Week 10 · Chapter

Chapter 10 — Causal Reasoning

Critical Thinking Free sample edition

The module's primary reading · ~35–45 minutes · Objective 5 (with Objective 4 carried forward) · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the Argument Workshop all build on it.

In plain language

This week is about claims that one thing makes another happen, and the question to ask of each one is what else could explain the pattern. Two things that tend to go together show a correlation (a pattern, not yet a cause). A causal claim says one of them brings about the other. You will check three other explanations, see why "it came first" is not enough, and learn when a chain of predicted steps deserves to be taken seriously. These tests help you decide whether a cause has been shown or only guessed.

Why you already care about this

Your friend has worn the same striped socks to 3 exams this term and passed all 3. Before the fourth exam, the socks go on again. A health headline you scroll past says people who log 5 workouts a week in a fitness app are fitter than people who do not (illustrative). Your roommate's theory: the 2 times the campus coffee cart closed early, the vending machine broke that afternoon.

Each of these turns a pattern into a cause. Sometimes that move is right, and sometimes acting on the wrong cause wastes money and time.

Last week you asked whether a sample stands for its population. This week you ask whether a pattern between two things shows that one makes the other happen.

By the end of this chapter you will be able to:

tell a correlation from a causal claim and name the three alternatives every correlation must be checked against;
run the confounding check on a small table by holding one factor fixed, and say the verdict in words;
spot reverse causation and coincidence in everyday claims;
identify after this, therefore because of this (post hoc), and tell it from a causal claim backed by a comparison;
judge a slippery slope by checking each link, and tell a supported slope from an unsupported one;
work Mill's methods of agreement and difference, and say what a controlled comparison adds.

Carry this line with you: a pattern is a question, not an answer — ask what else could explain it.

Where we are

Already covered: Week 1 taught you to find an argument, and Week 2 to put it in standard form. Week 3 taught strength and cogency for inductive arguments. Weeks 6 to 8 taught the fallacies of language, relevance, and presumption. Week 9 taught you to judge a sample and an analogy.
This week: you test causal claims, check a correlation for a confounding factor, reverse causation, and coincidence, and key two more fallacies from the course's set.
How they connect: a causal claim usually rests on a comparison of two groups, and each group is a sample, so Week 9's sampling check comes first. Week 7's habit of asking whether a premise bears on the conclusion returns in the post hoc and slippery-slope fallacies.
Where we're going next: Week 11 states causal and sampling claims in numbers, with percentages, risks, and base rates.

Week 9 handed you the idea you lean on most. A causal claim compares two groups, such as days with high sales and days with low sales, and each group is a sample. If the groups differ in some other way, that difference can produce the pattern. From Week 3, a causal argument is inductive: strong or weak, never valid.

1. Tell the pattern from the cause: correlation and causation

Picture a street where houses with more umbrellas in the hallway also have wetter doormats. Nobody thinks umbrellas make doormats wet. Rain does both. In most claims you will read, the "rain" is hidden, and that is where careful reasoning earns its keep.

A correlation is a pattern in which two things tend to go together, so that one shows up more often when the other does; a correlation by itself does not show that either one causes the other.

A causal claim is a claim that one thing makes another happen, which says more than that the two tend to go together and so needs more than a pattern to support it.

Every correlation has three rivals to the causal reading. Check all three before you accept a cause.

A confounding factor is a third factor that drives both of two things that go together, so that they move together even though neither causes the other, as hot weather drives both ice-cream sales and sunburn cases.
Reverse causation is the alternative in which the causal arrow runs the other way from the one claimed, so that the supposed effect is in fact the cause, as when fit people are the ones who choose a fitness app.
Coincidence is a pattern between two things that arises by chance rather than from any causal link, most likely when the pattern rests on only a few cases.

Worked example — the three questions (illustrative). Run the check on the roommate's theory: "The 2 times the coffee cart closed early, the vending machine broke the same afternoon, so the cart closing breaks the machine."

