No, that was my point: the problem is not solved. Do you have any idea how many murders are committed each year by killers who were not executed as punishment for their first murder? Since 1976 when SCOTUS reinstated the death penalty, even the most ardent opponents can identify only 8-10 individuals who were even possibly innocent. That’s over a period of 40 years. There as many innocents murdered by recidivist killers each year.
Now I will grant that coming up with that statistic is difficult, but here is a Bureau of Justice Statistics paper (pg 8) from 2002 that says this:
Within 3 years…1.2% of the 1,443 persons who had served time for homicide were rearrested for a homicide.
This was based on data from only 15 states but was calculated to represent about 2/3 of the total population of released prisoners. Doing the math this works out to about recidivist 9 murders per year, and this doesn’t include any of the various repeat murders that are committed by prisoners themselves or the murders that are committed on their behalf.
So: while the possibility exists that an innocent person has been executed, recidivist murders are not a theoretical possibility. It’s a measurable quantity. This is why I asked how many murders you are willing to tolerate to avoid the possibility of executing an innocent person?