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