Vacuous truth

In mathematics and logic, 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" will be true whenever 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 vacuously true, 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 Uluru 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. This notion has relevance in pure mathematics, as well as in any other field that 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 their 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 $P\Rightarrow Q$, where $P$ is known to be false.

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

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

Vacuous truth most commonly appears 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, if $P$ is false, $P\Rightarrow Q$ will yield vacuous truth in any logic that uses the material conditional; if $P$ is a necessary falsehood, then it will also yield vacuous truth under the strict conditional.

Other non-classical logics (for example, relevance logic) may attempt to avoid vacuous truths by using alternative conditionals (for example, the counterfactual conditional).

Examples

These examples, one from mathematics and one from natural language, illustrate the concept:

"For any integer x, if x > 5 then x > 3." – This statement is true non-vacuously (since some integers are greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 then 2 > 3".

"All my children are cats" is a vacuous truth when spoken by someone without children.

See also

• De Morgan's Laws – specifically the law that a universal statement is true just in case no counterexample exists: $\forall x\,P(x)\equiv \neg \exists x\,\neg P(x)$
• Empty sum and Empty product
• Paradoxes of material implication, especially the Principle of explosion
• Presupposition; Double question
• Tautology (logic) – another type of true statement that also fails to convey any substantive information
• Triviality (mathematics) and Degeneracy (mathematics)

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.
• Beer, Ilan; Ben-David, Shoham; Eisner, Cindy; Rodeh, Yoav (1997). "Efficient Detection of Vacuity in ACTL Formulas". Computer Aided Verification: 9th International Conference, CAV'97 Haifa, Israel, June 22–25, 1997, Proceedings. Lecture Notes in Computer Science. 1254. pp. 279–290. doi:10.1007/3-540-63166-6_28.

External links

• Conditional Assertions: Vacuous truth