# 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.

$\blacksquare$