Can I Notch a Double 2x12 Floor Joist Down to 2x6 Size (5.5 inches) if I need a 2x8 (7.25) per My Local Span Table
It’s not clear the spacing of the 2x12’s nor how many 2x12’s you’d like to notch. However, I’ll make a few assumptions and you can let me know if I’m wrong. 

The Code requires a Live Load of 20 psf for uninhabited attics and 30 psf for habitable attics. (See ICC Table 1607.1) Normal living spaces are required to have 40 psf. 

I’ll assume 20 psf Live Load (furniture, people, etc.) plus 10 psf for Dead Load (subfloor, joists, ceiling below, etc.) for a total load of 30 psf. 

I’ll also assume your 2x12 joists are 16” on center spanning 12’ and notched as shown in your sketch. 

Depending on the species and grade of lumber, a 2x6 at 16” on center spanning 12’ (the notch essentially reduced the 2x12’s to 2x6’s) can support about 37 psf (live and dead load).

Therefore, your double 2x6’s at 16” on center spanning 12’ can easily support a Code required 30 psf (live and dead load). 

Bearing is adequate with the joists resting on the 2x4 top plate. 

I’d be very careful of your vertical cut. Do not over cut and make a notch into the remainder of the joist. 

The weak link in this design is the bearing end. If the notched end piece can be “racked” it could split off and become detached. I’d insure that these joists do not move laterally by carefully extending the subfloor to the adjacent walls so it can’t move. 

If you are going to use the space for living space or if you’re going to store heavy items (like stacks of books, etc.) then my analysis is not correct. (I’d put my Christmas decorations in that corner.)