In my opinion, for Sony to be successful with the PS4 (and as a company in general) they have to be willing to make choices that maximize the benefit for the customer while minimizing the research, development and system costs for Sony. This is not a trivial problem. If Sony is spending money to include PS1 and PS2 backwards compatibility on the PS4 it would be a sign that Sony was willing to spend a lot of money for minimal impact on customers; and I would see this as being very problematic for the success of the PS4.

This isn't to say that supporting these consoles would "doom" the PS4, but if they're willing to spend money on this what else would they be spending money on; and dozens of these side projects with minimal value would add up to a very expensive system with limited added appeal.

PS3 can emulate the PS1 which is 2 gens before it. So naturally the PS4 should be able to emulate the PS2. It has the power to do it, all it needs is Sony's effort.

It really depends on just how much effort and resources this would require and would be reward be worth the cost. The PS3 doesn't have B/C for discs so anyone thats playing PS2 disc games right now is doing it on a PS2 and a few possibly only earlier PS3s. Anyone that has a PS2 now and activily uses it likely wouldn't mind just keeping that PS2 when the PS4 roles around. 

The need for PS2 B/C in the PS4; I just can't imagine it being a very big deal. 

EDITED

Yes. Originally it didn't, but I had a few stern words with Kaz and now he says he'll implement it. You're welcome guys!

Sony would much rather sell you all your PS 1,2,3 games to you again on their own version of a virtual console. They don't make money onyour playing your old games, but they do make money tricking people into paying for the convience of playing old games on new hardware.

Just do what I do. Keep all your old consoles just in case you ever feel like playing them again. I still play my Dreamcast, SNES, Genesis and original Gameboy often. I don't buy Virtual Console games because I still have the real consoles.

Are they forcing you to buy them? NO. If you are stupid/dedicated enough to rebuy games you already own then go ahead, that only encourages Sony even more.
Anyway all PS3s play PS1 discs fine.....

The money it costs to make a proper PS2 emulator for PS3 would probably be much cheaper than the cost to port these games to PS4 like they are doing with the PS2 Classics right now on PS3. Wouldn't it be better if they could just upload that original PS2 game to PSN and let people pay for it? YES IT WOULD.

As a dedicated gamer one would hope that Sony would make a separate BC to sell. Whether it be a different SKU or an addon. They could make a lot of money on this and it could tell them what fans really want (on which SKU sells more or if the add on sells at all).

PS4 should be powerful enough to emulate PS2 disks in the same manner PS3 emulates PS1 games, but I don't see this happening, especially when PS3 BC is rumored to be a $100 disk drive add on.

I wouldn't count on it.

Backwards compatibility isn't that important anyway. I buy new consoles to play new games, I've played the old ones already. Life too short to replay stuff, gotta move on to new experiences.

Nostalgia quickly goes off anyway, just watch some Lets Play on Youtube and I feel like I played that old game.

why bother now after 7 whole years? It's better this way and PSN is all what to expect from.