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The Problem of Induction

We generally think that the observations we make are able to justify some expectations or predictions about observations we have not yet made, as well as general claims that go beyond the observed. For example, the observation that bread of a certain appearance has thus far been nourishing seems to justify the expectation that the next similar piece of bread I eat will also be nourishing, as well as the claim that bread of this sort is generally nourishing. Such inferences from 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 of induction” is in Book 1, part iii, section 6 of A Treatise of Human Nature by David Hume, published in 1739 (Hume 1739). In 1748, Hume gave a shorter version of the argument in Section iv of An enquiry concerning human understanding (Hume 1748). Throughout this article we will give references to the Treatise as “T”, and the Enquiry as “E”.

Hume asks on what grounds we come to our beliefs about the unobserved on the basis of inductive inferences. He presents an argument in the form of a dilemma which appears to rule out the possibility of any reasoning from the premises to the conclusion of an inductive inference. There are, he says, two possible types of arguments, “demonstrative” and “probable”, but neither will serve. A demonstrative argument produces the wrong kind of conclusion, and a probable argument would be circular. Therefore, for Hume, the problem remains of how to explain why we form any conclusions that go beyond the past instances of which we have had experience (T. 1.3.6.10). Hume stresses that he is not disputing that we do draw such inferences. The challenge, as he sees it, is to understand the “foundation” of the inference—the “logic” or “process of argument” that it is based upon (E. 4.2.21). The problem of meeting this challenge, while evading Hume’s argument against the possibility of doing so, has become known as “the problem of induction”.

Hume’s argument is one of the most famous in philosophy. A number of philosophers have attempted solutions to the problem, but a significant number have embraced his conclusion that it is insoluble. There is also a wide spectrum of opinion on the significance of the problem. Some have argued that Hume’s argument does not establish any far-reaching skeptical conclusion, either because it was never intended to, or because the argument is in some way misformulated. Yet many have regarded it as one of the most profound philosophical challenges imaginable since it seems to call into question the justification of one of the most fundamental ways in which we form knowledge. Bertrand Russell, for example, expressed the view that if Hume’s problem cannot be solved, “there is no intellectual difference between sanity and insanity” (Russell 1946: 699).

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

1. Hume’s Problem

2. reconstruction, 3.1 synthetic a priori, 3.2 the nomological-explanatory solution, 3.3 bayesian solution, 3.4 partial solutions, 3.5 the combinatorial approach, 4.1 inductive justifications of induction, 4.2 no rules, 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, other internet resources, related entries.

Hume introduces the problem of induction as part of an analysis of the notions of cause and effect. Hume worked with a picture, widespread in the early modern period, in which the mind was populated with mental entities called “ideas”. Hume thought that ultimately all our ideas could be traced back to the “impressions” of sense experience. In the simplest case, an idea enters the mind by being “copied” from the corresponding impression (T. 1.1.1.7/4). More complex ideas are then created by the combination of simple ideas (E. 2.5/19). Hume took there to be a number of relations between ideas, including the relation of causation (E. 3.2). (For more on Hume’s philosophy in general, see Morris & Brown 2014).

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

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

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

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

In the Treatise , Hume says that

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

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

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

All reasonings may be divided into two kinds, namely, demonstrative reasoning, 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 in turn, and in each case argues that it is impossible for it to supply an argument for the Uniformity Principle.

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

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

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

Second, Hume argues that the reasoning also cannot be “such as regard 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 be conformable to the past”, in other words on the Uniformity Principle (E. 4.2.19).

Therefore, if the chain of reasoning is based on an argument of this kind it will again be relying on this supposition, “and taking that 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 then fails 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 the ultimate connexion of causes and effects, but even after experience has inform’d us of their constant conjunction , ’tis impossible for us to satisfy ourselves by our reason, why we shou’d extend that experience beyond those particular instances, which have fallen under our observation. (T. 1.3.6.11/91–2)

The conclusion then is that our tendency to project past regularities into the future is not underpinned by reason. The problem of induction is to find a way to avoid this conclusion, despite Hume’s argument.

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 inductive inferences are driven by, if not reason. In the Treatise Hume raises the problem of induction in an explicitly contrastive way. He asks whether the transition involved in the inference is produced

by means of the understanding or imagination; whether we are determin’d by reason to make the transition, or by a certain association 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 one object to the idea or belief of another, it is not determin’d by reason, but by certain principles, which associate together the ideas of these objects, and unite them in the imagination. (T. 1.3.6.12)

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

In the Enquiry , Hume suggests that the step taken by the mind,

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

That principle is “custom” or “habit”. The idea is that if one has seen similar objects or events constantly conjoined, then the mind is inclined to expect a similar regularity to hold in the future. The tendency or “propensity” to draw such inferences, is the effect of custom:

… having found, in many instances, that any two kinds of objects, flame and heat, snow and cold, have always been conjoined together; if flame or snow be presented anew to the senses, the mind is carried by custom to expect heat or cold, and to believe , that such a quality does exist and will discover itself upon a nearer approach. This belief is the necessary result of of placing the mind in such circumstances. It is an operation of the soul, when we are so situated, as unavoidable as to feel the passion of love, when we receive benefits; or hatred, when we meet with injuries. All these operations are a species of natural instincts, which no reasoning or process 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 course of nature is a kind of “pre-established harmony” (E. 5.2.21). It is a kind of natural instinct, which may in fact be more effective in making us successful in the world, than if we relied on reason to make these inferences.

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

Hume’s argument concerns specific inductive inferences such as:

All observed instances of A have been B .

The next instance of A will be B .

Let us call this “inference I ”. Inferences which fall under this type of schema are now often referred to as cases of “simple enumerative induction”.

Hume’s own example is:

All observed instances of bread (of a particular appearance) have been nourishing.

The next instance of bread (of that appearance) will be nourishing.

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

Consequences:

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 would eliminate the possibility of a deductive argument, and the second would eliminate the possibility of an inductive argument. However, under this interpretation, premise P3 would not hold, because it is possible for the conclusion of a deductive argument to be a non-necessary proposition. Premise P3 could be modified to say that a demonstrative (deductive) argument establishes a conclusion that cannot be false if the premises are true. But then it becomes possible that the supposition that the future resembles the past, which is not a necessary proposition, could be established by a deductive argument from some premises, though not from a priori premises (in contradiction to conclusion C1 ).

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

However, recent commentators have argued that in the historical context that Hume was situated in, the distinction he draws between demonstrative and probable arguments has little to do with whether or not the argument has a deductive form (Owen 1999; Garrett 2002). In addition, the class of inferences that establish conclusions whose negation is a contradiction may include not just deductively valid inferences from a priori premises, but any inferences that can be drawn using a priori reasoning (that is, reasoning where the transition from premises to the conclusion makes no appeal to what we learn from observations). It looks as though Hume does intend the argument of the first horn to rule out any a priori reasoning, since he says that a change in the course of nature cannot be ruled out “by any demonstrative argument or abstract reasoning a priori ” (E. 5.2.18). On this understanding, a priori arguments would be ruled out by the first horn of Hume’s dilemma, and empirical arguments by the second horn. This is the interpretation that I will adopt for the purposes of this article.

In Hume’s argument, the UP plays a central role. As we will see in section 4.2 , various authors have been doubtful about this principle. Versions of Hume’s argument have also been formulated which do not make reference to the UP. Rather they directly address the question of what arguments can be given in support of the transition from the premises to the conclusion of the specific inductive inference I . What arguments could lead us, for example, to infer that the next piece of bread will nourish from the observations of nourishing bread made so far? For the first horn of the argument, Hume’s argument can be directly applied. A demonstrative argument establishes a conclusion whose negation is a contradiction. The negation of the conclusion of the inductive inference is not a contradiction. It is not a contradiction that the next piece of bread is not nourishing. Therefore, there is no demonstrative argument for the conclusion of the inductive inference. In the second horn of the argument, the problem Hume raises is a circularity. Even if Hume is wrong that all inductive inferences depend on the UP, there may still be a circularity problem, but as we shall see in section 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 often formulated without invoking the UP.

Since Hume’s argument is a dilemma, there are two main ways to resist it. The first is to tackle the first horn and to argue that there is after all a demonstrative argument –here taken to mean an argument based on a priori reasoning—that can justify the inductive inference. The second is to tackle the second horn and to argue that there is after all a probable (or empirical) argument that can justify the inductive inference. We discuss the different variants of these two approaches in sections 3 and 4 .

There are also those who dispute the consequences of the dilemma. For example, some scholars have denied that Hume should be read as invoking a premise such premise P8 at all. The reason, they claim, is that he was not aiming for an explicitly normative conclusion about justification such as C5 . Hume certainly is seeking a “chain of reasoning” from the premises of the inductive inference to the conclusion, and he thinks that an argument for the UP is necessary to complete the chain. However, one could think that there is no further premise regarding justification, and so the conclusion of his argument is simply C4 : there is no chain of reasoning from the premises to the conclusion of an inductive inference. Hume could then be, as Don Garrett and David Owen have argued, advancing a “thesis in cognitive psychology”, rather than making a normative claim about justification (Owen 1999; Garrett 2002). The thesis is about the nature of the cognitive process underlying the inference. According to Garrett, the main upshot of Hume’s argument is that there can be no reasoning process that establishes the UP. For Owen, the message is that the inference is not drawn through a chain of ideas connected by mediating links, as would be characteristic of the faculty of reason.

There are also interpreters who have argued that Hume is merely trying to exclude a specific kind of justification of induction, based on a conception of reason predominant among rationalists of his time, rather than a justification in general (Beauchamp & Rosenberg 1981; Baier 2009). In particular, it has been claimed that it is “an attempt to refute the rationalist belief that at least some inductive 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 arguments from the premises to the conclusion of inference I , then inference I is not justified.

Such interpretations do however struggle with the fact that Hume’s argument is explicitly a two-pronged attack, which concerns not just demonstrative arguments, but also probable arguments.

The question of how expansive a normative conclusion to attribute to Hume is a complex one. It depends in part on the interpretation of Hume’s own solution to his problem. As we saw in section 1 , Hume attributes the basis of inductive inference to principles of the imagination in the Treatise, and in the Enquiry to “custom”, “habit”, conceived as a kind of natural instinct. The question is then whether this alternative provides any kind of justification for the inference, even if not one based on reason. On the face of it, it looks as though Hume is suggesting that inductive inferences proceed on an entirely arational basis. He clearly does not think that they do not succeed in producing good outcomes. In fact, Hume even suggests that this operation of the mind may even be less “liable to error and mistake” than if it were entrusted to “the fallacious deductions of our reason, which is slow in its operations” (E. 5.2.22). It is also not clear that he sees the workings of the imagination as completely devoid of rationality. For one thing, Hume talks about the imagination as governed by principles . Later in the Treatise , he even gives “rules” and “logic” for characterizing what should count as a good causal inference (T. 1.3.15). He also clearly sees it as possible to distinguish between better forms of such “reasoning”, as he continues to call it. Thus, there may be grounds to argue that Hume was not trying to argue that inductive inferences have no rational foundation whatsoever, but merely that they do not have the specific type of rational foundation which is rooted in the faculty of Reason.

