Back to the on-screen lesson ·
One case finishes a claim about every case and leaves a claim about how many exactly where it was.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will read a claim for the word that sets its scope, and use that to decide whether a single case settles it or leaves it untouched. You will also say what a counterexample earns once it has landed, and give the range a claim is left in when one case has been found and nothing else has been checked.
You have just seen that a story cannot settle how often. This lesson is the other side of it, and it is the more useful side: there is a whole family of claims that one case settles completely, and knowing which family is which saves an enormous amount of arguing.
| Term | What it means |
|---|---|
| Domain | The cases, objects, or occasions a claim actually covers. |
| Counterexample | A qualifying case that contradicts a universal claim. |
| Strict majority | More than half of the members of a defined group. |
| Composition | An unsupported transfer of a property from parts to the whole they form. |
| Division | An unsupported transfer of a property of a whole to its individual parts. |
A verified case can establish that something happened and can refute a claim that it never happens. 'Every locker on this corridor is empty' is false if a qualifying locker contains a coat. The counterexample must belong to the stated corridor and the relevant time: a coat in a different building does not contradict this claim.
A frequency claim usually requires more information. One late bus can coexist with a claim that most buses arrive on time. The effect of that case depends on the group size and what is already known. In a group of only two buses, one late bus would rule out a strict majority being on time. So inspect the scope word, the population, and the available count together.
After finding a counterexample, state only what it earns. Refuting 'every locker is empty' establishes at least one nonempty locker. It does not establish that most lockers are nonempty or that every locker is full.
Another way: picture
Picture the claim as a fence round every case there is. Every and never draw the fence round all of them, and one case outside it breaks the fence. Most draws a fence round more than half, and there is open ground inside the claim where exceptions are allowed to stand.
Another way: steps
Three steps:
Knocking out nobody has ever is not the same as establishing the opposite.
| The claim | One counterexample gives you |
|---|---|
| Nobody has ever been late | that somebody has been late, once |
| This lock cannot be picked | that it can be picked, once, by that person |
| Every locker is empty | that one locker is not empty |
None of those rows says often. A lock that has been picked once is a lock that can be picked; how easy it is remains completely unknown, and the person who says so the lock is useless has walked straight into the mistake of the previous lesson from the other direction.
The same arithmetic works on the range. Thirty people in the class, one found who walks: the number who walk is now somewhere from 1 to 30. The lower end has moved by one and the upper end has not moved at all.
A universal claim has a domain: the people, objects, places, or occasions it covers. 'Every locker on this corridor is empty at noon' concerns those lockers at that time. A full locker downstairs does not belong to the domain. A coat placed in a corridor locker at one o'clock does not contradict the noon claim. Before celebrating a counterexample, check that it satisfies the conditions under which the original claim was made. Otherwise you may have found an interesting case without finding a contradiction.
Next identify the exact property being asserted. Empty could mean no stored belongings, or it could be used loosely to mean no assigned user. A locker can be unassigned and still contain a forgotten coat. Clarify the meaning rather than moving between senses to manufacture a refutation. A good counterexample preserves the original meanings, belongs to the stated domain, and lacks the property that the claim attributes to every member. Those three checks are as important as the vividness of the case.
The logical opposite of all is not none. If not every locker is empty, at least one is not empty. Some may be empty and some not. Similarly, refuting 'nobody arrived late' establishes that somebody arrived late, not that everybody arrived late. Write the weakest conclusion forced by the evidence. Stronger claims might also be true, but they would require additional support. This keeps a successful criticism from becoming a new overgeneralization.
A finite group makes the limits calculable. Suppose 30 students are in a class and one is known to walk to school. If nobody else's travel is known, the count of walkers can be any whole number from 1 through 30. One confirmed nonwalker would lower the maximum to 29. Ten confirmed walkers and five confirmed nonwalkers would leave the possible walker count from 10 through 25. Each known case changes the relevant boundary; unknown cases remain genuinely open.
For a strict majority of 30, at least 16 must have the property. Sixteen confirmed walkers establish that most walk, even if the other 14 remain unknown. Fifteen confirmed nonwalkers rule out most walking because at most 15 could walk. One nonwalker does neither: as many as 29 could still walk. This is why the shorthand that one case never matters to a frequency claim is too crude. A case's force depends on the threshold, the population, and the count already established.
