Philosophy 152
Science & Reason
Spring 2006
Lecture Notes

Statistical Explanations and Hempel’s Model

I. Hempel’s I/S Explanations

A. Hempel came to realize that the D-N model was too restrictive - the stated conditions are not necessary for something to be a good explanation. Here’s an example:

            C1. Henrietta was in contact with Henry

            C2. Henry has measles.

            L. 99% of children in contact with measles’ patients subsequently get measles.

            E. Henry got measles

You could add details, and maybe you want to add that he is not immune.

See also the example on p. 40. These are similar to the D/N explanations, but the laws here are statistical. In response to this, Hempel added what he called “inductive-statistical”(I-S) explanations. The conditions on these explanations are:

S1) The explanandum follows from the explanans with high inductive probability.
S2) The explanans must contain at least one statistical law.
S3) The explanans must have empirical content
S4) The sentences in the explanans must be true.

Another element will be added shortly.

B. Non-deductive reasoning

Once people learn about deduction, they sometimes come to think that only deductively valid arguments have rational merit. But that’s mistake. Examples: There is no deductively valid argument whose premises I am sure of that has its conclusion that it will snow in Rochester next winter. But I have very good reason to believe that it will, and I’d be unreasonable not to believe it. Similarly, I have good reasons to believe that the my wife is in the next room when I hear a noise from there, but my reasons don’t provide a way to make a valid argument for that conclusion (or for a premise that supports that conclusion). (Display arguments.) So, some non-deductive reasoning is good reasoning.

This suggests that Hempel’s main idea – that explanations are good arguments that invoke laws – can sensibly be extended to cases like the one above. (Emphasize this.) What’s crucial, though, is that the generalization be a law and not merely some accidental non-universal generalization. Consider the proposed explanation of Smith’s getting a B or better since almost all students whose name begins with “S” got such a grade. This is no explanation either. This statistical generalization is not law-like. In contrast, the idea is that the statistical generalization in the measles explanation really is a law.

A complicated idea that we will return to a little later is that this sort of statistical explanation is thought not to be an explanation sketch to be filled out with some universal generalization.

C. Maximum Specificity

There is a complication. Statistical reasoning that is otherwise good can be defeated by background information. Deductive reasoning has no similar feature. If an argument is deductively valid, and you know that the premises are true, then you can know that the conclusion is true. No background information can spoil the argument.

To see how this works, first consider an example that will warm us up to the problem. I ask why it’s below 30 degrees today here and you explain by saying that it is usually - say 80% of the time - below 30 on days in Jan. Then suppose that it’s 50 degrees one day next week. You explain that by saying that it is usually - say 60% of the time - above 50 (year round) here. By picking different classes that the day is in, you can seemingly explain any outcome. Hempel realized that good explanations can’t be selective in this way. What seems clear is that temperatures in Jan. matter more in this case than temperatures year round.  

The same thing happens in the example on p. 41 about voting behavior. Some (possible) statistical laws about how people vote:

L1: 80% of voters vote for candidates politically similar to candidates that their same gender parent voted for.

L2: 90% of millionaires vote for right-wing candidates

L3: 75% of female self-made millionaires from Minnesota vote for left-wing candidates.

So the idea is that narrowest class wins. Obviously, the examples we’ve considered are very simple, but they do illustrate the idea. This led Hempel to add:

S5) The law must be “maximally specific”.

So Hempel’s claim is that (a) every correct scientific explanation conforms to either the D/N model or the I/S model and that (b) anything that conforms to either of these models is a correct scientific explanation.

We’ve already seen that (b) isn’t right.

II. Puzzles and Objections For I-S Explanations

A. Is High Probability Sufficient?

            C1. Jones had neurotic symptom N and underwent psychoanalysis

            L: Most people ... recover

            E: Jones recovered

But maybe the psychoanalysis had nothing to do with it. As we saw before - it looks like we need a causal condition. (If you don’t like this example, substitute your own.)

B. Is High Probability Necessary?

In a much discussed example, it was argued that we explain paresis in terms of untreated syphilis. But most people with the later don’t get the former. Perhaps the same is true of smoking and lung cancer. R. says the rate is 40%.

Again, perhaps causation matters here. We’ll think more about this a little later.

C. Puzzles about Maximum Specificity

[This point will probably be fairly obscure. Sorry, but it’s hard.]

Recall that Hempel’s idea is that good explanations are arguments with certain characteristics. But, importantly, those characteristics don’t make reference to what we know. The argument that explains the behavior of a planet, say, is the one that puts it under the right law (the covering law). The explanation didn’t become a good one when we figured it out. And a mistaken one that we accept isn’t a good explanation.

With respect to D-N explanations, then, we have the idea that there is one (or more) correct explanation of the explanandum and that we can have a better or worse understanding of what it is. It’s hard to see how there can be anything similar in the case of I-S explanations. The maximally specific law will be one that includes all the facts about the object in question. You will then have a D-N explanation. The only apparent way to avoid that is to take the maximally specific category to be the most specific one we know about (or have beliefs about). [See Rosenberg, p. 41 - that’s how he formulated the condition on explanations.]

Reconsider Rosenberg’s example about voting. We stopped at L3, and we said it’s a good explanation because L3 was the narrowest class we knew about. Had we not known about L2 and L3, we might have used L1. But there are unknown to us generalizations that are still narrower than L3. So the explanation using L3 is only narrowest among the ones we know about. So now there is an essential dependence upon what we know about in the account of what a good explanation is. And that wasn’t the intended idea.

It may be hard to see why this is such a big deal. But it is. It’s unclear that there is a way to spell out Hempel’s idea that makes good on the idea that what counts as a good explanation is not somehow relative to what we know.