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

There are a number of approaches which effectively, if not explicitly, take issue with premise P8 and argue that providing a chain of reasoning from the premises to the conclusion is not a necessary condition for justification of an inductive inference. According to this type of approach, one may admit that Hume has shown that inductive inferences are not justified in the sense that we have reasons to think their conclusions true, but still think that weaker kinds of justification of induction are possible ( section 5 ). Finally, there are some philosophers who do accept the skeptical conclusion C5 and attempt to accommodate it. For example, there have been attempts to argue that inductive inference is not as central to scientific inquiry 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 aimed at establishing that there is no demonstrative argument for the UP. There are several ways people have attempted to show that the first horn does not definitively preclude a demonstrative or a priori argument for inductive inferences. One possible escape route from the first horn is to deny premise P3 , which amounts to admitting the possibility of synthetic a priori propositions ( section 3.1 ). Another possibility is to attempt to provide an a priori argument that the conclusion of the inference is probable, though not certain. The first horn of Hume’s dilemma implies that there cannot be a demonstrative argument to the conclusion of an inductive inference because it is possible to conceive of the negation of the conclusion. For instance, it is quite possible to imagine that the next piece of bread I eat will poison me rather than nourish me. However, this does not rule out the possibility of a demonstrative argument that establishes only that the bread is highly likely to nourish, not that it definitely will. One might then also challenge premise P8 , by saying that it is not necessary for justification of an inductive inference to have a chain of reasoning from its premises to its conclusion. Rather it would suffice if we had an argument from the premises to the claim that the conclusion is probable or likely. Then an a priori justification of the inductive inference would have been provided. There have been attempts to provide a priori justifications for inductive inference based on Inference to the Best Explanation ( section 3.2 ). There are also attempts to find an a priori solution based on probabilistic formulations of inductive inference, though many now think that a purely a priori argument cannot be found because there are empirical assumptions involved (sections 3.3 - 3.5 ).

As we have seen in section 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 (Kant 1781). A classic example of an analytic proposition is “Bachelors are unmarried men”, and a synthetic proposition is “My bike tyre is flat”. For Hume, demonstrative arguments, which are based on a priori reasoning, can establish only relations of ideas, or analytic propositions. The association between a prioricity and analyticity underpins premise P3 , which states that a demonstrative argument establishes a conclusion whose negation is a contradiction.

One possible response to Hume’s problem is to deny premise P3 , by allowing the possibility that a priori reasoning could give rise to synthetic propositions. Kant famously argued in response to Hume that such synthetic a priori knowledge is possible (Kant 1781, 1783). He does this by a kind of reversal of the empiricist programme espoused by Hume. Whereas Hume tried to understand how the concept of a causal or necessary connection could be based on experience, Kant argued instead that experience only comes about through the concepts or “categories” of the understanding. On his view, one can gain a priori knowledge of these concepts, including the concept of causation, by a transcendental argument concerning the necessary preconditions of experience. A more detailed account of Kant’s response to Hume can be found in de Pierris and Friedman 2013.

The “Nomological-explanatory” solution, which has been put forward by Armstrong, BonJour and Foster (Armstrong 1983; BonJour 1998; Foster 2004) appeals to the principle of Inference to the Best Explanation (IBE). According to IBE, we should infer that the hypothesis which provides the best explanation of the evidence is probably true. Proponents of the Nomological-Explanatory approach take Inference to the Best Explanation to be a mode of inference which is distinct from the type of “extrapolative” inductive inference that Hume was trying to justify. They also regard it as a type of inference which although non-deductive, is justified a priori . For example, Armstrong says “To infer to the best explanation is part of what it is to be rational. If that is not rational, what is?” (Armstrong 1983: 59).

The a priori justification is taken to proceed in two steps. First, it is argued that we should recognize that certain observed regularities 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 just because of “chance”. Rather, we should infer to the better explanation that the coin has a certain bias. Saying that the coin lands heads not only for the observed cases, but also for the unobserved cases, does not provide an explanation of the observed regularity. 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 natural necessity (Armstrong 1983; Foster 2004).

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

The Nomological-Explanatory solution relies on taking IBE as a rational, a priori form of inference which is distinct from inductive inferences like inference I . However, one might alternatively view inductive inferences as a special case of IBE (Harman 1968), or take IBE to be merely an alternative way of characterizing inductive inference (Henderson 2014). If either of these views is right, IBE does not have the necessary independence from inductive inference to provide a non-circular justification of it.

One may also object to the Nomological-Explanatory approach on the grounds that regularities do not necessarily require an explanation in terms of necessary connections or robust metaphysical laws. The viability of the approach also depends on the tenability of a non-Humean conception of laws. There have been several serious attempts 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-Explanatory solution simply begs the question, even if it is taken to be legitimate to make use of IBE in the justification of induction. In the first step of the argument we infer to a law or regularity which extends beyond the spatio-temporal region in which observations have been thus far made, in order to predict what will happen in the future. But why could a law that only applies to the observed spatio-temporal region not be an equally good explanation? The main reply seems to be that we can see a priori that laws with temporal or spatial restrictions would be less good explanations. Foster argues that the reason is that this would introduce more mysteries:

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

Another way in which one can try to construct an a priori argument that the premises of an inductive inference make its conclusion probable, is to make use of the formalism of probability theory itself. At the time Hume wrote, probabilities were used to analyze games of chance. And in general, they were used to address the problem of what we would expect to see, given that a certain cause was known to be operative. This is the so-called problem of “direct inference”. However, the problem of induction concerns the “inverse” problem of determining the cause or general hypothesis, given particular observations.

One of the first and most important methods for tackling the “inverse” problem using probabilities was developed by Thomas Bayes. Bayes’s essay containing the main results was published after his death in 1764 (Bayes 1764). However, it is possible that the work was done significantly earlier and was in fact written in direct response to the publication of Hume’s Enquiry in 1748 (see Zabell 1989: 290–93, for discussion of what is known about the history).

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

Consider first the problem of direct inference. Given the proportion of white balls in the urn, what is the probability of various outcomes for a sample of observations of a given size? Suppose the proportion of white balls in the urn is \(\theta = 0.6\). The probability of drawing 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, such as drawing two white balls in a sample of two, using the rules of the probability calculus (see section 1 of Hájek 2011). Generally, the probability that \(n_w\) white balls are drawn in a sample of size N , 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 evidence E in a sample, on the assumption that a certain hypothesis H is true. Calculation of the sampling distribution can in general be done a priori , given the rules of the probability calculus.

However, the problem of induction is the inverse problem. We want to infer not what the sample will be like, with a known hypothesis, rather we want to infer a hypothesis about the general situation or population, based on the observation of a limited sample. The probabilities of the candidate hypotheses can then be used to inform predictions about further observations. In the case of the urn, for example, we want to know what the observation of a particular sample frequency 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 only to the events which constitute evidence, but also to hypotheses. One starts with a “prior probability” distribution over the relevant hypotheses \(p(H)\). On learning some evidence E , the Bayesian updates the prior \(p(H)\) to the conditional probability \(p(H\mid E)\). This update rule is called the “rule of conditionalisation”. The conditional probability \(p(H\mid E)\) is known as the “posterior probability”, and is calculated using Bayes’ rule:

Here the sampling distribution can be taken to be a conditional probability \(p(E\mid H)\), which is known as the “likelihood” of the hypothesis H on evidence E .

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

where the sum becomes an integral in cases where H is a continuous variable.

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

With the choice of uniform prior, the posterior probability and predictive distribution can be calculated. It turns out that the probability that the next ball will be white, given that \(n_w\) of N 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 the next ball being white is \(91/102=0.89\). It is quite conceivable that the next ball might be black. Even in the case, where all 100 balls have been white, so that the probability of the next ball being white is 0.99, there is still a small probability that the next ball is not white. What the probabilistic reasoning supplies then is not an argument to the conclusion that the next ball will be a certain color, but an argument to the conclusion that certain future observations are very likely given what has been observed in the past.

Overall, the Bayes-Laplace argument in the urn case provides an example of how probabilistic reasoning can take us from evidence about observations in the past to a prediction for how likely certain future observations are. The question is what kind of solution, if any, this type of calculation provides to the problem of induction. At first sight, since it is just a mathematical calculation, it looks as though it does indeed provide an a priori argument from the premises of an inductive inference to the proposition that a certain conclusion is probable.

However, in order to establish this definitively, one would need to argue that all the components and assumptions of the argument are a priori and this requires further examination of at least three important issues.

First, the Bayes-Laplace argument relies on the rules of the probability calculus. What is the status of these rules? Does following them amount to a priori reasoning? The answer to this depends in part on how probability itself is interpreted. Broadly speaking, there are prominent interpretations of probability according to which the rules plausibly have a priori status and could form the basis of a demonstrative argument. These include the classical interpretation originally developed by Laplace (1814), the logical interpretation (Keynes (1921), Johnson (1921), Jeffreys (1939), Carnap (1950), Cox (1946, 1961), and the subjectivist interpretation of Ramsey (1926), Savage (1954), and de Finetti (1964). Attempts to argue for a probabilistic a priori solution to the problem of induction have been primarily associated with these interpretations.

Secondly, in the case of the urn, the Bayes-Laplace argument is based on a particular probabilistic model—the binomial model. This involves the assumption that there is a parameter describing an unknown proportion \(\theta\) of balls in the urn, and that the data amounts to independent draws from a distribution over that parameter. What is the basis of these assumptions? Do they generalize to other cases beyond the actual urn case—i.e., can we see observations in 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 an urn, will not hold for other cases of inductive inference. Thus, the probabilistic solution to the problem of induction might be of relatively limited scope. At the least, there are some assumptions going into the choice of model here that need to be made explicit. Arguably the choice of model introduces empirical assumptions, which would mean that the probabilistic solution is not an a priori one.

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

The quest for an a priori argument for the assignment of the prior has been largely abandoned. For many, the subjectivist foundations developed by Ramsey, de Finetti and Savage provide a more satisfactory basis for understanding probability. From this point of view, it is a mistake to try to introduce any further a priori constraints on the probabilities beyond those dictated by the probability rules themselves. Rather the assignment of priors may reflect personal opinions or background knowledge, and no prior is a priori an unreasonable choice.

So far, we have considered probabilistic arguments which place probabilities over hypotheses in a hypothesis space as well as observations. There is also a tradition of attempts to determine what probability distributions we should have, given certain observations, from the starting point of a joint probability distribution over all the observable variables. One may then postulate axioms directly on this distribution over observables, and examine the consequences for the predictive distribution. Much of the development of inductive logic, including the influential programme by Carnap, proceeded in this manner (Carnap 1950, 1952).

This approach helps to clarify the role of the assumptions behind probabilistic models. One assumption that one can make about the observations is that they are “exchangeable”. This means that the joint distribution of the random variables is invariant under permutations. Informally, this means that the order of the observations does not affect the probability. For instance, in the urn case, this would mean that drawing first a white ball and then a black ball is just as probable as first drawing a black and then a white. De Finetti proved a general representation theorem that if the joint probability distribution of an infinite sequence of random variables is assumed to be exchangeable, then it can be written as a mixture of distribution functions from each of which the data behave as if they are independent random draws (de Finetti 1964). In the case of the urn example, the theorem shows that it is as if the data are independent 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 natural formalization of Hume’s assumption that the past resembles the future. This is intuitive because assuming exchangeability means thinking that the order of observations, both past and future, does not matter to the probability assignments.

However, the development of the programme of inductive logic revealed that many generalizations are possible. For example, Johnson proposed to assume an axiom he called the “sufficientness postulate”. This states that outcomes can be of a number of different types, and that the conditional probability that the next outcome is of type i depends only on the number of previous trials and the number of previous outcomes of type i (Johnson 1932). Assuming the sufficientness postulate for three or more types gives rise to a general predictive distribution corresponding to Carnap’s “continuum of inductive methods” (Carnap 1952). This predictive distribution takes the form:

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

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

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

Some think that although the problem of induction is not solved, there is 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 is sound, given appropriate premises ” (Howson 2000: 239, his emphasis). According to this view, there is no getting away from an empirical premise for inductive inferences, but we might still think of Bayesian conditioning as functioning like a kind of logic or “consistency constraint” which “generates predictions from the assumptions and observations together” (Romeijn 2004: 360). Once we have an empirical assumption, instantiated in the prior probability, and the observations, Bayesian conditioning tells us what the resulting predictive probability distribution should be.

