Elimination of all but 24 Categorical Syllogisms as Invalid

== Theorem ==
Of the $256$ different types of categorical syllogism, all but $24$ can be identified as invalid.
These are the $24$ patterns which may still be valid:
:$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} & EAE \\
\text{I} & EIO \\
\text{I} & AAI \\
\text{I} & EAO \\
\end{array}
\qquad
\begin{array}{rl}
\text{II} & EAE \\
\text{II} & AEE \\
\text{II} & AOO \\
\text{II} & EIO \\
\text{II} & EAO \\
\text{II} & AEO \\
\end{array}
\qquad
\begin{array}{rl}
\text{III} & AAI \\
\text{III} & AII \\
\text{III} & IAI \\
\text{III} & EAO \\
\text{III} & EIO \\
\text{III} & OAO \\
\end{array}
\qquad
\begin{array}{rl}
\text{IV} & AAI \\
\text{IV} & AEE \\
\text{IV} & EAO \\
\text{IV} & EIO \\
\text{IV} & IAI \\
\text{IV} & AEO \\
\end{array}$
== Proof ==
From Elimination of all but 48 Categorical Syllogisms as Invalid there are $12$ possible patterns of categorical syllogism per figure:
:$\begin{array}{cccccc}
AAA & AAI & AEE & AEO & AII & AOO \\
EAE & EAO & EIO & IAI & IEO & OAO \\
\end{array}$
=== Figure $\text I$ ===
Consider Figure $\text I$:
From Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise, the patterns:
:$AEE$, $AEO$, $AOO$ and $IEO$
can be eliminated as they all have a negative minor premise, and:
:$IAI$ and $OAO$
can be eliminated as they all have a particular major premise.
Thus the only patterns in Figure $\text I$ that may be valid are:
:$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} & EAE \\
\text{I} & EIO \\
\text{I} & AAI \\
\text{I} & EAO \\
\end{array}$
=== Figure $\text {II}$ ===
Consider Figure $\text {II}$:
From Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise, the patterns:
:$AAA$, $AAI$, $AII$ and $IAI$
can be eliminated as they all have an affirmative conclusion, and:
:$IEO$ and $OAO$
can be eliminated as they all have a particular major premise.
Thus the only patterns in Figure $\text {II}$ that may be valid are:
:$\begin{array}{rl}
\text{II} & EAE \\
\text{II} & AEE \\
\text{II} & AOO \\
\text{II} & EIO \\
\text{II} & EAO \\
\text{II} & AEO \\
\end{array}$
=== Figure $\text {III}$ ===
Consider Figure $\text {III}$:
From Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise, the patterns:
:$AAA$, $AEE$ and $EAE$
can be eliminated as they all have a universal conclusion, and:
:$AEO$, $AOO$ and $IEO$
can be eliminated as they all have a negative minor premise.
Thus the only patterns in Figure $\text {III}$ that may be valid are:
:$\begin{array}{rl}
\text{III} & AAI \\
\text{III} & AII \\
\text{III} & IAI \\
\text{III} & EAO \\
\text{III} & EIO \\
\text{III} & OAO \\
\end{array}$
=== Figure $\text {IV}$ ===
Consider Figure $\text {IV}$:
From Valid Syllogisms in Figure IV, the patterns:
:$AII$ and $EOO$
can be eliminated as they have an affirmative major premise and a particular minor premise.
:$IEO$ and $OAO$
can be eliminated as they have a negative conclusion and a particular major premise.
:$AAA$ and $EAE$
can be eliminated as they have a universal conclusion and an affirmative minor premise.
Thus the only patterns in Figure $\text {IV}$ that may be valid are:
:$\begin{array}{rl}
\text{IV} & AAI \\
\text{IV} & AEE \\
\text{IV} & EAO \\
\text{IV} & EIO \\
\text{IV} & IAI \\
\text{IV} & AEO \\
\end{array}$
It remains to be established whether these $24$ patterns actually do represent valid categorical syllogisms.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Categorical Syllogisms