A friend told me to solve TOUGH+TOUGH+TOUGH+TOUGH=PUZZLE, and when I asked for some help, he told me that even if he let me know any of the digits of PUZZLE, I wouldn't be able to solve this. I'm glad he's not an expert in Logic, because what he told me let me deduce the solution. What's the answer?

This metalogic puzzle is entirely the work of Federico Kereki, to whom all credit goes.