I understand the distinction between the legitimacy of the death penalty and the way this sentence is used in a particular set of rules.
But you obviously don't care. What's more, when I talk about the infallibility (or not) of the prosecution process, I'm referring to liberal democracies, as it should go without saying. The prosecution's case in totalitarian states is simply a joke.
Since you know that applying the death penalty will result in irreperable mistakes (which is not the case for other kinds of punishment)
That's utterly incorrect and this is where your whole argumentation collapses I'm afraid.
Unless you happen to own the secret of time travel, I'd like to know how you could "fix" the mistake of a 30 years sentence (for instance) served in jail by an innocent.
Actually, no matter the length of the imprisonment, any sentence is as "irreparable" as any other. You can't give back the days of this persons life, be it 1, 10 or 100. They're irremissibly lost and that's "irreparable" just the same.
In fact, I'd love to hear what your self-proclaimed moral superiority and good feelings could lead you to tell a poor fellow who had most of his life wrecked by such sentence. If I follow your logic, I imagine we could tell him to shut up and stop complaining since after all, he's still alive… How noble of you.
And please, don't tell me that it's a less serious offence, just because the poor bastard is alive to mourn his lost life.
We all die one day, but I'm far from being convinced that spending the rest of your life bitterly pondering on the unjust deprivation of the first half "“ or to actually die in prison, considering that if they're convicted on the charge of murder, the alternative to death sentence would be life in prison - is in any case a more enjoyable fate.
But anyway, let's push that a bit further and apply it to your moral dilemma equation:
1. One who broke the law shall be jailed 2. A decision of justice cannot result in jailing an innocent.
Now what do you propose? Let's empty the prisons, wipe out the penitentiary system and completely give up the ideas of justice and law?
The moral dilemma you impose onto the death penalty (and again, curiously enough, not to any other sentence) is in fact already taken into account by the judicial system of advanced countries. This is why there are extensive investigations and counter-investigations, with ever evolving techniques improved precisely to reduce the risk of error (Because here comes another Great Lesson of Life: zero-risk does not exist), this is why there are lawyers, witnesses, jury, habeas corpus (which actually deals with the other moral dilemma that you apparently don't care too much about: detention) and a whole set of appeal system relying on the principle that a man is innocent until proven guilty.
The way I can see it, your answer to that is: okay, considering that we may convict an innocent, then we shall not convict anybody. Including the guilty of course, since you assume that we can't really tell who is and who is not with 100% accuracy.
As a side note, I'll add that if we where to follow that kind of logic, it would reduce nearly all human activity to zero (risk is inherent to the nature of life. Dealing with it is always a question of balance between potential outcomes and the expected benefits or failures), but that's another debate.
It leaves you with a big problem anyway (well, several in fact) : what do you do with them? Even if I was to accept that to save an unknown and unpredictable number of potential innocents (that's 1. certainly marginal, thank to the "safety trigger" of the judicial process and 2. very likely to be decreasing according to the sophistication of investigating techniques) you shouldn't punish the murderers (Certainly a largely superior proportion, thank to the said sophistication), then you still have to face the fact that you're either going to jail innocents "“ in most case for life - or completely give up with justice.
Or do you declare a moratorium and… What? Lock everybody without trial - including the handful of innocents for that matter - until you're finished talking on a very touchy subject "“ which could take several years? (and end up leaving them to jail instead of executing them...)
A bit sadistic of you, if you ask me. And definitely unacceptable morally speaking.
It always strike me that the case that saw the final victory of the anti-death penalty faction in France (and the lasting fame of the Socialist lawyer who "won" it, Robert Badinter) was that of Patrick Henry who turned out to be genuinely and incontestably guilty of the murder of the 7 years old Philippe Bertrand, strangled before the request of the ransom, his body let to rot under the bed of an hotel room in the mid 70s.
Talk about a symbol of the protection of the innocents indeed…
(Coincidentally, I also notice that the morally superior power-that-be and that know better what's good for us, carefully avoided to consult the French people on such an important question. But of course, they know better than the rabble, right?)
I'm afraid that the moral dilemma you are facing is that either you arbitrarily declare that it's okay to punish the innocent in one way but not in another, or that the cause for concern that has to be questioned and improved (and is, actually) is the structural process leading to the sentence, not the nature of the sentence itself.
In short, the fact that the sentence can be wrongly pronounced doesn't make it wrong per se.