typically if the left shifter manages the 1-2 shift and then goes all floppy, the upshift pawl has stuck. WD40 or similar will usually free it off but some form of better lubrication will keep it good for longer.
You can usually remove a cover on the bottom of the shifter, and actually see the upshift pawl; this makes it easier to direct the WD40 correctly. Obviously if you still have a mech connected (check the cable isn't binding and that the mech itself is free.....) it should pull back at the workings so that it is easier to 'exercise' the shifter.
Occasionally a spring breaks inside an STI shifter but IME a sticky pawl is far more common. If a shifter is to go sticky, it will almost invariably be a front triple shifter, simply because it is used so infrequently.
cheers