Vacuous truth

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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" may be true simply because 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 true, and vacuously so, 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 Ayers Rock is in France, then the Eiffel Tower is in Bolivia".[note 1] 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.[1] This notion has relevance 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. (More accurate although awkward: "I didn't leave any vegetables on my plate (i.e. 'uneaten')" — vs. the implied assertion ("I ate x") of a non-event.)

Scope of the concept[edit]

A statement S is "vacuously true" if it resembles the statement P \Rightarrow Q, where P is known to be false.

Statements that can be reduced (with suitable transformations) to this basic form include the following:

  • \forall x: P(x) \Rightarrow Q(x), where it is the case that \forall x: \neg P(x).
  • \forall x \in A: Q(x), where the set A is empty.
  • \forall \xi: Q(\xi), where the symbol \xi is restricted to a type that has no representatives.

Vacuous truth is usually applied in classical logic, which in particular is two-valued. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.

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Notes[edit]

  1. ^ Such statements can also be used sarcastically, as in the line "If she's a lady, then I'm a Vermicious Knid!" spoken by the character Grandpa Joe from the film Willy Wonka & the Chocolate Factory.

References[edit]

  1. ^ Baldwin, Douglas L.; Scragg, Greg W. (2011), Algorithms and Data Structures: The Science of Computing, Cengage Learning, p. 261, ISBN 9781285225128 .

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