Socrates and the logicians of Induction Island

Far in the South Pacific lies the tropical paradise known as "Induction Island." Induction Island is inhabited entirely by logicians. Some logicians have brown eyes, others have blue. Nobody is allowed to know the color of their own eyes. In fact, there is an Island-wide ban on discussing eye color. Moreover, mirrors and other reflective surfaces are banned from the Island. The reason for the rule is that should someone discover that their eyes are blue they are exiled; in fact, they must leave Induction Island at noon the day after they find out.

Fortunately, with no mirrors and discussions of eye color banned, life on Induction Island was able to continue peacefully with little more disruption than messy hair and poorly applied make-up. That is until a truth-telling trouble maker named Socrates arrived on the Island. Nobody knew where Socrates had come from, but they all agreed he was insufferable: Socrates was always lecturing people and he had to win every argument. Finally, the Island inhabitants put him on trial for "making weaker arguments appear to be stronger" (logicians hated that), convicted him, and voted him off the Island.

All the logicians gathered to watch Socrates board the ship for Athens. Just before stepping aboard, Socrates turned and addressed the crowd: "Someone on this Island has blue eyes!" Then he cackled wildly, pulled up the gangplank, and set sail. The crowd was stunned. Although they all disliked Socrates, they knew he always told the truth.

Do any of the blue-eyed logicians have to leave Induction Island? If so, how many days after Socrates leaves for Athens must they go into exile?