The idea of a partial solution also arises in the context of the learning theory that grounds contemporary machine learning. Machine learning is a field in computer science concerned with algorithms that learn from experience. Examples are algorithms which can be trained to recognise or classify patterns in data. Learning theory concerns itself with finding mathematical theorems which guarantee the performance of algorithms which are in practical use. In this domain, there is a well-known finding that learning algorithms are only effective if they have ‘inductive bias’ — that is, if they make some a priori assumptions about the domain they are employed upon (Mitchell 1997).

The idea is also given formal expression in the so-called ‘No-Free-Lunch theorems’ (Wolpert 1992, 1996, 1997). These can be interpreted as versions of the argument in Hume’s first fork since they establish that there can be no contradiction in the algorithm not performing well, since there are a priori possible situations in which it does not (Sterkenburg and Grünwald 2021:9992). Given Hume’s premise P3 , this rules out a demonstrative argument for its good performance.

Premise P3 can perhaps be challenged on the grounds that a priori justifications can also be given for contingent propositions. Even though an inductive inference can fail in some possible situations, it could still be reasonable to form an expectation of reliability if we spread our credence equally over all the possibilities and have reason to think (or at least no reason to doubt) that the cases where inductive inference is unreliable require a ‘very specific arrangement of things’ and thus form a small fraction of the total space of possibilities (White 2015). The No-Free-Lunch theorems make difficulties for this approach since they show that if we put a uniform distribution over all logically possible sequences of future events, any learning algorithm is expected to have a generalisation error 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 on justifying learning algorithms when these algorithms are seen as ‘purely data-driven’ — that is as mappings from possible data to conclusions. However, learning algorithms may also be conceived as functions not only of input data, but also of a particular model (Sterkenburg and Grünwald 2021). For example, the Bayesian ‘algorithm’ gives a universal recipe for taking a particular model and prior and updating on the data. A number of theorems in learning theory provide general guarantees for the performance of such recipes. For instance, there are theorems which guarantee convergence of the Bayesian algorithm (Ghosal, Ghosh and van der Vaart 2000, Ghosal, Lember and van der Vaart 2008). In each instantiation, this convergence is relative to a particular specific prior. Thus, although the considerations first raised by Hume, and later instantiated in the No-Free-Lunch theorems, preclude any universal model-independent justification for learning algorithms, it does not rule out partial justifications in the form of such general a priori ‘model-relative’ learning guarantees (Sterkenburg and Grünwald 2021).

An alternative attempt to use probabilistic reasoning to produce an a priori justification for inductive inferences is the so-called “combinatorial” solution. This was first put forward by Donald C. Williams (1947) and later developed by David Stove (1986).

Like the Bayes-Laplace argument, the solution relies heavily on the idea that straightforward a priori calculations can be done in a “direct inference” from population to sample. As we have seen, given a certain population frequency, the probability of getting different frequencies in a sample can be calculated straightforwardly based on the rules of the probability calculus. The Bayes-Laplace argument relied on inverting the probability distribution using Bayes’ rule to get from the sampling distribution to the posterior distribution. Williams instead proposes that the inverse inference may be based on a certain logical syllogism: the proportional (or statistical) syllogism.

The proportional, or statistical syllogism, is the following:

• Of all the things that are M , \(m/n\) are P .

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

For example, if 90% of rabbits in a population are white and we observe a rabbit a , then the proportional syllogism says that we infer that a is white with a probability of 90%. Williams argues that the proportional syllogism is a non-deductive logical syllogism, which effectively interpolates between the syllogism for entailment

• All M s are P

Therefore, a is P .

And the syllogism for contradiction

Therefore, a is not P .

This syllogism can be combined with an observation about the behavior of increasingly large samples. From calculations of the sampling distribution, it can be shown that as the sample size increases, the probability that the sample frequency is in a range which closely approximates the population frequency also increases. In fact, Bernoulli’s law of large numbers states that the probability that the sample frequency approximates the population frequency tends to one as the sample size goes to infinity. Williams argues that such results support a “general over-all premise, common to all inductions, that samples ‘match’ their populations” (Williams 1947: 78).

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

• Most samples match their population
• S is a sample.

Therefore, S matches its population, with high probability.

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

The next step is to argue that if we observe that the sample contains a proportion of \(m/n\) F s, then we can conclude that since this sample with high probability matches its population, the population, with high probability, has a population frequency that approximates the sample frequency \(m/n\). Both Williams and Stove claim that this amounts to a logical a priori solution to the problem of induction.

A number of authors have expressed the view that the Williams-Stove argument is only valid if the sample S is drawn randomly from the population of possible samples—i.e., that any sample is as likely to be drawn as any other (Brown 1987; Will 1948; Giaquinto 1987). Sometimes this is presented as an objection to the application of the proportional syllogism. The claim is that the proportional syllogism is only valid if a is drawn randomly from the population of M s. However, the response has been that there is no need to know that the sample is randomly drawn in order to apply the syllogism (Maher 1996; Campbell 2001; Campbell & Franklin 2004). Certainly if you have reason to think that your sampling procedure is more likely to draw certain individuals than others—for example, if you know that you are in a certain location where there are more of a certain type—then you should not apply the proportional syllogism. But if you have no such reasons, the defenders claim, it is quite rational to apply it. Certainly it is always possible that you draw an unrepresentative sample—meaning one of the few samples in which the sample frequency does not match the population frequency—but this is why the conclusion is only probable and not certain.

The more problematic step in the argument is the final step, which takes us from the claim that samples match their populations with high probability to the claim that having seen a particular sample frequency, the population from which the sample is drawn has frequency close to the sample frequency with high probability. The problem here is a subtle shift in what is meant by “high probability”, which has formed the basis of a common misreading of Bernouilli’s theorem. Hacking (1975: 156–59) puts the point in the following terms. Bernouilli’s theorem licenses the claim that much more often than not, a small interval around the sample frequency will include the true population frequency. In other words, it is highly probable in the sense of “usually right” to say that the sample matches its population. But this does not imply that the proposition that a small interval around the sample will contain the true population frequency is highly probable in the sense of “credible on each occasion of use”. This would mean that for any given sample, it is highly credible that the sample matches its population. It is quite compatible with the claim that it is “usually right” that the sample matches its population to say that there are some samples which do not match their populations at all. Thus one cannot conclude from Bernouilli’s theorem that for any given sample frequency, we should assign high probability to the proposition that a small interval around the sample frequency will contain the true population frequency. But this is exactly the slide that Williams makes in the final step of his argument. Maher (1996) argues in a similar fashion that the last step of the Williams-Stove argument is fallacious. In fact, if one wants to draw conclusions about the probability of the population frequency given the sample frequency, the proper way to do so is by using the Bayesian method described in the previous section. But, as we there saw, this requires the assignment of prior probabilities, and this explains why many people have thought that the combinatorial solution somehow illicitly presupposed an assumption like the principle of indifference. The Williams-Stove argument does not in fact give us an alternative way of inverting the probabilities which somehow bypasses all 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’s dilemma might be tackled. But it is of course also possible to take on the second horn instead.

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

One way to tackle the second horn of Hume’s dilemma is to reject premise P6 , which rules out circular arguments. Some have argued that certain kinds of circular arguments would provide an acceptable justification for the inductive inference. Since the justification would then itself be an inductive one, this approach is often referred to as an “inductive justification of induction”.

First we should examine how exactly the Humean circularity supposedly arises. Take the simple case of enumerative inductive inference that follows the following pattern ( X ):

Most observed F s have been G s

Therefore: Most F s are G s.

Hume claims that such arguments presuppose the Uniformity Principle (UP). According to premises P7 and P8 , this supposition also needs to be supported by an argument in order that the inductive inference be justified. A natural idea is that we can argue for the Uniformity Principle on the grounds that “it works”. We know that it works, because past instances of arguments which relied upon it were found to be successful. This alone however is not sufficient unless we have reason to think that such arguments will also be successful in the future. That claim must itself be supported by an inductive argument ( S ):

Most arguments of form X that rely on UP have succeeded in the past.

Therefore, most arguments of form X that rely on UP succeed.

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

As we have seen in section 2 , some reject Hume’s claim that all inductive inferences presuppose the UP. However, the argument that basing the justification of the inductive inference on a probable argument would result in circularity need not rely on this claim. The circularity concern can be framed more generally. If argument S relies on something which is already presupposed in inference X , then argument S cannot be used to justify inference X . The question though is what precisely the something is.

Some authors have argued that in fact S does not rely on any premise or even presupposition that would require us to already know the conclusion of X . S is then not a “premise circular” argument. Rather, they claim, it is “rule-circular”—it relies on a rule of inference in order to reach the conclusion that that very rule is reliable. Suppose we adopt the rule R which says that when it is observed that most F s are G s, we should infer that most F s are G s. Then inference X relies on rule R . We want to show that rule R is reliable. We could appeal to the fact that R worked in the past, and so, by an inductive argument, it will also work in the future. Call this argument S *:

Most inferences following rule R have been successful

Therefore, most inferences following R are successful.

Since this argument itself uses rule R , using it to establish that R is reliable is rule-circular.

Some authors have then argued that although premise-circularity is vicious, rule-circularity is not (Cleve 1984; Papineau 1992). One reason for thinking rule-circularity is not vicious would be if it is not necessary to know or even justifiably believe that rule R is 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 as R is in fact reliable, one can form a justified belief in the conclusion of an argument relying on R , as long as one has justified belief in the premises.

If one is not persuaded by the externalist claim, one might attempt to argue that rule circularity is benign in a different fashion. For example, the requirement that a rule be shown to be reliable without any rule-circularity might appear unreasonable when the rule is of a very fundamental nature. As Lange puts it:

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

Proponents of this point of view point out that even deductive inference cannot be justified deductively. Consider Lewis Carroll’s dialogue between Achilles and the Tortoise (Carroll 1895). Achilles is arguing with a Tortoise who refuses to perform modus ponens . The Tortoise accepts the premise that p , and the premise that p implies q but he will not accept q . How can Achilles convince him? He manages to persuade him to accept another premise, namely “if p and p implies q , then q ”. But the Tortoise is still not prepared to infer to q . Achilles goes on adding more premises of the same kind, but to no avail. It appears then that modus ponens cannot be justified to someone who is not already prepared to use that rule.

It might seem odd if premise circularity were vicious, and rule circularity were not, given that there appears to be an easy interchange 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 is indeed a fundamental difference between being prepared to accept a premise stating a rule (the Tortoise is happy to do this), and being prepared to use that rule (this is what the Tortoise refuses to do).

Suppose that we grant that an inductive argument such as S (or S *) can support an inductive inference X without vicious circularity. Still, a possible objection is that the argument simply does not provide a full justification of X . After all, less sane inference rules such as counterinduction can support themselves in a similar fashion. The counterinductive rule is CI:

Most observed A s are B s.

Therefore, it is not the case that most A s are B s.

Consider then the following argument CI*:

Most CI arguments have been unsuccessful

Therefore, it is not the case that most CI arguments are unsuccessful, i.e., many CI arguments are successful.

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

Argument S can be used to support inference X , but only for someone who is already prepared to infer inductively by using S . It cannot convince a skeptic who is not prepared to rely upon that rule in the first place. One might think then that the argument is simply not achieving very much.

The response to these concerns is that, as Papineau puts it, the argument is “not supposed to do very much” (Papineau 1992: 18). The fact that a counterinductivist counterpart of the argument exists is true, but irrelevant. It is conceded that the argument cannot persuade either a counterinductivist, or a skeptic. Nonetheless, proponents of the inductive justification maintain that there is still some added value in showing that inductive inferences are reliable, even when we already accept that there is nothing problematic about them. The inductive justification of induction provides a kind of important consistency check on our existing beliefs.

It is possible to go even further in an attempt to dismantle the Humean circularity. Maybe inductive inferences do not even have a rule in common. What if every inductive inference is essentially unique? This can be seen as rejecting Hume’s premise P5 . Okasha, for example, argues that Hume’s circularity problem can be evaded if there are “no rules” behind induction (Okasha 2005a,b). Norton puts forward the similar idea that all inductive inferences are material, and have nothing formal in common (Norton 2003, 2010, 2021).

