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

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Theorem

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

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 $\text {III}$:

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

$\Box$


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.

$\Box$


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 $\text{Maj}$.

$\blacksquare$


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