Post hoc ergo propter hoc

The fern and the music

“I started playing classical music for my fern on Monday, and by Friday it had put out a new frond. Plants really do respond to music.”

The clear-cut case. Healthy ferns put out new fronds every so often whether or not anyone plays them music, so a new frond in any given week was likely anyway. Nothing in the story shows what the fern would have done in a silent week. The music might help, but one before-and-after can’t tell that apart from the fern’s ordinary growth.

The redesign that “boosted” sales

An online shop launches a new homepage design in mid-November. December sales are up 40% on November. The design team’s end-of-year report credits the redesign.

This one comes with numbers and sounds like a result, which is what makes it persuasive. But for most shops December is the busiest month of the year, so sales would have risen with the old homepage too. Other things may also have changed at the same moment: a holiday discount, an advertising push, a competitor’s stockout. Whatever else started around the same time is an equally good candidate, and the timing can’t choose between them. Comparing December with the previous December, or with a version of the site that kept the old design, would begin to.

The barometer that “brings” storms

A sailor notices that every time the barometer on his boat drops sharply, a storm arrives within a day. “When that needle falls, it pulls the weather in behind it.”

Here the sequence isn’t a one-off: it repeats reliably, and the barometer is genuinely useful for predicting storms. It’s still post hoc reasoning, because both the falling needle and the storm are produced by the same thing, a drop in air pressure, and the needle simply responds sooner. Tapping the glass to make the needle fall wouldn’t summon a storm. A reliable sequence can make a good sign without being a cause.

The missing premise is a claim about something nobody observed: what would have happened if A hadn’t occurred. Philosophers call this a counterfactual, and on counterfactual theories of causation, such as David Lewis’s, it is the heart of what “cause” means. The ways of supporting it are all ways of getting a look at the “without A” case: a comparison group, a baseline from before, or removing A and seeing whether B stops.

• Coincidence. A single pairing of events that were unrelated, like the socks and the win.
• It would have happened anyway. Many things change on their own: colds clear up, seasons turn, sales rise in December. Crediting whatever came just before is post hoc reasoning. When the starting point was unusually bad or good, so that a return toward average was likely, this is the regression fallacy, a close and well-studied relative.
• Something else changed at the same time. A second event came first too, and may be the real cause. The Stanford Encyclopedia’s example is “Unemployment decreased in the fourth quarter because the government eliminated the gasoline tax in the second quarter”, where the decrease “may have been due to other causes; perhaps new industrial machinery or increased international demand for products.”
• An earlier sign of a common cause. A comes before B because both are effects of C, and A shows up sooner, as with the barometer. The underlying structure is Confounding.
• One cause out of many. John Stuart Mill, as the Stanford Encyclopedia reports, treated post hoc as a fallacy that “tends to single out a single cause when there are in reality many contributing causes.” Hurley and Watson count this as a separate variety of false cause, oversimplified cause, which can resemble post hoc when the cause credited came first.

The family. Post hoc is the most frequently discussed of several fallacies about causes, according to Hans Hansen’s Stanford Encyclopedia entry; the others include confusing cause and effect and overlooking a common cause. Older logic books gathered causal errors under the heading non causa pro causa (“a non-cause for a cause”): the Port-Royal Logic discusses causal errors under that heading and names post hoc separately as a common fallacy, and Isaac Watts introduced “false cause” as another name for the heading. More recent references use “false cause” for the family, with post hoc as one member: Patrick Hurley and Lori Watson’s textbook lists it alongside oversimplified cause and the gambler’s fallacy, and Bradley Dowden’s Internet Encyclopedia of Philosophy entry alongside the regression fallacy. Post hoc’s closest sibling is cum hoc ergo propter hoc, which rests on occurring together rather than on order in time; that page sets out how the sources classify the two, and why this site treats them as siblings rather than one as a kind of the other. The general principle behind both is on the correlation vs. causation page.

The order of events is real evidence about causes, just not enough on its own. Reasoning from a sequence holds up when:

• It rules a cause out. A cause can’t come after its effect, so if B began before A, A didn’t start it. Austin Bradford Hill listed this temporality, “which is the cart and which the horse?”, among the things to check before concluding that an association is causal.
• There’s a comparison with the “without A” case. A similar group, place or period that didn’t get A, followed over the same time, shows what would likely have happened anyway.
• The sequence repeats when A is controlled. If B appears when A is introduced, stops when A is removed, and returns when A is brought back, coincidence and outside changes become very unlikely explanations. Hill’s “experiment” viewpoint asks exactly this: when an exposure is reduced, does the outcome change?
• There’s a known mechanism and no rival explanation for the timing. Touching a hot pan and immediately getting a burn needs no control group, because how heat burns skin is understood and nothing else plausibly happened in that second.
• The sequence is treated as a lead. “This started right after the update; let’s check whether the update did it” is a hypothesis, not a conclusion.

The test: what would most likely have happened next if A hadn’t happened, and how do you know?

The toaster and the smoke alarm

The smoke alarm in a small apartment goes off most mornings, always a minute or two after the toaster is switched on, though nothing is burning. The tenant moves the toaster to the counter on the far side of the kitchen, and for two weeks the alarm is silent. She moves it back under the alarm to check, and the next morning it goes off again.

This is “B followed A” reasoning, but it isn’t resting on one sequence. The tenant changed A deliberately, watched B stop, and then changed A back and watched B return. An outside cause would have to switch on and off at exactly those times, which is hard to believe. That’s the repeats when A is controlled condition, and there’s a plausible mechanism too: toasters give off heat and fine particles that can trigger a nearby alarm.

The image that slowed the page

A library’s website takes about eight seconds to load its homepage. A developer finds that a banner image added the previous week is 12 megabytes, far larger than the rest of the page combined. She replaces it with a compressed version; load times drop to under two seconds within minutes, while the library’s test copy of the site, which still has the large image, stays at eight seconds.

The improvement followed the change, as in a post hoc argument. What makes this sound is everything else: a specific mechanism (a large file takes longer to download) identified before the fix, a change in the outcome right when the fix went in, and an unchanged copy of the site showing what would have happened without it. That’s the comparison and known mechanism conditions together. Craig and colleagues, writing about evaluating public health policies, describe a bare before-and-after comparison as assuming “that outcomes change only as a result of exposure to the intervention”; the test copy is what removes the need for that assumption.

Causes do come before their effects, so a sequence looks like the beginning of a causal story, and often the rest of the story is easy to supply. What never appears is the alternative: nobody sees the week the fern would have had without music, or the December the shop would have had with its old homepage. The one sequence we witnessed feels like the whole of the evidence, a case of what you see is all there is.

Timing also isn’t random. People try a remedy when they feel worst, launch a fix when a problem peaks, and make changes at the start of a new season or year, exactly when things were about to change anyway. That lines up actions with natural turning points and makes post hoc conclusions feel confirmed by experience. Hansen remarks that causal fallacies such as confusing cause and effect “are perhaps better understood as faults of explanation than faults of arguments”, and post hoc often works the same way: the error lies in accepting the first explanation that fits, more than in a stated argument.

• Ask what would have happened anyway. Is there a reason to expect B without A: a trend, a season, a natural course, a return from an unusual low or high?
• List what else changed around the same time. A sequence supports every event that came before B equally well.
• Look for a comparison. A similar case that didn’t get A, or the same case at a comparable earlier time, is the quickest check.
• Where it’s safe and cheap, test it. Remove A and see whether B stops, then reintroduce it.
• Don’t over-correct. Showing that an argument is post hoc shows the timing isn’t enough, not that A had no effect. The music might help the fern.