Valid Syllogisms in Figure IV

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

Then it is a necessary condition that:


 * $(1): \quad$ Either:
 * the major premise of $Q$ be a negative categorical statement
 * or:
 * the minor premise of $Q$ be a universal categorical statement
 * or both.


 * $(2): \quad$ If the conclusion of $Q$ be a negative categorical statement, then the major premise of $Q$ be a universal categorical statement.


 * $(3): \quad$ If the conclusion of $Q$ be a universal categorical statement, then the minor premise of $Q$ be a negative categorical statement.

Proof
Consider Figure IV:

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 subject of $\text{Min}$.

We have:
 * Middle Term of Valid Categorical Syllogism is Distributed at least Once.

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

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

Both may be the case.

Thus $(1)$ is seen to hold.

Let $\text{C}$ be 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 $\text{Maj}$.

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

Thus $(2)$ is seen to hold.

Let $\text{C}$ be a universal categorical statement.

From Universal Categorical Statement Distributes its Subject:
 * $S$ is distributed in $\text{C}$.

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

From Negative Categorical Statement Distributes its Predicate:
 * $S$ is a negative categorical statement.

Thus $(3)$ is seen to hold.