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How I Solved Hume’s Problem and Why Nobody Will Believe Me

Eugene Earnshaw saves Western philosophy.

It was a few years ago that I solved the biggest problem in philosophy. I was teaching undergraduates, and I wanted to blow their minds a little, tear down their preconceptions. As one does. So I taught them Hume’s Problem, and then solved it for them, which I hadn’t expected.

Hume’s problem is probably less well known than Descartes’ skeptical argument (a being with Godlike powers could trick you about everything, so you don’t know anything, because there’s no way you can know you are not being thusly deceived). But despite not having inspired The Matrix, it is probably the more important and influential argument overall, for two reasons. One reason is that Kant devised his whole philosophy as a response to it, and I have it on good authority that Kant is important. The other is that the philosophy community today still treats Hume and Hume’s argument seriously in a way that it just does not treat Descartes. So although Descartes is probably even more useful for installing intellectual humility in undergrads, he doesn’t pose a huge, looming, unresolved problem that people try to ignore hanging over their head while it makes everything they do and say pointless nonsense. That’s pretty much just Hume.

Hume’s problem is that induction is unjustifiable. Induction is (narrowly) whenever we draw conclusions from particular experiences to a general case or to further similar cases. So, for example, I believe that tomorrow I will wake up in my bed with the Sun having risen in the east, based on the fact that this has always happened to me. I believe that if I don’t water my plants, they will die, and that if I hold a lit match to my curtains, I will set fire to my house. Life would be impossible without making inferences of this kind. But Hume argues that these inferences have no rational basis: they’re just a habit of mind and lack any real justification. My long history of successfully setting fire to things with matches is no reason at all to have an opinion about what will happen next time a lit match is applied to the upholstery.



Hume’s Fork?

Douglas Adams made the ruler of the galaxy in his Hitchhiker’s Guide to the Galaxy series someone who reasoned this way – that is, he absolutely refused to draw inferences from past to future events. For those of us forced to navigate a world full of poisons, precipices, and pregnancies it is impractical. Even if Hume were right, we would have to go on acting as if he is wrong – he said so himself. Fortunately for us all, while teaching it to undergrads I discovered the fatal flaw in his argument, thus saving human reason and, I think it is fair to say, all of Western philosophy.

Hume’s problem is sometimes called Hume’s Fork, as it literally presents a dilemma: two options neither of which is acceptable. The version of it which is taught in introductory philosophy courses goes like this: There are two types of reasoning: deductive and inductive. Deductive reasoning is certain: it proves things with necessity, and it is self-justifying: we can’t even imagine how a logical contradiction could be true. But our knowledge about the world isn’t certain knowledge like this. It seems to work via inferring necessities from particular instances. So we observe an instance of choking and being unable to breathe when we are submerged in water, and we conclude: hey, I can’t breathe water. From the one, or a few experiences, we draw a general conclusion with universal scope: no humans can breathe water.

These sorts of inferences are clearly fallible. If I concluded from my experience that no human can memorize a random deck of cards having looked at it just once then I would be wrong, even though I can’t do it and have never met anyone who can. It is possible that any pattern we have seen in the past might be disconfirmed in the future: the Sun might not rise (because the Earth is destroyed by Vogons later today), or a human might breathe water (artificial gills were installed in her neck). Not only are inductive inferences uncertain, they are also sometimes entirely unjustified. Almost everyone I know speaks English, but it would be wrong to infer that almost everyone speaks English, because the people I know are a biased sample of people generally.

There seems to be a kind of gap in induction between the evidence for the conclusion and the conclusion itself: the conclusion goes beyond what the premises can prove (otherwise the conclusion would be certain). Hume said that this gap is filled by an assumption, an assumption that the future will be like the past, or that nature is uniform. If the future will be like the past, I can reasonably use my past experience with fires and with children to infer that I should take a lit candle away from a two year old. But if in the future nature does not proceed along like it used to, any conclusions we draw from past events will be misleading. So how do we know that the future will be like the past? Hume’s problem is that we can’t.

We cannot deductively prove that the future will be like the past. It is possible that things will be different than how they have been, and we can’t deductively prove something to be true if it’s possibly false. But inductively proving that the future will be like the past seems promising to unwary undergraduates. We can come up with lots of examples of how past events have been very useful to us for predicting things. After all, the sky gets cloudy before it rains very consistently, and we predict that it will get cold in Toronto every January, and every January we are distressingly correct. And yes, my poor, dear undergraduate, it is true, I can recall many past instances where the past was used to accurately predict the future. So, should we then conclude, based on our many past experiences of the future being like the past, that the future will be like the past? Well, this inference seems fair, if we may assume that our past experiences of successful inductive prediction will continue to be borne out in the future. We can therefore triumphantly induce that the future will be like the past, but only on the assumption that the future will be like the past.

This is the satisfying moment when the trap is sprung and the undergraduates who have actually been paying attention realize they were doomed from the beginning. You can’t use induction to prove something that induction relies on to function. It is, as Hume says, evidently arguing in a circle.

So, since a justification of the assumption that the future must be like the past must be either inductive or deductive, and as we have seen it can’t be either of those, there can be no such justification. Therefore we cannot have any good reason to believe the future will be like the past, therefore induction is unjustified, and therefore I have no good reason to think there is anything particularly dangerous about lighting a cigarette while my body is soaked in gasoline.

The reader may at this moment be feeling the electric, almost sensual frisson that accompanies superlative philosophical reasoning. Many people mistake this sensation for intense annoyance, and hastily conclude that philosophers merely enjoy making people feel stupid and that philosophy is all about proving things true that are obviously false out of a basically sadistic contrarianism. There is a fair amount of evidence to support this view, but as we just saw, that doesn’t give us any reason to think it is correct.

That last point was just a joke, of course, because as I already mentioned, Hume was wrong. That Hume was wrong must be almost the most obvious thing that anybody ever said, because that we may justifiably draw inferences from past events to future predictions is literally more certain than that the Sun will rise in the east and that all humans are mortal. Indeed, the simplest rejoinder to Hume is a version of the simple argument to confound skepticism of any kind:

P1: If Hume’s argument is correct, then inferences from past events to future predictions are unjustified. P2: It is not the case that inferences from past events to future predictions are unjustified. C: It is not the case that Hume’s argument is correct.

There you go, as tidy as you please: two true premises forming a deductively valid argument, hence we can be absolutely certain that Hume was wrong. However, as pretty and satisfying as this argument may be, it does have the disadvantage that it begs the question, since one of its premises asserts the very point under dispute.

Despite the fact that Hume’s argument is certainly wrong, one would never get that impression from your standard intro philosophy course. So, in service of the possibility of knowledge and the ability of philosophers to solve problems, here’s my explanation of why it is wrong:

Hume demands we prove the truth of a statement in order to justify induction: a statement such as “the future will be like the past” or “the course of nature continues always uniform”. But that is not what we actually need to do to justify induction. To justify an argument, we have to show there is a certain sort of relation between the premises and the conclusion. In the case of deductive arguments, the premises have to necessitate the conclusion: for inductive arguments, the premises have to make the conclusion probable – at least, more probable than not. If we argue like Hume wants us to…

P1: In the past, holding a lit match to a curtain had the effect that the house caught fire. P2: The effects of holding a lit match to a curtain will be the same in the future as in the past. C: In the future, holding a lit match to a curtain will have the effect that the house will catch fire.

… we make our argument into a deductive one. If the future will be like the past, we may draw certain conclusions about the future. But inductive arguments are supposed to be probable, not certain. And by making an inductive argument into a deductive argument with an extra premise, we do nothing to improve or justify the argument.