A confounding factor? Could one thing cause both? A power outage in the building could shut the cart's espresso machine and the vending machine at once.
Reverse causation? Could the arrow run the other way, with the broken machine somehow closing the cart? Nothing in the theory rules that out.
Coincidence? The pattern rests on 2 afternoons. Two matches can happen by chance.

In words: the roommate has a correlation built on 2 cases, and each of the three rivals is open, so the causal claim is weak until someone checks what else happened those afternoons.

Misconception check — a strong pattern proves a cause. Students often think that if two things go together often enough, one must cause the other. That is tempting, because causes really do produce patterns. But confounding factors produce patterns too, and they can be every bit as steady. A pattern tells you where to look. It does not tell you what you will find.

Memory hook: check the rivals first; a third thing, the reverse, or chance.

2. Hold one factor fixed: the hot-weather table

A beach clinic keeps a summer log. It records whether ice-cream sales near the beach were high or low that day, and whether the clinic saw many sunburn cases or few. A reader looks at the log and asks: does ice cream have something to do with sunburn?

This is the course's hot-weather table. It is illustrative, built for teaching, and every number in it is pinned.

The illustrative hot-weather table: 100 summer days sorted by temperature, ice-cream sales, and sunburn cases at a beach clinic
Days	High sales, many sunburns	High sales, few	Low sales, many sunburns	Low sales, few
Hot (40)	24	6	8	2
Mild (60)	4	16	8	32
All (100)	28	22	16	34

Worked example — the confounding check on the hot-weather table (illustrative). Find the pattern, then hold temperature fixed.

The days. 40 hot days + 60 mild days = 100 days.
The two groups of days. High-sales days are 28 with many sunburn cases + 22 with few = 50. Low-sales days are 16 with many + 34 with few = 50.
The overall pattern. On high-sales days, 28 ÷ 50 × 100 = 56 percent had many sunburn cases. On low-sales days, 16 ÷ 50 × 100 = 32 percent did. Many sunburn cases go with high ice-cream sales more often, so the table shows a correlation.
Hot days held fixed. High-sales hot days are 24 + 6 = 30, and 24 ÷ 30 × 100 = 80 percent had many sunburn cases. Low-sales hot days are 8 + 2 = 10, and 8 ÷ 10 × 100 = 80 percent did.
Mild days held fixed. High-sales mild days are 4 + 16 = 20, and 4 ÷ 20 × 100 = 20 percent had many sunburn cases. Low-sales mild days are 8 + 32 = 40, and 8 ÷ 40 × 100 = 20 percent did.

In words: overall, 56 percent of high-sales days had many sunburn cases versus 32 percent of low-sales days. Once you compare only hot days with hot days, the rates are 80 percent and 80 percent. Once you compare only mild days with mild days, they are 20 percent and 20 percent. The gap disappears, so hot weather is the confounding factor behind both.

Worked example — why the gap showed up at all (illustrative). Look at which days landed in each group.

High-sales days. 24 + 6 = 30 of the 50 high-sales days were hot, and 30 ÷ 50 × 100 = 60 percent.
Low-sales days. 8 + 2 = 10 of the 50 low-sales days were hot, and 10 ÷ 50 × 100 = 20 percent.

In words: high-sales days are mostly hot days, and hot days bring sunburn whatever the ice-cream sales. The two groups of days were not alike in temperature, so the comparison was unfair from the start.

Holding a factor fixed did the work. You compared only days alike in temperature, so temperature could not explain a difference, and no difference was left for ice cream to explain.

Misconception check — a confounding factor is an error in the data. Students often hear "confounded" and think someone counted wrong. That is tempting, because the overall numbers seem to say something false. But every count in the table is accurate. The 56 percent and the 32 percent are real. The flaw is in reading them as a cause, because the two groups of days differed in a third factor. Confounding is a flaw in the inference, not in the counting.

An honesty note: holding one factor fixed rules out that one factor. A different hidden factor could still be at work, and a real study would check more than temperature.

Memory hook: compare like with like; then see if the gap survives.

