Middle Term of Valid Categorical Syllogism is Distributed at least Once

Theorem

Let $M$ be the middle term of a valid categorical syllogism $Q$.

Then $M$ is distributed in at least one of the premises of $Q$ in which it appears.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

To violate this rule is to commit the Fallacy of Undistributed Middle.

• Rules of Quantity
• Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise

Historical Note

This rule of quantity was axiomatic at the time of Aristotle, when the theory of the categorical syllogism was initially developed.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercises: $\text{A}$: Rules of Quantity: $2$