predictively optimal” if itsucceeds best in making successful predictions out of all competingmethods, no matter what data is received. Schurz brings to bearresults from the regret-based learning framework in machine learningthat show that there is a meta-inductive strategy that is predictivelyoptimal among all predictive methods that are accessible to anepistemic agent (Cesa-Bianchi and Lugosi 2006, Schurz 2008, 2017,2019). This meta-inductive strategy, which Schurz calls“wMI”, predicts a weighted average of the predictions ofthe accessible methods, where the weights are“attractivities”, which measure the difference between themethod’s own success rate and the success rate of wMI.

The main result is that the wMI strategy is long-run optimal in thesense that it converges to the maximum success rate of the accessibleprediction methods. Worst-case bounds for short-run performance canalso be derived. The optimality result forms the basis for anapriori means-ends justification for the use of wMI. Namely, thethought is, it is reasonable to use wMI, since it achieves the bestsuccess rates possible in the long run out of the given methods.

Schurz also claims that thisa priori justification of wMI,together with the contingent fact that inductive methods have so farbeen much more successful than non-inductive methods, gives rise to ana posteriori non-circular justification of induction. SincewMI will achieve in the long run the maximal success rate of theavailable prediction methods, it is reasonable to use it. But as amatter of fact, object-inductive prediction methods have been moresuccessful than non-inductive methods so far. Therefore Schurz says“it is meta-inductively justified to favor object-inductivisticstrategies in the future” (Schurz 2019: 85). This justification,he claims, is not circular because meta-induction has anapriori independent justification. The idea is that since it isa priori justified to use wMI, it is alsoa priorijustified to use the maximally successful method at the object level.Since it turns out that that the maximally successful method isobject-induction, then we have a non-circulara posterioriargument that it is reasonable to use object-induction.

Schurz’s original theorems on the optimality of wMI apply to thecase where there are finitely many predictive methods. One point ofdiscussion is whether this amounts to an important limitation on itsclaims to provide a full solution of the problem of induction. Thequestion then is whether it is necessary that the optimality resultsbe extended to an infinite, or perhaps an expanding pool of strategies(Eckhardt 2010, Sterkenburg 2019, Schurz 2021a).

Another important issue concerns what it means for object-induction tobe “meta-inductively justified”. The meta-inductivestrategy wMI and object-induction are clearly different strategies.They could result in different predictions tomorrow, if OI would stopworking and another method would start to do better. In that case, wMIwould begin to favour the other method, and wMI would start to comeapart from OI. The optimality results provide a reason to follow wMI.How exactly does object-induction inherit that justification? At most,it seems that we get a justification for following OI on the nexttime-step, on the grounds that OI’s prediction approximatelycoincides with that of wMI (Sterkenburg 2020, Sterkenburg(forthcoming)). However, this requires a stronger empirical postulatethan simply the observation that OI has been more successful thannon-inductive methods. It also requires something like that “asa matter of empirical fact, the strategy OI has been so much moresuccessful than its competitors, that the meta-inductivist attributesit such a large share of the total weight that its prediction(approximately) coincides with OI’s prediction”(Sterkenburg 2020: 538). Furthermore, even if we allow that theempirical evidence does back up such a strong claim, the issue remainsthat the meta-inductive justification is in support of following thestrategy of meta-induction, not in support of thestrategy offollowing OI (Sterkenburg (2020), sec. 3.3.2).

6. Living with Inductive Skepticism

So far we have considered the various ways in which we might attemptto solve the problem of induction by resisting one or other premise ofHume’s argument. Some philosophers have however seen hisargument as unassailable, and have thus accepted that it does lead toinductive skepticism, the conclusion that inductive inferences cannotbe justified. The challenge then is to find a way of living with sucha radical-seeming conclusion. We appear to rely on inductive inferenceubiquitously in daily life, and it is also generally thought that itis at the very foundation of the scientific method. Can we go on withall this, whilst still seriously thinking none of it is justified byany rational argument?

One option here is to argue, as does Nicholas Maxwell, that theproblem of induction is posed in an overly restrictive context.Maxwell argues that the problem does not arise if we adopt a differentconception of science than the ‘standard empiricist’ one,which he denotes ‘aim-oriented empiricism’ (Maxwell2017).

Another option here is to think that the significance of the problemof induction is somehow restricted to a skeptical context. Humehimself seems to have thought along these lines. For instance hesays:

Nature will always maintain her rights, and prevail in the end overany abstract reasoning whatsoever. Though we should conclude, forinstance, as in the foregoing section, that, in all reasonings fromexperience, there is a step taken by the mind, which is not supportedby any argument or process of the understanding; there is no danger,that these reasonings, on which almost all knowledge depends, willever be affected by such a discovery. (E. 5.1.2)

Hume’s purpose is clearly not to argue that we should not makeinductive inferences in everyday life, and indeed his whole method andsystem of describing the mind in naturalistic terms depends oninductive inferences through and through. The problem of inductionthen must be seen as a problem that arises only at the level ofphilosophical reflection.

Another way to mitigate the force of inductive skepticism is torestrict its scope. Karl Popper, for instance, regarded the problem ofinduction as insurmountable, but he argued that science is not in factbased on inductive inferences at all (Popper 1935 [1959]). Rather hepresented a deductivist view of science, according to which itproceeds by making bold conjectures, and then attempting to falsifythose conjectures. In the simplest version of this account, when ahypothesis makes a prediction which is found to be false in anexperiment, the hypothesis is rejected as falsified. The logic of thisprocedure is fully deductive. The hypothesis entails the prediction,and the falsity of the prediction refutes the hypothesis by modustollens. Thus, Popper claimed that science was not based on theextrapolative inferences considered by Hume. The consequence then isthat it is not so important, at least for science, if those inferenceswould lack a rational foundation.

Popper’s account appears to be incomplete in an important way.There are always many hypotheses which have not yet been refuted bythe evidence, and these may contradict one another. According to thestrictly deductive framework, since none are yet falsified, they areall on an equal footing. Yet, scientists will typically want to saythat one is better supported by the evidence than the others. We seemto need more than just deductive reasoning to support practicaldecision-making (Salmon 1981). Popper did indeed appeal to a notion ofone hypothesis being better or worse “corroborated” by theevidence. But arguably, this took him away from a strictly deductiveview of science. It appears doubtful then that pure deductivism cangive an adequate account of scientific method.

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Other Internet Resources

• Vickers, John, “The Problem of Induction,”Stanford Encyclopedia of Philosophy (Spring 2018 Edition),Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2018/entries/induction-problem/>. [This was the previous entry on the problem of induction in theStanford Encyclopedia of Philosophy — see theversion history.]
• Teaching Theory of Knowledge: Probability and Induction, organization of topics and bibliography by Brad Armendt (ArizonaState University) and Martin Curd (Purdue).
• Forecasting Principles, A brief survey of prediction markets.

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Acknowledgments

Particular thanks are due to Don Garrett and Tom Sterkenburg forhelpful feedback on a draft of this entry. Thanks also to DavidAtkinson, Simon Friederich, Jeanne Peijnenburg, Theo Kuipers andJan-Willem Romeijn for comments.