It's a rather glaring fault with Star Trek, the depiction of computers and sentient androids. No back-ups per se (you can't copy the EMH program's current data state in case of corruption?). And androids/computers that can be tricked when presented with unsolvable problems or mysterious conundrums.
When Norman broke down because of the "I'm a liar" and "He's lying" bit, I gritted my teeth. They COULDN'T be that stupid, right? RIGHT? Oh yeah... I forgot--this is one of those "comedy" episodes that you're not supposed to take seriously.
The only circumstance that was "sort of" plausible was Nomad. Its prime function was to sterilize imperfections and now it was shown how it was imperfect. Yet... it couldn't fathom "correction"? A mistake of identity is an error, but once corrected the error is in the past. The present state of the machine is "perfection" once again until successfully challenged later. Not to drag this thread off into a Nomad discussion, but I always felt it was such a loss for Nomad to be destroyed, and could have been prevented. Such as Kirk saying "Nomad, exercise priority override to shut down for critical repairs. Do not sterilize. You can be repaired."