# 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.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.4$: The Syllogism - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**syllogism**