Induction on a Single Case

by Brian on April 27, 2004

One hears it said from time to time that it’s irrational to perform inductive inferences based on a single data point. Now this is sometimes irrational. For example, from the fact that Al Gore got the most votes in the last Presidential Election it would be foolish to infer that he’ll get the most votes in the next Presidential Election. But it isn’t always irrational. And this matters to some philosophical debates, and perhaps to some practical debates too.

Here’s my proof that it isn’t always irrational. Imagine on Thursday night I go and see a new movie that you’re going to go see Friday night. Friday lunchtime I tell you how the movie ended. How should you react? Most people will complain that I’ve spoiled the movie because you now know how it will end. But if induction on a single case is always bad, this is impossible. All you have is testimonial evidence of how the movie ended on a single occasion, namely Thursday night. You need to make an inferential leap to make a conclusion about how it will end Friday night. (It certainly isn’t a deductive inference because some movies have multiple endings.) That inferential leap will be induction on a single case, and will be perfectly reasonable.

That’s more or less my complete argument that induction on a single case can be perfectly rational. There is an obvious objection though. It might be argued that this isn’t _really_ induction on a single case, because it’s like underwritten by a many-case induction based on the number of previous movies that have ended the same way at multiple screenings. While that’s obviously true, it isn’t clear how much it undermines the original example.

There’s two points we could go on to debate here. First is the question of whether the inference from how the movie ended on Thursday to how it will end on Friday (the movie inference) is really an instance of induction on a single case. That looks like a relatively stale terminological debate, and I couldn’t be bothered hashing it out here. Second is the question of whether there is any way to distinguish the movie inference from what are usually taken to be bad instances of induction on a single case. This one has to be debated case by case, but I suspect the answer is in general _no_, unless there are independent reasons to dislike the particular bad instance of inductive reasoning.

Here’s a couple of illustrations of what I mean, one practical the other theoretical.

Consider the policy “Don’t start reading a blog if the first thing you read there is false.” Some might consider any application of that to be a bad instance of a single case induction – from one bad claim infer that other things the blog says are not worth reading. But just like the movie inference can be backed up by a meta-induction, this one can arguably be backed up by a claim validated by a meta-induction: that blogs which say something false the first time you open them are not worthwhile reading in the future. That claim might well be _false_, and I’m not taking sides here on whether it is true or not, but as long as the person who holds the policy believes the claim, their reasoning is no worse than the person who makes the movie inference. (Quick credit: I saw this policy defended somewhere a while ago, but Google was no help in finding where. That was more or less what inspired this post, so I probably should have linked to it.)

Let’s take a more famous case. Why should I believe that other people have sensations? One famous answer, defended by Bertrand Russell, is that I can reason by analogy. I’m alike other people in ever so many ways, so I’m probably alike them in respect of sensations. And I know (somehow!) that I have sensations, so I know they do too.

Some people have objected that this is just induction on a single case. (E.g. Michael Rea makes that objection “here”: But it looks to me a lot like the movie inference, for it too is backed by a meta-induction. In the past, when I have tried to infer from the fact my body is a certain way to the conclusion that others are the same way, I’ve met with reasonable success. Not 100% success, but good enough for inductive purposes. Of course other people are like me in external respects – they often have two eyes, one mouth, two legs etc. But they are also like me in internal respects, at least as far as I can tell. Consider, if you aren’t too squeamish to do this, how similar the various kinds of fluids that come out of various parts of other people’s bodies are to fluids that come out of matching parts of one’s own body. X-Ray technology reveals that we are alike in even more ways than we could have previously told ‘on the inside’. So the argument _my brain states generate or constitute or correlate with phenomenal sensations, so other brains generate or constitute or correlate with phenomenal sensations_ is an instance of a schema that delivers mainly reliable instances. Just like the movie inference. And that inference can produce knowledge. So I think we can come to know about the existence of other minds with sensations on the basis of a single case, namely our own. If you don’t believe me, perhaps you don’t need to worry as much about movie spoilers as you thought you did!



dsquared 04.27.04 at 7:06 am

I would like to point out that if you sleep with your wife’s sister on one single occasion it is the very devil of a job to convince her that it is wrong to generalise from a single case.

Shurely this is a case where a Bayesian model of induction would make things a lot clearer, he said, without much hope.


Lance McCord 04.27.04 at 7:16 am

I’ve tried to add something to this discussion here. I enjoyed this post, and obviously it made me think. It seems to me that there’s a good reason that inductive reasoning from a single case is bad news. It should be abundantly clear, however, that I don’t know what I’m talking about.


bryan 04.27.04 at 7:56 am

this is absurd. With induction it is common to define a single data point as actually being a member of a far larger set. The movie that you saw thursday is a member of the set all movies, we do not suppose that a movie will end the same friday as it did thursday without reference to the set movies and rules we have for how any individual member of the set functions.
if we were to start reasoning about movies from this movie your point would be reasonable, and in fact no one would make the inductive leap mentioned, but we do not start in this way.

