Valid Syllogisms in Figure IV

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Theorem

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

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 $\text {IV}$:

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

$\Box$


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.

$\Box$


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.

$\blacksquare$


Sources

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