The connection between the executioner and the death sentenced murderer is the same like the connection between the murderer and the victim - there is death at the end. And it goes on and on...
But it can stop if the people and the state decide not to kill more.
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Death is not justice

Our 2010 project asked for the universal abolition of the death penalty.
For us it’s simple. The death penalty is a violation of human rights.
Article 5 of the Universal Declaration of Human Rights states: “No one
shall be subjected to torture or to cruel, inhuman or degrading
treatment or punishment.”
How does that fit with beheading, stoning, hanging, lethally injecting
or shooting someone?