Consider the following three statements:

(i) Some roses are red.

(ii) All red flowers fade quickly.

(iii) Some roses fade quickly.

Which of the following statements can be logically inferred from the above statements?

To determine which statement can be logically inferred, let us analyze the relationships between the statements under the assumption that both (i) and (ii) are true.

First, let statement (i) be true:
"Some roses are red."
This means there exists at least one rose that is red. In terms of set theory, if $R$ is the set of all roses and $D$ is the set of all red things, then their intersection is non-empty:
$R\cap D\ne \varnothing$.

Second, let statement (ii) be true:
"All red flowers fade quickly."
Since a red rose is a red flower, any member of the set of red roses must belong to the set of things that fade quickly. In terms of sets, if $F$ represents the set of things that fade quickly, statement (ii) implies that:
$D\subseteq F$.

Now, let us combine these two statements:
We know there is at least one object $x$ such that $x\in R$ and $x\in D$ (from statement i).
Since $x\in D$ and $D\subseteq F$ (from statement ii), it must be true that $x\in F$.
Therefore, $x\in R$ and $x\in F$, which means $R\cap F\ne \varnothing$.

This translates back to:
"Some roses fade quickly," which is precisely statement (iii).
Thus, if both (i) and (ii) are true, then (iii) must logically be true.