How, if at all, can the traditional problem of induction be solved?

1. Introduction

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.

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).

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.

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


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