The apparent paradox is that the logical deduction of the statement allows for the truth. This professor can make this bold statement because of the way humans think. Most people can make a guess at the unknown, but they are not sure of it, and they do not know the truth. The know-it-all who attempts to use the above logic then knows only that the test will not be administered.
Imagine a class of 19 students, identified as A through S, for short. On Monday, at the beginning of class, the professor makes his statement. Assume:
The professor then administers the test 5 minutes after his announcement (on Monday), having given student S sufficient time to work out the logic.
After the test was given, students E-R are surprised, but also impressed that the professor tricked them. Student S is utterly shocked, as his prized logic has failed him. Students A-D, however, protest. They tell the professor that they knew it would be on Monday. He asks them how they knew, and they reply, "Well, we just kinda guessed, really." The professor is then vindicated, as some assumed it would be on Monday, but none knew that it would be on Monday.
The important factor in this problem is human behavior. Unlike computers, humans rarely think in absolute terms, and the declaration that one knows something does not actually mean that one does know something. Thus, even if 5 students each say that they know the day of the test, and each claims a different day, they do not actually know that the test will be on their chosen day; they are lying to attempt to cheat the professor.
tdent says re: Paradox of the surprise examination: The usual definition of knowledge is a justified true belief. Since S's belief is not true, and (of course) isn't really justified, it can't be called knowledge.
I don't agree. Many people knew
that the earth
was flat. It turns out that it was a false belief, but in the context of the ancient world, it might be called knowledge. Regardless, the professor still "wins".