LCD vs. No LCD

Hopefully I can convey my question right, so here goes. On another thread someone mentioned that hooking up a monitor to the camera will reduce the resolution. I didn't want to hijack the thread, but I want to know:

Does hooking up an LCD to a DSLR (T2i and T3i specifically) reduce the recording resolution or the output resolution to the monitor?
If it does reduce the resolution of the actual recording, why do people do it?
 
it does not affect the footage that is recorded in the flash card, at all

what happens is that when you start recording on the card, the monitor receives a low-res signal
 
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