Proponents of such views have attacked Hume’s claim that there is a UP on which all inductive inferences are based. There have long been 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 been under 40 years old. This does not give me a reason to expect that I will be under 40 years old on my next birthday. There seems then to be a major lacuna in Hume’s account. He might have explained or described how we draw an inductive inference, on the assumption that it is one we can draw. But he leaves untouched the question of how we distinguish between cases where we extrapolate a regularity legitimately, regarding it as a law, and cases where we do not.

Nelson Goodman is often seen as having made this point in a particularly 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 time t and blue otherwise. Goodman considers a thought experiment in which we observe a bunch of green emeralds before time t . We could describe our results by saying all the observed emeralds are green. Using a simple enumerative inductive schema, we could infer from the result that all observed emeralds are green, that all emeralds are green. But equally, we could describe the same results by saying that all observed emeralds are grue. Then using the same schema, we could infer from the result that all observed emeralds are grue, that all emeralds are grue. In the first case, we expect an emerald observed after time t to be green, whereas in the second, we expect it to be blue. Thus the two predictions are incompatible. Goodman claims that what Hume omitted to do was to give any explanation for why we project predicates like “green”, but not predicates like “grue”. This is the “new riddle”, which is often taken to be a further problem of induction that Hume did not address.

One moral that could be taken from Goodman is that there is not one general Uniformity Principle that all probable arguments rely upon (Sober 1988; Norton 2003; Okasha 2001, 2005a,b, Jackson 2019). Rather each inductive inference presupposes some more specific empirical presupposition. A particular inductive inference depends on some specific way in which the future resembles the past. It can then be justified by another inductive inference which depends on some quite different empirical claim. This will in turn need to be justified—by yet another inductive inference. The nature of Hume’s problem in the second horn is thus transformed. There is no 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 rests on a quantifier shift fallacy (Sober 1988; Okasha 2005a). Hume says that there exists a general presupposition for all inductive inferences, whereas he should have said that for each inductive inference, there is some presupposition. Different inductive inferences then rest on different empirical presuppositions, and the problem of circularity is evaded.

What will then be the consequence of supposing that Hume’s problem should indeed have been a regress, rather than a circularity? Here different opinions are possible. On the one hand, one might think that a regress still leads to a skeptical conclusion (Schurz and Thorn 2020). So although the exact form in which Hume stated his problem was not correct, the conclusion is not substantially different (Sober 1988). Another possibility is that the transformation mitigates or even removes the skeptical problem. For example, Norton argues that the upshot is a dissolution of the problem of induction, since the regress of justifications benignly terminates (Norton 2003). And Okasha more mildly suggests that even if the regress is infinite, “Perhaps infinite regresses are less bad than vicious circles after all” (Okasha 2005b: 253).

Any dissolution of Hume’s circularity does not depend only on arguing that the UP should be replaced by empirical presuppositions which are specific to each inductive inference. It is also necessary to establish that inductive inferences share no common rules—otherwise there will still be at least some rule-circularity. Okasha suggests that the Bayesian model of belief-updating is an illustration how induction can be characterized in a rule-free way, but this is problematic, since in this model all inductive inferences still share the common rule of Bayesian conditionalisation. Norton’s material theory of induction postulates a rule-free characterization of induction, but it is not clear 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 the possibility of justifying inference I , via a premise such as P8 , though as we have seen in section section 2 , some have questioned whether Hume is best interpreted as drawing a conclusion about justification of inference I at all. In this section we examine approaches which question in different ways whether premise P8 really does give a valid necessary condition for justification of inference I and propose various alternative conceptions of justification.

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

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

One might not, for instance, think that there even needs to be a chain of reasoning in which each step or presupposition is supported by an argument. Wittgenstein took it that there are some principles so fundamental that they do not require support from any further argument. They are the “hinges” on which enquiry turns.

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

Some philosophers have set themselves the task of determining a set or sets of postulates which form a plausible basis for inductive inferences. Bertrand Russell, for example, argued that five postulates lay 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 to different inductive methods (Burks 1953, 1955).

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

Rather than allowing undefended empirical postulates to give normative support to an inductive inference, one could instead argue for a completely 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 a justification of inductive inferences (Strawson 1952). This has become known as the “Ordinary language dissolution” of the problem of induction.

Strawson points out that it could be meaningful to ask for a deductive justification of inductive inferences. But it is not clear that this is helpful since this is effectively “a demand that induction shall be shown to be really a kind of deduction” (Strawson 1952: 230). Rather, Strawson says, when we ask about whether a particular inductive inference is justified, we are typically judging whether it conforms to our usual inductive standards. Suppose, he says, someone has formed the belief by inductive inference that All f ’s are g . Strawson says that if that person is 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 come across innumerable cases of f and never a case of f which wasn’t a case of g ”. In saying this, he is clearly claiming to have inductive 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 wide variety of conditions.

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

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

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

According to Strawson,

It is an analytic proposition that it is reasonable to have a degree of belief in a statement which is proportional to the strength of the evidence in its favour; and it is an analytic proposition, though not a proposition of mathematics, that, other things being equal, the evidence for a generalisation is strong in proportion as the number of favourable instances, and the variety of circumstances in which they have been found, is great. So to ask whether it is reasonable to place reliance on inductive procedures is like asking whether it is reasonable to proportion the degree of one’s convictions to the strength of the evidence. Doing this is what “being reasonable” means in such a context. (Strawson 1952: 256–57)

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

The ordinary language philosophers do not explicitly argue against Hume’s premise P8 . But effectively what they are doing is offering a whole different story about what it would mean to be justified in believing the conclusion of inductive inferences. What is needed is just conformity to inductive standards, and there is no real meaning to asking for any further justification for those.

The main objection to this view is that conformity to the usual standards is insufficient to provide the needed justification. What we need to know is whether belief in the conclusion of an inductive inference is “epistemically reasonable or justified in the sense that …there is reason to think that it is likely to be true” (BonJour 1998: 198). The problem Hume has raised is whether, despite the fact that inductive inferences have tended to produce true conclusions in the past, we have reason to think the conclusion of an inductive inference we now make is likely to be true. Arguably, establishing that an inductive inference is rational in the sense that it follows inductive standards is not sufficient to establish that its conclusion is likely to be true. In fact Strawson allows that there is a question about whether “induction will continue to be successful”, which is distinct from the question of whether induction is rational. This question he does take to hinge on a “contingent, factual matter” (Strawson 1952: 262). But if it is this question that concerned Hume, it is no answer to establish that induction is rational, unless that claim is understood to involve or imply that an inductive inference carried out according to rational standards is likely to have a true conclusion.

Another solution based on an alternative criterion for justification is the “pragmatic” approach initiated by Reichenbach (1938 [2006]). Reichenbach did think Hume’s argument unassailable, but nonetheless he attempted to provide a weaker kind of justification for induction. In order to emphasize the difference from the kind of justification Hume sought, some have given it a different term and refer to Reichenbach’s solution as a “vindication”, rather than a justification of induction (Feigl 1950; Salmon 1963).

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

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

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

Reichenbach argued that even if Hume is right to think that we cannot be justified in thinking for any particular application of the rule that the conclusion is likely to be true, for the purposes of practical action we do not need to establish this. We can instead regard the inductive rule as resulting in a “posit”, or statement that we deal with as if it is true. We posit a certain frequency f on the basis of our evidence, and this is like making a wager or bet that the frequency is in fact f . One strategy for positing frequencies is to follow the rule of induction.

Reichenbach proposes that we can show that the rule of induction meets his weaker justification condition. This does not require showing that following the inductive principle will always work. It is possible that the world is so disorderly that we cannot construct series with any limits. In that case, neither the inductive principle, nor any other method will succeed. But, he argues, if there is a limit, by following the inductive principle we will eventually find it. There is some element of a series of observations, beyond which the principle of induction will lead to the true value of the limit. Although the inductive rule may give quite wrong results early in the sequence, as it follows chance fluctuations in the sample frequency, it is guaranteed to eventually approximate the limiting frequency, if such a limit exists. Therefore, the rule of induction is justified as an instrument of positing because it is a method of which we know that if it is possible to achieve the aim of inductive inference we shall do so by means of this method (Reichenbach 1949: 475).

One might question whether Reichenbach has achieved his goal of showing that following the inductive rule is a necessary condition of success. In order to show that, one would also need to establish that no other methods can also achieve the aim. But, as Reichenbach himself recognises, many other rules of inference as well as the straight rule may also converge on the limit (Salmon 1966: 53). In fact, any method which converges asymptotically to the straight rule also does so. An easily specified class of such rules are those which add to the inductive rule a function \(c_n\) in which the \(c_n\) converge to zero with increasing n .

Reichenbach makes two suggestions aimed at avoiding this problem. On the one hand, he claims, since we have no real way to pick between methods, 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 “smallest risk” is following the inductive rule (Reichenbach 1938 [2006: 355–356]).

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

In assessing this argument, it is helpful to distinguish between levels at which the principle of induction can be applied. Following Skyrms (2000), we may distinguish between level 1, where candidate methods are applied to ordinary events or individuals, and level 2, where they are applied not to individuals or events, but to the arguments 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 that another method might do better than object-induction at level 1. It only shows that the success of that other method may be recognised by a meta-induction at level 2 (Skyrms 2000). Nonetheless, Reichenbach’s thought was later picked up and developed into the suggestion that a meta-inductivist who applies induction not only at the object level to observations, but also to the success of others’ methods, might by those means be able to do as well predictively as the alternative method (Schurz 2008; see section 5.5 for more discussion of meta-induction).

Reichenbach’s justification is generally taken to be a pragmatic one, since though it does not supply knowledge of a future event, it supplies a sufficient reason for action (Reichenbach 1949: 481). One might question whether a pragmatic argument can really deliver an all-purpose, general justification for following the inductive rule. Surely a pragmatic solution should be sensitive to differences in pay-offs that depend on the circumstances. For example, Reichenbach offers the following analogue to his pragmatic justification:

We may compare our situation to that of a man who wants to fish in an unexplored part of the sea. There is no one to tell him whether or not there are fish in this place. Shall he cast his net? Well, if he wants to fish in that place, I should advise him to cast the net, to take the chance at least. It is preferable to try even in uncertainty than not to try and be certain of getting nothing. (Reichenbach 1938 [2006: 362–363])

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

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

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

Formal learning theory is particularly concerned with showing that methods are “logically reliable” in the sense that they arrive at the truth given any sequence of data consistent with our background knowledge (Kelly 1996). However, it goes further than this. As we have just seen, one of the problems for Reichenbach was that there are too many rules which converge in the limit to the true frequency. Which one should we then choose in the short-run? Formal learning theory broadens Reichenbach’s general strategy by considering what happens if we have other epistemic goals besides long-run convergence to the truth. In particular, formal learning theorists have considered the goal of getting to the truth as efficiently, or quickly, as possible, as well as the goal of minimising the number of mind-changes, or retractions along the way. It has then been argued that the usual inductive method, which is characterised by a preference for simpler hypotheses (Occam’s razor), can be justified since it is the unique method which meets the standards for getting to the truth in the long run as efficiently as possible, with a minimum number of retractions (Kelly 2007).

Steel (2010) has proposed that the Principle of Induction (understood as a rule which makes inductive generalisations along the lines of the Straight Rule) can be given a means-ends justification by showing that following it is both necessary and sufficient for logical reliability. The proof is an a priori mathematical one, thus it allegedly avoids the circularity of Hume’s second horn. However, Steel also does not see the approach as an attempt to grasp Hume’s first horn, since the proof is only relative to a certain choice of epistemic ends.

