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Universal quantifier elimination

If we know that something is true of everything, then we should be able to conclude that it is true of a particular individual \bgroup\color{red}$t$\egroup. This is called universal elimination because we go from statements of the form \bgroup\color{red}$\forall \; x \, . \, F$\egroup to \bgroup\color{red}$F[x\mapsto t]$\egroup, thereby removing the universal quantifier. In NDL this is performed by specialize, whose semantics are as follows:

\begin{displaymath}\bgroup\color{red}\mbox{$\beta$}\cup \{\mbox{\tt (forall x }\...
...x{\tt with } t \mbox{$\:\longrightarrow\;$}F[x\mapsto t]\egroup\end{displaymath}

It is an error if the universal quantification \bgroup\color{red}$\mbox{\tt (forall x }F\mbox{\tt )}$\egroup is not in the assumption base. As an example, if \bgroup\color{red}$\beta$\egroup contains (forall x male(father(x))), then

\begin{displaymath}\bgroup\color{red}\mbox{\tt specialize (forall x male(father(x))) with ann}\egroup\end{displaymath}

will produce the conclusion male(father(ann)).

2004-08-06