# 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: $\map {\mathbf \Phi_3} {S, P}$

$AEE$, $AEO$, $AOO$ 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: $\map {\mathbf \Phi_3} {S, P}$

$AAA$, $AAI$, $AII$ 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: $\map {\mathbf \Phi_3} {S, P}$

$AAA$, $AEE$ 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 a negative 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.

$\blacksquare$