As with other results in formal learning theory, this solution is also only valid relative to a given hypothesis space and conception of possible sequences of data. For this reason, some have seen it as not addressing Hume’s problem of giving grounds for a particular inductive inference (Howson 2011). An alternative attitude is that it does solve a significant part of Hume’s problem (Steel 2010). There is a similar dispute over formal learning theory’s treatment of Goodman’s riddle (Chart 2000, Schulte 2017).

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

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

The main result is that the wMI strategy is long-run optimal in the sense that it converges to the maximum success rate of the accessible prediction methods. Worst-case bounds for short-run performance can also be derived. The optimality result forms the basis for an a priori means-ends justification for the use of wMI. Namely, the thought is, it is reasonable to use wMI, since it achieves the best success rates possible in the long run out of the given methods.

Schurz also claims that this a priori justification of wMI, together with the contingent fact that inductive methods have so far been much more successful than non-inductive methods, gives rise to an a posteriori non-circular justification of induction. Since wMI will achieve in the long run the maximal success rate of the available prediction methods, it is reasonable to use it. But as a matter of fact, object-inductive prediction methods have been more successful than non-inductive methods so far. Therefore Schurz says “it is meta-inductively justified to favor object-inductivistic strategies in the future” (Schurz 2019: 85). This justification, he claims, is not circular because meta-induction has an a priori independent justification. The idea is that since it is a priori justified to use wMI, it is also a priori justified to use the maximally successful method at the object level. Since it turns out that that the maximally successful method is object-induction, then we have a non-circular a posteriori argument that it is reasonable to use object-induction.

Schurz’s original theorems on the optimality of wMI apply to the case where there are finitely many predictive methods. One point of discussion is whether this amounts to an important limitation on its claims to provide a full solution of the problem of induction. The question then is whether it is necessary that the optimality results be 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 to be “meta-inductively justified”. The meta-inductive strategy wMI and object-induction are clearly different strategies. They could result in different predictions tomorrow, if OI would stop working and another method would start to do better. In that case, wMI would begin to favour the other method, and wMI would start to come apart 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 next time-step, on the grounds that OI’s prediction approximately coincides with that of wMI (Sterkenburg 2020, Sterkenburg (forthcoming)). However, this requires a stronger empirical postulate than simply the observation that OI has been more successful than non-inductive methods. It also requires something like that “as a matter of empirical fact, the strategy OI has been so much more successful than its competitors, that the meta-inductivist attributes it 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 the empirical evidence does back up such a strong claim, the issue remains that the meta-inductive justification is in support of following the strategy of meta-induction, not in support of the strategy of following OI (Sterkenburg (2020), sec. 3.3.2).

So far we have considered the various ways in which we might attempt to solve the problem of induction by resisting one or other premise of Hume’s argument. Some philosophers have however seen his argument as unassailable, and have thus accepted that it does lead to inductive skepticism, the conclusion that inductive inferences cannot be justified. The challenge then is to find a way of living with such a radical-seeming conclusion. We appear to rely on inductive inference ubiquitously in daily life, and it is also generally thought that it is at the very foundation of the scientific method. Can we go on with all this, whilst still seriously thinking none of it is justified by any rational argument?

One option here is to argue, as does Nicholas Maxwell, that the problem of induction is posed in an overly restrictive context. Maxwell argues that the problem does not arise if we adopt a different conception of science than the ‘standard empiricist’ one, which he denotes ‘aim-oriented empiricism’ (Maxwell 2017).

Another option here is to think that the significance of the problem of induction is somehow restricted to a skeptical context. Hume himself seems to have thought along these lines. For instance he says:

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

Hume’s purpose is clearly not to argue that we should not make inductive inferences in everyday life, and indeed his whole method and system of describing the mind in naturalistic terms depends on inductive inferences through and through. The problem of induction then must be seen as a problem that arises only at the level of philosophical reflection.

Another way to mitigate the force of inductive skepticism is to restrict its scope. Karl Popper, for instance, regarded the problem of induction as insurmountable, but he argued that science is not in fact based on inductive inferences at all (Popper 1935 [1959]). Rather he presented a deductivist view of science, according to which it proceeds by making bold conjectures, and then attempting to falsify those conjectures. In the simplest version of this account, when a hypothesis makes a prediction which is found to be false in an experiment, the hypothesis is rejected as falsified. The logic of this procedure is fully deductive. The hypothesis entails the prediction, and the falsity of the prediction refutes the hypothesis by modus tollens. Thus, Popper claimed that science was not based on the extrapolative inferences considered by Hume. The consequence then is that it is not so important, at least for science, if those inferences would 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 by the evidence, and these may contradict one another. According to the strictly deductive framework, since none are yet falsified, they are all on an equal footing. Yet, scientists will typically want to say that one is better supported by the evidence than the others. We seem to need more than just deductive reasoning to support practical decision-making (Salmon 1981). Popper did indeed appeal to a notion of one hypothesis being better or worse “corroborated” by the evidence. But arguably, this took him away from a strictly deductive view of science. It appears doubtful then that pure deductivism can give an adequate account of scientific method.

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• Sterkenburg, Tom, 2019, “The meta-inductive justification of induction: the pool of strategies”, Philosophy of Science , 86: 981–992.
• –––, 2020, “The meta-inductive justification of induction”, Episteme , 17: 519–541.
• –––, forthcoming, “Explaining the success of induction”, British Journal for the Philosophy of Science , https://doi.org/10.1086/717068.
• Sterkenburg, Tom and Peter Grünwald, 2021, “The no-free-lunch theorems of supervised learning”, Synthese , 199: 9979–10015.
• Stove, David C., 1986, The Rationality of Induction , Oxford: Clarendon Press.
• Strawson, Peter Frederick, 1952, Introduction to Logical Theory , London: Methuen.
• Tooley, Michael, 1977, “The Nature of Laws”, Canadian Journal of Philosophy , 7(4): 667–698.
• White, Roger, 2015, “The problem of the problem of induction”, Episteme , 12: 275–290.
• Will, Frederick L., 1948, “Donald Williams’ Theory of Induction”, Philosophical Review , 57(3): 231–247.
• Williams, Donald C., 1947, The Ground of Induction , Harvard: Harvard University Press.
• Wittgenstein, Ludwig, 1953, Philosophical Investigations , New Jersey: Prentice Hall.
• Wolpert, D. H., 1997, “No free lunch theorems for optimization”, IEEE Transactions on Evolutionary Computation , 1: 67–82.
• –––, 1992, “On the connecton between in-sample testing and generalization error”, Complex Systems , 6: 47–94.
• –––, 1996, “The lack of a priori distinctions between learning algorithms”, Neural Computation 8: 1341–1390.
• Worrall, John, 2010, “For Universal Rules, Against Induction”, Philosophy of Science , 77(5): 740–53.
• Wright, Crispin, 2004, “Wittgensteinian Certainties”, in Denis McManus (ed.), Wittgenstein and Scepticism , London: Routledge, pp. 22–55.
• Zabell, Sandy L., 1988, “Symmetry and Its Discontents”, in Brian Skyrms (ed.), Causation, Chance and Credence , Dordrecht: Springer Netherlands, pp. 155–190.
• –––, 1989, “The Rule of Succession”, Erkenntnis , 31(2–3): 283–321.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• 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 the Stanford Encyclopedia of Philosophy — see the version history .]
• Teaching Theory of Knowledge: Probability and Induction , organization of topics and bibliography by Brad Armendt (Arizona State University) and Martin Curd (Purdue).
• Forecasting Principles , A brief survey of prediction markets.

Bayes’ Theorem | belief, formal representations of | confirmation | epistemology, formal | Feigl, Herbert | Goodman, Nelson | Hume, David | Kant, Immanuel: and Hume on causality | laws of nature | learning theory, formal | logic: inductive | Popper, Karl | probability, interpretations of | Reichenbach, Hans | simplicity | skepticism | statistics, philosophy of | Strawson, Peter Frederick

Acknowledgments

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

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• Inductive vs. Deductive Research Approach | Steps & Examples

Inductive vs. Deductive Research Approach | Steps & Examples

Published on April 18, 2019 by Raimo Streefkerk . Revised on June 22, 2023.

The main difference between inductive and deductive reasoning is that inductive reasoning aims at developing a theory while deductive reasoning aims at testing an existing theory .

In other words, inductive reasoning moves from specific observations to broad generalizations . Deductive reasoning works the other way around.

Both approaches are used in various types of research , and it’s not uncommon to combine them in your work.

Table of contents

Inductive research approach, deductive research approach, combining inductive and deductive research, other interesting articles, frequently asked questions about inductive vs deductive reasoning.

When there is little to no existing literature on a topic, it is common to perform inductive research , because there is no theory to test. The inductive approach consists of three stages:

• A low-cost airline flight is delayed
• Dogs A and B have fleas
• Elephants depend on water to exist
• Another 20 flights from low-cost airlines are delayed
• All observed dogs have fleas
• All observed animals depend on water to exist
• Low cost airlines always have delays
• All dogs have fleas
• All biological life depends on water to exist

Limitations of an inductive approach

A conclusion drawn on the basis of an inductive method can never be fully proven. However, it can be invalidated.

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When conducting deductive research , you always start with a theory. This is usually the result of inductive research. Reasoning deductively means testing these theories. Remember that if there is no theory yet, you cannot conduct deductive research.

The deductive research approach consists of four stages:

• If passengers fly with a low cost airline, then they will always experience delays
• All pet dogs in my apartment building have fleas
• All land mammals depend on water to exist
• Collect flight data of low-cost airlines
• Test all dogs in the building for fleas
• Study all land mammal species to see if they depend on water
• 5 out of 100 flights of low-cost airlines are not delayed
• 10 out of 20 dogs didn’t have fleas
• All land mammal species depend on water
• 5 out of 100 flights of low-cost airlines are not delayed = reject hypothesis
• 10 out of 20 dogs didn’t have fleas = reject hypothesis
• All land mammal species depend on water = support hypothesis

Limitations of a deductive approach

The conclusions of deductive reasoning can only be true if all the premises set in the inductive study are true and the terms are clear.

• All dogs have fleas (premise)
• Benno is a dog (premise)
• Benno has fleas (conclusion)

Many scientists conducting a larger research project begin with an inductive study. This helps them develop a relevant research topic and construct a strong working theory. The inductive study is followed up with deductive research to confirm or invalidate the conclusion. This can help you formulate a more structured project, and better mitigate the risk of research bias creeping into your work.

Remember that both inductive and deductive approaches are at risk for research biases, particularly confirmation bias and cognitive bias , so it’s important to be aware while you conduct your research.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square goodness of fit test
• Degrees of freedom
• Null hypothesis
• Discourse analysis
• Control groups
• Mixed methods research
• Non-probability sampling
• Quantitative research
• Inclusion and exclusion criteria

Research bias

• Rosenthal effect
• Implicit bias
• Cognitive bias
• Selection bias
• Negativity bias
• Status quo bias

Inductive reasoning is a bottom-up approach, while deductive reasoning is top-down.

Inductive reasoning takes you from the specific to the general, while in deductive reasoning, you make inferences by going from general premises to specific conclusions.

Inductive reasoning is a method of drawing conclusions by going from the specific to the general. It’s usually contrasted with deductive reasoning, where you proceed from general information to specific conclusions.

Inductive reasoning is also called inductive logic or bottom-up reasoning.

Deductive reasoning is a logical approach where you progress from general ideas to specific conclusions. It’s often contrasted with inductive reasoning , where you start with specific observations and form general conclusions.

Deductive reasoning is also called deductive logic.

Exploratory research aims to explore the main aspects of an under-researched problem, while explanatory research aims to explain the causes and consequences of a well-defined problem.

Explanatory research is used to investigate how or why a phenomenon occurs. Therefore, this type of research is often one of the first stages in the research process , serving as a jumping-off point for future research.

Exploratory research is often used when the issue you’re studying is new or when the data collection process is challenging for some reason.

You can use exploratory research if you have a general idea or a specific question that you want to study but there is no preexisting knowledge or paradigm with which to study it.

