Critical thinking exercises
Critical thinking exercises are short, self-contained problems that force one specific move: finding a missing premise, naming a flaw, checking a sample, or reading a number that was arranged to be misread. Ten of them are set out below, with the answers worked through underneath each one. They are exercises rather than illustrations, which means the answer is covered until you have committed to yours.
Reading about reasoning does almost nothing on its own. The gap between recognizing a flaw when it is labeled and spotting it when it is not is enormous, and it only closes by repetition. Dunning and Kruger reported in 1999 that people who perform worst on tests of reasoning tend to overestimate their performance most, a finding later argued to be partly a statistical artifact of how the data are plotted; the useful residue is unchanged, which is that your own sense of how well you reason is not evidence about how well you reason. A scored exercise is.
How to work these exercises
To work these exercises properly, follow four rules. They matter more than the exercises themselves.
- Write your answer down in a full sentence before reading the worked answer. An answer you only thought is an answer you will believe you had.
- Name the move, not the verdict. "That's wrong" scores nothing. "The sample was drawn only from people already doing the thing being measured" scores.
- Attempt the repair. For every flawed argument, write the version that would have worked, or state what evidence would be needed.
- Return after a week to the ones you missed. Recognition fades much faster than the feeling of having understood.
Exercises 1 to 3: find the missing premise
Each argument below is missing an unstated premise it cannot work without. Name it.
Exercise 1
The shop should stay open until eight, because the bus from the industrial estate reaches the high street at seven forty.
Answer. The missing premise is that a useful number of people on that bus would shop if the shop were open. The stated premise establishes only that people arrive at seven forty. The argument treats presence as demand, and the check is a count: how many people leave that bus, and how many of them currently walk past a lit shop window without going in.
Exercise 2
The new delivery route is better, because it is nine miles shorter.
Answer. The missing premise is that shorter distance produces the outcome actually wanted, which is usually time or cost rather than miles. Nine miles of town center with fourteen junctions can take longer than eighteen miles of dual carriageway. The word doing all the work here is better, and it is undefined. An argument whose conclusion contains an undefined comparative cannot be checked until somebody says better at what.
Exercise 3
Rebuild the canal footbridge in steel, because the last three timber footbridges on this stretch each needed replacing within twelve years.
Answer. Two missing premises, and both are contestable. First, that the timber was the cause of the failures rather than the mounting, the traffic or the water level. Second, that steel would perform differently under those same conditions. The evidence given is about timber failing; it is silent on steel, and a conclusion that names a material nobody has tested is reaching past its premises. Mapping arguments into premises and conclusion like this is the core drill, and Logical reasoning sets out the anatomy the drill depends on.
Exercises 4 to 6: name the flaw
Exercise 4
Every member of the cycling club who fitted the new tire got a puncture within a month. The tire is defective.
Answer. Two flaws, and the second is the deeper one. The sample is unrepresentative: a cycling club that rides gravel tracks is not a general population of riders. More importantly there is no comparison group, so nothing tells you the puncture rate for the old tire over the same month on the same tracks. Without a comparison, a rate is just a number. The repair is to ask what the club's puncture rate was last month on the old tire, and if nobody counted, the honest answer is that the question has not been investigated.
Exercise 5
The bakery repainted its front in July and sales rose eleven percent that month. The new front is bringing in customers.
Answer. Sequence is being read as cause, which is the failure named post hoc. The specific alternative here is easy to name: July is when the local schools break up, and footfall on that street changes for reasons that have nothing to do with paint. The test is a comparison that separates the two: what did sales do in July of the previous two years, and what did the unpainted bakery three streets away do this July.
Exercise 6
The timetable is reliable, because it is produced by the scheduling office, and the scheduling office produces reliable timetables.
Answer. Circular. The conclusion is smuggled in as a premise, so anyone who doubts the conclusion has equal reason to doubt the premise, and the argument moves nobody. Note what it is not: it is not invalid. A circular argument is impeccably valid, since its premises cannot be true while its conclusion is false. It fails on a different axis entirely, and Valid vs sound arguments is where that split is worked out.
Exercises 7 to 9: read the number and the chart
Exercise 7
A shop's monthly sales rose fifty percent in March and fell forty percent in April. Sales are up ten percent over the two months.
