Inductive Reasoning

1. Introduction

Can Inductive Reasoning Be Justified Without Using Induction?

Hume outlined the traditional problem; in sum it poses the question as to whether any justification of induction is possible that is not invalidated by being required to assume that induction is already valid.

Inductive reasoning is less rigorous than deduction, because it does not entail its conclusions when its premises are correct. This is because it by definition involves an extrapolation from the observed to the unobserved. The latter may be distinct from the former in a number of ways, including being future examples of a previous observed kind of item, or future events of a similar type to previous events.

The structure of an inductive argument will be as follows: All observed F’s are G, therefore All F’s are G. Note that there are few if any examples of all items of type F being observed, and even if they had been, there would probably remain epistemic doubt as to whether this was the case. So it is always possible that even if many thousands of white swans have been observed, a black swan will exist. And of course if all F’s had been observed, there would be no application of any inductive knowledge to any further F’s. We would always need a ‘uniformity of nature’ principle holding that the future will resemble the past, and arguments to show the principle would rely on it.

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One is naturally convinced that seeing that may F’s are G is excellent evidence for all of them being so, but this is a direct parallel to Hume’s sceptical problem and why he is also prominent in this area. The fact that we seem naturally to have no choice but to accept the existence of the external world and the validity of inductive reasoning does not mean we are ipso facto released from the question as to how these things are possible.

2. Mellor

Mellor appeals to Ramsey’s encapsulation of the reason for our belief in induction. This says that we use induction because it leads to true beliefs that we can then use with success to complete actions – this is a pragmatic theory of truth. He defines induction as “something that can warrant anticipating an observation” and then introduces a non-standard fallible definition of observation by noting that false beliefs can be acquired through sensory data. So on this line, observation “no more entails the truth of the beliefs it gives us than induction does. How then can it warrant them?”.

Mellor also notes that observation is generally much more powerful than induction: if I observe something highly surprising, I do not dismiss the observation merely because I have never seen an elephant in Piccadilly Circus before. I assume on the other hand that something unusual is happening. Therefore observation must be able to meet any standard that induction can. Mellor will argue that since observation cannot pass what might be termed a ‘uniformity of observation’ test, neither can induction and we therefore need a different test of warranted inductive beliefs.

It is frequently argued that there must be a KK principle viz. I can only know something if I know that I know it. Arguments for this would include the idea that I could only act successfully based on a piece of knowledge if I recognise that piece of knowledge as such. Yet there are strong counterarguments to KK including the myriad of unconsidered background conditions that I frequently assume to be the case without conscious consideration. For example, I will know there is an elephant in Piccadilly Circus if I see one even though I do not know that there is not a rare optic disorder that causes one to see illusory elephants and that I do not have it.

Mellor combines this type of attack with the other idea that if KK, then the first K must be self-intimating. I should not have to consider whether I know that I know something: if I know it, that should already be enough to settle the higher orders. Yet there is an infinite regress here. If true warranted belief if enough for knowledge, as Mellor claims, then by KK I will need to have a true warranted belief that K in order to have K. And I will also need a true warranted belief that KK and so on. Even if these infinities are acceptable, the phenomenology does not allow the idea that they are self-intimating any plausibility.

There is a nod in the direction of Sartre in Mellor’s observation that the attraction of the KK principle may lie in our strong but false intuitions that we know ourselves and that there is a fixed self to know. Abandoning KK permits the adoption of a pragmatic theory of truth; truth becomes “the attribute of beliefs which ensures that the actions they cause will succeed”. So a warrant will be anything that works as opposed to being KK. Note that these are not necessarily incompatible: I could have self-intimating KK that works – it is just that KK does not lead to beliefs that work. And Mellor also claims that KK produces the problem of induction because it introduces the regress.

We can acquire a habit of inductive reasoning because it works. It then continues to work. Even if it evolves randomly, organisms that have it will be immensely more successful than those without. And the position would be so much the worse for those that draw counterinductive conclusions. So without KK, I can use induction because it works and I do not need the first K, and I cannot have it – I do not need to know the principle of induction to use inductive reasoning successfully and therefore I can in fact inductively reason.

3. Dancy

Dancy considers a number of responses in the literature to the problem.

3.1. Circularity not Vicious

Black states that the problematic argument takes the following form:

“Inductive reasoning has proved reliable in the past. Therefore inductive reasoning is (generally reasonable).”

This is an example of inductive reasoning. The conclusion is needed to move from the premise to the conclusion. However – it is not needed as a premise, but as a rule of procedure. If it were included as a premise, the argument would clearly be problematically circular. Yet we do not adopt that approach. There is no premise in deductive syllogisms to the effect that syllogisms entail their conclusions: this is taken as read. We should also bear in mind that inductive arguments are not required to be deductive and so neither is an inductive argument for induction.

