In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property. For example, the statement "all cell phones in the room are turned off" will be true whenever 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".
More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is...