Counterexample in logic: meaning and examples

A counterexample is one assignment of values that refutes a claim: for an argument, a row where every premise is true and the conclusion is false; for a tautology claim, a row where the formula is false.

A counterexample is the shortest possible disproof. A universal claim (“always true”, “the conclusion follows”, “these two formulas agree”) fails as soon as one case breaks it, and that case is the counterexample. It is the mirror image of a witness, which is one case that establishes an existential claim. The engine reports every violated verdict with a counterexample row, written as an assignment such as A = FALSE, B = TRUE, and replays it through the independent evaluator before showing it.

The artefact on this page is the fallacy of affirming the consequent: premises A -> B and B, conclusion A. The validity check is violated with the counterexample A = FALSE, B = TRUE. On that row A -> B is true, because its antecedent is false, B is true, and the conclusion A is false, so both premises hold and the conclusion fails; the summary says exactly that. The premise-consistency check holds with the same assignment as its witness, since the premises can be true together.

Counterexamples are how logic is used day to day: a reviewer who doubts a rule does not re-derive it, they look for the case that breaks it, and a failing test is a counterexample to the claim that the code is right. In mathematics a single counterexample ends a conjecture. The term is confused with an exception that is “tolerated”; in logic there is no tolerated counterexample, one is enough. It is also confused with a witness, which proves rather than refutes.

Example

p1 — A -> B · premisep2 — B · premiseconclusion — A · conclusion

Exact checks

Does the conclusion follow from the premises? No Counterexample: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Can all premises be true together? Yes Example assignment: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (4 rows)

Counterexample in logic: meaning and examples
#	A	B	p1	p2	conclusion
0	FALSE	FALSE	TRUE	FALSE	FALSE
1	FALSE	TRUE	TRUE	TRUE	FALSE
2	TRUE	FALSE	FALSE	FALSE	TRUE
3	TRUE	TRUE	TRUE	TRUE	TRUE

Plain-language reading

BeginnerConclusion: A is true.

IntermediateConclusion: A is true.

AdvancedConclusion: A is true.
Formal notation: A

The common slip

Looking for a counterexample among the rows where the conclusion is true. A counterexample to an argument needs all premises true and the conclusion false; rows that make the conclusion true can never be one, however many premises they violate.

Questions

Is one counterexample really enough?

Yes. A claim of the form “in every case” is refuted by any one case where it fails, and no number of confirming cases restores it. That is why the engine shows one counterexample and does not count the others.

Why is the counterexample on this page also the consistency witness?

The premise-consistency check wants a row where all premises are true, and any counterexample row has that property by definition. On this example the same row, A = FALSE, B = TRUE, serves both purposes.

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.