A research project is an academic, scientific, or professional undertaking to answer a research question . Research projects can take many forms, such as qualitative or quantitative , descriptive , longitudinal , experimental , or correlational . What kind of research approach you choose will depend on your topic.

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Induction and Its Benefits for Employees Essay

• To find inspiration for your paper and overcome writer’s block
• As a source of information (ensure proper referencing)
• As a template for you assignment

Purposes of Induction

Benefits of induction to individuals, benefits of induction to your organization, works cited.

• It is necessary to mention that the primary purpose of induction is to make sure that a new employee is introduced to a workplace environment
• Another aspect that should not be overlooked is that individuals are provided with standard information that would help them to start working (Randhawa 108).
• It is also imperative to mention that it can be used to make sure that new employees do not have to deal with difficulties that would affect the process of work or may lead to other significant complications.
• Another purpose of induction that needs to be noted is that it ensures that employees can settle into an environment that is new, and adaptation to the process of work can also be critical in most cases (Sutherland and Canwell 135).
• The fact that it helps to ensure that new employees are aware of the unique traditions and culture of an organization also should not be disregarded, and it allows to avoid possible conflicts or disagreements.
• The first benefit for an individual that should be noted is that it helps new employees to socialize, and it is an essential part of the process of work that should not be overlooked because relationships in the workplace can be incredibly valuable in most cases.
• Also, it is necessary to say that an employee may be provided with a range of benefits at the start of the process, such as gifts and others, and it increases their satisfaction levels dramatically (El-Shamy 12).
• Also, an opportunity to be introduced to new technologies also needs to be regarded as beneficial, and the knowledge that is gained can be vital to the process of work (Frater 99).
• Another benefit for an individual that needs to be mentioned is that any possible hazards are mentioned during the process, and it helps a person to get an understanding of what aspects of a particular job may be dangerous.
• Also, the fact that a new worker may ask numerous questions during the process can also be viewed as an advantage because the information that is gained can be valuable in most cases (Gallagher 236).
• One of the biggest advantages for an organization that needs to be mentioned is that the process of induction helps to ensure that new employees are properly introduced to all the necessary aspects of a particular job and have the knowledge and skills to start working.
• Another aspect that is incredibly beneficial is that performance levels of individuals are also improved significantly as a result of induction in most cases, and this is a factor that needs to be taken into account.
• Also, it is necessary to say that it helps to establish a positive perception of an organization, and the image of the company is of utmost importance.
• It is also paramount to note that it can be beneficial from the financial point of view because it helps to reduce the number of possible risks that would be incredibly costly for an organization most of the time (Thompson 46).
• Another benefit to an organization that should be listed is that the process of induction helps to make sure that employees stay because they understand that they are valued by the company.

The Best Practice

It is imperative to understand that an induction program needs to be well developed to ensure that it is as efficient as possible. Also, such aspects as scheduling and expenses on activities should be paid most attention to because it would help to reduce the possibility of complications.

El-Shamy, Susan. Dynamic Induction: Games, Activities and Ideas to Revitalize Your Employee Induction Process . Farnham, UK: Gower Publishing, 2012. Print.

Frater, Glynis. Business and Communication Systems . Cheltenham, UK: Nelson Thornes, 2003. Print.

Gallagher, Kevin. Skills Development for Business and Management Students: Study and Employability . Oxford, UK: OUP Oxford, 2013. Print.

Randhawa, Gurpreet. Human Resource Management . New Delhi, IN: Atlantic Publishers, 2007. Print.

Sutherland, Jonathan, and Diane Canwell. Key Concepts in Human Resource Management . Basingstoke, UK: Palgrave Macmillan, 2004. Print.

Thompson, Neil. People Management . Basingstoke, UK: Palgrave Macmillan, 2013. Print.

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• Academic Writing / Online Writing Instruction

Inductive vs. Deductive Writing

by Purdue Global Academic Success Center and Writing Center · Published February 25, 2015 · Updated February 24, 2015

Dr. Tamara Fudge, Kaplan University professor in the School of Business and IT

There are several ways to present information when writing, including those that employ inductive and deductive reasoning . The difference can be stated simply:

• Inductive reasoning presents facts and then wraps them up with a conclusion .
• Deductive reasoning presents a thesis statement and then provides supportive facts or examples.

Which should the writer use? It depends on content, the intended audience , and your overall purpose .

If you want your audience to discover new things with you , then inductive writing might make sense.   Here is n example:

My dog Max wants to chase every non-human living creature he sees, whether it is the cats in the house or rabbits and squirrels in the backyard. Sources indicate that this is a behavior typical of Jack Russell terriers. While Max is a mixed breed dog, he is approximately the same size and has many of the typical markings of a Jack Russell. From these facts along with his behaviors, we surmise that Max is indeed at least part Jack Russell terrier.

Within that short paragraph, you learned about Max’s manners and a little about what he might look like, and then the concluding sentence connected these ideas together. This kind of writing often keeps the reader’s attention, as he or she must read all the pieces of the puzzle before they are connected.

Purposes for this kind of writing include creative writing and perhaps some persuasive essays, although much academic work is done in deductive form.

If your audience is not likely going to read the entire written piece, then deductive reasoning might make more sense, as the reader can look for what he or she wants by quickly scanning first sentences of each paragraph. Here is an example:

My backyard is in dire need of cleaning and new landscaping. The Kentucky bluegrass that was planted there five years ago has been all but replaced by Creeping Charlie, a particularly invasive weed. The stone steps leading to the house are in some disrepair, and there are some slats missing from the fence. Perennials were planted three years ago, but the moles and rabbits destroyed many of the bulbs, so we no longer have flowers in the spring.

The reader knows from the very first sentence that the backyard is a mess! This paragraph could have ended with a clarifying conclusion sentence; while it might be considered redundant to do so, the scientific community tends to work through deductive reasoning by providing (1) a premise or argument – which could also be called a thesis statement, (2) then evidence to support the premise, and (3) finally the conclusion.

Purposes for this kind of writing include business letters and project documents, where the client is more likely to skim the work for generalities or to hunt for only the parts that are important to him or her. Again, scientific writing tends to follow this format as well, and research papers greatly benefit from deductive writing.

Whether one method or another is chosen, there are some other important considerations. First, it is important that the facts/evidence be true. Perform research carefully and from appropriate sources; make sure ideas are cited properly. You might need to avoid absolute words such as “always,” “never,” and “only,” because they exclude any anomalies. Try not to write questions: the writer’s job is to provide answers instead. Lastly, avoid quotes in thesis statements or conclusions, because they are not your own words – and thus undermine your authority as the paper writer.

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Essay on the induction of employees in an organization.

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Read this essay to learn about the Induction of Employees in an Organisation. After reading this essay you will learn about: 1. Definition of Induction 2. Objectives of Induction 3. Procedure 4. Contents of Induction Programme 5. Elements of Good Induction Programme 6. Problems 7. Practices 8. Induction Training In India.

• Essay on the Induction Training In India

Essay # Definition of Induction :

Inductions may be viewed as the socialising process by which the organisation seeks to make an individual its agent for the achievement of its objectives and the individual seeks to make an agency of the organisation for the achievement of his personal goals.

A few definitions of induction are as follows:

According to Edwin B. Flippo, “Induction is the welcoming process to make the new employee feel at home and generate in him a feeling of belongingness to the organisation.”

According to Michael Armstrong, “Orientation or induction is the process of receiving and welcoming an employee when he first joins a company and giving him the basic information he needs to settle down quickly and happily and start work.”

After selecting compatible personnel the organisation must communicate to the new employees its philosophy, policies, customs and practices. Planned induction helps the new employee creates a good attitude, reduces labour turnover and the employee feels at home right from the very beginning.

“Orientation or induction is thus the process of indoctrination, welcoming, acclimatisation, acculturation and socialisation.”

Essay # Objectives of Induction :

An organisation especially a large one should have a systematic induction process to achieve the following aims:

1. To promote a feeling of belonging and loyalty to the organisation among new comers so that they may not form false impression regarding the company because first impression is the last impression.

2. To build up the new employee’s confidence in the organisation and in himself so that he may become an efficient employee.

3. To bring an agreement between the organisation goals and the personal goals of the organisation.

4. To give the new employee information regarding company (its structure, product, policies, rules and regulations) and facilities provided by the company such as cafeterias, locker room, break time, leave rules etc.

5. To introduce the new worker to the supervisor and the fellow workers with whom he has to work.

6. To create a sense of security for the worker in his job by impressing upon the idea that fairness to the worker is the inherent policy of the organisation.

7. To lessen or reduce the cost of replacing the worker in the early impressionable period because of lack of information or incorrect business impressions.

Essay # Procedure for Induction :

There is no model induction procedure. Each industry develops its own induction procedure as per its needs.

The procedure should basically follow the following steps:

1. The new person should be given a definite time and place to report.

2. A very important step is that the supervisor or the immediate boss should meet and welcome the new employee to the organisation.

3. Administrative work should be completed as early as possible. Such items as vacations, probationary period, medical leave, suggestion systems etc. should be conveyed to the employee.

4. Departmental orientation should be conducted. This should include a get acquainted talk, introduction to the department, explanation of the functions of the department, job instructions and to whom he should look for help and guidance when he has any problem.

5. Verbal explanations are, usually, supplemented by a wide variety of printed material, employee hand book, employee manuals, house journals, picture stories, pamphlets etc., along with short guided four around the plant.

Orientation programme usually covers things like employee compensation benefits, personnel policies, employee’s daily routine, company organisation and operations, safety measures and regulations. The supervisor should ensure that he covers all the necessary orientation steps.

Essay # Contents of Induction Programme :

Every organisation has an obligation to make integration of the individual into it as smooth and comfortable as possible. Small organisations may do it through informal orientation by the employee’s immediate supervisor whereas large organisations usually develop formal orientation programmes.

The range of information that may be covered under orientation training is as follows:

(i) Company’s history, philosophy and operations

(ii) Products and services of the company

(iii) Company’s organisation structure

(iv) Location of departments and employee services

(v) Personnel policies and practices

(vi) Employee’s activities

(vii) Rules and regulations

(viii) Grievance procedure

(ix) Safety measures

(x) Standing orders

(xi) Terms and conditions of service

(xii) Benefits and services for employees

(xiii) Opportunities for training, promotions, transfers etc.

Essay # Elements of Good Induction Programme :

A good induction programme has three main elements:

(I) Introductory Information :

Introductory information regarding the history of the company and company’s products, its organisational structure, policies, rules and regulations etc. should be given informally or in group session in the personnel department. It will help the candidate to understand the company and the organisational policies and standards well.

(II) On the Job Information :

Further information should be given to the new employee by the department supervisor in the department concerned where he is placed on the job about departmental facilities and requirements such as nature of the job, the extent of his liability and employee’s activities such as recreational facilities, safety measures, job routine etc.

(III) Follow up Interview :

A follow up interview should be arranged several weeks after the employee has been on the job by the supervisor or a representative of the personnel department to answer the problems that a new employee may have on the job.

Essay # Problems in Induction :

An orientation programme can go wrong for a number of reasons.

The HR department should try to avoid following errors:

1. The supervisor who has to induct the employee may not be trained or may be too bossy.

2. Employee is overwhelmed with too much information in a short time

3. Employee is confused with a wide variety of forms to be filled

4. In the initial stages, employee is given only manual jobs that discourage job interest and company loyalty

5. Employee is asked to perform challenging jobs where there are high chances of failure that could needlessly discourage the employee.

6. Employee is given only a sketchy induction under the mistaken belief that “trial and error” method is the best induction.

7. Employee is forced to balance between a broad orientation by the HR department and a narrow orientation at the departmental level.

8. Employee is thrown into action too soon. His mistakes can damage the company.

9. Employee may be asked to work on a number of jobs and he may develop wrong perceptions because of short periods spent on each job.

Essay # Practices of Induction :

Different induction practices which are generally used in an industry are:

(I) Induction Guide:

Such guide books are prepared by the personnel department with information on what induction steps have been taken and what are still to be covered various steps to be taken and by whom and when the instructions are to be given are listed in the guide book.

In some large concerns guide books containing the information regarding the company and its various personnel policies are distributed among the new comers.

(II) Counselling :

The supervisor may induct the new employees working under him by introducing and counselling them by reassuring and reinforcing the confidence and guarding against false impression.

On the basis of this interview, personnel department can take action to know the employee’s feelings and to remove the difficulties faced by him through personal talks, guidance and counselling. It may be coordinated by the joint efforts of job supervisor and the personnel department. Periodic following is required to ensure that the employee is properly placed and feels at home.

The best method of induction training is talk plus pictures followed by printed materials. A tour of the plant and the department should be arranged to acquaint the new employee with the overall operations of the company.

Essay # Induction Training In India :

Examples of the induction training programmes followed by a few companies in India are as follows:

1. Maruti Udyog:

Maruti Udyog has different types of induction programmes for different fields.

For engineers the programme is offered in four parts:

i. Familiarise with various functions and meet division heads

ii. Work on shop floor

iii. Work at various other departments

iv. Work finally in those departments for about 2 months, where they eventually have to work.

The company takes its new employees through a one day structured induction training programme.

The programme includes:

i. Briefing on the company’s market position

ii. Business of the company

iii. Functioning style

iv. Organisational structure

v. HR policies

In additional to this, six month behavioural training is offered in team building, self development, customer sensitivity etc. Finally the new employees are put through an appraisal process to judge the progress.

3. Standard Chartered Bank:

The Bank picks up management trainees from premier B-Schools and gives them induction training for about six months. During this period, the trainees spend time in various divisions of the bank to get a brief view of the bank’s operations and get a chance to meet each of the business heads.

Afterwards, a two day session on team building is also conducted. After taking charge of the jobs, the new employees have to attend a review session about the job itself.

4. Sony India:

This company does not follow any uniform policy for acclimatisation and there is no specific time frame given to the newcomers. The company, however, gives enough opportunities to them to understand the process, the culture and the systems of the organisation. In the case of junior managers, new recruits are given less time as compared to the entrants who need to supervise, chalk out strategies and delegate work.

Overall, Sony tries to bring out the best in a person, thus, allowing the individuals to develop their abilities.

Related Articles:

• Induction: Meaning and Importance of Induction
• How To Make An Induction Programme More Effective?

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Induction Process

The induction process is a crucial aspect of integrating new employees into an organization. It encompasses a series of activities and initiatives designed to familiarize newcomers with the company culture, policies, procedures, and expectations. A well-executed induction process sets the stage for a positive employee experience, enhances retention rates, and facilitates productivity. In this essay, we will delve into the components of an effective induction process and its significance in the modern workplace.

Firstly, the induction process begins even before the new employee's first day on the job. It typically involves sending welcome emails or letters, providing information about the company's history, values, and mission, as well as logistical details such as the location, parking, and dress code. This pre-arrival communication helps alleviate anxiety and builds excitement for the new role.

Once the employee joins the organization, the formal induction process kicks off. This phase often includes an orientation session conducted by HR personnel or senior management. During orientation, employees learn about the organization's structure, key personnel, departmental functions, and relevant policies. They may also receive training on essential tools, software systems, and safety protocols, depending on their role.

Following orientation, the induction process continues with on-the-job training and mentorship. This hands-on approach allows new employees to apply theoretical knowledge in real-world scenarios, gain practical skills, and understand their role within the team and the broader organizational context. Mentors or buddy systems provide invaluable support and guidance, helping newcomers navigate challenges and integrate smoothly into the workplace culture.

Moreover, the induction process should be personalized to cater to the individual needs and learning styles of each employee. Some may require more time and support to grasp complex concepts or adapt to new environments, while others may thrive with minimal supervision. Flexibility and responsiveness are key to tailoring the induction experience and ensuring that every employee feels valued and supported from day one.

In conclusion, the induction process plays a pivotal role in shaping the employee experience and setting the stage for long-term success within an organization. By investing time and resources in a comprehensive and well-structured induction program, companies can foster a sense of belonging, accelerate employee acclimatization, and ultimately drive performance and retention rates. As the workplace continues to evolve, prioritizing the induction process remains essential for building resilient and high-performing teams.

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Discuss the Value and Importance of Employee Induction in the Modern Workplace

Introduction.

In the modern workplace, employee induction programs are very important.  The main reason why induction is important for an organization is that it helps to integrate new employees into the business and show them the procedures, systems, culture, and values of an organization. It also familiarizes them with the new environment. A well-performed induction communicates to employees that the business values and cares about them (Bornman, 2014). A proper induction reduces the number of accidents and mistakes at work and improves the quality of work and ensures that the customers are satisfied. However, various benefits arise from  employee inductions to the company.

Importance of Employee Induction in the Modern Workplace

Reduces Costs and Turnover

The well-structured induction training program is an important and natural development of a recruitment process. The programs are important to ensure the success of a worker, ensure that they adapt easily to their new obligations. Every business intends to get its return on investment. Therefore, someone leaving in a half a year is not a good business result and not proper for an organization’s culture, productivity and morale. While a business is keen to get through the backlog within the shortest time possible, the new member of staff requires some time to arrive at an organization’s specific culture. They should also know their obligation in the business and become conversant with ‘why’ and ‘how’ to do such things, and see their queries regarding the company being addressed (Patwardhan, 2020).

Reduces Risks and Ensures Efficiency

New workers need to be across any legal and compliance needs related to a business, and the procedures and processes of “how” to do business. They should understand the culture, mission, vision, and goals of an organization. For them to operate efficiently and be involved in their job, they should be educated on the organizational policies. This includes understanding their duties and responsibilities as employees. It is important if they sign off on such policies to ensure that they understand, which is a good practice for risk management for possible confusion down the track. It cannot be assumed that because they know where the organization’s manual is on the internet, they understood it (Patwardhan, 2020).

Leads to a Smooth Changeover

With the correct form of induction, employees can clearly understand an organization’s corporate prospects and ensure that the new hires do not pick up a second-hand partisan view. A good induction program does not have to be wide or huge, but it is organized and simply rolled out to every new worker. Induction increases a worker’s initial experience with the business. It reveals that they are maintained, cared for and that the business is devoted to their success. From a practical viewpoint, it ensures that the transition into a business is measured, smooth, and relaxed (Patwardhan, 2020).

Gives the New Members of a Team Confidence in their Business Practices 

Normally, new workers should feel anxious or insecure about the new responsibility and how they fit in the organization’s business practices and culture. The process of onboarding is a good business opportunity to make a perfect first impression. Induction enables new workers to clearly understand how an organization works, where it is, where it foresees itself in the future, and how they, as new workers can contribute to making such vision a reality. Inductees will have an understanding of the business’s background, values and culture, policies, learning and development, benefits, and health and safety guidelines, among others (Powers, 2019). Ultimately, the process of induction is a good opportunity to make the new workers proud of their new business. Inductees will feel included and valued because they will understand that their contributions to the business are valued.

Sets the Scene for New Role of Inductees

The induction process familiarizes the new workers with the organization and their main roles and responsibilities. After the induction process, the new staff should clearly understand their role in the business and have the information needed to prepare for the new role (Powers, 2019).

Shows the Professionalism of a Business 

Induction is a good opportunity for a business to build a good impression. The process needs careful preparation and is a formal way of welcoming new employees into the business. This shows an organization’s commitment to observing the professional values as far as performing its business and handling its existing workers and new employees (Powers, 2019).

Gives the New Members a Structure to Settle

Inducting new workers gives a way for the business to give a structure and help them settle in their role and make the process of integration seamless. Typically, it includes a road map for new employees, which includes relevant training and their long-term goals (Powers, 2019).

Makes Sure that the Vital Elements of Workers and Practices are Well-Defined 

Induction allows the new workers to meet and become familiar with other staff and the managerial team and socialize and begin to build relationships (Powers, 2019). Also, it ensures that inductees understand the rules and regulations, code of conduct, and employee expectations. A well-organized process of orientation is a witness to the business’s commitment to making its workers feel valued. Induction enables new employees to feel welcome and get rid of their worries and confusion. In the end, the business gains from a well-thought-out process of induction. This includes improved job satisfaction, performance, and improved employee retention.

Establishes Good Communication

Induction training helps new workers establish good communication with an organization. As part of the training program, the new workers are introduced to their direct supervisor, other workers, leads, and managers of an organization. This makes them calmer when communicating with them later (Antonacopoulou & Güttel, 2010).

Employer Brand Building Opportunity

Each organization wants to employ the best talent in the market. Brand building rotates on packaging a company to stimulate job applicants or potential workers to an organization. Each organization is looking for the best talent in the market and must prove that as an organization they have the best processes, systems, benefits/perks, employee development opportunities that might be employed in the organization. Induction makes a company excite the new worker about its products, systems, services, structures, growth opportunities, and line management support (Antonacopoulou & Güttel, 2010). Once new workers are excited, they would recommend such an organization to their friends, family, school mates, among other people, and endorse the brand of that specific organization. More people may apply to such an organization and then the business can recruit the very best talents.

Research by Brandon Hall Group recommends that firms with strong onboarding processes boost their new hire rate of retention by 82 percent and their worker productivity by over 70 percent. The need for employee induction cannot be overstated. Robert Half found out that 59 percent of hiring managers in Australia have had workers resign during their probation time because of poor processes for onboarding and 43 percent of managers lost new workers within one month of hiring them (Bryson, 2018). Despite such alarming statistics, it appears that managers are missing a beat regarding the efficiency of their induction efforts. The Robert Half survey above shows that 28 percent of employment managers believe their present onboarding process is ‘excellent’, 51percent consider their process as ‘good’, and 16 percent believe what they do in the induction space is ‘sufficient’ (Bryson, 2018). Such disparity in manager insight and retaining figures proposes there is an important detach between how managers believe they perform from an induction viewpoint and what is happening when a new worker joins an organization. The paradigm is just simple, to ensure the success and retention of new employees, an organization must ensure their induction experience is consistent, structured, and positive.

Organizations that do not value induction training often risk personnel having the feeling of being drowned. They may also feel that they are wasting their time learning about the systems, protocols, and processes that everybody else appears to know about, and are scared to ask others (everybody appears so busy, leaving them to feel unskilled and misused). New workers should instead focus on tasks with a stronger financial outcome (Edwards, 2005). Therefore, rather than consider inductions a wastage of time, it is important to view a good induction program as a process of helping people become more productive. A good program should comprise system training and if the system used by an organization is complex, then there should be a schedule for formal training. A good induction program should not necessarily be a long process but if the work is complex, possibly consider running it for some time to prevent information burden for the workers. Inductions are important for organizational employees and the health of an organization.

Each new worker must go through induction to get the right impression on the business. Induction can be an advantage for each organization to reduce turnover of employees, boost efficiency and help the organization to be an employer of choice for extremely skilled employees. Also, new workers will adjust and mix quickly into the company and perform to their best. A well-performed induction communicates to employees that the business values and cares about them. How a new worker fails or succeeds at work may depend on the process of induction.

Reference List

Antonacopoulou, E. P., & Güttel, W. H. (2010). Staff induction practices and organizational socialization: A review and extension of the debate.  Society and business review .

Bornman, L. (2014).  Exploring Realistic Job Previews in the modern workplace: an employee perspective  (Doctoral dissertation, University of Pretoria).

Bryson, A. (2018). Mutual gains? The role of employee engagement in the modern workplace. In  Rethinking Entrepreneurial Human Capital  (pp. 43-62). Springer, Cham.

Edwards, M. R. (2005). Employer and employee branding: HR or PR.  Managing human resources: personnel management in transition ,  4 , 266-286.

