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 $\text {III}$.
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:
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 $\text{Maj}$.
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercise $\text{(f)}$