Vacuous truth

In mathematics and logic, a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied. For example, the statement "all cell phones in the room are turned off" will be true even if 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". For that reason, it is sometimes said that a statement is vacuously true only because it does not really say anything.
More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if London is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. In essence, they are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true or false).
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 that 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 their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In addition, vacuous truth is often used colloquially with absurd statements about the speaker, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle"), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the Queen of England").
Scope of the concept.
A statement formula_1 is "vacuously true" if it resembles the statement formula_2, where formula_3 is known to be false.
Statements that can be reduced (with suitable transformations) to this basic form include the following universally quantified statements:
Vacuous truth most commonly appears in classical logic with two truth values. However, vacuous truth can also appear in, for example, intuitionistic logic, in the same situations as given above. Indeed, if formula_3 is false, then formula_2 will yield vacuous truth in any logic that uses the material conditional; if formula_3 is a necessary falsehood, then it will also yield vacuous truth under the strict conditional.
Other non-classical logics, such as relevance logic, may attempt to avoid vacuous truths by using alternative conditionals (such as the case of the counterfactual conditional).
Examples.
These examples, one from mathematics and one from natural language, illustrate the concept: