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=                           Vacuous truth                            =
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                             Introduction
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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 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.


                         Scope of the concept
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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 universally quantified statements:
* \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 most commonly appears 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, if P is false, P \Rightarrow Q will yield vacuous truth in any
logic that uses the material conditional; if P is a necessary
falsehood, then it will also yield vacuous truth under the strict
conditional.

Other non-classical logics (for example, relevance logic) may attempt
to avoid vacuous truths by using alternative conditionals (for
example, the counterfactual conditional).


                               Examples
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These examples, one from mathematics and one from natural language,
illustrate the concept:

"For any integer x, if x > 5 then x > 3." - This statement is
true non-vacuously (since some integers are 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 2 >
5 then 2 > 3".

"All my children are cats" is a vacuous truth when spoken by someone
without children.


                               See also
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* De Morgan's laws - specifically the law that a universal statement
is true just in case no counterexample exists: \forall x \, P(x)
\equiv \neg \exists x \, \neg P(x)
* Empty sum and Empty product
* Paradoxes of material implication, especially the Principle of
explosion
* Presupposition; Double question
* State of affairs (philosophy)
* Tautology (logic) - another type of true statement that also fails
to convey any substantive information
* Triviality (mathematics) and Degeneracy (mathematics)


                             Bibliography
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* Blackburn, Simon (1994). "vacuous," 'The Oxford Dictionary of
Philosophy'. Oxford: Oxford University Press, p. 388.
* David H. Sanford (1999). "implication." 'The Cambridge Dictionary of
Philosophy', 2nd. ed., p. 420.
*


                            External links
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*
[https://web.archive.org/web/20080415192548/http://abstractmath.org/MM/MMConditi
onal.htm#_Toc133208747
Conditional Assertions: Vacuous truth]


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