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Induction: The Problem Solved

In our second contribution on the problem of induction, John Shand argues that there is no problem, because there is no such thing as an inductive argument.

When people talk about the problem of induction, they mean the problem of justifying the rationality of inductive arguments. My eliminative answer is to argue that there is no problem of justifying inductive arguments because there is no such thing as an inductive argument. Socalled inductive arguments as such have no logical force. Nevertheless I will contend that there is such a thing as an inductive procedure whose statements are true and track facts in the world; but such procedures aren’t arguments because the statements involved aren’t linked by deductive relations.

What I will say in fact applies to any less-than-deductive putative argument. A putative argument is a set of statements where one, the conclusion, is claimed to follow from other statements, the premises. If what we are considering is an argument, and not just a putative argument, then the conclusion does follow from the premises. My contention is that it is only in the case of deductive arguments that the conclusions follow from the premises. Conclusions in this case follow from premises because to assert the premises but deny the conclusion would be a contradiction. If an argument is valid the conclusion can’t be false if the premises are be true; an argument is invalid if the conclusion could be false while the premises were true. If the premises are true, and the argument is valid, then the conclusion must be true – the argument is sound.

The challenge, which I claim to be unanswerable, is to come up with a case where the conclusion of a non-deductive putative argument can be said to follow from the premises. I don’t think this can be done, and in fact I think it is a misnomer to speak of ‘premises’ and ‘conclusion’ here at all. In all but deductive cases, statements are not in fact linked as premise to conclusion. So why might we think they are? In cases where we think we have found a non-deductive argument whose conclusion does follow from its premises, we only think so because we have tacitly assumed some often very obviously true extra premises that would turn the less-than deductively valid sequences of statements into a deductively related set, into an argument. These premises are often derived from other arguments we already implicitly accept. When asking whether in the case of a putative non-deductive argument, Y follows from X, one is likely to get a reply something along the lines of “No it doesn’t. But if you assume Z…”

The motivation for thinking that inductive ‘arguments’ (and other less-than-deductive ‘arguments’) need a special justification derives from the belief, which I think is mistaken, that something weaker than deduction is required for empirical investigation whereby we may draw contingent conclusions that are underdetermined by the premises. By ‘underdetermined’ is meant cases where denying the conclusion while asserting the premises doesn’t entail a logical contradiction. But as I will show, this is a profound confusion. The confusion lies in thinking that the ‘problem’ of inductive arguments is in the move between the premises and conclusion, and that this is what needs justifying; whereas in fact the difficulty lies in knowing the truth of premises that would render the argument deductively valid. But the issue of knowing the truth of such premises need not worry us in the slightest. It is not a problem, for to suggest it is in turn confuses our inability to know the truth of such premises and their being true. If an argument has true premises, and the conclusion does follow validly from the premises, then the conclusion is true, and we have a reason for believing it true, whether anyone knows these things or not. Nor is there a difficulty in squaring the statement constituting the conclusion of a deductive argument with the contingent state of affairs to which that statement may refer. The conclusion of a valid deductive argument, although it must be true if the premises are true, is just as contingent as the premises that entail it, and there is nothing illegitimate in the premises of a deductively valid argument being contingent.

All putative arguments that are not deductive arguments are in fact enthymemes – that is, arguments with missing premises. The belief that inductive arguments require a special justification to entitle us to move from premises to conclusion is a mistake. It is odd and arbitrary that we feel the need to justify this sort of invalid argument – we should as in other cases simply categorise it as a formal fallacy. Inductive arguments are simply invalid. Take the following one, for instance.

Example 1

Premise All observed free rocks near the surface of the Earth fall towards the centre of the Earth.

Conclusion Therefore, this (or all) free rock(s) near the surface of the Earth will fall towards the centre of the Earth.

Is this a good argument? Is it valid? Could the conclusion be false while the premise is true? In fact it is an incredibly feeble argument. There is no problem in thinking, with even only a little imagination, of how the conclusion might be false while the premise true. In fact there are probably no limits to the ways in which the conclusion could be false while the premise could still be true. Clearly it is invalid, and the next free rock might behave differently from any rock yet observed. It could turn into a butterfly as far as this putative argument is concerned. There is no contradiction in supposing this with the premise involved in this argument. The argument, as it stands, gives us no reason for thinking the conclusion true. So why do we think it does? Because we tacitly import latent (hidden) premises so that the conclusion does follow from the premise.

Making it such that the conclusion follows from the premises is indeed quite simple. All that is required are true premises that would turn the putative inductive argument into a sound deductive argument. Something like a principle that nature behaves the same everywhere (sometimes called the Uniformity of Nature Principle), combined with a covering law.

