Contradiction in logic: meaning and examples

A contradiction is a formula that is false on every row of its truth table: no assignment of values to its variables makes it true. A AND NOT A is the simplest one.

A contradiction is also called an unsatisfiable formula, an inconsistent formula, or in Boolean algebra a function identically equal to 0. It is the negation of a tautology: NOT (A AND NOT A) is A OR NOT A by De Morgan. A set of premises is contradictory, or inconsistent, when their conjunction is a contradiction, and that is the case the premise-consistency check looks for.

The artefact on this page is the two-row table of A AND NOT A. Both rows read false. The contradiction check holds with the summary that f is false on every assignment, and the satisfiable check is violated with the summary that f can never be true on any assignment; neither has a witness, because a witness would be a row that makes f true and there is none. The tautology check is violated with the counterexample A = FALSE, the first row of the table.

In practice a contradiction is usually a mistake that hides. A rule that can never fire, a feature flag that no combination of settings turns on, a database constraint no row can satisfy, or a test precondition no input meets: all are contradictions, and all look fine until someone enumerates the cases. The term is often confused with contingency, a formula that is false on some rows but not all, and with a false statement, which is just one row of one formula.

Example

f := A AND NOT A · output

Exact checks

Always true (tautology)? No Counterexample: A = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? Yes · Exhaustively checked and solver cross-checked · 2 rows

Can be true? No · Exhaustively checked and solver cross-checked · 2 rows

Truth table (2 rows)

Contradiction in logic: meaning and examples
#	A	f
0	FALSE	FALSE
1	TRUE	FALSE

Plain-language reading

Beginnerf is true exactly when A is true and A is false.
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.

Intermediatef is true exactly when A is true and A is false.
In every other case, f is false.

Advancedf is true exactly when A is true and A is false [conjunction ∧].
In every other case, f is false.
Formal notation: A ∧ ¬A

The common slip

Confusing a contradiction with a formula that is merely false in the case at hand. A AND B is false when A is false, but it is true when both are true, so it is a contingency. A contradiction has no true row at all.

Questions

Why does the satisfiable check fail on a contradiction?

Satisfiable means at least one row makes the formula true. A contradiction has no such row, so the satisfiable check is violated and the contradiction check holds; they are the same question asked from two sides, and the engine reports both.

Practice

Tautology, contradiction or contingency: exercises with answers

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.