3. Check the other two rivals: reverse causation and coincidence

A fitness app's ad says its users are fitter than people who do not use it (illustrative). The ad wants you to hear "the app makes you fit." Before you agree, turn the arrow around.

Worked example — reverse causation (illustrative). Run the reverse question on two claims.

The fitness app. Claim: using the app makes people fitter. Reverse: people who are already fit, or already training, are the ones who download a workout app. The arrow could run from fitness to the app.
The front row. Claim: sitting in the front row raises a student's grades. Reverse: students who already care about their grades are the ones who choose the front row. The arrow could run from caring about grades to the seat.

In words: in both claims the pattern also fits an arrow running the other way. The pattern alone cannot tell you which way the cause runs, so the ad's claim is weak.

Coincidence is the third rival. It is most likely when a pattern rests on a few cases.

Worked example — coincidence (illustrative). A city softball team wins all 3 games that a new mascot attends, and a fan concludes the mascot brings wins.

Count the cases. The pattern rests on 3 games.
Ask whether chance could produce it. A team that wins some of its games can win 3 in a row with no help at all.
Ask what would test it. Many more games, with and without the mascot, would show whether the pattern holds.

In words: 3 games is too few to rule out chance, so the mascot claim is a guess, not a shown cause. This is Week 9's lesson again: too small a sample can land on an unusual run by luck.

Misconception check — if the cause came second, it cannot be a cause. Students often check reverse causation and then think a claim is proved once the order is right. That is tempting, because a cause does have to come before its effect. But the right order rules out only one rival. A confounding factor and coincidence can both produce a pattern with the "right" order. Check all three.

Memory hook: flip the arrow; then count the cases.

4. Key the order fallacy: after this, therefore because of this

Back to the striped socks. Your friend wore them, then passed. The socks came first. That order is the friend's whole case.

After this, therefore because of this (post hoc) is the fallacy of concluding that one event caused another only because it came first, with no comparison that rules out the other explanations.

Worked example — the lucky socks (illustrative). "I wore my lucky socks to the exam and passed, so the socks helped me pass."

Find the support. The only reason given is that the socks came before the passing grade.
Run the rivals. Studying, sleep, and the exam's difficulty all came before the grade too, and any of them could explain it.
Ask for the comparison. Nothing compares exams taken with the socks against exams taken without them, with the other factors alike.

In words: the only support is that the socks came first, so the passage commits after this, therefore because of this (post hoc).

As Week 7 taught you, a fallacy is a flaw in the reasoning, not a false premise. The premise here is true: the socks were worn, and the exam was passed. The flaw is the step from order to cause.

The non-fallacious neighbor is a causal claim backed by a comparison that rules out the rivals. A clinic that compares people who got a treatment with people alike in other ways who did not, and finds a difference, has more than an order of events.

Misconception check — after this means because of this. Students often accept a cause because it came right before the effect: "I changed my phone's wallpaper, and the next day the battery lasted longer." That is tempting, because causes do come first, and the timing feels like evidence. But many things come before any event, and most of them did nothing. Battery life changes from day to day for many reasons. Timing is where a causal question starts, never where it ends.

Memory hook: first is not the same as because.

5. Check each link: the slippery slope

A city council member warns that one budget cut will set off a chain of problems. A student senator warns that one rule change will lead step by step to disaster. You can neither dismiss nor accept either warning on sight. You check the chain.

A slippery slope is an argument that predicts one step will lead through a chain of further steps to a bad outcome. It is reasonable when each link in the chain is backed by evidence and a fallacy when the links are not.

An unsupported slippery slope is the fallacy of predicting that one step will set off a chain of further steps ending in a bad outcome when the links in that chain are not backed by evidence.

A slippery slope is a string of causal claims. Each link says one step causes the next. So you test it the way you test any causal claim, one link at a time.

Worked example — the coffee-to-grills slope (illustrative). "If we let students bring coffee into the library, next they'll bring full meals, then grills, and soon the building will burn down."