It is actually quite easy to switch back and forth between inductive and deductive arguments that establish the same conclusion. If we are reasoning about the real world, there will always be some uncertainty about our conclusion. By constructing an argument with a deductive pattern, we embed that uncertainty into the premises; whereas an argument with an inductive pattern embeds the uncertainty into the inference from the premises to the conclusion. So, for example:

P1: If the Heat fail to win the Championship next year, the author will be very happy. P2: The Heat will fail to win the Championship next year. C: The author will be very happy.

This is deductive and valid, but it is not conclusive because we may justifiably be dubious about the truth of premise 2. If we replace premise 2 with

P2a: The Heat will probably not win the Championship next year.

then the argument becomes an inductive one, because it doesn’t follow from the fact that the Heat will probably fail that the author will necessarily be happy. They have to actually fail, but they might not. The two arguments are equally good because they are based on precisely the same information about the same set of possible events, but the first argument’s failure to be entirely conclusive resides entirely in our uncertainty about the second premise, whereas the second argument’s uncertainty is primarily in the inference from the premises to the conclusion.

So the problem of justifying induction is not the problem of proving that the future will be like the past. The statement “the future will be like the past” is only useful for transforming certain types of inductive arguments into deductive arguments, and doing that doesn’t actually do anything to make our arguments better. It just takes the uncertainty in the inference from

P1: In the past, dropping a wineglass on the tile floor has had the effect of breaking it; to C: in the future, dropping a wineglass on the tile floor will have the effect of breaking it.

and embeds that uncertainty in a further premise:

P2: the future will be like the past with respect to the effects of dropping a wineglass on the tile floor.

Justifying induction actually requires us to reason about the connection between the premises and the conclusions of inductive arguments. There is no general justification of inductions, since some inductive arguments are bad ones, but there can potentially be justifications of particular inductive arguments, or even entire kinds of inductive arguments, that is, inductive arguments that have a certain structure to them. These justifications are deductive, and hence the project of justifying induction is not circular; such arguments show that it is necessarily the case that the conclusion of an argument follows with probability from its premises. Here is one such justification. Consider arguments of the form:

P1: An individual will be randomly selected from the population. P2: Most members of the population blow the bugle beautifully. C: It is probable that the individual randomly selected from the population will blow the bugle beautifully.

Now, we may have our doubts about whether the method we use to select an individual is really as random as all that, or whether the population is really full of such great buglers. But those worries are beside the point. That argument is deductively valid. It is necessarily the case that, if those premises are true, the conclusion is true. This means that if we substitute the conclusion:

C*: The individual randomly selected from the population will blow the bugle beautifully.

it is necessarily the case that if those premises are true, the conclusion is probably true. Therefore we have justified inductions of that general form.

More broadly, any valid deductive argument that has as its conclusion “it is probable that waddle-waddle is the case” justifies the corresponding inductive argument that argues from the same premises to the conclusion “waddle-waddle is the case”.



Hume statue, Edinburgh

© Kim Traynor 2013

You are welcome, Western philosophy.

It is possible that you may have already drawn the conclusion that the author did not in fact save Western philosophy. And I must confess, the evidence would seem to make that conclusion highly probable. You see, in the course of doing the necessary research to prepare his brilliant smackdown of Hume for academic publication, the author came to the realization that he had been beaten to the punch by about sixty years. The substance of his argument can be found in a 1947 book by D.C. Williams, and the subsequent author D.C. Stove ran with it further. There are numerous differences in how the idea is expressed, of course, but the basic view that one can deductively justify various types of inductive arguments and that Hume was demanding in a confused way that inductive arguments be turned into deductive arguments by a supplemental premise is all there.

So if Hume’s problem was solved seventy years ago, why doesn’t anybody know or care? Well, one might conclude that there is some flaw with my and Stove and Williams arguments. The case for that can certainly be made (unless Hume was right). But is it really so plausible that an argument with an obviously false conclusion which has been subjected to intense scrutiny and consideration has remained unrefuted by anybody? How likely is it that the community of philosophers might just fail to recognize a proof that was staring them in the face?

Pretty likely, I think. I even have an argument.