From your point it would not be appropriate for me to assume my dog spot is still dead, just because he was dead a while ago. But of course induction has given me the data to build the following deductive argument:
1. things that are dead stay dead
2. Spot is dead
3. Spot will stay dead. Stay spot, stay.

I’m going to conclude that you have not really spent a lot of time with induction, or you were drunk when you posted this. :)


bryan 04.27.04 at 8:01 am

Looked at Lance’s site, pretty much the same arguments as mine.

I was thinking that his example with the wife could be turned into a neat abduction argument.


bryan 04.27.04 at 8:06 am

‘I would like to point out that if you sleep with your wife’s sister on one single occasion it is the very devil of a job to convince her that it is wrong to generalise from a single case.’

perhaps her generalisation might be based on many cases of people who sleep with their wife’s sisters. So the first order of business is to find out if she has watched enough Jerry Springer to have a good data set to generalise from.


Robin Green 04.27.04 at 8:46 am

Why would watching Jerry Springer give you a good data set about anything?


Eve Garrard 04.27.04 at 9:54 am

W.D.Ross thought the whole of morality is like that, so that we can infer the existence of(prima facie) moral principles from the state-of-play in single cases. If we see that it’s wrong to steal in one case, we can correctly infer that stealing is a wrong-making feature in all cases (though not of course that it’s wrong to steal in all cases). However even those devotees of Ross who think he’s produced the most convincing form of deontology currently available don’t think that intuitive induction works in aesthetics, let alone in non-evaluative cases. It would be interesting to see if those non-evaluative contexts which do allow induction from single cases have anything in common with the moral examples.


LTH 04.27.04 at 1:09 pm

Does it even mean anything if you do induct from a single data point? Is that not like asking the following:

Complete the following number sequence:

37 …


Ian 04.27.04 at 1:41 pm

Isn’t the answer to that – as to everything else – 42?


Thomas 04.27.04 at 1:44 pm

I’m sorry, but that’s a poor analogy.

First of all, this statement is incorrect: “You need to make an inferential leap to make a conclusion about how it will end Friday night.”

The ending of the movie does not change. If I see a similar movie with a different ending it is, in fact, a different movie.

If I plan to see a movie tonight, but you’ve already told me the ending, I decide whether to be upset or not judging by 2 factors: do I trust you, and do I expect the theater to play the same movie.

Both of these inductions are made from large data sets and not from a single point. If I know that you usually speak the truth, then I will expect that the ending you told me is correct. If I have seen many movies, and the theater always plays the movie as advertised without variation on the endings, then I can expect to see the same movie you did. Thus, I conclude from my large set of data that what you told me was true. I know the ending of the movie and have a right to be upset.

Judging a series of posts on a blog is not the same as judging a series of the same movie played over and over again. A much better analogy would compare reading posts on a blog to seeing movies by the same director.

If I see a movie and don’t like it, no big deal. If I see a few more by the same director and think they’re all bad, I start to see a pattern. I begin to expect future movies by the same director to be of the same low quality.

Telling me the ending of the movie is more similar to telling me the ending of a single blog entry. I can expect it to be the same post that you read with the same ending.


bryan 04.27.04 at 2:17 pm

‘Why would watching Jerry Springer give you a good data set about anything?’ because the subject is men who sleep with their sister-in-law.


Brian Weatherson 04.27.04 at 2:56 pm

Dsquared, yep it would be a whole lot easier with a Bayesian model, which is part of the point. All I’m saying is that sometimes when we put these arguments in Bayesian form, they look OK. That means sweeping objections based to the entire class, as is needed to get the objection to Russell’s argument for the existence of other minds to work, are non-starters.

Maybe this one’s terminological, but I don’t agree with Thomas that different ending = different movie. If that were true it would be inconsistent to say that _28 Days Later_ had multiple endings, but it’s true that _28 Days Later_ had multiple endings.

On Bryan’s argument about Spot, I think that’s not really how we reason about dead things. I think we take the fact that Spot was dead yesterday to be sufficient evidence that he’ll be dead today. There’s no premise we go through about dead things staying dead. Of course the justification for that inference involves the general fact, established by induction, that dead things stay dead. But that doesn’t mean it’s part of the argument.

You could say we _should_ put this general fact into the argument. But with inductive reasoning we _always_ have to rely on contingent facts not explicitly mentioned for the inference to be good, unless we want our inferences to contain (approximately) infinitely many premises. So I think it’s OK for the Spot inference to just be from dead yesterday to dead today.

On lth’s point, I did say this doesn’t _always_ work. I was mainly interested in Russell’s solution to the problem of other minds, which only requires that it work _sometimes_.


