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Inmathematics andlogic, avacuous truth is aconditional oruniversalstatement (specifically a universal statement that can be converted to a conditional statement) that is true because theantecedent cannot besatisfied.^[1] It is sometimes said that a statement is vacuously true because it does not really say anything.^[2] For example, the statement "all cell phones in the room are turned off" (alternatively said "for all x in this room,if x is a cellphone then x is turned off") will betrue when 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 theconjunction of the two: "all cell phones in the room are turned onand all cell phones in the room are turned off", which would otherwise be incoherent and false.

More formally, a relativelywell-defined usage refers to a conditional statement (or a universal conditional statement) with a falseantecedent.^[1][3][2][4] One example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia".

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 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 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.

Inpure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs bymathematical induction.^[5] 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 aboutqualified objects 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.

A statement${\displaystyle S}$ is "vacuously true" if itresembles amaterial conditional statement${\displaystyle P\Rightarrow Q}$, where theantecedent${\displaystyle P}$ is known to be false.^[1][3][2]

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

• ${\displaystyle \mathrm{\forall }x:P(x)\Rightarrow Q(x)}$, where it is the case that${\displaystyle \mathrm{\forall }x:\mathrm{\neg }P(x)}$.^[4]
• ${\displaystyle \mathrm{\forall }x\in A:Q(x)}$, where theset${\displaystyle A}$ isempty.
  • This logical form${\displaystyle \mathrm{\forall }x\in A:Q(x)}$ can be converted to the material conditional form in order to easily identify theantecedent. For the above example${\displaystyle S}$ "all cell phones in the room are turned off", it can be formally written as${\displaystyle \mathrm{\forall }x\in A:Q(x)}$ where${\displaystyle A}$ is the set of all cell phones in the room and${\displaystyle Q(x)}$ is "${\displaystyle x}$ is turned off". This can be written to a material conditional statement${\displaystyle \mathrm{\forall }x\in B:P(x)\Rightarrow Q(x)}$ where${\displaystyle B}$ is the set of all things in the room (including cell phones if they exist in the room), the antecedent${\displaystyle P(x)}$ is "${\displaystyle x}$ is a cell phone", and theconsequent${\displaystyle Q(x)}$ is "${\displaystyle x}$ is turned off".
• ${\displaystyle \mathrm{\forall }\xi :Q(\xi )}$, where the symbol${\displaystyle \xi }$ is restricted to atype that has no representatives.

Vacuous truths most commonly appear inclassical logic withtwo truth values. However, vacuous truths can also appear in, for example,intuitionistic logic, in the same situations as given above. Indeed, if${\displaystyle P}$ is false, then${\displaystyle P\Rightarrow Q}$ will yield a vacuous truth in any logic that uses thematerial conditional;^[6] if${\displaystyle P}$ 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).

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, thearray methodevery 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 theevery method on an empty array will return true for any condition.^[7]
• InPython, the built-inall() function returnsTrue only when all of the elements of an iterable (in this example, a list) areTrue or the iterable is empty:all([1,1])==True; all([1,1,0])==False; all([])==True.^[8] A less ambiguous way to express this is to sayall() returns True when none of the elements areFalse.
• InRust, theIterator::all function accepts an iterator and a predicate and returnstrue only when the predicate returnstrue for all items produced by the iterator, or if the iterator produces no items.^[9]
• In SQL, the function, the functionANY_VALUE can differ depending on the RDBMS's behaviour relatingNULLs to vacuous truth. Some RDBMS might returnnull even if there are non-null values.^[10] Some DBMS might not allow for its use infilter(...) orover(.. ) clauses.
• InKotlin, the collection methodall returnstrue when the collection is empty.
• InC#, the Linq methodAll returnstrue when the collection is empty.
• InC++, thestd::all_of function template returnstrue for an empty collection.^[11]
• InAgda, an empty type (for example,⊥, which is defined with no constructors) is 'false' at the type level, following theCurry–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 ofex falso quodlibet can be defined this way as a functionefq : ∀ {n} {a : Set n} → ⊥ → a. The functionefq is then a proof of the vacuously true proposition⊥ → a for every proposition (i.e. type)a. For example, it is a proof of⊥ → ⊥.

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

• "For any integerx, ifx > 5 thenx > 3."^[12] – This statement istrue non-vacuously (since someintegers are indeed greater than 5), but some of its implications are only vacuously true: for example, whenx is the integer 2, the statement implies the vacuous truth that "if2 > 5 then2 > 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.

• Definite description
• De Morgan's laws – specifically the law that a universal statement is true just in case no counterexample exists:${\displaystyle \mathrm{\forall }xP(x)\equiv \mathrm{\neg }\mathrm{\exists }x\mathrm{\neg }P(x)}$
• Empty sum andempty 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)

• Mathematical logic
• Informal fallacies
• Logical truth

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