Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise
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:
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.
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercise $\text{(e)}$