Suppose X is a set and P is a property defined as follows:

X has property P if and only if
Y[ Y satisfies condition 1] Y satisfies condition 2

where condition 1 and condition 2 define the property. If condition 1 is never satisfied then X satisfies property P vacuously.


  1. 1.

    If X is the set {1,2,3} and P is the property defined as above with condition 1= Y is a infinite subset of X, and condition 2= Y contains 7. Then X has property P vacously; every infinite subset of {1,2,3} contains the number 7 [1].

  2. 2.

    The empty setMathworldPlanetmath is a Hausdorff space (vacuously).

  3. 3.

    Suppose property P is defined by the statement :
    The present King of France does not exist.
    Then either of the following propositionsPlanetmathPlanetmathPlanetmath is satisfied vacuously.
    The present king of France is bald.
    The present King of France is not bald.


  • 1 Wikipedia on Vacuous truth.
Title vacuous
Canonical name Vacuous
Date of creation 2013-03-22 14:42:27
Last modified on 2013-03-22 14:42:27
Owner matte (1858)
Last modified by matte (1858)
Numerical id 9
Author matte (1858)
Entry type Definition
Classification msc 00A20
Synonym vacuously
Synonym vacuously true
Synonym vacuous truth