Patwardhan, S. (2020). Employee volunteering programs: an emerging dimension of modern workplaces.  Mandated Corporate Social Responsibility , 215-243.

Powers, K. (2019). Stress and the Modern Workplace.  Workplace Psychology .

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The place of induction in science.

Published online by Cambridge University Press:  14 March 2022

The place of induction in the framing and test of scientific hypotheses is investigated. The meaning of ‘induction’ is first equated with generalization on the basis of case examination. Two kinds of induction are then distinguished: the inference of generals from particulars (first degree induction), and the generalization of generalizations (second degree induction). Induction is claimed to play a role in the framing of modest empirical generalizations and in the extension of every sort of generalizations—not however in the invention of high-level hypotheses containing theoretical predicates. It is maintained, on the other hand, that induction by enumeration is essential in the empirical test of the lowest-level consequences of scientific theories, since it occurs in the drawing of “conclusions” from the examination of empirical evidence. But it is also held that the empirical test is insufficient, and must be supplemented with theorification, or the expansion of isolated hypotheses into theories. Refutation is not viewed as a substitute for confirmation but as its complement, since the very notion of unfavorable case is meaningful only in connection with the concept of positive instance. Although the existence of an inductive method is disclaimed, it is maintained that the various patterns of plausible reasoning (inductive inference included) are worth being investigated. It is concluded that scientific research follows neither the advice of inductivism nor the injunction of deductivism, but takes a middle course in which induction is instrumental both heuristically and methodologically, although the over-all pattern of research is hypothetico-deductive.

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Slightly modified version of a paper read at the 6th Inter American Congress of Philosophy, Buenos Aires, August-September, 1959.

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• Volume 27, Issue 3
• Mario Bunge (a1)
• DOI: https://doi.org/10.1086/287745

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• Emma Stanton , psychiatry specialist registrar, seconded to the chief medical officer 1 ,
• Claire Lemer , paediatric registrar 2
• 1 Department of Health
• dremmastanton{at}gmail.com

Emma Stanton and Claire Lemer look at how the NHS does induction, and how it should change

Induction to the NHS is a rite of passage for each of the 15 000 doctors starting new jobs every August. At present, induction is formulaic and uninspiring. A valuable opportunity is being lost to introduce new colleagues to the vision for the NHS, to develop insight into the local healthcare organisation and population health needs, and to create a sense of teamwork and institutional loyalty.

A review of hospital induction programmes in the Anglia region showed that trainees valued being given certain information when they started a new job. 1 Box 1 shows the topics that junior doctors wished to be included in the early stages of an induction programme.

Box 1: Topics to cover on day one of an induction programme 1

Service processes and procedures—Ordering investigations; hospital forms and notes; service departments; admission and transfer procedures; discharge drugs and procedures

Understanding hospital services—Bleeps; switchboard; car parking; ID; support staff

Personal and comfort requirements—Accommodation; housekeeping; catering facilities

Orientation to the new environment—Mess and hospital facilities; timetables; rota

Essential practical skills—Cardiopulmonary resuscitation; hospital computer

Professional and financial concerns—P45; contracts; indemnity.

A questionnaire completed by 107 consultants and 121 specialist registrars from two teaching hospitals in the East Midlands found junior doctors “were seen as not prepared for starting work, especially in regard to clinical and practical skills and the more challenging communication skills.” 2 The experiences of six foundation year doctors describing their first day at work in August 2009 suggests that there is more the NHS could do to improve the quality of induction. 3

This article reviews what induction means for Generation Y and presents the findings of an online survey comparing the induction experiences of junior doctors and NHS management trainees.

Generation Y style

Most junior doctors are from Generation Y (aged up to 28 years). Research on what “Gen Y” want from the workplace, from a 26 question survey of 2521 respondents, is shown in figure 1 ⇓ . 4 This suggests that Gen Y think differently to Generation X (aged up to 42 years) and the baby boomers (aged up to 62 years) about work, learning, and relationships. Up to 80% of Gen Y who were surveyed said that a good induction was important when starting a new job. Induction was considered more important than earning lots of money or fast promotion.

How Gen Y learn best is when it is fun or through mentoring or coaching (fig 2 ⇓ ). When considering redesigning induction programmes, it is important to note that computer based learning and classroom style learning are unpopular with Gen Y. The survey described below showed that 55% of junior doctors received online training as part of their induction. Negative comments received about this were that such modules were frequently “poorly written,” “irrelevant,” and “long winded and took up large amounts of personal time.”

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Junior doctors’ experiences of induction compared with NHS management trainees’

In September 2009, 42 junior doctors completed an online survey about their most recent experiences of induction (from BAMMbino, www.bamm.co.uk ). Sixty two NHS management trainees completed the same questions, shown in box 2, in November 2009.

Box 2: Induction survey questions

1. On a scale of 0 to 10, with 0 being the worst and 10 being the best, how would you score your most recent induction programme? Why?

2. What percentage of your most recent induction programme was spent on functional training such as fire safety?

3. Was patient safety discussed at your induction programme?

4. Please tick if you met either of these people during your induction programme?

Medical director

Chief executive

5. Did you receive any online training either before or as part of your induction programme? If yes, what did this consist of?

6. Did your induction programme teach you more about your organisation and the patients you will be working for so that you can work more effectively?

7. Do you feel part of a team as a result of your induction programme?

8. Do you feel more inspired about your job as a consequence of your induction programme? Why?

9. Does the organisation you are working for have a vision that was clearly stated during your induction programme?

10. Please state what you think the point of your most recent induction was.

The findings are stark—no attempts to inspire and create institutional loyalty; limited attempts to explore the local health organisation and population health needs; and an overall cynical reaction to induction. Of note, medical trainees seem more detached from the process than their managerial colleagues, perhaps because the managers are more likely to meet a senior executive face to face during induction. Only 19% of junior doctors met the chief executive of their trust at induction, whereas more than 50% of management trainees reported meeting the chief executive during their induction.

Figure 3 ⇓ shows that NHS management trainees are more positive about their most recent induction experiences when asked to score them out of 10, with 0 being the worst and 10 being the best. The most frequent score given by management trainees was 8 out of 10 (21% of responders), whereas the most frequent score given by junior doctors was 3 out of 10 (24% of responders). Induction for NHS graduate management trainees introduces participants to all aspects of the organisation. This includes acute and mental health care, primary care trusts, and the ambulance service. This provides new management trainees with an overview of the whole organisation, which is not currently part of junior doctors’ induction.

Forty nine per cent of junior doctors and 66% of management trainees spent less than one fifth of their induction on functional training, such as fire safety. Despite recent high profile cases of poor patient safety at Mid Staffordshire and Basildon and Thurrock Foundation Trusts, 40% of junior doctors surveyed did not recall patient safety being discussed at their most recent induction. More than 80% of management trainees, however, did discuss patient safety at their induction.

On the basis of the findings of this small survey, the majority of junior doctors (57%) were not taught about their organisation and the patients they would be working for, compared with only 21% of management trainees. Disappointingly, more than two thirds of junior doctors did not feel part of a team as a result of their induction programme. Nearly half of management trainees did not feel part of a team following their induction experience.

More than 80% of junior doctors and 47% of management trainees did not feel more inspired about their jobs after their induction programme. More than two thirds of junior doctors and management trainees were unable to recall a vision for the organisation they work for clearly stated during their induction. When asked about the point of induction, both sets of responses were infused with cynicism: “tick box exercise,” “to ‘protect’ the trust rather than to help patients or make me feel more welcome.” Online training was described as “very long winded and annoying induction modules.”

Onboarding: learning from other organisations

The NHS has yet to embrace the potential benefits of induction, although there are signs that this is changing. North Middlesex University Hospital is moving towards unified corporate induction, where all staff are inducted together rather than in professional silos.

Other industries have focused time and attention on this area because of the relationship between induction, absenteeism and staff retention, and employee productivity. The Aberdeen Group (a leading provider of research on global technology) found that “90 per cent of employees make their decision to stay at the company within the first 6 months.” 5 High performing companies turn these crucial months into a positive experience before the employee even sets foot in the company. The Aberdeen Group found that high performing organisations were 35% more likely to have a “new hire programme.” Within this programme there were four key components for success: beginning before day one (that is, the process of introducing the environment should begin before the doctor starts work); continuing for at least six months; making “socialisation” part of the process; and identifying the important business issues and making these part of the programme. The Aberdeen Group call this process of immersion and induction “onboarding.”

It is clear that there are key differences between this approach and induction into the NHS. Successful onboarding focuses on reaffirming an employee’s decision to join an organisation—the findings of the above survey show that the NHS appears to be having the opposite effect. Onboarding seeks to immerse the new recruit into the institutional culture—it is clear from the survey that those experiencing NHS induction, however, lack information about the corporate culture of the NHS. Furthermore, onboarding seeks to make process issues, such as paperwork, easy, and links induction to appraisal and assessment or personal development: both of which lack emphasis in NHS induction. 5

Achieving these core features does not need to rely on complex technology or expensive interventions. It requires a shift from a procedure driven process to one where the aim is to welcome new employees, to capture hearts and minds, and to enthuse.

Many organisations have made inroads into this with simple strategies such as issuing welcoming DVDs, providing books and information on the history of the organisation, ensuring access to executives, particularly in less formal settings, and, most importantly, celebrating the new recruits. Box 3 shows the experiences of a junior doctor’s induction to a secondment at McKinsey and Company management consultants.

Box 3: Induction to McKinsey and company

My induction to McKinsey and Company began in the board room. A small group of new arrivals spent over a week immersing ourselves in the corporate culture, being provided with the tools of the trade—a laptop, a BlackBerry, and a credit card—and learning about the values of the company alongside the practical skills needed to start work. At the end of long days spent grappling with Excel were opportunities to socialise, meet key individuals, and be made to feel welcome and special.

Recommendations for the future

The findings of this small survey suggest that induction programmes for junior doctors could learn from the induction programmes provided for NHS management trainees. The NHS could also look to other organisations to learn about successful approaches to engaging employees. The Gen Y research shows that induction is important and that Gen Y-ers learn best when it is fun and multisensory, and through their peers.

At the heart of improving induction is a shift towards a culture where institutional loyalty is sought. Following the example of corporate induction in other industries, the NHS could make a number of small alterations to achieve this.

Following the corporate example means starting induction early, as well as changing the language of job offers from a bureaucratic function to an enthusiastic welcome, and doing so as soon as possible to allow doctors to make the necessary changes to their lives.

Learning from John Lewis, the department store, one further practical suggestion for improving junior doctors’ engagement with the NHS, and to do so early on, is to create an induction DVD. This would introduce new employees to their organisation, clearly stating the background and aims of the NHS through brief interviews with key senior leaders such as David Nicholson, chief executive of the NHS; Bruce Keogh, medical director of the NHS; and Liam Donaldson, chief medical officer. Sending such a DVD before junior doctors’ induction would inspire them and help to develop a wider awareness of the environment in which they will be working. It also allows for the induction process to focus on core business issues, reminding doctors of what is important for the NHS.

Following the example of the North West Deanery, making the deanery the sole employer would reduce the unnecessary and time consuming paperwork, such as repeatedly filling out occupational health forms. This would considerably streamline the induction process.

Finally, following the lead of organisations such as the North Middlesex, integrating staff induction not only improves teamwork and a sense of belonging, but, by reducing the number of inductions required, means that there is possibly more likelihood of senior executives attending.

Too often, induction for junior doctors consists of a single day of formal didactic presentations that alienate our valuable workforce before they have set foot on the wards. This article presents the case for a radical overhaul of the way in which junior doctors are inducted into the NHS. There is a need to make induction fun, inspiring, and to cover crucial aspects of quality, including patient safety and teamwork, together with providing an opportunity to meet senior leaders within the local organisation, such as the medical director and the chief executive.

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• ↵ Martin K, Saba J. All aboard: effective onboarding techniques and strategies. The Aberdeen Group, 2008.