Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise

Theorem
Let $Q$ be a valid categorical syllogism in Figure III.

Then it is a necessary condition that:
 * The conclusion of $Q$ be a particular categorical statement

and:
 * If the conclusion of $Q$ be a negative categorical statement, then so is the major premise of $Q$.

Proof
Consider Figure III:

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 subject 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 Universal Categorical Statement Distributes its Subject: $\text{Min}$ must be universal.

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

Then by No Valid Categorical Syllogism contains two Negative Premises:
 * $\text{Maj}$ is an affirmative categorical statement.

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 Negative Categorical Statement Distributes its Predicate:
 * $P$ is undistributed in $\text{Maj}$.

From Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise:
 * $P$ is distributed in $\text{Maj}$.

That is, $P$ is both distributed and undistributed in $\text{Maj}$.

From this Proof by Contradiction it follows that $\text{Min}$ is an affirmative categorical statement.

Thus from Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative:
 * if $\text{C}$ is a negative categorical statement, then so is $\text{Maj}$

We have that $\text{Min}$ is an affirmative categorical statement.

Hence from Negative Categorical Statement Distributes its Predicate:
 * $S$ is undistributed in $\text{Min}$.

From Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise:
 * $S$ is undistributed in $\text{C}$.

So from Universal Categorical Statement Distributes its Subject:
 * $\text{C}$ is a particular categorical statement.

Hence, in order for $Q$ to be valid:
 * $\text{C}$ must be a particular categorical statement
 * if $\text{C}$ is a negative categorical statement, then so is $\text{Maj}$.