Answer. No. Percentages compound rather than add. Starting from 100, March gives 150, and forty percent off 150 is 60, leaving 90. Sales are down ten percent, not up ten. The general rule worth memorizing: a rise of x percent followed by a fall of x percent always ends below where it started.
Exercise 8
A scanner on a production line flags defective parts. Suppose it catches ninety percent of genuinely defective parts and wrongly flags five percent of sound ones, and suppose one percent of parts on this line are defective. A part has just been flagged. Is it probably defective?
Answer. No, and it is not close. Work it on two thousand parts.
| Group | Parts | Flagged by scanner |
|---|---|---|
| Genuinely defective, one percent | 20 | 18 |
| Sound | 1,980 | 99 |
| Total flagged | 2,000 | 117 |
Of 117 flagged parts, 18 are actually defective, so a flagged part is defective about fifteen percent of the time. The scanner is accurate and the flag is still usually wrong, because the sound parts are so much more numerous that five percent of them swamps ninety percent of the defective ones. The move that produces the right answer every time is to ignore the percentages until you have written down how many items are in each group.
Exercise 9
A bar chart shows on-time arrivals for three depots at 97, 98 and 99 percent. The bars are in the ratio one to two to three, and the depot at 99 looks three times better than the depot at 97.
Answer. The vertical axis does not start at zero; it starts at 96. Truncating an axis magnifies a small difference into a visual claim the numbers do not support. The difference between the best and worst depot is two percentage points. The test to run on any chart: cover the picture and read the axis labels first, then look at the bars and see whether your impression survives.
Exercise 10: the four card problem
Peter Wason introduced this task in the 1960s and it remains the most instructive single exercise in reasoning, because most people who get it wrong are confident while getting it wrong.
Four cards lie on a table. Each has a letter on one side and a number on the other. The visible faces read A, K, 4 and 7. The rule to be tested: if a card has a vowel on one side, it has an even number on the other. Which cards must you turn over to find out whether the rule is broken?
Answer. Turn A and 7. The card showing A can break the rule directly, by having an odd number behind it. The card showing 7 can break it too, by having a vowel behind an odd number. The card showing 4 is the one most people pick and it is useless: the rule says nothing about what must be behind an even number, so whatever is there, the rule survives. The card showing K is irrelevant for the same reason.
The pattern of error is the point. People reach for the case that would confirm the rule, the vowel and the even number, when only the cases that could refute it carry any information. That instinct is the reason a habit of asking what would show me wrong is worth more than any amount of gathering evidence in favor.
Critical thinking, critical reasoning and deduction
Critical thinking and critical reasoning are used interchangeably in most curricula, with one difference of emphasis: critical reasoning usually names the narrower skill of evaluating an argument's structure, while critical thinking includes the wider work of deciding what evidence is relevant, whose account to trust and when to suspend judgment. Some admissions tests use the narrow sense and label a section critical reasoning; treat the label as a description of the item type rather than a claim about the discipline.
Deduction is a component rather than a rival. Deductive reasoning covers the forms that guarantee a conclusion, and exercises 1 to 3 above are partly deductive work, since finding a missing premise means finding what would make the step follow. Deductive vs inductive reasoning covers the other half, the inferences behind exercises 4 and 8, where the conclusion is only ever probable.
Building the habit outside a worksheet
To build the habit at work or in study, attach the drill to something you already read. Three practical scenarios, in ascending difficulty:
- Take one claim a week from a report you receive anyway, write the conclusion in one line and the premises underneath, and mark the weakest link.
- Rewrite an objection you disagree with in the form its holder would accept, before answering it. If they would not recognize your version, you are answering something else.
- Argue the side you reject for five minutes in a meeting, out loud, and note which of your own reasons stopped working while you did it.
Professional training programs put the same drills under local names, such as clinical judgment in nursing curricula or hazard analysis in engineering, which is why the exercises transfer between fields more easily than the vocabulary does. Whatever the setting, the single question that does the most work is this one:
If you had to argue the opposite conclusion tomorrow, what is the first piece of evidence you would go looking for?
Whatever you name is evidence that exists, that bears on the question, and that you have not yet gone and found. That is the exercise, and unlike the ten above it never runs out of items.