I have a special affinity for logic systems and this is predicate logic. I also realize this is quite the narcissistic joke and yet I still find it funny. Here is how to decipher:
(Ǝx) means there is at least one thing
Px means that thing is a person
Pj means a person named Joel
(y) means all things in the universe
(Py -> x=y)) means if y is a person then x and y are the same person
Full translation: There is only one Joel.
For the picky logicians out there, I make the following identity assumption: In using Pj I am referring to a specific person and not to all people with the same name.