And in addition, insertion of such a premise leads to a regress in the familiar way: the premise would need support if explicitly included.

However, the problem with this argument is that it only preaches to the converted. There is no reason to accept the premise unless one already does accept it. This can be illustrated by an inversion of the form ‘induction has succeeded in the past but it may not in the future, therefore induction is not an appropriate form of reasoning’. This seems equally persuasive, especially when one notes that the conclusion is doubly supported by argument from the premise and also by the premise considered alone.

3.2. Argument Analytic

Dancy cites Edwards and Strawson as arguing that solely the meanings of the term evidence can justify induction: the “statement that observation does constitute evidence is true because of what we mean by ‘evidence’ ”. This seems weak for a number of reasons; Quine indeed does not accept the notion of analyticity at all.

But it also seems to load to much on stipulation. It is somewhat similar to Mellor’s redefinition of ‘observation’ to include what we might alternatively call ‘unsuccessful observation’ as discussed above. We can allow Mellor his stipulation because it does not represent an attempt to change the world by fiat. However, while it is clear that a definition of ‘evidence’ can be allowed to only refer to ‘successful evidence’, that does nothing to change the availability in the real world of such evidence.

It is certainly the case on this definition that if we have evidence, we have good grounds for believing the thesis supported by that evidence. But how would we know when we had evidence?

There is a further argument due to Urmson which notes that we learn what is good evidence from approved specialists or those with more experience, but that we can later form different views that are totally at variance with those we have learnt. In fact, this is not just possible, it seems to frequently be the case. And simply posing the problem of induction as Hume did is an example. Thus it seems hard to allow that the notion of evidence is analytic.

3.3. Coherentism

Coherentism is one of three possible exits from Agrippa’s trilemma. The justification for a belief will normally be an appeal to another belief. There are three ways this process can develop: there may (i) be a final core of beliefs that do not have and do not require similar justification (foundationalism); there may (ii) be no end to the chain of beliefs (infinitism) or the best picture may be obtained by (iii) allowing that the chains can loop back into themselves (coherentism).

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Here the key idea is that the beliefs all cohere together and make each other more plausible. The extension to the problem of induction is simply to claim that use of a working principle of induction is justified because it leads to a more coherent set of beliefs. This approach of course relies on the truth of coherentism for its effectiveness.

Hume’s problem of induction stems from his views on causation: there is no necessary connection between events. We just come to expect them by constant observation. Realists about the future who hold that we can have knowledge about future events can deny this and insist that there must be a necessary connection. We could not know that the brick will break the window when it is flying towards it and is an inch away unless there is in fact a necessary connection between the two events of the brick hitting the window and the window breaking.

Again, this line of reasoning may succeed in answering the problem of induction, but again it requires coherentism. It also requires realism about the future when anti-realism is appealing in that context; the most natural statement in relation to statements about the future is that there are no facts of the matter yet and so no truth values so far for such statements.

If anti-realism about the future is true, then there is a reduction in the scope of the problem of induction because there is no knowledge about the future in any case. But some variants might remain. If I have myself observed 90% of the swans in the world and found them all to be white, I may well use an inductive approach to decide what my Australian colleague, who has observed the remaining 10% but not yet reported his results, has already found.

4. Reichenbach

Reichenbach has a further pragmatic approach to the problem. His paper is somewhat redolent of Valberg on scepticism, in that it notes the contrast between some problematic reasoning (purporting to show that there is no justification for induction) and our everyday behaviour of use of induction. This contrast is just as relevant to those familiar with the reasoning as to those not.

Since Reichenbach has a probabilistic view of causation, whereby causes make their effects more probable as opposed to necessitating them, he states the target formulation of the aim of induction as being to “find series of events whose frequency of occurrence converges towards a limit”. Note that the limit is not supposed to be 1.0. The idea is that if a fair coin is thrown, and there is a 50% chance of heads, the percentage difference between the proportion of heads will approach 50% as more throws are made. This is good enough to defeat the problem for Reichenbach: with sufficient observations to confirm that the coin is landing on heads 50% of the time, I can make that statement abut future throws.

It is admitted that the world must be predictable for this to be the case. If the world is destroyed, there will be no truth-values in relation to coins thereafter. Equally, if the laws of physics may fluctuate dramatically, we would not be able to observe the limits of series. But Reichenbach insists that while we cannot know if there can be success, if any method will permit the determination of the limit of the series, then induction will be one such method. An analogy is given of a doctor who justifies an operation without knowing whether it will work on the basis that if anything will work, then it is an operation.

