Backwards compatibility while simultaneously changing architectures is not the norm.
They cannot endure the loss - they lost money on the PS3 for several years in a row, they're losing money on the PSVita and they need to start making money STAT if they want to stay in the game.
Releasing a PS4 is somehow going to fix it? There are just way too many alternatives. The majority of the people who are going to buy a PS4 are PS3 people. Streaming PS1 ... PS3 games is a solution, but we don't even have details how this is going to be done or if it's going to be successful. They only have their reputation to go off of which is majorly influenced by it's userbase.
Why do you think the Vita is doing bad, it lacked an appeal to it's PSP customers. When they cut the ties from their userbase, they just expect it to go back. This was true decades ago because there weren't too many alternatives
It's no doubt that they are going to take a loss by adding BC as Rydian pointed out with his Math (although I think the proportion they lose with BC compared to without BC is way to high which is why the numbers were so far apart). However, the loss they lose could probably come from investors if they are willing to contribute the difference.
If I were to buy a Wii-U, eventually I'll start buying Wii-u games (cheaper than now). If I were to buy a Wii, there is no way I'm going to buy wii-u yet alone games.
Similar analogy to PS4. If I were to buy another PS3 (because PS4 simply won't play the games I still want to play on the PS3), there is no way in hell I'll buy a PS4 yet alone PS4 games. This is that connection they are killing.
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CONCLUSION:
By not including BC, Sony's reputation is all they have for PS4 sales. Just like the Vita, this is a huge disadvantage. It may be popular for parents who want to buy their children expensive things, but they will put a dent into their userbase because of people like me.
Years ago, it was possible to create a new system without BC because nearly the entire userbase carried forward. Now is not the case. There are soooo many alternatives to gaming, if they don't appeal to the previous userbase, they are going to have a bad time.
PS. I have had a huge hangover this morning. My thought process is cloudy and may be inconsistent. If you take my post and start replying to each paragraph, you may find inconsistencies. This is a disclaimer.
Edit: Taken off the word used, counter productive to my argument.