We can play a limited collection of PS2 titles on PS4. Sony could include a selection of PS3 games on PS4 as well. A leaked picture of the PS Store shows a section called “Best of PS3”:

Best PS3 [Rumor] PS3 games coming to PS4 | VGLeaks 2.0

I highly doubt that they are going to introduce full backwards compatibility on PS4, I would bet for a collection of PS3 games, like the action performed with PS2 games on the mentioned system. We will see.

Thanks, Reddit.