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Existential quantifier elimination

Oftentimes when we know a statement of the form \bgroup\color{red}$\exists \; x \, . \, F$\egroup, we reason as follows: ``We know that there is something for which \bgroup\color{red}$F$\egroup holds. Let \bgroup\color{red}$w$\egroup be a name for that something, so that \bgroup\color{red}$F$\egroup holds for \bgroup\color{red}$w$\egroup. Then \bgroup\color{red}$\cdots$\egroup'' where \bgroup\color{red}$\cdots$\egroup is a derivation of some conclusion \bgroup\color{red}$G$\egroup, presumably with the help of the assumption that \bgroup\color{red}$F$\egroup holds for \bgroup\color{red}$w$\egroup, i.e., the premise \bgroup\color{red}$F[x \mapsto w]$\egroup. It is customary to refer to \bgroup\color{red}$w$\egroup as a witness, and to the formula \bgroup\color{red}$F[x \mapsto w]$\egroup as the witness premise. No special assumptions about the witness \bgroup\color{red}$w$\egroup must be made. It must serve only as a ``dummy''--any other name should do just as well. In particular, the witness should not occur in the final conclusion \bgroup\color{red}$G$\egroup. This form of reasoning is captured in NDL with deductions of the form

\begin{displaymath}\bgroup\color{red}\mbox{\tt pick-witness } w \,\mbox{\tt for (exists $x$ }F\mbox{\tt )} \;\; D\egroup\end{displaymath}

Their semantics are as follows:
\bgroup\color{red}$\;\;$\egroup \bgroup\color{red}$\mbox{$\beta$}\cup \{F[x\mapsto v]\} \mbox{$\:\vdash\:$}\{w \mapsto v\}D \mbox{$\:\longrightarrow\;$}G$\egroup \bgroup\color{red}$\;$\egroup
\bgroup\color{red}$\;\:$\egroup
\bgroup\color{red}$\;\;$\egroup
\bgroup\color{red}$\mbox{$\beta$}\mbox{$\:\vdash\:$}\mbox{\tt pick-witness } w \...
...r (exists $x$} \;\: F\mbox{\tt )} \;\;
D \mbox{$\:\longrightarrow\;$}G$\egroup
provided \bgroup\color{red}$v$\egroup does not occur in \bgroup\color{red}$\beta$\egroup, in \bgroup\color{red}$D$\egroup, or in \bgroup\color{red}$G$\egroup.
\bgroup\color{red}$\;$\egroup
Here \bgroup\color{red}$v$\egroup is a freshly generated variable, which helps to ensure that the witness is used only as a dummy.
next up previous contents
Next: Examples Up: Quantifier reasoning Previous: Existential quantifier introduction   Contents
2004-08-06