This seems plausible because if the world is predictable, induction is fine – the argument so far has been that we cannot know non-inductively that the world is predictable.

So under these circumstances, induction is the best bet because it minimises risk. The risk that the world is unpredictable cannot be addressed. But using induction at least minimises what might be termed ‘estimation risk’. If I see a coin landing on heads approximately 50% of the time, there are two ways I can be in error in using that as my estimate of future results. The first way is that I depart from induction and choose 75% as my estimate. The second way is that I choose 50% but the world is not predictable.

This is the only justification of induction that can be given. Hume’s objections succeed in showing that no full proof is available as a justification. The only remaining approach is to do what works, and Reichenbach has shown that nothing works better than induction.

See Also:

What Is “Theory Of Mind?”

‘Thoughts’ and Sense For Frege And Burge

Does Nietzsche Favor Master Morality Over Slave Morality?

Does The Observation That Knowledge Ascriptions Are Context-Sensitive Provide The Basis For A Satisfactory Response To Scepticism?


D Hume, An Enquiry Concerning Human Understanding, eds L A Selby-Bigge, P H Nidditch, Clarendon Press, 1975, s4.
D H Mellor, Matters of Metaphysics, Cambridge University Press, 1991, ch. 15
op. cit. p. 263
J Dancy, Introduction to Contemporary Epistemology, Blackwell Publishing, 1985, ch.13 (“ICE”)
Cited by Dancy as M Black, Problems of Analysis, Routledge, 1954
Dancy’s formulation of Black’s position.
Cited by Dancy as P Edwards, Russell’s Doubts About Induction, Mind 68
Cited by Dancy as P Strawson, Introduction to Logical Theory, Methuen, 1952
ICE, p. 203
Cited by Dancy as J Urmson, Some Questions Concerning Validity, Revue Internationale de Philosophie, 25
H Reichenbach, The Pragmatic Justification of Induction, edited selection in S Bernecker & F Drestske (eds), Knowledge: readings in contemporary epistemology, Oxford University Press, 2000, (“BD”)
J Valberg, The puzzle of experience, in T Crane (ed), The contents of experience, Cambridge University Press, 1992


Hume On Causation


I will discuss Hume on Causation and raise some questions.

Hume gives two definitions of cause:

“We may define a CAUSE to be ‘An object precedent and contiguous to another, and where all the objects resembling the former are plac’d in like relations of precedency and contiguity to those objects that resemble the latter’ ” ;


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“A CAUSE is an object precedent and contiguous to another, and so united with it, that the idea, of the one determines the mind to form the idea of the other, and the impression of the one to form a more lively idea of the other” .

Hume On Causation: Problems

As Noonan points out; there are a number of immediate difficulties here. The two definitions do not appear to be coextensive. Also we may ask whether Hume’s use of the word ‘determination’ in the second definition does not include the concept of causation, thus rendering the definition circular.

Hume himself however believes that the two definitions are two ways of looking at the same object. He is led to this by his theory of mind. We are led by habit formed by repeated observations to assume that there is a necessary connection between events. It is part of Hume’s scepticism to insist that this is an illusion, albeit an unavoidable one.

What is common to both definitions is the tripartite structure and the first two elements thereof. The use of the term ‘precedent’ simply means that Hume does not accept the idea of backward causation; any cause must be prior in time to its effect. Further, he does not allow the idea of simultaneous causation:

“Some pretend that ’tis not absolutely necessary a cause shou’d precede its effect; but that any object or action, in the very first moment of its existence, may exert its productive quality, and give rise to another object or action, perfectly co-temporary with itself.”

No Simultaneous Causation

Hume argues against simultaneous causation on two grounds, the first being that it is inconsistent with our observations. His more detailed argument is that if an object is fully existent then that entails it is existent with its full causal powers. If it then fails to have an effect, then it must require an additional partial cause or organising principle to bring about the effect; it is therefore not the sole cause of the effect.

There is on the other hand a contradiction in the notion of a sole cause that does not exert itself. It must lack some element of causal power. Thus a sole cause must act instantaneously if it can do so. It must be able to do so if the notion of a sole cause is rightly understood. And this outcome is contradictory, because it would immediately populate the universe with effects and destroy any idea of succession.

Contiguity simply rules out the idea of action at a distance.

“nothing can operate in a time or place, which is ever so little remov’d from those of its existence”

Again, here Hume relies on the observational evidence that whenever we see a cause in action, it acts locally, and he further supposes that if we do not observe this, it is because the cause is invisible to us rather than that it does not exist, as noted by Zuboff:

“Thus Hume will sometimes speak of ‘secret springs’ of necessity in nature, which are responsible in ways we could never understand for the observed regularities of causation” .