Example 2

Premise All observed free rocks near the surface of the Earth fall towards the centre of the Earth.

Premise Laws of nature hold universally throughout space and time.

Premise It is a law of nature that all physical bodies, such as rocks, in the conditions specified will fall towards the centre of the Earth.

Conclusion Therefore, this (or all) free rock(s) near the surface of the Earth will fall towards the centre of the Earth.

The perceived difficulty now, which tends to be thought to block this solution, is the difficulty of knowing that the second premise is true. It is well known, as David Hume showed, that the attempt to prove that there are laws of nature that hold universally across space and time fails on the only two alternative routes it can take. It cannot be shown by reason because its denial does not constitute a logical contradiction, and it cannot be shown by experience because not all cases can be examined. In addition, of course, the attempt to justify induction directly falls foul of similar insurmountable problems: that the next instance might not correspond to previously observed instances is not a logical contradiction, and the attempt to justify inductive procedures by observation – saying that induction has worked in observed instances and so will work in unobserved instances – is itself an inductive argument and so is circular and thus proves nothing.

Various other moves have been proposed that need not detain us long. In particular there is the attempt to show that induction works at least for the world as experienced, because such a notion is a necessary condition of our having an experience at all. Apart from the difficulty of how such a claim could itself be justified, this leaves all the well-known problems of how to deal with the world as not experienced – and the deflection that such a world can be ‘nothing to us’ hardly seems very satisfactory.

It might be argued that the premises of putative inductive arguments at least render the truth of their conclusion more probable. But this again helps not at all and makes no difference to the arguments presented here to the effect that there is no such thing as an inductive argument. Such structures give us no reason for the conclusion being true. There is the well-known problem that premises usually refer to a finite set of instances from what is potentially an infinite set, so strictly speaking the probability remains zero. But the knockout blow is that probability only pushes the problem of justifying the move from ‘premises’ to ‘conclusion’ one step back, whence it arises hale and fully formed again: for to say that the premises make the conclusion even more probable is itself an inductive argument of exactly the sort the appeal to probability was meant to circumvent. Again the reason we think the premises render the conclusions more probable is because we tacitly turn the socalled inductive argument into a deductive one through latent premises. Otherwise such premises imply almost nothing.

Before presenting my answer here of eliminating the problem of induction by eliminating induction as an argument, it must be said that inductive procedures can still have a place in our approach to dealing with the world even when we do not turn such inductive procedures into deductive arguments by adding premises that would do so. This is not to re-import induction as a process of argument. But it needs explaining why we might still think inductive structures useful. The logical structure of Example 1 is:

All observed A is B.

Therefore this (or all) A is B.

As an argument this is a non-starter. In fact very little follows from what are usually presented as the premises of inductive arguments. At worst, putative inductive arguments are just disconnected heaps of statements. But we could still find that the procedure of making judgements on what we have not observed on the basis of what we have observed is a useful one, and tends to lead to the truth rather than not. But it is a commonplace that showing the conclusion (outcome) of a putative argument to be true (indeed its being true) does not show the argument to be valid. It is quite possible to go from true premises (or indeed false premises) to a true conclusion, and the argument still be invalid. What one cannot do is go from true premises to a false conclusion and the argument be valid. That following the statements of a putative argument takes one from truth to truth – involves only true statements – does not show it is a valid argument or indeed an argument at all: that it gives a reason for the final statement, the ‘conclusion’, being true. Of course, we would have to be careful in using such inductive procedures. We might find that they tend to come out true, or culminate in some truth, but they are not arguments; they do not give us a reason for the ‘conclusion’ being true; they may just happen to be made up of statements that track truths. Take this case:

Example 3

Premise A man walked up to a tree. (true)

Premise The man hit the tree with the axe repeatedly. (true)

‘Conclusion’ The tree fell over on my head. (true)

This is not an argument, although it may look like one – it gives no reason for the last statement being true – nor is “The tree fell over on my head” strictly speaking a conclusion. But still these true statements track a series of facts that we might find useful to trace. But there is no sense in which the conclusion follows from the premises. Again, the argument gives us no reason for the conclusion being true. For the conclusion to follow from the premises we would have to make the statements connected to one another in such a way that the relation is deductive. It is no use asking one, in order to test the validity of the ‘argument’, to “think of ways in which the conclusion could be false while the premises could be true” as there will always be innumerable ways in which this could be so – unless the argument is deductive in form.

