Why Can't a Shadow Explain the Height of a Pole?
A Puzzle in the Sun
It’s a sunny afternoon. You measure the shadow of a flagpole: exactly 4 meters. You also know the angle of the sun. Using a simple formula from physics, you calculate the pole’s height: 6 meters. That feels like a tidy scientific explanation — you started with the pole, the light, and the laws of optics, and you deduced the shadow length. But now turn it around: can you explain the pole’s height from its shadow? The same formula works in reverse. So why does it seem like a worse explanation? This puzzle launched a long argument among philosophers about what a scientific explanation really is. Their fight wasn’t just about flagpoles — it was about the heart of science itself.
Hempel’s Recipe for a Perfect Explanation
In the mid-20th century, philosopher Carl Hempel (1905–1997) proposed a crisp model. He called it the Deductive-Nomological model, or DN model for short. “Deductive” means the explanation is a logically valid argument: if the premises are true, the conclusion must be true. “Nomological” comes from the Greek word for law, nomos — so the premises must include at least one law of nature. The thing you want to explain is the explanandum, and the sentences that do the explaining are the explanans. For an explanation to be good, the explanandum must be derivable from the explanans in a sound deductive argument where a law plays an essential role.
Think of a planet. You can predict its future position from Newton’s laws of motion, the law of gravity, and its current position and velocity. Hempel said that’s not just a prediction — it’s an explanation. It shows that, given the laws and starting conditions, the planet’s future position was to be expected with certainty. He called this idea nomic expectability: a phenomenon is explained when we see it as expected on the basis of laws. Even for statistical laws, like the probability of recovery from an infection after taking medicine, he developed a version where the explanation makes the outcome highly probable. In the DN/IS world, explanation was all about lawful expectation.
But there was a hidden challenge: what counts as a law of nature? Hempel thought laws were exceptionless regularities — like “all gases expand when heated under constant pressure” rather than just an accidental truth like “all members of the 1964 Greensbury school board are bald” (if that happened to be true). Distinguishing laws from mere coincidences proved so tricky that Hempel himself called it “highly recalcitrant.” This question remains unsettled today.
When the Recipe Goes Wrong
Hempel’s recipe seemed neat, but philosophers soon found examples that didn’t taste right. One famous problem was explanatory asymmetry. You can deduce the length of a shadow from the height of a pole and the sun’s angle using the laws of light. That’s a perfect DN explanation. However, you can also deduce the pole’s height from the shadow and the sun’s angle using the same laws — and that deduction meets all DN requirements too. Yet nobody thinks the shadow explains the pole’s height. The explanation has a direction: causes explain effects, not the other way around. The DN model couldn’t capture that.
Another problem was explanatory irrelevance. Imagine a man named John Jones who takes birth control pills. He never gets pregnant. You can state a law: “All males who take birth control pills fail to get pregnant.” From this law and the fact that Jones is male and takes pills, you can deduce that he won’t get pregnant. DN criteria satisfied. But it’s a terrible explanation — the pills are completely irrelevant to his not getting pregnant. Something was missing: the explanation didn’t cite the actual cause.
A similar puzzle crops up with silly laws: it might be true (as an exceptionless regularity) that all table salt touched by a witch’s wand dissolves in water. From that “law” and the fact that some salt was hexed, you can deduce it dissolved. Yet the hexing is beside the point. These examples showed that nomic expectability alone isn’t enough. You can have a derivation that makes an outcome expected but still be a bad explanation if it picks the wrong direction or includes irrelevant facts.
Salmon’s New Recipe: It’s All About Relevance
Philosopher Wesley Salmon (1925–2001) took a different tack. He first developed the Statistical Relevance model (SR). The key idea: an explanation is not a deductive argument; it’s a collection of information that is statistically relevant to the outcome you want to explain. A factor is statistically relevant if the probability of the outcome changes depending on whether that factor is present. Salmon said that to explain why a particular person with strep throat recovered quickly, you need to list all the factors that affect the probability of recovery — for example, whether they took penicillin and whether the infection was resistant. You then show how these factors partition the population into groups with different recovery probabilities, making sure you’ve included everything relevant and nothing else.
Crucially, Salmon dropped the high-probability requirement. You can explain even a very unlikely outcome as long as you’ve given all the relevant factors correctly. So the same set of facts can explain both recovery and non-recovery, as long as it accurately describes the relevant probability landscape. This model captured the idea that explanation cares about relevance, not just logical form — irrelevancies that are harmless in a deductive argument can be fatal in an explanation.
But Salmon soon realized that statistical relevance alone couldn’t pinpoint causes. Different causal structures can produce the same statistical patterns. So he later built the Causal Mechanical model (CM). Here, an explanation traces causal processes — physical processes like a moving billiard ball that can transmit a mark continuously in space and time. A genuine causal process can carry a scratch from one place to another; a shadow cannot. An explanation also shows causal interactions, where two processes meet and change each other, like billiard balls colliding. Salmon thought this captured the “something more” that the DN model missed.
Yet the CM model still had a blind spot. A billiard ball transmits both its momentum and any chalk mark on it. The chalk mark moves through space just like a scratch, but it’s irrelevant to where the ball goes after a collision. So tracing processes and interactions alone doesn’t tell you why momentum matters and chalk marks don’t. Salmon himself later acknowledged that even this model needed extra information about relevance.
Or Maybe It Depends on What You’re Asking?
Other philosophers, like Bas van Fraassen (born 1941), thought the whole project was misguided from the start. He argued that explanation isn’t a straightforward relationship between a theory and a fact — it’s a pragmatic affair that always involves the asker’s interests and the context. When you ask “Why is this conductor warped?” you might mean “Why is it warped (instead of straight)?” or “Why is this conductor warped (instead of that one)?” The contrast class and the relevance relation depend on what you want to know.
Van Fraassen told a story about a tower whose height is “explained” by the fact that it was designed to cast a specific shadow for a sundial. In that context, answering “because the architect wanted this shadow” seems like a legitimate explanation of the tower’s height — but it’s not a causal explanation in the usual sense. So the asymmetry between explaining shadow from pole and pole from shadow might be a matter of what question we’re really asking, not a deep feature of the world.
Critics pushed back: if any relevance relation counts depending on context, can’t anything explain anything? Van Fraassen replied that scientific context restricts acceptable relevance relations — astral influence, for example, isn’t allowed in modern science. But many worry that his view doesn’t give enough guidance for telling good scientific explanations from bad ones. The debate remains open, and no single model has won universal agreement.
Why This Still Matters for Every Question You Ask
These philosophical arguments aren’t just for physics textbooks. Every time you hear that a new medicine works, that crime rates dropped because of a policy, or that a historical event happened for a certain reason, you’re facing the same questions: Is this a genuine explanation or just a description? Has the right cause been identified? Are the factors cited relevant? What would count as a satisfying answer changes depending on what you already know and what you’re curious about.
Next time a friend tells you “I didn’t do my homework because I was tired,” that might be true but not very explanatory; you’d want to know why they were tired. The puzzles about shadows and birth control pills remind us that explanation is a demanding business. It’s not about being certain or even about probability — it’s about showing the why. Philosophers still disagree about exactly what that means, but they’ve given us sharper questions. And asking sharper questions is the first step toward better answers.
Think about it
- Your friend says, “The window broke because the rock hit it.” That’s true, but would a scientist consider it a full explanation? What else might you want to know?
- Imagine someone says, “I did well on the test because my lucky socks were on.” You can find no law in science that connects socks to test scores. Does that mean it’s not an explanation? Why or why not?
- A weather app predicts rain with 90% certainty using a complex computer model. Is that prediction also an explanation? What more would you need to feel you understood why it will rain?
