Well, I already admitted I'm not a logician: if A & B => C, (!A | !B) => !C looks implicit to me, and while this isn't the first time I've had my nose rubbed in the fact that it's not, I never could understand why not.
That's the same as [ A => B ] and therefore [ !A => !B ]. This can easily be shown to be an invalid inference by instantiating A and B appropriately.
A = "It's Raining"
B = "Water is falling from the sky"
The first is clearly true. If it's raining, then water is definitely falling from the sky. The second is clearly false. Just because water is falling from the sky, that does not necessarily mean that it's raining.
You can probably invent your own, somewhat more humorous examples.
Just because A & B get you C doesn't mean that having A and B is the only way to get to C.
So you can't conclude that because you don't have A or B you don't have C.
It's like saying "Going down the A20 and the M20 (roads in the UK) gets you to Rochester (I've no idea if it does). I've not been down the A20 or the M20, thus I'm not in Rochester." The second sentence there isn't really valid, maybe you took a different route.
