From what I've read, Persona 4 is a totally new game with new characters and a new storyline, so I don't think the previous games are required in order to move along with the story of this one.
And just whaaaaaaaaaatttttttt is so Final about Final Fantasy?!! They're always making new Final Fantasy games! nothhhiiiiiiinnnnnggggg!!!!!. Nothing, I tell you, is final in Final Fantasyyyyyyyy!!!!! /satirical rant
Well, for those who don't know, back in the mid 80's Square was about to go bankrupt, but they had just enough money to make one last game. This game would make or break their company. They decided to make a JRPG set in a fantasy world. It was possibly their final game. Final Fantasy.