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.
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercise $\text{(c)}$