Vacuous truth

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.