List the links. Coffee leads to full meals. Full meals lead to grills. Grills lead to a fire.
Ask what backs each link. The passage offers no evidence for any of them. Nothing shows that a coffee rule changes what else people bring, and grills are a different kind of object from a cup.
Say the verdict. No link is backed, so the chain is a prediction with nothing under it.

In words: no link from coffee to meals to grills to fire is backed by evidence, so the passage commits the unsupported slippery slope.

Worked example — the bus route (illustrative, discussed and never keyed). A warning says cutting the late-night bus route leaves late-shift workers without a ride, so some drive instead and late-night traffic rises. Each step is backed by the city's own ridership counts.

List the links. The cut leaves riders without a ride. Some of them drive instead. More drivers means more late-night traffic.
Ask what backs each link. The city's own ridership counts back each step.
Say the verdict. Every link is supported, so it is a reasonable slope.

In words: the bus-route warning predicts consequences, as the coffee passage does, but it backs each link, so it is a reasonable argument you would weigh rather than a fallacy you would name.

That gives this week's tally from the course's fallacy set. 1 post hoc + 1 unsupported slippery slope = 2 Week 10 passages. Through Week 9 you keyed 17, so 17 + 2 = 19 passages are keyed through Week 10. The set holds 20, and 20 − 19 = 1 passage is still to come in a later week.

Misconception check — every slippery slope is a fallacy. Students often learn the name and then call any warning about consequences a slippery slope. That is tempting, because "slippery slope" is used as an insult. But predicting consequences is how planning works. A warning with each link backed by evidence is good reasoning. Name the fallacy only when a link has nothing behind it.

Memory hook: test the slope one link at a time, not all at once.

6. Find the cause by comparing cases: Mill's methods

At a neighborhood potluck, several guests get sick that night. Everyone wants to know which dish did it. The way you would sort that out has a long history.

Mill's methods are the methods for finding causes set out in Mill's A System of Logic (1843): agreement, difference, residues, concomitant variation, and a joint method of agreement and difference.

Mill (1843), A System of Logic, Book III, Chapter 8, names all five. Mill's own chapter speaks of four methods while stating five, so this course works and grades only the first two. Here they are, with the other three named so you will recognize them.

The method of agreement is the method of finding a likely cause by looking for the one factor that every case of an effect shares while the cases differ in their other factors.
The method of difference is the method of finding a likely cause by comparing a case with the effect and a case without it that are alike in every factor but one, which singles out that one factor.
The method of residues is the method of subtracting the parts of an effect already explained by known causes and treating what is left over as the effect of the remaining factor.
The method of concomitant variation is the method of linking two things causally because one changes in step with the other. It is open to the same confounding check as any correlation.
The joint method uses agreement and difference together.

Worked example — agreement at the potluck (illustrative). The guests who got sick ate different main dishes, different desserts, and different drinks.

List the cases of the effect. Every guest who got sick.
List what each one ate. The main dishes, desserts, and drinks vary from guest to guest.
Find the shared factor. Every sick guest ate the potato salad.

In words: the cases differ in every factor but one, so the method of agreement singles out the potato salad as the likely cause, a lead a health inspector would then test.

Worked example — difference on the windowsill (illustrative). Two plants of the same kind sit in the same soil, get the same water on the same days, and came from the same store. One sits on the kitchen windowsill and stays green. The other sits on the bedroom windowsill, above a heating vent, and wilts.

List the case with the effect and the case without it. The bedroom plant wilts; the kitchen plant does not.
List the factors they share. Same kind, same soil, same water, same store.
Find the one factor that differs. The bedroom windowsill sits above a heating vent.

In words: the two plants are alike in every factor listed but one, so the method of difference singles out the heat from the vent as the likely cause of the wilting.

Ice-cream sales and sunburn cases rise and fall together, which is concomitant variation. That made the table a correlation, and it failed the confounding check.

