Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise

Theorem

Let $Q$ be a valid categorical syllogism in Figure $\text I$.

Then it is a necessary condition that:

The major premise of $Q$ be a universal categorical statement

and

The minor premise of $Q$ be an affirmative categorical statement.

Proof

Consider Figure $\text I$:

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

Let the major premise of $Q$ be denoted $\text{Maj}$.

Let the minor premise of $Q$ be denoted $\text{Min}$.

Let the conclusion of $Q$ be denoted $\text{C}$.

$M$ is:

the subject of $\text{Maj}$

the predicate of $\text{Min}$.

So, in order for $M$ to be distributed, either:

$(1): \quad$ From Universal Categorical Statement Distributes its Subject: $\text{Maj}$ must be universal

or:

$(2): \quad$ From Negative Categorical Statement Distributes its Predicate: $\text{Min}$ must be negative.

Suppose $\text{Min}$ is a negative categorical statement.

Then by Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative:

$\text{C}$ is a negative categorical statement.

From $(2)$:

$P$ is distributed in $\text{C}$.

From Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise:

$P$ is distributed in $\text{Maj}$.

From Negative Categorical Statement Distributes its Predicate:

$\text{Maj}$ is a negative categorical statement.

Thus both:

$\text{Min}$ is a negative categorical statement

$\text{Maj}$ is a negative categorical statement.

But from No Valid Categorical Syllogism contains two Negative Premises, this means $Q$ is invalid.

Thus $\text{Min}$ is not a negative categorical statement in Figure $\text I$.

As $\text{Min}$ needs to be an affirmative categorical statement, $M$ is not distributed in $\text{Min}$.

From Middle Term of Valid Categorical Syllogism is Distributed at least Once, this means $M$ must be distributed in $\text{Maj}$.

As $M$ is the subject of $\text{Maj}$ in Figure $\text I$, it follows from $(1)$ that:

$\text{Maj}$ is a universal categorical statement.

Hence, in order for $Q$ to be valid:

$\text{Maj}$ must be a universal categorical statement

$\text{Min}$ must be an affirmative categorical statement.

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercise $\text{(d)}$