I'd wager on both companies working on emulators -- perhaps not with full support for the entire back-catalog, but the popular titles plus whatever is lucky enough to come along for the ride, as the 360 did with the original XBox.
On one hand, emulating contemporary consoles on a PC is difficult because of architecture differences, and most especially because of the lack of suitably low-level interfaces to map the console interfaces to. I suspect that an API like Mantle would ease some of that pain on the PC, and that console-bound emulators get to sidestep at least that issue. Still no cake-walk to close the gap, but either setup reduces the distance between hither and yon pretty significantly I'd imagine.
Interestingly, but perhaps coincidentally, if you do the math on the 4 extra compute-only GPU clusters in the PS4, it works out to be exactly the same theoretical throughput as all 8 SPEs in the Cell processor. I think with relatively similar programming models, it probably wouldn't be beyond the realm of feasibility to write a dynamic recompiler to translate those SPE programs to vectorized GPU code. With that processing burden off the PS4 CPUs, there would seem to be plenty of horsepower left to emulate the PS3s single, in-order, dual-threaded PPC CPU.
If Sony can offload the SPE code in this way, they've probably got an easier task of emulating their previous console on the new one than Microsoft does (because Microsoft has 3x the PPC CPUs to emulate on an equal number of x86 cores as PS4, each with an extended altivec unit to make up for.)