COMPO they will understand he lied for the better of the public. I mean come on Gordon is one of few cops who was never corrupt and risked his life saving the mayor who Joker tried to kill. So hey man people will understand.
Will they?
I mean, let's analyze; if they can understand the whole situation then that means they can understand both Gordon (and Batman) lying AND the fact that Dent was corrupted and became a murderer.
And since they can understand both things together, wouldn't it have been better if they were honest from the very beginning, to begin with? I mean, there wasn't a real reason to lie if people is so understanding.
But in general I don't think people are too keen on authorities lying to them no matter how “good” the reason could seem. Because if they take the liberty of lying - and it’s not like a temporary lie, Gordon and Batman planned to keep that lie forever – then who’s to say they won’t be lying again? And who’s to say the next time the reason to lie will be this “good”? When someone think he got away with a lie then nothing’s there to keep him from lying again. Which leads me the next point; are people in general always agreed on things so we can say they will
all understand? Is every Gothamite going to say “Yes, that’s a good reason to lie, I condone this”? I mean, in a world where people’s opinions are usually divided in topics like presidential candidates or - specially - about ethics (Should we accept homosexuals? Should we bomb this or that country?)
I really doubt people will just “understand” what Gordon and Batman did by lying. If they did, as I said, then there’s no reason to lie to begin with.
For me this is a far more difficult “dead end” plot to solve than, say, Superman’s son and Lois’ reationship with Richard in SR (which is usually said to be “a dead end that’s impossible to fix”

. Because they either believe Batman to be a criminal forever (and Batman spends the rest of his life a fugitive) or they realize that Batman is able to lie to them and he ceases to be this ‘inspiring symbol’ he’s trying to be.
I was thinking the same way about Rachel. The deaths of Rachel and Harvey in the Dark Knight are (for a lack of a better word) "vague" to me but in a good way. It makes you think that they died, but question maybe they didn't (ie. because of no funeral scenes, morgue scenes and stuff like that etc.).
There are simply dozens and dozens of movies where they don’t show morgue or funeral scenes in order to state someone’s dead. When things are so evident specially, like in this case. That’s like saying I doubt Gordon has biological functions since I never saw a scene of him having a pee so it’s all “vague.”
That said, what exactly from the shot of Rachel’s face being violently surrounded by the flames do you call “vague”? Or are we supposed to think that someone – in this web of lies about who’s dead and who’s not the “heroes” of Gotham are always weaving – thought for some unfathomable reason that it would be a good idea to fake Rachel’s death even to Batman himself?