Vacuous truth (nonfiction)

Vacuous truth.

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 "if Uluru is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. They are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true).

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 which uses classical logic.

Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his or her parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with.

Nonfiction cross-reference

• Logic (nonfiction)
• Mathematics (nonfiction)
• Principle of explosion (nonfiction)

External links:

• Vacuous truth @ Wikipedia