*The next Friday* or *Next Friday* is **practically** nonmeaningful. I mean what's **next** doing there? If Friday comes, it'll be just *Friday*, why next? To be precise, you should call it **this Friday** which'll mean that you are talking about the Friday which is coming. Something like *this week*, *this month* or *this year* which means following whatsoever period. 

Let's consider this case - Today is June 9 (Monday). So, I'll refer June 13 as **This Friday.** You'll find this good use of phrase on TV channels. They always advertise or notify the viewers that way *(Say...The Matrix THIS FRIDAY on AXN)* However, if you want to refer to Friday falling on June 20, **then putting next in the same sentence** makes sense. I'd call it this way:

> My exam is on **this** Friday (13th) and Paul's on **next** Friday (20th). 

**BUT** whatever it is, to stay clear, **always mention the date** to avoid ambiguity especially if you are talking on a serious deal. :)