Misconception check — Mill's methods prove a cause. Students often treat the method of agreement as a machine: find the shared factor, and the case is closed. That is tempting, because the method feels mechanical. But the method finds the shared factor among the factors you listed. If the sick guests also all sat near the same warm serving table, a factor you never listed could be the real cause. Mill's methods are tools for organizing judgment, not machines that settle a real case.

A caveat: each method is only as good as your own list of factors. That limitation is exactly what the next section's tool repairs.

Memory hook: agreement finds what the cases share; difference finds what the pair does not.

7. Control the comparison: holding factors fixed or randomizing them

Suppose a tutoring center wants to know whether its evening review sessions raise exam grades (illustrative). Students who chose to attend did better than students who did not. You already know the rivals: the students who chose to attend may have been the most motivated, which is a confounding factor.

A controlled comparison is a comparison that holds other factors fixed, or spreads them evenly by random assignment, so that a difference in outcome can be traced to the one factor that differs.

Random assignment is deciding by chance which cases get a factor and which do not, so that other factors, including ones no one thought to measure, end up spread evenly between the groups.

Worked example — designing the fair test (illustrative). The tutoring center wants a comparison that rules out the rivals.

Hold what you can fixed. Use one course, one exam, and the same review material.
Assign by chance. Among students who sign up, a lottery decides who attends the sessions this month and who attends next month.
Compare the outcomes. Compare exam grades between the two groups.

In words: because a lottery decided who attended, motivation and every other factor should be spread about evenly between the groups, so a difference in grades could be traced to the sessions themselves.

This is the step past Mill's methods. Agreement and difference work only on the factors you thought to list. Random assignment handles the factors you never thought of, because chance spreads them across both groups. It also cures the front-row claim: assign seats by lottery.

Misconception check — a bigger comparison fixes confounding. Students often think that comparing more people solves the problem. That is tempting, because Week 9 showed that size matters. But if motivated students choose the sessions, a comparison of 10,000 students is confounded in the same way as a comparison of 100. Size cannot fix how your groups were formed. Assignment by chance can.

Memory hook: let chance pick the groups; hidden factors then split evenly.

Key terms

correlation — A pattern in which two things tend to go together, so that one shows up more often when the other does; a correlation by itself does not show that either one causes the other. (first used: 1. Tell the pattern from the cause: correlation and causation)
causal claim — A claim that one thing makes another happen, which says more than that the two tend to go together and so needs more than a pattern to support it. (first used: 1. Tell the pattern from the cause: correlation and causation)
confounding factor — A third factor that drives both of two things that go together, so that they move together even though neither causes the other, as hot weather drives both ice-cream sales and sunburn cases. (first used: 1. Tell the pattern from the cause: correlation and causation)
reverse causation — The alternative in which the causal arrow runs the other way from the one claimed, so that the supposed effect is in fact the cause, as when fit people are the ones who choose a fitness app. (first used: 1. Tell the pattern from the cause: correlation and causation)
coincidence — A pattern between two things that arises by chance rather than from any causal link, most likely when the pattern rests on only a few cases. (first used: 1. Tell the pattern from the cause: correlation and causation)
after this, therefore because of this (post hoc) — The fallacy of concluding that one event caused another only because it came first, with no comparison that rules out the other explanations. (first used: 4. Key the order fallacy: after this, therefore because of this)
slippery slope — An argument that predicts one step will lead through a chain of further steps to a bad outcome. It is reasonable when each link in the chain is backed by evidence and a fallacy when the links are not. (first used: 5. Check each link: the slippery slope)
unsupported slippery slope — The fallacy of predicting that one step will set off a chain of further steps ending in a bad outcome when the links in that chain are not backed by evidence. (first used: 5. Check each link: the slippery slope)
Mill's methods — The methods for finding causes set out in Mill's A System of Logic (1843): agreement, difference, residues, concomitant variation, and a joint method of agreement and difference. (first used: 6. Find the cause by comparing cases: Mill's methods)
method of agreement — The method of finding a likely cause by looking for the one factor that every case of an effect shares while the cases differ in their other factors. (first used: 6. Find the cause by comparing cases: Mill's methods)
method of difference — The method of finding a likely cause by comparing a case with the effect and a case without it that are alike in every factor but one, which singles out that one factor. (first used: 6. Find the cause by comparing cases: Mill's methods)
method of residues — The method of subtracting the parts of an effect already explained by known causes and treating what is left over as the effect of the remaining factor. (first used: 6. Find the cause by comparing cases: Mill's methods)
method of concomitant variation — The method of linking two things causally because one changes in step with the other. It is open to the same confounding check as any correlation. (first used: 6. Find the cause by comparing cases: Mill's methods)
controlled comparison — A comparison that holds other factors fixed, or spreads them evenly by random assignment, so that a difference in outcome can be traced to the one factor that differs. (first used: 7. Control the comparison: holding factors fixed or randomizing them)
random assignment — Deciding by chance which cases get a factor and which do not, so that other factors, including ones no one thought to measure, end up spread evenly between the groups. (first used: 7. Control the comparison: holding factors fixed or randomizing them)

