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Whatever gate you have, if all the inputs go low, it's dead... output is low. Even if the gate is supposed to be high on all inputs low, it'll go output low. That's because it's powered by the inputs... no power, no output.
Now, that aside, the negated logic will not 'power' anything but another gate, so using one means you'd need to buffer the output with an AND gate, where the previous gate output is input 1 and a live power feed is input 2. You may find that you have to buffer some inputs in the same manner.
You'll have to get creative... you may need to grab a sheet of paper and grid out the various (il)logic tables if you want something complex.
Oh, and btw, gate feedback is right out. Don't even attempt feeding back to an input further back up the chain, because it'll just bug out and break switches (they get in wierd states that won't go away until they're scrapped and rebuilt).
I've posted on the bethesda forums various ways to fix these issues without too much work on their part, no idea if that'll ever appear in a patch or not. There are mods that provide advanced switches that work properly, if you're not afraid of such things.
AND gate
OR gate
NOT gate
NAND gate
NOR gate
EOR gate
ENOR gate.
?
XOR and XNOR. They *did*... but then they introduced an ancillary condition that results in a broken table. That's part of what I wrote on their forum, letting the things "work" on radiant power would fix everything wrong with the tables, without needing to re-make the graphics and meshes to accomodate a new/third power input pole.
It's not that complicated. You just take the regular table, and if there's a condition where both inputs are low (or for more than two inputs, *no* inputs are high), and output was supposed to be high, scratch it out... it'll be low.
http://www.nexusmods.com/fallout4/mods/11165/?