With only two people in the group, one confirmed nonwalker is enough to refute 'most of them walk', because strict most requires both. With a large group, the same type of exception usually leaves a majority possible. Always state the group size when it matters. For vague words such as often or usually, ask what threshold and time period are intended before pretending there is a precise numerical test. Clarifying the claim is part of evaluating it.
Another change of scope occurs when reasoning moves between members and the collection they form. 'Each book in this box is light enough for one child to lift, therefore the full box is light enough for one child to lift' moves from parts to a whole. It may fail because the books' weights accumulate. This is a fallacy of composition when the property is transferred without a valid connecting reason. The word each does not by itself establish a matching property of the assembled collection.
The reverse move can fail too. 'The team completed the project in ten hours, so every team member worked ten hours' moves from a fact about the whole undertaking to each member. Some members may have worked shorter shifts, and others may have contributed at different times. This is a fallacy of division when a property of the whole is assigned to its parts without support. A successful team does not imply every member performed every role equally well.
Do not turn these labels into a ban on every inference between parts and wholes. Some properties transfer under suitable rules. If six separate packages each weigh exactly two kilograms and the total being discussed excludes all additional packaging, their combined mass is twelve kilograms. The conclusion follows from an additive relationship that has been stated. It does not follow because the word weight automatically behaves like every other property. The relevant rule supplies the bridge.
Likewise, a twelve-kilogram total for six packages gives a mean of two kilograms per package. It does not establish that each package weighs two kilograms unless equality is separately known. One distribution could be one, one, two, two, three, and three kilograms. The total remains twelve while individual values differ. This resembles the waiting-time example: an aggregate measure constrains the parts but does not normally fix each individual part's value.
Some collective properties arise from relationships between parts. A group can coordinate effectively even though no single member can perform the entire task. Individually strong players can form a poorly coordinated team. To evaluate such claims, ask what mechanism relates individual qualities to the collective result. Merely repeating successful at both levels does not show that the same property is involved. The argument may need information about coordination, role distribution, communication, or constraints.
Use a scope diagram with two levels when an argument shifts between a group and its members. Write the observed claim beside the level it describes. Draw an arrow toward the proposed conclusion's level, then label the rule that would justify the transfer. If no rule is supplied, try a distribution or arrangement that preserves the original claim while defeating the conclusion. A heavy box of light books and an uneven division of labor make the missing link visible without implying that every part-to-whole inference must fail.
A fictional club has 30 helmets assigned to a cupboard. Its notice says, 'Every helmet in this cupboard has been checked this month.' An audit finds one helmet with a reliable record showing its last check was two months ago and no current check. Under the case's stated records, this is a counterexample: it belongs to the cupboard and lacks the claimed current inspection. The universal notice is therefore incorrect.
The audit has not shown that all 30 helmets are unchecked. If the other 29 records have not yet been examined, the number lacking a current check could range from one to 30. If the auditor then confirms 24 current checks, the number without one can be at most six. The remaining five unknown records still need examination; they should not be silently counted as either checked or unchecked.
The storekeeper also says, 'The whole cupboard was audited, so every helmet must have been individually inspected for damage.' That conclusion changes the meaning of the collective activity. An audit of records is not automatically a physical inspection of every item. The procedure must state what was done to each helmet before that stronger claim is justified. The word whole cannot supply the missing methodological detail.
A careful report therefore distinguishes the universal claim refuted, the confirmed current checks, the records still unknown, and the type of audit performed. It can recommend completing the remaining record review and following the club's inspection procedure without making unsupported claims about any helmet's physical safety. One counterexample has done a precise job; the broader equipment decision requires the rest of the relevant evidence.
Refuting all by concluding none. One nonempty locker refutes every locker empty but leaves the others open.
Ignoring the domain. A case outside the stated time, place, or group does not contradict the claim.
Treating most like all. An exception may coexist with a majority; calculate the relevant bound when the group is finite.
Moving a group property to every member. A total or average does not establish identical individual values without an additional equality premise.
State the domain and time.
Every locker on this corridor is empty at noon.
The claim concerns specified lockers at a specified time.