Phenomenal Vs Noumenal

This is highly suggestive of Kant’s later division of the world into phenomenal and noumenal aspects. It must be part of what he meant when he wrote that Hume had awoken him from his ‘dogmatic slumbers’.

The final elements of the two tripartite definitions may both be understood as variants of the concept of necessary connection. In the first definition, the final elements may be viewed as holding that all events of type A are prior and contiguous to all events of type B, then we can say that A causes B. The final element of the second definition means similarly that the idea of A leads us to the idea of B. Also the impression of A leads to the idea of B. This is in accordance with Hume’s theory on the association of ideas.

Hume On Causation: Complete Definitions

Hume sees the definitions of cause he gives as complete and exhaustive i.e. this is all we can say on the topic. It will be apparent that this is a quite reductionist approach, which is in keeping with his general scepticism. His view on causation is in keeping with his view on the closely related topic of induction.

Hume’s statement of the problem of induction holds that in order to justify inductive reasoning, we need to assert a uniformity of nature principle. This would mean that we could use the past to predict the future. However, there is no way to demonstrate the uniformity of nature principle non-inductively. We can never use our experience to go beyond our experience. Likewise, we can never use our experience of causation to go beyond our experience thereof. Thus what we see is all there is.

Hume is an empiricist and subscribes to the maxim ‘nothing in the intellect not first in the senses’. In his terminology, an impression is a perception of the senses. An idea is a later memory or representation of such an impression.

Hume considers whether we can prove that a cause is always necessary for any event. He examines four arguments for this and dismisses them all.

Arguments For The Necessity Of A Cause

The first argument can be phrased after the scholastic fashion ‘ex nihilo nihil fit’, or nothing comes from nothing. However, Hume demands a falsification of the proposition that “any thing can ever begin to exist without some productive principle” and holds that this is both impossible and also consistent with the scholastic phrasing, or “it is a mistake to think that this defeats the logical possibility of a thing simply existing without being produced at all” .

The second argument asserts that nothing can be a cause of itself. Again Hume accepts this, but does not agree that if anything lacks a cause, it must cause itself: “An object, that exists absolutely without any cause, certainly is not its own cause” . To assert otherwise begs the question.

The third argument, which Hume ascribes to Locke, is that anything produced without a cause is produced by nothing; and this impossible because nothing cannot have any causal power. However, Hume again points out that this begs the question because it has already assumed that every event must have a cause.

Finally Hume defeats a semantic argument from the proposition that every effect must have a cause. This again is analytically true in virtue of the meaning of ‘effect’, but does not entail that every event has a cause:

“this does not prove, that every being must be preceded by a cause; no more than it follows, because every husband must have a wife, that therefore every man must be marry’d” .

Hume On Causation: Why We Experience Causation

Hume is now in a position to conclude:

“Since it is not from knowledge or any scientific reasoning, that we derive the opinion of the necessity of a cause to every new production, that opinion must necessarily arise from observation and experience” .

The example given is of a person being fully constituted with faculties but lacking experience. If this person were to observe a billiard ball approaching another, he would have no reason to expect that their collision would result in the struck ball moving off. It could just as well disappear in a puff of smoke or combine with the first ball. It is only our repeated observations of the constant conjunctions of events of type ‘first ball hits second ball’ with events of the type ‘second ball moves’ that makes us think that there is something in the properties of the objects themselves that makes this a necessary connection. As Hume later comments,

“there […] is nothing in any object, consider’d in itself, which can afford us a reason for drawing a conclusion beyond it”


However, it is not the case that we are really justified in using our experience to make this argument. As Reid points out,

“Necessary connection, he concludes, is a mental association that we project onto objects, rather than something that we discover through our experience of them”


“Our experience of constant conjunction leads our imagination to associate events of the conjoined kinds, and to expect one when presented with the other” .

Again this is in good harmony with Hume’s general scepticism as to what we can achieve with limited human intellects, and what is perforce inappropriately but unavoidably foisted on us.

See Also:

Husserl’s Phenomenological Reduction: What Is It And Why Does Husserl Believe It To Be Necessary?

What Is “Theory Of Mind?”

Anscombe: Modern Moral Philosophy — Summary

Nozick’s Claim That Knowledge Is Truth-tracking: A Critical Evaluation


D Hume, A Treatise of Human Nature, Eds. L A Selby-Bigge and P H Nidditch, Oxford University Press 1978 (hearafter “THN”)
H W Noonan, Hume on Knowledge, Routledge Philosophy Guidebooks 1999