A little room to wonder

Categories & Counterexamples

A counterexample satisfies a claim’s starting conditions while defeating its conclusion. One good case can refute an all-claim without settling every nearby question.

What would a genuine exception have to show?

Several paperclips show repeated loops, curved ends and different surface coatings.
The paperclips vary in visible colour and overlap. They are an illustration of possible grouping features, not the four fictional cards or proof of an all-claim.Dietmar Rabich · Source · CC BY-SA 4.0

Identify the claim before collecting cases

An all-claim concerns every member of a stated category. A some-claim needs at least one member with the stated feature. Examples have different jobs in these two claims. Finding one blue tile supports some tiles are blue; it does not establish that every tile is blue.

A counterexample must lie inside the claim’s scope. A yellow triangle does not refute all four-sided tiles are blue unless it is also four-sided. Carefully name the category and the promised feature before selecting a test.

Worked example: an exception in a fictional collection

A paper collection contains card A: blue triangle; B: blue quadrilateral; C: yellow quadrilateral; D: blue circle. Someone claims every four-sided card in this collection is blue. B agrees with the claim, while C refutes it: C has four sides but is yellow.

D is not a relevant counterexample because it has no straight sides. Three blue cards do not cancel C. A narrower claim, at least one four-sided card is blue, is supported by B. These conclusions concern only the supplied collection, not every card that could ever be designed.

Revise without hiding the exception

You can change a failed claim, but show what changed. Replacing every with some narrows its reach. Excluding C by an independently meaningful rule changes the domain. Quietly discarding C because it is inconvenient does not supply a reason for the original claim.

Many confirming cases may help suggest a pattern, yet a universal statement requires reasoning that covers its whole scope. A counterexample shows the proposed statement fails; it does not automatically provide the best replacement statement.

Keep uncertainty visible

A card with a covered edge might be a candidate counterexample, but its membership is unresolved. Record what would need checking. Unknown is a different result from confirmed exception or confirmed match.

This exercise tests statements about invented shapes. It is not a method for placing people into categories. A careful final note names the original claim, the decisive feature of C and the narrower statement still supported.

Only if you feel like trying

Find a relevant exception

  1. Write the all-claim and underline its category and promised property.
  2. Explain why C is decisive and D is irrelevant, despite both differing from some blue examples.
  3. Invent a new all-claim about the same four cards and either refute it or explain the limits of your test.

Optional paper or spoken reasoning with supplied fictional examples. No personal beliefs, experiences, accounts, purchases or public sharing are required.