Elimination of all but 48 Categorical Syllogisms as Invalid

== Theorem ==
Of the $256$ different types of categorical syllogism, all but $48$ can immediately be identified as invalid by consideration of the Rules of Quantity and the Rules of Quality.
== Proof ==
There are $64$ patterns of categorical syllogism per figure:
:$\begin{array}{cccc}
AAA & AAE & AAI & AAO \\
AEA & AEE & AEI & AEO \\
AIA & AIE & AII & AIO \\
AOA & AOE & AOI & AOO \\
\end{array} \qquad
\begin{array}{cccc}
EAA & EAE & EAI & EAO \\
EEA & EEE & EEI & EEO \\
EIA & EIE & EII & EIO \\
EOA & EOE & EOI & EOO \\
\end{array}$
:$\begin{array}{cccc}
IAA & IAE & IAI & IAO \\
IEA & IEE & IEI & IEO \\
IIA & IIE & III & IIO \\
IOA & IOE & IOI & IOO \\
\end{array} \qquad
\begin{array}{cccc}
OAA & OAE & OAI & OAO \\
OEA & OEE & OEI & OEO \\
OIA & OIE & OII & OIO \\
OOA & OOE & OOI & OOO \\
\end{array}$
From No Valid Categorical Syllogism contains two Negative Premises, all those whose patterns start $EE$, $EO$, $OE$ and $OO$ can be eliminated:
:$\begin{array}{cccc}
AAA & AAE & AAI & AAO \\
AEA & AEE & AEI & AEO \\
AIA & AIE & AII & AIO \\
AOA & AOE & AOI & AOO \\
\end{array} \qquad
\begin{array}{cccc}
EAA & EAE & EAI & EAO \\
 & & & \\
EIA & EIE & EII & EIO \\
 & & & \\
\end{array}$
:$\begin{array}{cccc}
IAA & IAE & IAI & IAO \\
IEA & IEE & IEI & IEO \\
IIA & IIE & III & IIO \\
IOA & IOE & IOI & IOO \\
\end{array} \qquad
\begin{array}{cccc}
OAA & OAE & OAI & OAO \\
 & & & \\
OIA & OIE & OII & OIO \\
 & & & \\
\end{array}$
From No Valid Categorical Syllogism contains two Particular Premises, all those whose patterns start $II$, $IO$ and $OI$ and $OO$ can be eliminated:
:$\begin{array}{cccc}
AAA & AAE & AAI & AAO \\
AEA & AEE & AEI & AEO \\
AIA & AIE & AII & AIO \\
AOA & AOE & AOI & AOO \\
\end{array} \qquad
\begin{array}{cccc}
EAA & EAE & EAI & EAO \\
 & & & \\
EIA & EIE & EII & EIO \\
\end{array}$
:$\begin{array}{cccc}
IAA & IAE & IAI & IAO \\
IEA & IEE & IEI & IEO \\
\end{array} \qquad
\begin{array}{cccc}
OAA & OAE & OAI & OAO \\
\end{array}$
From Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative, further patterns can be eliminated:
:$\begin{array}{cccc}
AAA & & AAI & \\
 & AEE & & AEO \\
AIA & & AII & \\
 & AOE & & AOO \\
\end{array} \qquad
\begin{array}{cccc}
 & EAE & & EAO \\
 & & & \\
 & EIE & & EIO \\
\end{array}$
:$\begin{array}{cccc}
IAA & & IAI & \\
 & IEE & & IEO \\
\end{array} \qquad
\begin{array}{cccc}
 & OAE & & OAO \\
\end{array}$
From No Valid Categorical Syllogism with Particular Premise has Universal Conclusion, all those whose patterns match that condition can be eliminated:
:$\begin{array}{cccc}
AAA & & AAI & \\
 & AEE & & AEO \\
 & & AII & \\
 & & & AOO \\
\end{array} \qquad
\begin{array}{cccc}
 & EAE & & EAO \\
 & & & \\
 & & & EIO \\
\end{array}$
:$\begin{array}{cccc}
 & & IAI & \\
 & & & IEO \\
\end{array} \qquad
\begin{array}{cccc}
 & & & OAO \\
\end{array}$
Thus there are $12$ patterns remaining.
Each one may apply to any one of the $4$ figures
Thus there are no more than $48$ valid patterns of categorical syllogism.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercise $\text{(c)}$
Category:Categorical Syllogisms