Question

Assume the given statements to be true, even if they seem to be at variance from the commonly known facts. Decide which of the given conclusions logically follow the given statements.
Statement:
- Some Directors are Principals.
- All Principles are Professors.
Conclusions:
(I) No Director is Professor.
(II) All Professors, who are Directors are Principals.

• A. Only Conclusion (I) follows.
• B. Only Conclusion (II) follows.
• C. Both Conclusions (I) and (II) follows.
• D. Neither Conclusion (I) nor Conclusion (II) follows.

Hint:

In syllogisms, a conclusion only "follows" if it is true in all possible Venn diagram configurations.
If you can draw even one valid diagram where a conclusion is false, then that conclusion does not logically follow.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This is a syllogism problem from deductive logic.
We are given two statements and must determine which of the proposed conclusions logically and unconditionally follow from them.
Venn diagrams are the most reliable tool to visualize set relationships and verify these conclusions.

Step 2: Detailed Explanation:

• Analyze the Statements using Sets:
Let $D$ represent the set of Directors, $P$ represent the set of Principals, and $Pr$ represent the set of Professors.
- Statement 1: "Some Directors are Principals."
This means the intersection of $D$ and $P$ is non-empty:
\[ D \cap P \neq \emptyset \]
- Statement 2: "All Principals are Professors."
This means the set of Principals is a subset of the set of Professors:
\[ P \subseteq Pr \]

• Combine the Statements:
Since some Directors are Principals ($D \cap P \neq \emptyset$) and all Principals are Professors ($P \subseteq Pr$), it is a logical necessity that those Directors who are Principals must also be Professors.
Thus, the intersection of $D$ and $Pr$ is definitely non-empty:
\[ D \cap Pr \neq \emptyset \quad \implies \quad \text{"Some Directors are Professors" is True.} \]

• Evaluate Conclusion (I) ("No Director is Professor"):
Since we established that "Some Directors are Professors" is definitely true, the statement "No Director is Professor" is false.
Therefore, Conclusion (I) does not follow.

• Evaluate Conclusion (II) ("All Professors, who are Directors are Principals"):
The set of "Professors who are Directors" is $Pr \cap D$.
While we know $P \cap D \subseteq Pr \cap D$, there is no rule stating that $Pr \cap D \subseteq P$.
It is possible to have a Director who is a Professor but not a Principal (since some Professors might lie outside the set of Principals).
Therefore, Conclusion (II) is not a logical necessity and does not follow.

Step 3: Final Answer:
Neither Conclusion (I) nor Conclusion (II) follows.
Hence, the correct option is (D).