In mathematics and logic, a vacuous truth is a conditional or universal statement (specifically a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. An example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia". It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" (alternatively said "for all x in this room, if x is a cellphone then x is turned off") will be true when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true. The conjunction of the two: "all cell phones in the room are turned on and all cell phones in the room are turned off", can only be true vacuously, and it implies "there are no cell phones in the room". Vacuous statements are also used as a rhetorical device for creating verbal irony. A common example is the "Queen of England" retort; For example, "I'm a great swimmer". "If you're a great swimmer, then I'm the Queen of England", using a transparent false conclusion to imply the statement is vacuous and thus the premise is false.
Definitions These 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, a conditional statement that is based on the material conditional, is true when the antecedent ("Tokyo is in Spain" in the example) is false regardless of whether the conclusion or consequent ("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in that way. Examples common to everyday speech include conditional phrases used as idioms of improbability like "when hell freezes over ..." and "when pigs can fly ...", indicating that not before the given (impossible) condition is met will the speaker accept some respective (typically false or absurd) proposition. 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 in the form of a vacuous truth, while logically valid, can nevertheless be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true.
Scope of the concept A statement S {\displaystyle S} is "vacuously true" if it resembles a material conditional statement P ⇒ Q {\displaystyle P\Rightarrow Q} , where the antecedent P {\displaystyle P} is known to be false. Vacuously true statements that can be reduced (with suitable transformations) to this basic form (material conditional) include the following universally quantified statements:
∀ x : P ( x ) ⇒ Q ( x ) {\displaystyle \forall x:P(x)\Rightarrow Q(x)} , where it is the case that ∀ x : ¬ P ( x ) {\displaystyle \forall x:\neg P(x)} .
∀ x ∈ A : Q ( x ) {\displaystyle \forall x\in A:Q(x)} , where the set A {\displaystyle A} is empty. This logical form ∀ x ∈ A : Q ( x ) {\displaystyle \forall x\in A:Q(x)} can be converted to the material conditional form in order to easily identify the antecedent. For the above example S {\displaystyle S} "all cell phones in the room are turned off", it can be formally written as ∀ x ∈ A : Q ( x ) {\displaystyle \forall x\in A:Q(x)} where A {\displaystyle A} is the set of all cell phones in the room and Q ( x ) {\displaystyle Q(x)} is " x {\displaystyle x} is turned off". This can be written to a material conditional statement ∀ x ∈ B : P ( x ) ⇒ Q ( x ) {\displaystyle \forall x\in B:P(x)\Rightarrow Q(x)} where B {\displaystyle B} is the set of all things in the room (including cell phones if they exist in the room), the antecedent P ( x ) {\displaystyle P(x)} is " x {\displaystyle x} is a cell phone", and the consequent Q ( x ) {\displaystyle Q(x)} is " x {\displaystyle x} is turned off".
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