Vacuous truth: a property that holds because nothing satisfies the assumptions

From the premises A and NOT A, anything follows, including B. The engine calls that argument valid, and then warns that it is valid only vacuously: no row makes all the premises true, so the conclusion was never tested.

The argument below has premises A and NOT A and conclusion B. Validity means there is no row where all premises hold and the conclusion fails. There is no row where both premises hold at all, so there is no counterexample, so the argument is valid. The engine says so and adds its warning in the same breath: the premises can never all be true, so anything would follow. The separate consistency check reports that no assignment satisfies all the premises.

The same thing happens with assumptions on a decision model. A rule checked under assumptions that contradict each other has no admissible case; every property holds of it, tautology and contradiction alike, and the engine marks each verdict vacuous and the scope inconsistent rather than printing a row of green ticks. The pitfall page on inconsistent assumptions shows that case.

Vacuous truth is not a bug in logic; it is what “valid” means. The practical lesson is to read the warning. A pass with a vacuous label means the test exercised nothing, and the thing to fix is the premises or the assumptions, not the conclusion.

p1 — A · premisep2 — NOT A · premiseconclusion — B · conclusion

Does the conclusion follow from the premises? Yes · vacuous: no admissible case exists · Exhaustively checked and solver cross-checked · 4 rows

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

Truth table

Vacuous truth: a property that holds because nothing satisfies the assumptions
#	A	B	p1	p2	conclusion
0	FALSE	FALSE	FALSE	TRUE	FALSE
1	FALSE	TRUE	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE	FALSE	FALSE
3	TRUE	TRUE	TRUE	FALSE	TRUE

Plain-language reading

BeginnerConclusion: B is true.

IntermediateConclusion: B is true.

AdvancedConclusion: B is true.
Formal notation: B

The common slip

Reading a vacuous pass as evidence. “No counterexample” sounds reassuring; when it is because no case was admissible, it establishes nothing about the conclusion, and the engine's label says so.

Questions

Why does logic allow anything to follow from a contradiction?

Because validity is defined by the absence of a counterexample, and a contradiction leaves no row on which to build one. The lesson on arguments has a step on exactly this case.

How do I know whether my premises are consistent?

The premise-consistency check answers it directly, with a witness row when they are and the verdict “inconsistent” when they are not. The worked example on inconsistent premises shows both verdicts together.

Worked examples

Inconsistent premises: A and NOT A prove anything (vacuous validity)

Tools

Pitfalls

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.