Vacuous truth

These 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, a conditional statement that is based on the material conditional, is true when the antecedent ("Tokyo is in Spain" in the example) is false regardless of whether the conclusion or consequent ("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in that way. Examples common to everyday speech include conditional phrases used as idioms of improbability like "when hell freezes over ..." and "when pigs can fly ...", indicating that not before the given (impossible) condition is met will the speaker accept some respective (typically false or absurd) proposition. 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 in the form of a vacuous truth, while logically valid, can nevertheless be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true. == Scope of the concept ==

A statement S is "vacuously true" if it resembles a material conditional statement P \Rightarrow Q, where the antecedent P is known to be false. if P is a necessary falsehood, then it will also yield a 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). == In computer programming ==

Many programming environments have a mechanism for querying if every item in a collection of items satisfies some predicate. It is common for such a query to always evaluate as true for an empty collection. For example: • In JavaScript, the array method every executes a provided callback function once for each element present in the array, only stopping (if and when) it finds an element where the callback function returns false. Notably, calling the every method on an empty array will return true for any condition. • In Python, the built-in all() function returns True only when all of the elements of an iterable (in this example, a list) are True or the iterable is empty: all([1,1])==True; all([1,1,0])==False; all([])==True. A less ambiguous way to express this is to say all() returns True when none of the elements are False. • In Rust, the Iterator::all function accepts an iterator and a predicate and returns true only when the predicate returns true for all items produced by the iterator, or if the iterator produces no items. • In SQL, the function, the function ANY_VALUE can differ depending on the RDBMS's behaviour relating NULLs to vacuous truth. Some RDBMS might return null even if there are non-null values. Some DBMS might not allow for its use in filter(...) or over(.. ) clauses. • In Kotlin, the collection method all returns true when the collection is empty. • In C#, the Linq method All returns true when the collection is empty. • In C++, the std::all_of function template returns true for an empty collection. • In Agda, an empty type (for example, ⊥, which is defined with no constructors) is 'false' at the type level, following the Curry–Howard correspondence. A parameter of such a type can be matched against an 'absurd' pattern and an equation containing such a pattern has no right hand side. The principle of ex falso quodlibet can be defined this way as a function efq : ∀ {n} {a : Set n} → ⊥ → a. The function efq is then a proof of the vacuously true proposition ⊥ → a for every proposition (i.e. type) a. For example, it is a proof of ⊥ → ⊥. == Examples ==

These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: • "For any integer x, if then ." – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if then ". • "All my children are goats" is a vacuous truth when spoken by someone without children. Similarly, "None of my children is a goat" would also be a vacuous truth when spoken by the same person. == See also ==