Contingency in logic: meaning and examples

Q: Can a contingency be true on exactly one row?

Yes. A AND B on this page is true on one row of four. A minterm is always true on exactly one row and false on the rest, and it is still a contingency.

A contingency is a formula that is true on some rows of its truth table and false on others: neither a tautology nor a contradiction. Almost every formula that does real work is one.

A contingent formula, or contingency, depends on its inputs: its truth value changes as they change. Logic textbooks sort every formula into three classes, tautology, contradiction and contingency, and the classification is decided by the whole table, not by any single row. The checks on this page apply the classification mechanically: a contingency is exactly a formula on which both the tautology check and the contradiction check are violated.

The artefact on this page is the four-row table of A AND B. The output column reads false, false, false, true, so the formula is false on rows 0 to 2 and true on row 3. The tautology check is violated with the counterexample A = FALSE, B = FALSE, and the contradiction check is violated with the witness A = TRUE, B = TRUE. The satisfiable check holds with that same witness. One row of each kind is all a contingency needs; the table has three false rows and one true row.

Contingencies are the normal case. A business rule, an access condition, a gate's output and a premise in an argument are all contingencies; that is what makes them informative, since a condition that cannot fail says nothing. In testing, a contingency is the only kind of condition worth a test case, because both outcomes are reachable. The term is confused with a contradiction when a formula is false in the case someone happens to be looking at.

Example

f := A AND B · output

Exact checks

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

Always false (contradiction)? No Counterexample: A = T, B = T · Exhaustively checked and solver cross-checked · 4 rows

Can be true? Yes Example assignment: A = T, B = T · Exhaustively checked and solver cross-checked · 4 rows

Truth table (4 rows)

Contingency in logic: meaning and examples
#	A	B	f
0	FALSE	FALSE	FALSE
1	FALSE	TRUE	FALSE
2	TRUE	FALSE	FALSE
3	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when both A and B are true.
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.

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

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

The common slip

Expecting a formula to be a contingency because it looks complicated. (A XOR B) <-> (NOT A <-> B) looks contingent and is a tautology; only the table decides, which is why the engine runs both checks instead of guessing from the shape of the expression.

Questions

Can a contingency be true on exactly one row?

Yes. A AND B on this page is true on one row of four. A minterm is always true on exactly one row and false on the rest, and it is still a contingency.

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.