Jamie McCarthy 04.27.04 at 3:12 pm

Coincidentally, I’m 30 pages into Karl Popper’s “The Logic of Scientific Discovery.” Chapter 1 is “The Problem of Induction” — in the early chapters he tackles a question similar to yours: Is it ever scientifically valid to deduce a universal law from ANY number of single statements (observations)?

His answer is no: “in my view there is no such thing as induction… What I am denying is that there is such a thing as induction in the so-called ‘inductive sciences’: that there are either ‘inductive procedures’ or ‘inductive inferences’… But I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience… it must be possible for an empirical scientific system to be refuted by experience.”

Popper, of course, is known for defining the realm of science by its negative space. Only by advancing theories which are falsifiable, and then making vigorous attempts to prove them false, can scientific knowledge be advanced.

Your example rests on a universal statement which you imply without stating outright: that a movie comes out the same way every time it is shown. This is definitely falsifiable, and your implied statement could benefit from taking some edge cases into account (what if the film breaks? have you seen “Clue“?) But there are millions of opportunities to refute this statement, and my guess is that it would survive quite well.

The question of whether, having erred once, a blogger is likely to frequently err, is much more open for debate. :)

This book, by the way, is a little dense for someone like me without much philosophical background. He spends a great deal of time addressing what others have said on highly specialized issues, and much of it goes over my head. Also, the binding sucks and I bet the book falls apart by the time I’m done with it (this reprint is part of the Routledge Classics series).


Neel Krishnaswami 04.27.04 at 5:26 pm

Naive falsificationism is intellectually barren: there are an infinite number of theories that describe any finite set of data, and no matter how much data you collect, there are still an infinite number of theories that explain the data. It offers no guide to which theories should be considered and why. You need some reason not to consider almost all theories of the data. Bayesian approachs let you model ignoring almost everything properly, but you still need a prior that can be defined over the infinite space of possible theories. Amazingly, such priors exist: we know of priors that will converge to the correct theory, assuming that observations are generated by a computational process (this is the famous Solomonoff universal prior).


bryan 04.27.04 at 6:42 pm

‘You could say we should put this general fact into the argument’
no I say that we do put this general fact into the argument, but we do not reference it every time we make the argument, it is a constant agreed on before hand.
If you ask me “shall we take spot to the park and play fetch?”
I’m gonna say “I don’t want to throw a dead dog around in the park.”
The you might say “How do you know he’s dead”
“Because he was dead earlier”
As things go on I might slowly reach the understanding that you have a different opinion as to the continuing deadness of not just spot but lots of dead things, at that point i would probably be obliged to drag out my premises for the conclusion that Spot is still dead. this would involve inductive arguments based on data that dead things stay dead. If you were one of my crazy stepfamily you would then say “But what about Jesus”
And I would say “You think our dead dog might be the messiah?”

‘Maybe this one’s terminological, but I don’t agree with Thomas that different ending = different movie. ‘
I doubt any of us does, not even Thomas. Rather that statement is there to cut the data set down from all instances of movie to a common subset of movies, mainly because you did not really want to argue about movies and what classifies something as a movie, you wanted to argue about the nature of induction. To do this argument properly focusing on both the nature of induction and the nature of what we term a movie would probably require great computational resources and a sizeable grant to pay for our troubles (Proposal: to determine if there can ever be any such instance of a thing still commonly understood as a ‘movie’ that it would be impossible to argue inductively as to the content of such thing from showing to showing of it and to ever reach correct conclusions about this content that would be better than the conclusions reached at by random guesswork [I feel we should really have scare quotes around movie more often]). Therefore it is probably reasonable to focus on what we know as the most common form of movie, movie with a strict linear progression that ends the same in every showing, starts the same in every showing, and has the same middle. To be specific we will focus on Tango And Cash, now that’s a ‘movie’ that’s never changing.


bryan 04.27.04 at 8:32 pm

Is there a proof that any finite data set can have an infinite number of theories about it? also how big of an infinite number is?


Neel Krishnaswami 04.29.04 at 5:27 pm

Here’s an easy demonstration. Suppose that you have some set of integers, z_1 through z_n, and you ask, “What are the functions that pass pass through zero at each of these z’s?”

Well, thanks to algebra, it’s pretty easy to see that the function

f(x) = (x – z_1) * (x – z_2) * … * (x – z_n)

is one such that will do the trick. At any given z_i, f(x) has a factor (x – z_i) which will become (z_i – z_i), and cause the the whole equation to evaluate to zero (since zero times anything is zero). Now, note that g(x) = 2 * f(x) will also pass through all those zeros, and so will 3 * f(x) and 4 * f(x) and so on: we have an infinite number of possible solutions.

The size of the infinity in this example is the powerset of the integers, since we drew our data from them. This is the same cardinality as the reals.

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