Tautology in logic: meaning and examples

A tautology is a formula that is true on every row of its truth table, whatever values its variables take. A OR NOT A is the simplest one.

A tautology is also called a valid formula or a logical truth; in Boolean algebra it is a function identically equal to 1, and in a circuit it is an output stuck at TRUE. The opposite is a contradiction, false on every row, and a formula that is neither is a contingency. Every law of Boolean algebra can be stated as a tautology by joining its two sides with a biconditional: (A AND B) <-> (B AND A) is one.

The artefact on this page is the two-row table of A OR NOT A. Both rows read true, so the tautology check holds, with the summary that f is true on every assignment; no witness is needed, because there is no row left to show. The contradiction check is violated, and the satisfiable check holds, both with the witness A = FALSE: that one row already makes the formula true. The engine's plain-language reading says the same thing in words: f is true when A is true or A is false.

Tautologies matter wherever a condition is supposed to constrain something and does not. A guard clause that is a tautology lets everything through; a test assertion that is a tautology cannot fail; a rule whose output is a tautology does not depend on its inputs at all. In a logic course, showing an argument valid is the same as showing that the implication from its premises to its conclusion is a tautology. The word is also used loosely for a redundant phrase, which is a different thing.

Example

f := A OR NOT A · output

Exact checks

Always true (tautology)? Yes · Exhaustively checked and solver cross-checked · 2 rows

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

Can be true? Yes Example assignment: A = F · Exhaustively checked and solver cross-checked · 1 rows

Truth table (2 rows)

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

Plain-language reading

Beginnerf is true exactly when A is true or A is false (or both).
In every other case, f is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.

Intermediatef is true exactly when A is true or A is false (or both).
In every other case, f is false.

Advancedf is true exactly when A is true or A is false (or both) [inclusive disjunction ∨].
In every other case, f is false.
Formal notation: A ∨ ¬A

The common slip

Treating a formula that is true in every case you tried as a tautology. Checking a few rows shows nothing about the others; a tautology is decided by the whole table, which is why the exact check reads every row before it reports that the property holds.

Questions

Is a tautology the same as a valid argument?

They are two views of one fact. The argument from premises P1 and P2 to conclusion C is valid exactly when (P1 AND P2) -> C is a tautology. The argument validity checker and the tautology check therefore agree on every example.

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.