Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise

Theorem
Let $Q$ be a valid categorical syllogism in Figure $\text{II}$.

Then it is a necessary condition that:
 * The major premise of $Q$ be a universal categorical statement

and
 * The conclusion of $Q$ be a negative categorical statement.

Proof
Consider Figure $\text{II}$:

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 predicate of $\text{Maj}$
 * the predicate of $\text{Min}$.

So, in order for $M$ to be distributed, either:
 * $(1): \quad$ From Negative Categorical Statement Distributes its Predicate: $\text{Maj}$ must be negative

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

Note that from No Valid Categorical Syllogism contains two Negative Premises, it is not possible for both $\text{Maj}$ and $\text{Min}$ to be negative.

From Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative:
 * $\text{C}$ is a negative categorical statement.

From Negative Categorical Statement Distributes its Predicate:
 * $P$ 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 Universal Categorical Statement Distributes its Subject:
 * $\text{Maj}$ is a universal categorical statement.

Hence, in order for $Q$ to be valid:
 * $\text{Maj}$ must be a universal categorical statement
 * Either $\text{Maj}$ or $\text{Min}$, and therefore $\text{C}$, must be a negative categorical statement.