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

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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 $\text{II}$:

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

$\blacksquare$


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