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 I
Consider Figure 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 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 II
Consider Figure 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 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 III
Consider Figure 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 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 IV
Consider Figure 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 a negative minor premise.

Thus the only patterns in Figure 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 syllogism.