How can I find a counterexample to a claim that says “always”?
Quick answer
Write down what the claim says must always be true, including any conditions or exceptions. Then look for one case that meets those conditions but makes the claimed result false. That case is a counterexample: one valid counterexample can show a universal claim is false. A case outside the stated scope does not test the claim as written. [S1]
The question, the details, the next step.
I keep seeing claims with words like “always” and “every.” How do I start looking for a counterexample without just guessing?
Treat the claim as a rule with two parts: the cases it covers, and what it says about those cases. A counterexample must fit the first part and contradict the second. One valid in-scope case is enough to show a universal claim is false; a case outside its stated scope does not challenge it as written. [S1]
Start by rewriting the claim in a checkable form. For example, suppose a puzzle states: “Every tile in this set marked with a star is a triangle.” The cases are tiles in this set that have a star. The required result is that each such tile is a triangle. Look among those tiles for one that is not triangular. You are not looking for any unusual tile—only one meeting the claim’s conditions.
What should I do with words that quietly limit the claim, like “in this set” or “under these rules”?
Keep every stated limit attached to the claim. Under the puzzle example, a star-marked tile from a different set is not a counterexample, even if it is a square: it is outside the claim’s scope. Likewise, if the claim applies only under a particular rule, a case where that rule is not in effect cannot refute the claim as written.
Try this scope check before accepting a candidate:
- Does this case belong to the named group or situation?
- Does it meet every condition in the claim?
- Does it actually contradict what the claim says must happen?
If any answer is no, keep searching or revise the claim’s wording. Be careful not to add a condition that the original claim did not state.
Can you show a copyable prompt I could use to search a claim more systematically?
Use this prompt, filling in the brackets with the exact claim and its scope:
“Claim: [write the claim exactly]. It applies to: [people, objects, cases, or conditions covered]. It says that: [what must always be true]. Find a possible case that meets every stated condition but makes that result false. Check the candidate against each condition. If you cannot find one, suggest a sensible next place to look; do not claim that the rule is proven true just because this search found nothing.”
Worked example:
- Claim: “Every tile in this set marked with a star is a triangle.”
- Search: Check only star-marked tiles in this set.
- Candidate: A star-marked square tile in this set.
- Check: It is in the set, it has a star, and it is not a triangle.
- Conclusion: If the candidate really is a tile in the described set, it is a counterexample to the claim as written.
The example is a hypothetical puzzle, not a report about a real set of tiles. The key is checking the candidate, not merely imagining an exception.
If I can’t find a counterexample, can I say the claim is true? And what if the wording is vague?
Say only what your search supports: “I haven’t found a counterexample in the cases I checked.” Not finding one in a limited search does not establish that the claim is true. The counterexample test explains how one valid case can refute a universal claim; it does not, by itself, guarantee that a search is complete. [S1]
For vague wording, list the unclear terms and ask what counts as belonging to the claimed group or satisfying its conditions. If different reasonable interpretations change which cases count, state the interpretation you used before judging a candidate. You can then ask the person who made the claim to clarify its scope, rather than treating an out-of-scope case as a refutation.
A practical checklist
Copy the claim’s exact wording, especially “always,” “every,” and any limits.
State which cases the claim covers and what result it says must hold.
Search for one candidate that meets every stated condition.
Check that the candidate actually contradicts the required result.
Report a counterexample only if it is in scope; otherwise, describe what remains unclear.
A few more useful answers
Does one counterexample disprove an “always” claim?
Yes, if it genuinely meets the claim’s stated conditions and contradicts the result the claim says must always hold. A case outside the scope is not a counterexample to that claim as written. [S1]
Does failing to find a counterexample prove the claim?
No. It means only that your search did not find one; the cases you checked may not cover every case the claim includes.
What if the claim has an exception?
Include the exception when defining the claim’s scope. A candidate that falls within an exception does not test the claim in the same way as a case to which the rule applies.
Sources
References used for this guide.
- [S1] Chapter 5 Summary — Introduction to Philosophy openstax.org · Checked