Vacuous truth

In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied.[1][2] For example, the statement "all cell phones in the room are turned off" will be true when 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 because it does not really say anything.[3]

More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent.[1][2][4][3][5] 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, a conditional statement, that is based on the material conditional, is true when the antecedent ("London is in France" 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 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.[6][1] 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 this case, the parent can believe that the child has actually eaten all the vegetable on this plate (misleading) but this is not a fact. In addition, a vacuous truth is often used colloquially with absurd statements, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle" to strongly claim that the dog was red), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the Queen of England" to disagree a previously made statement).

Scope of the conceptEdit

A statement   is "vacuously true" if it resembles a material conditional statement  , where the antecedent   is known to be false.[2][4][3]

Vacuously true statements that can be reduced (with suitable transformations) to this basic form (material conditional) include the following universally quantified statements:

  •  , where it is the case that  .[5]
  •  , where the set   is empty.
    • This logical form   can be converted to the material conditional form in order to easily identify the antecedent. For the above example   "all cell phones in the room are turned off", it can be formally written as   where   is the set of all cell phones in the room and   is "  is turned off". This can be written to a material conditional statement   where   is the set of all things in the room (including cell phones if they exist in the room), the antecedent   is "  is a cell phone", and the consequent   is "  is turned off".
  •  , where the symbol   is restricted to a type that has no representatives.

Vacuous truths most commonly appear in classical logic with two truth values. However, vacuous truths can also appear in, for example, intuitionistic logic, in the same situations as given above. Indeed, if   is false, then   will yield a vacuous truth in any logic that uses the material conditional; if   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 programmingEdit

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.[7]

ExamplesEdit

These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths:

  • "For any integer x, if x > 5 then x > 3."[8] – 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 2 > 5 then 2 > 3".
  • "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 alsoEdit

ReferencesEdit

  1. ^ a b c "The Definitive Glossary of Higher Mathematical Jargon — Vacuously true". Math Vault. 2019-08-01. Retrieved 2019-12-15.
  2. ^ a b c "Vacuously true". web.cse.ohio-state.edu. Retrieved 2019-12-15.
  3. ^ a b c "Vacuously true - CS2800 wiki". courses.cs.cornell.edu. Retrieved 2019-12-15.
  4. ^ a b "Definition:Vacuous Truth - ProofWiki". proofwiki.org. Retrieved 2019-12-15.
  5. ^ a b Edwards, C. H. (January 18, 1998). "Vacuously True" (PDF). swarthmore.edu. Retrieved 2019-12-14.
  6. ^ Baldwin, Douglas L.; Scragg, Greg W. (2011), Algorithms and Data Structures: The Science of Computing, Cengage Learning, p. 261, ISBN 978-1-285-22512-8.
  7. ^ Mozilla Developer Network: Array.prototype.every()
  8. ^ "What precisely is a vacuous truth?".

BibliographyEdit

External linksEdit