Valid Patterns of Categorical Syllogism
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
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): syllogism
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): syllogism