My actual area of expertise is the philosophy of evolution by natural selection. So I will tell you a story about the natural selection of problems.

Some problems are harder to solve than others. Many problems require special techniques or apparatus to be solved – mathematical methods, or technologies like the telescope. Problems that are easy to solve stop being problems, and what remains are the most insoluble difficulties.

All of the sciences started out as branches of philosophy, and they stopped being philosophy once they developed methods of empirical research that could command widespread agreement. By which I mean, you could show people just how wrong they were by waving the facts in their faces. But that is not an option in philosophy: you have to wave logical arguments in their faces, which is only possible metaphorically. And being convinced by a metaphor (or a logical argument) requires a certain attitude of co-operation. People have to keep an open mind. That is the point of teaching Hume and Descartes in the first place: it pries people’s minds open to the possibility that they might sometimes be wrong, which makes them more willing to judge logical arguments on their merits, rather than by whether they already agree with the conclusion.

But judging arguments on their logical merits is not something that human reason is especially well suited to do. And while some of the big problems in philosophy may be difficult to solve, the status of Hume’s problem points to another possibility: it may be that the big problems in philosophy are the ones that it is difficult to convince anyone that you have solved. If some correct philosophical arguments are basically too tricky, complex, or counterintuitive for the philosophical community to be convinced of their correctness, then the problem will simply coexist with its solution, with the true believers trying and failing to convince their sophisticated but mistaken critics.

D.C. Williams felt himself to be in this unfortunate position. He writes that a critic of Hume’s problem “may choose what weapons he will … ordinary language, the arithmetic of large numbers, the logic of probability, the history of science, whatever – the result is still the same. Not only are all actual criticisms of inductive scepticism felt to be hopelessly beside the mark: not one philosopher in a thousand can tell you what possible kind of criticism would prove it false.” (p.87) His book includes a chapter entitled “Why these arguments do not convince”; the reasons he discusses are rhetorical and psychological. The entry in the Stanford Encyclopedia of Philosophy on Hume’s problem, which is probably as close as we can get to settled philosophical opinion, ignores Williams’ and Stove’s refutation of Hume’s argument, and focuses on the technical difficulties of their attempts to prove that some types of inductive arguments are deductively justified. But justifying induction is just a postscript if Hume’s argument is flawed; without Hume’s argument no-one would feel the need to justify it at all.

I think that this situation is exactly what we should expect. The arguments which survive the process of philosophical disputation ought to be the ones whose solution is unconvincing to professional philosophers. And what could philosophers be less disposed to accept than that one of the acknowledged great philosophers’ great arguments is a pretty simple fallacy? Not an obvious fallacy, to be sure, but a simple one. There are a great many ways to be wrong in philosophy, which means there are very many plausible but mistaken arguments that can be marshalled in defence of one’s favourite conclusion. So it is hardly difficult for a motivated philosopher to avoid being convinced by an argument, regardless how correct.

And the intellectual community of philosophers is just not set up to solve intellectual problems. It is set up to elaborate and develop disputes. Continuous publication requires an ongoing debate; what is best is a back and forth series of refinements, quibbles, clarifications, and rebuttals. In a quite real sense everyone loses out if the matter gets definitively settled. I do not mean to imply that philosophers do this deliberately, but the pressure to publish means that they gravitate to areas where there is, as they say, a ‘live debate’. And while it may be that some live debates persist because nobody has yet found the truth, or the conclusive argument remains unasserted, it works just as well if one side just can’t be convinced by objectively good arguments.

Maybe this means that the progress of philosophy can’t just rely on logic; it must resort to rhetoric. The arguments have to be dressed up prettily, or the objectively confused must be socially shamed and pressured. Or something: I don’t really know. I do think that Hume’s problem won’t be solved by publishing a book or a paper for philosophers, though, and I’ve got some solid evidence to support that induction.

© Dr Eugene Earnshaw 2017

When he is not saving Western philosophy, Eugene Earnshaw teaches at Seneca College in Toronto.