Vacuous truth

In mathematics and logic, a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied. For example, the statement "all cell phones in the room are turned off" will be true even if 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". For that reason, it is sometimes said that a statement is vacuously true only because it does not really say anything.

More formally, a relatively well-defined usage refers to a conditional statement (or a universal 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 truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. In essence, they are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true or false).

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 with two truth values. However, vacuous truth can also appear in, for example, intuitionistic logic, in the same situations as given above. Indeed, if $P$ is false, then $P\Rightarrow Q$ 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, such as relevance logic, may attempt to avoid vacuous truths, by using alternative conditionals (such as the case of 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 indeed 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. Similarly, "None of my children are cats" would also be a vacuous truth, when spoken by the same person.

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
• State of affairs (philosophy)
• 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. Vol. 1254. pp. 279–290. doi:10.1007/3-540-63166-6_28. ISBN 978-3-540-63166-8.

External links

• Conditional Assertions: Vacuous truth