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.
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, as would the conjunction of the two: "all cell phones in the room are turned on and all cell phones in the room are turned off", which would otherwise be incoherent and false.