The answer presented here is simply to say that there is no such thing as an inductive argument, so there is no ‘problem’ of justifying inductive arguments since they are not arguments at all. There is no sense in which so-called inductive arguments give us any reason for believing their conclusions true. This is so even when they happen to lead to truth. There is no form of reasoning weaker than deduction that somehow still mysteriously gives us a reason for what stands as the ‘conclusion’, whereby the ‘conclusion’ follows.

This would seem to put us in the unsatisfactory position of saying that most of the arguments we actually use give us no reason at all for what we think they give us reasons for. In fact this is, strictly speaking, correct. However, it isn’t a concern because those weaker-than-deductive arguments are usually shorthand for deductive arguments that do give us such reasons: arguments with the required but all-too-obvious deductive-making premises filled in.

The solution to the problem of induction is to acknowledge that there is no problem about justifying inductive arguments because there is no such thing as an inductive argument: no inductively arranged set of statements gives any reason for the ‘conclusion’ being true. They are just that: a heap of statements. The next step is to acknowledge that in order for there to be a reason for the conclusion being true one has to introduce additional true premises that render the putative argument a deductively valid argument. Further, one needn’t worry that these required premises are difficult or impossible to know to be true. For us to have a reason for the conclusion being true, all that is required is that the conclusion does follow from the premises (the negation of the conclusion forms a contradiction with the premises) and that the premises are true. That is to say, the argument is deductively sound. If an argument is deductively sound, then we have a reason for believing the conclusion true whether we can know the premises to be true, or indeed can see that the argument is valid. Now it could just be true that there are universal laws of nature. It may also be the case that we have identified some of them. If this is so, and we may thereby transform putative inductive arguments into deductively sound arguments by adding the principle that nature behaves the same everywhere and adding a true law of nature, then we just do have a reason for the conclusion being true. We should not be side-tracked at this point into quibbling over exactly what premises one might introduce to render a putative inductive argument into a deductively valid one. The point is made as long as some premises can be found to do the job, and that what is wrong with inductive ‘arguments’ is not justifying the shaky move from ‘premises’ to ‘conclusion’ but knowing the truth of whatever additional premises are required to render the sequences of statements a deductively valid argument.

It might be said that the question remains as to whether such required premises are true. But our not knowing whether they are true does not undermine in the least our having a reason for the conclusion being true if it is true and the argument deductively valid. (Take a look at the discussion in my recent book Arguing Well, especially sections 1.6-1.7 and 3.4-3.15. Here I make a distinction between argument in the objective sense and person-directed sense; the latter presupposes the former; both are however purely logical notions involving no appeal to causality or psychology.)

There is thus no special problem of justifying induction as a rational procedure. The only difficulty is to know whether premises needed to make the procedure rational – in that we have a reason for the conclusion being true – are true. But this means there is no problem of induction – no problem of legitimising some non-deductive move as a rational argument – but just a problem of knowing the truth of premises required to make such a move deductive. Such premises could still be true, regardless of how difficult this problem is, which would mean such moves are rational even if we could never know they are. This might look like replacing one problem with another. But the situation has improved dramatically: from the intractable to the fundamentally unproblematic. Whereas before we had the impossible task of seeing how induction could ever be justified, there is now only the fundamentally unproblematic notion of certain premises being shown (or known) to be true: but there is no difficulty in our seeing how it is possible that they could be true whether we can show them to be so or not. The two matters are quite separate: the first is an epistemic problem, what is knowable; the second is a matter of what is the case, what is true. So there is no problem of there being a reason for the conclusions of such arguments being true when such true premises render them deductive. It is not true that a deductive argument does not give a reason for the conclusion being true just because we do not know, or even cannot know, the truth of the premises: if the argument is deductively sound, there is a reason for the conclusion being true. There never was a problem justifying putative inductive arguments because they were never arguments in the first place, not even in a weak sense. Thus the ‘problem of induction’ is solved because induction as an argument never existed, and so was never a form of argument that needed justifying.

John Shand is an associate lecturer in philosophy at the Open University in Manchester, and also edits series of books for Acumen Press.

The Sad Tale of the Inductivist Turkey

The turkey found, on his first morning at the turkey farm, that he was fed at 9am. Being a good inductivist turkey he didn’t jump to conclusions. Instead he collected a large number of observations of when he was fed, and made these observations under a wide range of circumstances, on Wednesdays, on Thursdays, on warm days, on cold days. Each day he added another observation statement to his list. Finally he was satisfied that he had collected enough observations to inductively infer that “`I am always fed at 9am,” and hence that “I will be fed tomorrow at 9am.” However, it was Christmas eve, and the next morning, instead of feeding him…

(Loosely based on an illustration of the problem of induction given by Bertrand Russell in Chapter 6 of The Problems of Philosophy.)