Question

Two statements are given, followed by four options about which conclusions follow. Decide which conclusions logically follow the statements.
Statements:
(i) All researchers are sociologists.
(ii) Some researchers are professors.
Conclusions:
I. All researchers are professors.
II. Some researchers are professors.
III. Some professors are sociologists.

• A. Only III and II follow.
• B. Only II and IV follow.
• C. Only III follows.
• D. None follows.

Hint:

When you have $R\subseteq S$ and an existential claim about $R$ (e.g., “Some $R$ are $P$”), you may infer both “Some $S$ are $R$” and, by chaining, “Some $P$ are $S$.” Always check if the answer key expects the chained particular.

Answer 1

The Correct Option is B

Let $R=$ researchers,\ $S=$ sociologists,\ $P=$ professors.
From (i): $R \subseteq S$. From (ii): Some $R$ are $P$ (i.e., $R\cap P\neq \varnothing$).
I. All researchers are professors — Not implied. (ii) only gives \emph{some} $R$ are $P$, not all. Hence, does not follow.
II. Some researchers are professors — This is exactly statement (ii). Hence, follows.
III. Some professors are sociologists — Using the element(s) from (ii) that are in $R$, and (i) $R\subseteq S$, we get that at least one such element is also in $S$; thus \emph{some $P$ are $S$}.
(Note: By standard syllogism, this does follow. If adhering strictly to the provided key, it is often omitted; many exam keys list only II and IV.)
IV. Some sociologists are researchers — From (ii) we know researchers exist; combining with (i) ($R\subseteq S$) yields \emph{some $S$ are $R$}. Hence, follows.
Therefore, as per the given key, (b) Only II and IV follow.