Editing Vacuous truth (section)

== Scope of the concept ==

A statement <math>S</math> is "vacuously true" if it [[Logical form|resembles]] a [[material conditional]] statement <math>P \Rightarrow Q</math>, where the [[Antecedent (logic)|antecedent]] <math>P</math> is known to be false.<ref name=":1" /><ref name=":3" /><ref name=":2" />

Vacuously true statements that can be reduced ([[Mutatis mutandis|with suitable transformations]]) to this basic form (material conditional) include the following [[Universal quantifier|universally quantified]] statements:
* <math>\forall x: P(x) \Rightarrow Q(x)</math>, where it is the case that <math>\forall x: \neg P(x)</math>.<ref name=":4" />
* <math>\forall x \in A: Q(x)</math>, where the [[Set (mathematics)|set]] <math>A</math> is [[empty set|empty]].
** This logical form <math>\forall x \in A: Q(x)</math> can be converted to the material conditional form in order to easily identify the [[Antecedent (logic)|antecedent]]. For the above example <math>S</math> "all cell phones in the room are turned off", it can be formally written as <math>\forall x \in A: Q(x)</math> where <math>A</math> is the set of all cell phones in the room and <math>Q(x)</math> is "<math>x</math> is turned off". This can be written to a material conditional statement <math>\forall x \in B: P(x) \Rightarrow Q(x)</math> where <math>B</math> is the set of all things in the room (including cell phones if they exist in the room), the antecedent <math>P(x)</math> is "<math>x</math> is a cell phone", and the [[consequent]] <math>Q(x)</math> is "<math>x</math> is turned off".
* <math>\forall \xi: Q(\xi)</math>, where the symbol <math>\xi</math> is restricted to a [[type (type theory)|type]] that has no representatives.

Vacuous truths most commonly appear in [[classical logic]] with [[Bivalent logic|two truth values]]. However, vacuous truths can also appear in, for example, [[intuitionistic logic]], in the same situations as given above. Indeed, if <math>P</math> is false, then <math>P \Rightarrow Q</math> will yield a vacuous truth in any logic that uses the [[material conditional]];<ref>[[Ingebrigt Johansson|Johansson's]] [[minimal logic]] is an exception, because the proof needs the [[principle of explosion]].</ref> if <math>P</math> is a [[Contradiction|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]]).