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A short rule is useful when its conditions are retained. It becomes misleading when “under this model” disappears and the statement is repeated as universal. A small counterexample can reveal the missing condition without requiring a large dataset.
Write the claim and its conditions separately
Identify the exact conclusion being asserted. Then list the premises that supposedly make it follow. To refute a universal statement, the counterexample must satisfy those premises while the conclusion fails. A case outside the stated domain does not accomplish that task.
| Claim | What is supposed to be true in every allowed case? |
| Domain | What conditions restrict the statement? |
| Construction | Does this example satisfy those conditions? |
| Check | Does the claimed conclusion fail? |
| Repair | What narrower statement is supported? |
Example: a falling ratio does not force a falling numerator
Emissions 100 to 120 and denominator 10 to 15 give intensity 10 to 8. The ratio fell, but the numerator rose. This refutes the unqualified claim that lower intensity proves lower absolute emissions. It does not refute the correctly stated ratio identity.
The ESG chart-review article shows how coverage and denominator definitions add further conditions before interpreting an actual reported trend.
Example: equal unit chances do not specify every sample set
For labelled units A, B, C and D, select either AC or BD with probability one half each. Every unit has inclusion chance one half. But four of the six size-two subsets are impossible, so this is not the fixed-size simple-random design that makes every size-two subset equally likely.
| Uniform simple random | 1/2 | All six, each probability 1/6 |
| AC or BD | 1/2 | Only two, each probability 1/2 |
The sample-set explanation connects the definition with estimator uncertainty. The small construction refutes an incomplete definition, not the usefulness of a properly specified probability design.
Example: half a two-sided probability is not every planned tail
For a standard-normal statistic −2, the symmetric two-sided p-value is about 0.0455. Half is about 0.02275, which is the lower tail. The planned upper-sided p-value is about 0.97725. The sign and chosen alternative matter.
Read the opposite-direction test example before using the shortcut. Choosing the favorable direction after observing the sign changes the analysis procedure.
Repair the rule without overcorrecting
After finding a counterexample, identify a supported qualified statement. The symmetric two-sided value doubles the smaller tail in the stated normal case. A simple-random design requires uniform probabilities over its fixed-size subsets. Intensity depends on both numerator and denominator.
Do not replace one overbroad rule with another. A counterexample to a shortcut does not prove every competing method is preferable, nor that the original topic has no useful structure. Keep the exact conditions and numerical check in the correction record.
Use the version-control workflow to retain that history. This habit helps study notes remain compact without losing the assumptions that make them true.
Frequently asked questions
Can one example disprove a universal rule?
Yes, if it satisfies the claimed domain and the asserted conclusion fails. The numerical and premise checks both matter.
Does a failed shortcut prove the entire method is wrong?
No. Identify the missing condition and retain the qualified relationship. A counterexample refutes the overbroad statement, not every legitimate application.
Sources
- NIST: mean uncertainty and test direction · itl.nist.gov · Retrieved
- GHG Protocol: published Scope 2 Guidance · ghgprotocol.org · Retrieved
