The counterexample
from 350 BCE As old as universal claims; central to method from the 1900s
One case that the claim says cannot exist, produced and shown to exist, which is the end of the claim.
- Active
- from 350 BCE
- Where
- Athens, later Cambridge
- Status
- In use
- Tradition
- Analytic tools
What it is
A counterexample is a single instance that a universal claim rules out and that nevertheless obtains. Because the claim covers every case, one exception destroys it — not weakens it, destroys it.
The asymmetry with confirmation is total. A thousand supporting instances leave a universal claim exactly as vulnerable as it was; one counterexample finishes it.
How it works
State the claim in its universal form, so that what would refute it is clear. Look for a case that satisfies the antecedent and fails the consequent. Check that the case is genuine rather than a misreading of the claim. Present it.
The third step is where most counterexamples die. The commonest reply is that the case has been misdescribed, and the commonest bad reply is to narrow the claim until the case falls outside it.
The form of the inference
- The claim is that all A are B
- Here is a case that is an A
- This case is not a B
- Therefore not all A are B, and the claim is false as stated
The argument
Proposition It is not true that all our overdue accounts are from the one client.
- The claim on which the collections policy was built is that every account more than sixty days overdue belongs to the Mullaway group.
- Account 3312 is ninety-four days overdue on the same ageing report the claim was drawn from.
- Account 3312 is billed to a different entity with no ownership connection to the Mullaway group.
- So there is an account over sixty days overdue that does not belong to that group.
Therefore The claim is false, and a policy built on it will not do what it was designed to do.
One account out of four hundred. The finding is not that the claim is mostly right; it is that the claim as stated is wrong, and anything built on its universality is unsound.
What it is good for
Testing any universal claim, which includes most definitions, most rules and most policies. It is the fastest way to find out whether a general statement has been thought through.
Where it fails
It invites the ad hoc rescue. A claim under threat can be narrowed to exclude the offending case — all A are B, except in circumstances like these — and each rescue leaves the claim technically alive and progressively emptier. The repairs to the analysis of knowledge after Gettier are the standard cautionary tale.
It also does not work against claims that were never universal. A statistical or a hedged claim is untouched by a single case, and treating a tendency as a universal in order to refute it is a straw man rather than a counterexample.
Common misconceptions
That a counterexample shows a claim is usually wrong. It shows the claim is not universally true, which may be compatible with its being nearly always right and still useful.
Where it came from
As old as the universal claim itself, but it becomes a self-conscious method with Moore and Russell and then the central move of mid-century analytic philosophy.
What became of it
Never superseded. It remains the standard reply to any proposed analysis, and the standard test of any general rule.
Lasting influence
Gettier's cases, Chisholm's, Goodman's grue — a striking share of twentieth-century philosophy consists of counterexamples and the repairs they forced.
Why it still matters
Universal. It is the first thing anyone should do to a general claim, including their own.
Put another way
One key that opens a lock the manufacturer said only one key opens.
Texts
Edmund Gettier, Is Justified True Belief Knowledge?
Two counterexamples, three pages, and an analysis that had stood for two millennia was over.
Where to start
Take any confident general claim you hold and spend ten minutes trying to break it. The exercise is more instructive than any reading.
The surprising bit
It is the only method on this list that can be decisive in a single sentence, and the only one that requires no argument at all — just a case.
In their own words
No amount of experimentation can ever prove me right; a single experiment can prove me wrong
Who held it
Roderick Chisholm
Connections
- Relies on
- Modus tollens
- Set against
- Induction
- Influenced
- Conceptual analysis, The method of cases
- Parallel to
- Modus tollens
The verdict
The most economical refutation available, and the one most easily evaded by a determined author.