I believe that the 2 Engines Inoperative case has been well covered here in consideration of the En-Route, and Handling capability cases.
Mad (Flt) Scientist has correctly indicated that only OEI (One Engine Inoperative) is considered for 2,3, and 4 engined aircraft at the Takeoff phase, with the 4 engined aircraft requiring an additional 0.5% OEI Climb Gradient compared to the 2 engined aircraft. So, the question arises, "Does this incremental 0.5% capability allow for a second engine failure at Takeoff?" - ABSOLUTELY NOT! In short, you're dead. The El Al B747 double engine failure on Takeoff accident amply demonstrates this. If the 4 engined aircraft were taking off at a weight well under typical takeoff weights, there MIGHT be a fighting chance of survival, and at typical weights, fuel cannot be dumped fast enough to make it possible.
So, which aircraft is the safest? Consider this -
(1) A 4 engined aircraft has twice the probability of engine failure as does a twin,
(2) A 4 engined aircraft in normal all engine operations has much less performance reserve than does a twin (see (3) below), thus requiring much longer application of high thrust levels, increasing cumulative engine stress, rendering them more likely to suffer engine failure.
(3) The One Engine Inoperative (OEI) performance for a 2 or 4 engined aircraft is APPROXIMATELY the same (slightly in the 4 engined aircraft's favour) at Takeoff. That is, the twin achieves the same on 1 engine as does the quad on 3 engines. Now, in the 99.999% of cases where an engine does not fail on Takeoff, the twin is operating with 200% of the thrust required, whilst the quad is operating with only 133% of the thrust required. Thus for normal operations (the 99.999% case), the twin has a huge excess of performance compared to the 4 engined aircraft.
For me, I'll stick to twins. I'll opt for the aircraft giving me the greatest performance reserve on 99.999% of my flights, whilst offering perfectly acceptable performance on the other 1 or 2 days of my career that I lose an engine. There's no jealousy here, I've flown 1,2,3, and 4 engined aircraft, and I know when I feel the safest.
Regards,
Old Smokey