### Vacuously true

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In mathematics and mathematical truth, a **vacuous truth** is a statement that asserts that all members of the empty set have a certain property. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned *on*" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on *and* turned off".

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Ayers Rock is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. They are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true).

In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.^{[1]} This notion has relevance as well as in any other field which uses classical logic.

Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his or her parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with.

## Scope of the concept

A statement $S$ is "vacuously true" if it resembles the statement $P\; \backslash Rightarrow\; Q$, where $P$ is known to be false.

Statements that can be reduced (with suitable transformations) to this basic form include the following:

- $\backslash forall\; x:\; P(x)\; \backslash Rightarrow\; Q(x)$, where it is the case that $\backslash forall\; x:\; \backslash neg\; P(x)$.
- $\backslash forall\; x\; \backslash in\; A:\; Q(x)$, where the set $A$ is empty.
- $\backslash forall\; \backslash xi:\; Q(\backslash xi)$, where the symbol $\backslash xi$ is restricted to a type that has no representatives.

Vacuous truth is usually applied in classical logic, which in particular is two-valued. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.

## See also

- Counterfactual conditional
- Degeneracy (mathematics)
- Logical truth
- Supervaluationism
- Tautology (logic)
- Trivial (mathematics)
- Principle of explosion

## References

## Bibliography

- Blackburn, Simon (1994). "vacuous,"
*The Oxford Dictionary of Philosophy*. Oxford: Oxford University Press, p. 388. - David H. Sanford (1999). "implication."
*The Cambridge Dictionary of Philosophy*, 2nd. ed., p. 420. -

## External links

- Conditional Assertions: Vacuous truth