Valid Patterns of Categorical Syllogism

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Theorem

The following categorical syllogisms are valid:

$\begin{array}{rl} \text{I} & AAA \\ \text{I} & AII \\ \text{I} & EAE \\ \text{I} & EIO \\ * \text{I} & AAI \\ * \text{I} & EAO \\ \end{array} \qquad \begin{array}{rl} \text{II} & EAE \\ \text{II} & AEE \\ \text{II} & AOO \\ \text{II} & EIO \\ * \text{II} & EAO \\ * \text{II} & AEO \\ \end{array} \qquad \begin{array}{rl} \dagger \text{III} & AAI \\ \text{III} & AII \\ \text{III} & IAI \\ \dagger \text{III} & EAO \\ \text{III} & EIO \\ \text{III} & OAO \\ \end{array} \qquad \begin{array}{rl} \S \text{IV} & AAI \\ \text{IV} & AEE \\ \dagger \text{IV} & EAO \\ \text{IV} & EIO \\ \text{IV} & IAI \\ * \text{IV} & AEO \\ \end{array}$


In the above:

$\text{I}, \text{II}, \text{III}, \text{IV}$ denote the four figures of the categorical syllogisms.
$A, E, I, O$ denote the universal affirmative, universal negative, particular affirmative and particular negative respectively: see Shorthand for Categorical Syllogism.
Syllogisms marked $*$ require the assumption that $\exists x: \map S x$, that is, that there exists an object fulfilling the secondary predicate.
Syllogisms marked $\dagger$ require the assumption that $\exists x: \map M x$, that is, that there exists an object fulfilling the middle predicate.
Syllogisms marked $\S$ require the assumption that $\exists x: \map P x$, that is, that there exists an object fulfilling the primary predicate.


Proof

From Elimination of all but 24 Categorical Syllogisms as Invalid, all but these $24$ patterns have been shown to be invalid.

It remains to be shown that these remaining syllogisms are in fact valid.


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Sources

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