Summary

A correlation is a pattern; a causal claim says one thing makes another happen.
Check every correlation against three rivals: a confounding factor, reverse causation, and coincidence.
Hold a factor fixed: in the hot-weather table, 56 versus 32 percent becomes 80 and 80, then 20 and 20.
Hot weather, not ice cream, is behind the sunburns.
Post hoc infers a cause from order alone, as your friend's striped socks show.
A slippery slope is a causal chain; it is a fallacy only when its links lack support.
Mill's method of agreement finds the shared factor; the method of difference finds the one that differs.
A controlled comparison holds factors fixed or assigns by chance, so hidden factors split evenly.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A café owner's log covers 40 winter days (illustrative). On the 20 days with high hot-chocolate sales, the morning bus ran late on 11. On the 20 days with low hot-chocolate sales, it ran late on 7. On snowy days, the bus ran late on 9 of the 12 high-sales days and 3 of the 4 low-sales days. On clear days, it ran late on 2 of the 8 high-sales days and 4 of the 16 low-sales days. Find each percent and name what is behind the pattern.
A restaurant review site notes that restaurants with more servers on shift serve more customers (illustrative). The site concludes that hiring more servers brings in more customers. Which rival should you check first, and why?
Two loaves of bread are made from the same recipe, in the same oven, by the same baker, on the same day (illustrative). One rises and one stays flat. The only listed difference is that the flat loaf used yeast from an older packet. Which of Mill's methods is this, and what does it single out?
A gym manager says (illustrative): "If we stay open one hour later, more members will come after work, the late crowd will need a second staff member, and the payroll will rise." Each step is backed by the gym's own sign-in counts and staffing rules. Is this an unsupported slippery slope? Explain.

Answers (worked reasoning — read after attempting)

Overall: 11 ÷ 20 × 100 = 55 percent of high-sales days had a late bus, against 7 ÷ 20 × 100 = 35 percent of low-sales days. Snowy days: 9 ÷ 12 × 100 = 75 percent and 3 ÷ 4 × 100 = 75 percent. Clear days: 2 ÷ 8 × 100 = 25 percent and 4 ÷ 16 × 100 = 25 percent. The gap disappears once weather is held fixed, so snow is the confounding factor: it raises hot-chocolate sales and slows the bus.
Reverse causation. A restaurant that already draws many customers schedules more servers to handle them, so the arrow may run from customers to servers. The pattern fits both directions, and it cannot tell you which way the cause runs.
The method of difference. The two loaves are alike in every listed factor but one, the yeast packet, so the method singles out the older yeast as the likely cause of the flat loaf. It is a lead, not proof: an unlisted difference could still matter.
No. It predicts a chain of consequences, but each link is backed by the gym's own sign-in counts and staffing rules. A slope with every link supported is a reasonable argument to weigh, not a fallacy.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Numbers in the News, the causal test meets risks and base rates stated in numbers.

Built with The Course Maker · samples.thecoursemaker.com