Predicate Logic · step 12 of 13

Disproving with a single counterexample

Because ¬∀x P(x) is ∃x ¬P(x), a universal claim is destroyed by a single element where it fails. This is the cheapest disproof in mathematics — but only if the counterexample is fully verified. Naming a candidate without showing the hypotheses hold and the conclusion fails earns nothing, because half of "counterexamples" offered in exams do not actually satisfy the hypotheses.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Write the claim in ∀ formIdentify precisely what is being asserted for all x, including every hypothesis in the antecedent.
Hunt near the boundaryTry 0, 1, negatives, the empty set, and the smallest structure permitted. Most false claims break at an edge case rather than a typical one.
Verify the hypotheses holdShow your candidate genuinely satisfies the antecedent. This is the half students skip.
Verify the conclusion failsCompute both sides explicitly. State the verdict: "so the claim is false."

Worked example

Disprove: "For all integers a, b, if a divides bc then a divides b or a divides c."

Take a = 4, b = 2, c = 6.
Hypothesis: bc = 12 and 4 ∣ 12, so the antecedent holds.
Conclusion part one: 4 ∤ 2.
Conclusion part two: 4 ∤ 6. So the disjunction fails.

Answer. False. With a = 4, b = 2, c = 6 the hypothesis holds but neither disjunct does. (The claim is true when a is prime — that is Euclid’s lemma.)

Where marks get dropped

These are the specific errors that cost credit on disproving with a single counterexample questions — QED's rubric penalises each of them separately.

Offering a candidate that fails the hypotheses. If the antecedent is false the implication is vacuously true, so nothing has been refuted.
Asserting the conclusion fails without computing it. "Clearly 4 does not divide 6" is fine; leaving it unstated is not.
Trying to disprove an existential with a counterexample. To refute ∃x P(x) you must show P fails for every x — one failure proves nothing.

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Disproving with a single counterexample — frequently asked questions

How many counterexamples do I need?

Exactly one, fully verified. A second adds no logical force, though it can be worth mentioning if it clarifies why the claim fails.

What if I cannot find one?

That is weak evidence the claim is true — switch to attempting a proof. Often the point where a proof attempt gets stuck reveals exactly the case that breaks it.

Can I disprove a ∀∃ statement with one example?

Yes: to refute ∀x ∃y R(x, y) you give a specific x and then argue no y works for it. The second half is a universal claim, so it needs an argument, not an example.

The rest of Predicate Logic

Quantifiers, predicates, binding, and validity. Each subtopic below has its own method, worked example and mark-losing traps.

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