Vacuous truth

Inmathematicsandlogic,avacuous truthis aconditionaloruniversalstatement(a universal statement that can be converted to a conditional statement) that is true because theantecedentcannot besatisfied. It is sometimes said that a statement is vacuously true because it does not really say anything.For example, the statement "all cell phones in the room are turned off" will betruewhen no cell phones are present in the room. In this case, the statement "all cell phones in the room are turnedon"would also be vacuously true, as would theconjunctionof the two: "all cell phones in the room are turned onandturned off ", which would otherwise be incoherent and false.

More formally, a relativelywell-definedusage refers to a conditional statement (or a universal conditional statement) with a falseantecedent.One example of such a statement is "if Tokyo is in Spain, 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 theconsequent.In essence, a conditional statement, that is based on thematerial conditional,is true when the antecedent ( "Tokyo is in Spain" in the example) is false regardless of whether the conclusion orconsequent( "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 asidioms of improbabilitylike "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.

Inpure mathematics,vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs bymathematical induction.This notion has relevance inpure mathematics,as well as in any other field that usesclassical logic.

Outside of mathematics, statements in the form of a vacuous truth, while logically valid, can nevertheless be misleading. Such statements make reasonable assertions aboutqualifiedobjects whichdo 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 itresemblesamaterial conditionalstatement$P\Rightarrow Q$,where theantecedent$P$is known to be false.

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

• $\forall x:P(x)\Rightarrow Q(x)$,where it is the case that$\forall x:\neg P(x)$.
• $\forall x\in A:Q(x)$,where theset$A$isempty.
  • This logical form$\forall x\in A:Q(x)$can be converted to the material conditional form in order to easily identify theantecedent.For the above example$S$"all cell phones in the room are turned off", it can be formally written as$\forall x\in A:Q(x)$where$A$is the set of all cell phones in the room and$Q(x)$is "$x$is turned off ". This can be written to a material conditional statement$\forall x\in B:P(x)\Rightarrow Q(x)$where$B$is the set of all things in the room (including cell phones if they exist in the room), the antecedent$P(x)$is "$x$is a cell phone ", and theconsequent$Q(x)$is "$x$is turned off ".
• $\forall \xi:Q(\xi )$,where the symbol$\xi$is restricted to atypethat has no representatives.

Vacuous truths most commonly appear inclassical logicwithtwo truth values.However, vacuous truths can also appear in, for example,intuitionistic logic,in the same situations as given above. Indeed, if$P$is false, then$P\Rightarrow Q$will yield a vacuous truth in any logic that uses thematerial conditional;if$P$is anecessary falsehood,then it will also yield a vacuous truth under thestrict conditional.

Other non-classical logics, such asrelevance logic,may attempt to avoid vacuous truths by using alternative conditionals (such as the case of thecounterfactual 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:

• InJavaScript,thearraymethodeveryexecutes 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 theeverymethod on an empty array will return true for any condition.
• InPython,theallfunction returnsTrueif all of the elements of the given iterable areTrue.The function also returnsTruewhen given an iterable of zero length.
• InRust,theIterator::allfunction accepts an iterator and a predicate and returnstrueonly when the predicate returnstruefor all items produced by the iterator, or if the iterator produces no items.

Examples

These examples, one frommathematicsand one fromnatural language,illustrate the concept of vacuous truths:

• "For any integerx,ifx> 5thenx> 3."– This statement istruenon-vacuously (since someintegersare indeed greater than 5), but some of its implications are only vacuously true: for example, whenxis the integer 2, the statement implies the vacuous truth that "if2 > 5then2 > 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 also

• Definite description
• 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 sumandempty product
• Empty function
• Paradoxes of material implication,especially theprinciple 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)anddegeneracy (mathematics)

Bibliography

• 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.
• Beer, Ilan; Ben-David, Shoham; Eisner, Cindy; Rodeh, Yoav (1997). "Efficient Detection of Vacuity in ACTL Formulas".Computer Aided Verification: 9th International Conference, CAV'97 Haifa, Israel, June 22–25, 1997, Proceedings.Lecture Notes in Computer Science.Vol. 1254. pp. 279–290.doi:10.1007/3-540-63166-6_28.ISBN978-3-540-63166-8.

External links

• Conditional Assertions: Vacuous truth