In mathematics and logic, a vacuous truth asserts a property on all of the elements of the empty set.
An example of a vacuous truth is, “All of the elephants in the kitchen speak English.” This statement is true whenever there are no elephants in the kitchen. Although the state of no elephants in the kitchen tells us nothing of the truth value of what language they would speak if there were.
A more formal mathematical definition is that a vacuous truth is a material conditional, ie., implication, in which the antecedent is always false. In propositional logic this is:
p → q, p ≡ false
The definition of an implication states that if the antecedent is false then the consequent is always true regardless of whether or not the the conclusion of the implication is true.
The example above can be rewritten “If there are elephants in the kitchen (proposition p) then the elephants speak English (proposition q).” While the implication p →q is true since p is false (there are no elephants in the kitchen), it is vacuous since the antecedent hasn’t provided any evidence to the value of the consequent (the language elephants in the kitchen would speak).
Statements in predicate, or first order logic, that can be reduced to the form p → q, p ≡ false are vacuous as well:
- ∀x: P(x) → Q(x) where ∀x: ¬P(x)
- ∀x: A → Q(x) where A is the empty set
- ∀ε: Q(ε) where ε is a type with no representatives