# Explanation is asymmetric

We all regularly reason in terms of the concept of explanation, but rarely think hard about what exactly we mean by this explanation. What constitutes a scientific explanation? In this post, I’ll point out some features of explanation that may not be immediately obvious.

Let’s start with one account of explanation that should seem intuitively plausible. This is the idea that to explain X to a person is to give that person some information I that would have allowed them to predict X.

For instance, suppose that Janae wants an explanation of why Ari is not pregnant. Once we tell Janae that Ari is a biological male, she is satisfied and feels that the lack of pregnancy has been explained. Why? Well, because had Janae known that Ari was a male, she would have been able to predict that Ari would not get pregnant.

Let’s call this the “predictive theory of explanation.” On this view, explanation and prediction go hand-in-hand. When somebody learns a fact that explains a phenomenon, they have also learned a fact that allows them to predict that phenomenon.

To spell this out very explicitly, suppose that Janae’s state of knowledge at some initial time is expressed by

K1 = “Males cannot get pregnant.”

At this point, Janae clearly cannot conclude anything about whether Ari is pregnant. But now Janae learns a new piece of information, and her state of knowledge is updated to

K2 = “Ari is a male & males cannot get pregnant.”

Now Janae is warranted in adding the deduction

K’ = “Ari cannot get pregnant”

This suggests that added information explains Ari’s non-pregnancy for the same reason that it allows the deduction of Ari’s non-pregnancy.

Now, let’s consider a problem with this view: the problem of relevance.

Suppose a man named John is not pregnant, and somebody explains this with the following two premises:

1. People who take birth control pills almost certainly don’t get pregnant.
2. John takes birth control pills regularly.

Now, these two premises do successfully predict that John will not get pregnant. But the fact that John takes birth control pills regularly gives no explanation at all of his lack of pregnancy. Naively applying the predictive theory of explanation gives the wrong answer here.

You might have also been suspicious of the predictive theory of explanation on the grounds that it relied on purely logical deduction and a binary conception of knowledge, not allowing us to accommodate the uncertainty inherent in scientific reasoning. We can fix this by saying something like the following:

What it is to explain X to somebody that knows K is to give them information I such that

(1) P(X | K) is small, and
(2) P(X | K, I) is large.

“Small” and “large’ here are intentionally vague; it wouldn’t make sense to draw a precise line in the probabilities.

The idea here is that explanations are good insofar as they (1) make their explanandum sufficiently likely, where (2) it would be insufficiently likely without them.

We can think of this as a correlational account of explanation. It attempts to root explanations in sufficiently strong correlations.

First of all, we can notice that this doesn’t suffer from a problem with irrelevant information. We can find relevance relationships by looking for independencies between variables. So maybe this is a good definition of scientific explanation?

Unfortunately, this “correlational account of explanation” has its own problems.

Take the following example. This flagpole casts a shadow of length L because of the angle of elevation of the sun and the height of the flagpole (H). In other words, we can explain the length of the shadow with the following pieces of information:

I1 =  “The angle of elevation of the sun is θ”
I2 = “The height of the lamp post is H”
I3 = Details involving the rectilinear propagation of light and the formation of shadows

Both the predictive and correlational theory of explanation work fine here. If somebody wanted an explanation for why the shadow’s length is L, then telling them I1, I2, and I3 would suffice. Why? Because I1, I2, and Ijointly allow us to predict the shadow’s length! Easy.

X = “The length of the shadow is L.”
(I1 & I2 & I3) ⇒ X
So I1 & I2 & I3 explain X.

And similarly, P(X | I1 & I2 & I3) is large, and P(X) is small. So on the correlational account, the information given explains X.

But now, consider the following argument:

(I1 & I3 & X) ⇒ I2
So I1 & I3 & X explain I2.

The predictive theory of explanation applies here. If we know the length of the shadow and the angle of elevation of the sun, we can deduce the height of the flagpole. And the correlational account tells us the same thing.

But it’s clearly wrong to say that the explanation for the height of the flagpole is the length of the shadow!

What this reveals is an asymmetry in our notion of explanation. If somebody already knows how light propagates and also knows θ, then telling them H explains L. But telling them L does not explain H!

In other words, the correlational theory of explanation fails, because correlation possesses symmetry properties that explanation does not.

This thought experiment also points the way to a more complete account of explanation. Namely, the relevant asymmetry between the length of the shadow and the height of the flagpole is one of causality. The reason why the height of the flagpole explains the shadow length but not vice versa, is that the flagpole is the cause of the shadow and not the reverse.

In other words, what this reveals to us is that scientific explanation is fundamentally about finding causes, not merely prediction or statistical correlation. This causal theory of explanation can be summarized in the following:

An explanation of A is a description of its causes that renders it intelligible.

More explicitly, an explanation of A (relative to background knowledge K) is a set of causes of A that render X intelligible to a rational agent that knows K.