Vacuous truth

related topics
{math, number, function}
{theory, work, human}
{law, state, case}
{woman, child, man}
{build, building, house}
{food, make, wine}
{@card@, make, design}

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true simply because 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 considered true, and vacuously so.

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Uluru is in France, then the Eiffel tower is in Bolivia". Such statements are considered vacuous because the falsity of the antecedent prevents one from using the conditional to infer the consequent. They are true because a material conditional is defined to be true when the antecedent is false (or the conclusion is true).

This notion has relevance in pure mathematics, as well as in any other field which 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 his parents "I ate every vegetable on my plate," when there were no vegetables on the child’s plate to begin with.


Full article ▸

related documents
Stochastic process
Multiplication algorithm
Lie algebra
Support vector machine
Taylor series
Uniform space
Truth table
Subset sum problem
Quadratic equation
General linear group
Relational database
Exponentiation by squaring
Finite set
Imaginary unit
Lp space
P = NP problem
Fermat number
Sorting algorithm
Busy beaver
Halting problem
Control flow
Category theory
Dylan (programming language)
Ackermann function
Mathematical constant
BCH code