Verify the candidate belongs to the domain.
Locker 3 is on this corridor and is checked at noon.
A different place or time would not test this universal claim.
Record the conflicting property.
Locker 3 contains a coat.
Containing a coat contradicts empty in the stated sense.
Reject the universal claim.
Not every locker is empty.
One qualifying counterexample defeats every.
State the supported positive result.
At least one locker is nonempty.
The other lockers' contents have not been established by this case.
Fix the population.
30 students.
A numerical bound needs a known group size.
Record the known walkers.
10 students definitely walk.
These cases set a minimum of ten walkers.
Record the known nonwalkers.
5 students definitely do not walk.
These cases exclude five people from the maximum walker count.
Calculate the remaining maximum.
30 − 5 = 25 walkers at most.
Every currently unknown student could still walk.
Report the integer range and majority status.
10 through 25 walkers; a majority is not yet settled.
The range includes counts below and above the strict-majority threshold of 16.
Record the aggregate information.
Six packages weigh 12 kilograms in total, excluding other packaging.
This supplies a whole-collection mass and a package count.
Compute the mean.
12 ÷ 6 = 2 kilograms per package on average.
The average divides the combined mass by the number of packages.
Identify the stronger individual claim.
Every package weighs exactly 2 kilograms.
This assigns the mean to each part separately.
Construct a different compatible distribution.
1, 1, 2, 2, 3, 3 kilograms.
These six values need not be equal.
Check the total.
1 + 1 + 2 + 2 + 3 + 3 = 12 kilograms.
The counterexample preserves all the aggregate information.
Diagnose the unsupported transfer.
The each-package conclusion commits division without an equality premise.
A total and mean constrain the parts without fixing every individual mass.
State the individual measurements.
20 books each weigh 1 kilogram.
Each individual book can be described by its own mass.
Calculate their combined mass.
20 × 1 = 20 kilograms, before the box itself.
Mass accumulates even though each book is individually light.
Reject the unsupported ease-of-lifting conclusion.
Four claims about the same fire door. Match each to what it would take to show it false.
| One case is enough to show it false | It would take a count | |
|---|---|---|
| The fire door is always propped open. | ||
| The fire door is usually propped open. | ||
| The fire door is never propped open. | ||
| The fire door is propped open about half the time. |
In a class of 30, ten students are confirmed walkers and five confirmed nonwalkers. The others are unknown. Complete the minimum, maximum, and strict-majority threshold.
Use the confirmed walkers for the lower bound.
Minimum walkers: minimum.
Unknown cases cannot remove the people already confirmed to walk.
Exclude confirmed nonwalkers from the upper bound.
30 − 5 = maximum walkers at most.
Every remaining person could still be a walker.
Find the first whole number above half the class.
30 ÷ 2 + 1 = majority.
A strict majority requires more than half rather than exactly half.
The claim is: *This lock cannot be opened without its key.* The case is: *Wen opened it on Tuesday with a piece of wire.* Does the case settle the claim?
Somebody says nobody in the class of thirty walks to school. Mira walks. Nobody else has been asked. Give the range the number of walkers could now be in, as a set.
This task has no paper form; do it on a device.
Six packages have a verified combined mass of 12 kilograms, excluding any extra packaging. Separately, a list of twenty books gives each book's mass as one kilogram. Construct only supported links. Do not infer equal package masses or that a child can lift the whole box merely from lifting books individually.
This task has no paper form; do it on a device.
Four replies to a manufacturer's claims about a kettle, rather than to claims about a club. Mark the one reply whose single case settles nothing.
This task has no paper form; do it on a device.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A group has 24 members. Five are confirmed to cycle to meetings and another five are confirmed not to cycle. The remaining members have not been asked. Give the inclusive interval bounds for the possible number k who cycle. Counts inside those bounds are whole numbers.
This task has no paper form; do it on a device.
You can tell a claim one case refutes from a claim that needs a count, and say what a counterexample entitles you to conclude and what it does not. Explain why finding one person who walks changes the smallest possible number and not the largest. Next: two things going up and down together.
13. Light books inside a heavy box, step 3
One person lifting each book separately does not show they can lift the full box.
Ease of lifting the parts does not automatically transfer to the assembled whole.