Question

In 2022, June Huh was awarded the Fields medal, which is the highest prize in Mathematics.
When he was younger, he was also a poet. He did not win any medals in the International Mathematics Olympiads. He dropped out of college.
Based only on the above information, which one of the following statements can be logically inferred with certainty?

• A. Every Fields medalist has won a medal in an International Mathematics Olympiad.
• B. Everyone who has dropped out of college has won the Fields medal.
• C. All Fields medalists are part-time poets.
• D. Some Fields medalists have dropped out of college.

Hint:

Translate options into quantifiers: “Every/All” $⇒ \forall$, “Some” $⇒ \exists$. Check if the passage provides a counterexample (to kill $\forall$) or a witness (to prove $\exists$).

Answer 1

The Correct Option is D

Let the universe be “all people.” Define predicates:
$F(x)$: “$x$ is a Fields medalist$.$”
$M(x)$: “$x$ has won an International Mathematics Olympiad medal$.$”
$P(x)$: “$x$ is a poet$.$”
$D(x)$: “$x$ has dropped out of college$.$”
Let $j$ denote June Huh. From the passage we have the facts:
1) $F(j)$ (June Huh is a Fields medalist.)
2) $\neg M(j)$ (He did not win any IMO medals.)
3) $D(j)$ (He dropped out of college.)
4) $P(j)$ (He was also a poet.)
Now evaluate each option using these facts alone.
Option (A): “Every Fields medalist has won an IMO medal.”
Formal form: $\forall x\,\big(F(x)⇒ M(x)\big)$.
A single counterexample falsifies a universal statement. We know $F(j)$ and $\neg M(j)$, hence $j$ is a counterexample.
Therefore (A) is False.
Option (B): “Everyone who has dropped out of college has won the Fields medal.”
Formal form: $\forall x\,\big(D(x)⇒ F(x)\big)$.
From the passage we only know $D(j)$ and $F(j)$ for one person. This does not justify a universal rule about all dropouts.
Construct a model consistent with the passage but making (B) false: add a person $u$ with $D(u)$ and $\neg F(u)$. None of the given facts mention $u$, so the passage remains true while (B) fails.
Hence (B) cannot be inferred; it is Not entailed.
Option (C): “All Fields medalists are part-time poets.”
Formal form: $\forall x\,\big(F(x)⇒ P(x)\big)$.
Again, one instance $F(j)\wedge P(j)$ is insufficient to conclude a universal rule. We can conceive a person $v$ with $F(v)$ and $\neg P(v)$ without contradicting any given fact.
Therefore (C) is Not entailed.
Option (D): “Some Fields medalists have dropped out of college.”
Formal form: $\exists x\,\big(F(x)\wedge D(x)\big)$.
This is an existential statement and is directly witnessed by $x=j$ since $F(j)$ and $D(j)$ are both true. No additional assumptions are needed.
Therefore (D) is Certainly True.
Key logical lesson:
- Universal claims ($\forall$) require evidence about all members; a single counterexample suffices to refute them.
- Existential claims ($\exists$) require evidence about at least one member; a single confirmed instance proves them.
- The passage gives one concrete instance (June Huh) with properties $F$, $D$, $\neg M$, $P$. Thus only the existential statement in (D) is logically forced by the data.
\[ \boxed{\text{(D) Some Fields medalists have dropped out of college.}} \]