vacuous
Vacuous2
2004-10-09 12:21:10
2008-02-23 19:39:07
Definition
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Suppose $X$ is a set and $P$ is a property defined as follows:
\begin{eqnarray*}
& & \mbox{$X$ has property $P$ if and only if} \\
& & \mbox{$\forall Y[$ $Y$ satisfies condition $1] \Rightarrow$ $Y$ satisfies condition $2$ }
\end{eqnarray*}
where condition $1$ and condition $2$ define the property.
If condition $1$ is never satisfied then $X$ satisfies property $P$
\emph{vacuously}.
\subsubsection*{Examples}
\begin{enumerate}
\item 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 $7$ \cite{wiki}.
\item The empty set is a Hausdorff space (vacuously).
\item Suppose property $P$ is defined by the statement\,: \\
\emph{The present King of France does not exist.}\\
Then either of the following propositions is satisfied vacuously. \\
\emph{The present king of France is bald.}\\
\emph{The present King of France is not bald.}
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{wiki}Wikipedia \PMlinkexternal{entry on Vacuous truth}{http://en.wikipedia.org/wiki/Vacuous_truth}.
\end{thebibliography}