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 $\text I$:

Major Premise:	$\map {\mathbf \Phi_1} {M, P}$
Minor Premise:	$\map {\mathbf \Phi_2} {S, M}$
Conclusion:	$\map {\mathbf \Phi_3} {S, P}$

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}$

$\Box$

Figure $\text {II}$

Consider Figure $\text {II}$:

Major Premise:	$\map {\mathbf \Phi_1} {P, M}$
Minor Premise:	$\map {\mathbf \Phi_2 } {S, M}$
Conclusion:	$\map {\mathbf \Phi_3} {S, P}$

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}$

$\Box$

Figure $\text {III}$

Consider Figure $\text {III}$:

Major Premise:	$\map {\mathbf \Phi_1} {M, P}$
Minor Premise:	$\map {\mathbf \Phi_2} {M, S}$
Conclusion:	$\map {\mathbf \Phi_3} {S, P}$

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}$

$\Box$

Figure $\text {IV}$

Consider Figure $\text {IV}$:

Major Premise:	$\map {\mathbf \Phi_1} {P, M}$
Minor Premise:	$\map {\mathbf \Phi_2} {M, S}$
Conclusion:	$\map {\mathbf \Phi_3} {S, P}$

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}$

$\Box$

It remains to be established whether these $24$ patterns actually do represent valid categorical syllogisms.

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism