No. Most old games aren't compatible (yet) with the PS4. In the case where a game has been released again on the PS4 (such as with Flower, The Unfinished Swan, other PSN games etc) they become free if you've purchased the older version. Other PSN games become available as time goes by, but the vast majority of those that have been released are remasters.
So for example, I own FFIX from the PSN and I've got it on my PS3, PSP and Vita. It's not playable on the PS4. With PS devices it's usually just a matter of time until these things are implemented to allow you to redownload (the Vita didn't have PS support for the first few months) however it's very likely they're trying to move toward Playstation Now for older games, so I'm not sure how they're going to implement playing older games you already own from PSN later (they might not).