Our leaders argued in the Cold War that nuclear weapons kept the peace between the West and the USSR. The corollary to this is that if we gave a nuke to every nation on earth, world peace would ensue. Why do we not do this? Because some fool would let one off, and that would probably trigger off a total nuclear war.
Which is evidence that MAD is not infallible.
Now, if the consequence of a breakdown of a system is infinitely negative, we can use that system if and only if the probability of its failure is zero. In the case of nuclear deterrence it is not zero, (see above) and therefore there is no logical case for using the nuclear deterrent system.
So we should go for total abolition, logically.
Which is interesting, but a bit of a waste of time, because the nuclear weapons system stems not from logic, but from an irrational state of mind marked by mutual paranoia ("Paranoid mutualis caesarium") and obsessionality.