Vacuous truth

A statement ${\displaystyle S}$ is "vacuously true", if it resembles the statement ${\displaystyle P\Rightarrow Q}$, where ${\displaystyle P}$ is known to be false.[2][4][3]

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

• ${\displaystyle \forall x:P(x)\Rightarrow Q(x)}$, where it is the case that ${\displaystyle \forall x:\neg P(x)}$.[5]
• ${\displaystyle \forall x\in A:Q(x)}$, where the set ${\displaystyle A}$ is empty.
• ${\displaystyle \forall \xi :Q(\xi )}$, where the symbol ${\displaystyle \xi }$ 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 ${\displaystyle P}$ is false, then ${\displaystyle P\Rightarrow Q}$ will yield vacuous truth in any logic that uses the material conditional; if ${\displaystyle P}$ 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).

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

• "For any integer x, if x > 5 then x > 3."[7] – 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.

• De Morgan's laws – specifically the law that a universal statement is true just in case no counterexample exists: ${\displaystyle \forall x\,P(x)\equiv \neg \exists x\,\neg P(x)}$
• 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)

• 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. ISBN 978-3-540-63166-8.

• Conditional Assertions: Vacuous truth