Definition:Figure of Categorical Syllogism/III

Definition

The third figure of a categorical syllogism, traditionally denoted figure $\text {III}$, is the pattern where:

in the major premise, the middle term is placed first

in the minor premise, the middle term is placed first.

Let $P$ denote the primary term, $S$ denote the secondary term and $M$ denote the middle term of a categorical syllogism.

Then figure $\text {III}$ can be tabulated as:

Major Premise:	$\map {\mathbf \Phi_1} {M, P}$
Minor Premise:	$\map {\mathbf \Phi_2} {M, S}$
Conclusion:	$\map {\mathbf \Phi_3} {S, P}$

where $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ each denote one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$.

Also known as

The third figure of a categorical syllogism is classically known as the figure of examples, from the fact that in order to be valid, its conclusion needs to be particular.

Also see

• Results about the third figure of a categorical syllogism can be found here.

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