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\noindent\textbf{Rule of Universal Specification (US) (pg. 59):} If a formula $S$ results from a formula $R$ by substituting a term $t$ for every free occurrence of a  variable $v$ in $R$ then $S$ is derivable from $(\forall v)R$.  \\
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\textbf{Def (flagged):} A variable that is a free variable in a premise is called flagged. They are denoted by listing them when they are used in the right column (as in the deduction on pg. 61) \\
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\textbf{Rule of Universal Generalization (UG) (pg. 60)}: From a formula $S$ we may derive $(\forall v)(S)$, provided the variable $v$ is not flagged in $S$.\\
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\textbf{Def (ambiguous name):} An ``ambiguous name" (which is unfortunately not really \textit{defined} in the book... but its nature is discussed through pages 80-85) is a ``constant" and is \textbf{\textit{not}} a ``variable". We will use Greek letters $\alpha, \beta, \gamma, \delta, \epsilon$, etc to denote ambiguous names. \\
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\noindent\textbf{Rule of Existential Specification (ES) (pg. 83):} If a formula $S$ results from a formula $R$ by substituting for every occurrence of a variable $v$ in $R$ an ambiguous name which has not previously been used in the derivation, then $S$ is derivable from $(\exists v)R$. \\
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\textbf{Rule of Existential Generalization (EG) (pg. 83):} If a formula $S$ results from a formula $R$ by substituting a variable $v$ for every occurrence in $R$ of some ambiguous name, then $(\exists v) S$ is derivable from $R$. \\
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A question was raised in class about the use of rule E.G. Allow me to describe by first writing part of a deduction: \\
\begin{center}\begin{tabular}{|l|l|l|}
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$\{1\}$&(1) $(\exists x)(Px)$ & Premise \\
$\{1\}$&(2) $P\alpha$ & 1 ES  \\
\hline
\end{tabular}\end{center}
The question was whether or not the following would be a valid use of rule UG following the above:
\begin{center}\begin{tabular}{|l|l|l|}
\hline
$\{1\}$ & (3) $(\forall x)(Px)$ & 2 UG \\
\hline
\end{tabular}\end{center}
because clearly $\alpha$ is not a ``flagged variable" and so it appears that UG could apply.  \\
\hrule
In fact it is not valid since the ambiguous name $\alpha$ is not a variable, which UG would require. Moreover if we had a premise in the argument with ambiguous name $\beta$ of the form 
\begin{center}
\begin{tabular}{|l|l|l|}
\hline 
$\{4\}$ & (4) $R\beta$ & Premise \\
\hline
\end{tabular}
\end{center}
we would not be listing $\beta$ as a ``flagged variable", again because it is not a variable! We must make this distinction because of the nature of our quantifiers $\forall$ and $\exists$: \\
\begin{itemize}
\item $\forall$ is the ``universal quantifier" because it says ``all objects in the universe have the property that..."
\item $\exists$ is the ``existential quantifier" because all that it says ``there is at least one object in the universe such that..."
\end{itemize}
What happened above in line (3) above was that we converted the formula in line (1) involving an existential quantifier into a formula involving the universal quantifier. If this were possible, we would be entirely erasing the distinction between the two quantifiers!
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