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\textheight=8.0in
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\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\I{\mathop{\cal I}\nolimits}
\def\A{\mathop{\cal A}\nolimits}
\def\B{\mathop{\cal B}\nolimits}
\def\N{\mathop{\cal N}\nolimits}
\def\F{\mathop{\cal F}\nolimits}
\def\G{\mathop{\cal G}\nolimits}
\def\La{\mathop{\cal L}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\gra{\mathop{\alpha}\nolimits}     
\def\grS{\mathop{\Sigma}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\implies{\mathop{\to}\nolimits}     
\def\deduces{\mathop{\vdash}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


\begin{document}
\begin{center} {\bf The Deductive Calculus}\\
Math 609  \end{center}
\bigskip\bigskip


\noindent{\bf RULES OF INFERENCE.} \bigskip
\begin{description}
\item[Modus Ponens.] From $\grf$ and $\grf \implies \psi$ infer $\psi.$
\end{description}
\bigskip
\bigskip
\noindent{\bf LOGICAL AXIOMS.} \bigskip

\noindent If $\grf$ is a logical axiom, then $\forall x \grf$ is also a logical axiom.

\begin{enumerate}
\item Every tautology is a logical axiom.
\item $\forall x \grf (x) \implies \grf (t)$ where $t$ is substitutible
for $x$ in $\grf.$ 
\item $\forall x (\grf \implies \psi) \implies (\forall x \grf \implies
\forall x \psi)$ for all formulas $\grf$ and $\psi.$
\item $\grf \implies \forall x \grf$ where $x$ does not occur freely in $\grf.$
\item $x \doteq x.$
\item $x \doteq y \implies (\gra \implies \gra')$ where $\gra$ is atomic
and $\gra'$ is obtained \\ by replacing some occurences of $x$ in $\gra$ by
$y.$
\end{enumerate}


\end{document}