And that's where the difference lies in what people perceive as justice and what the state institutes to hold the society together -- a reactive system which weighs anti-social behaviour against the cost to the social contract of treating it one way or the other; all forms of punishment, containment and rehabilitation have a social cost. It's a balanced compromise; satisfaction isn't really a part of it, as this is subjective, malleable and can be affected by the language of argument deployed in any given case.
The victim of crime seeks closure. The state seeks to minimise, deter, recover from and prevent if possible the sort of aberrant behaviour it defines as a risk to its members by precedent and reasonable evidence, including re-offending. Hence the removal of the emotional response of the victim in the moment from the equation and our process of mediating justice through the cases of the prosecution, the defence and the jury vote, not to forget the judge's legal interpretation; also the need to prove guilt beyond reasonable doubt, especially in cases of circumstantial evidence, wrong identity, faulty witness statements (patchy recall of events alone is a big issue), etc, as we currently have. It's not perfect, emotions fly any which way still, but the facts, however much people might not like them, do get considered and likewise for the long-term effect on our shared law and its interpretation, which is more important than any particular case.
As far as punishment goes, you take your pick between what actually has an effect and what simply creates a violent society, a slippery slope and a murder industry (which interestingly may inflate costs of incarceration and processing -- people have to be kept until they are executed, and you have to pay for whatever means of execution is adopted en masse; a similar argument crops up regarding the current role of sanctions in the realm of social security). For someone wanting to hurt a criminal back at any cost, no price is too high and damn the consequence; whilst for someone who still wants to keep the whole society ticking over, violence and its applications give room for pause.
Yes, this means the Daily Mail, the Spectator or some such rag will get to harp on about some heinous devil getting a release after X number of years on a life sentence, but if it continues to reduce the murder rate at the current pace, it's a price worth paying.
The reverse: killing people, including innocents, in the vain hope it eventually gets or dissuades all the killers in society, without a similarly associated drops in murder rates, but leaves the occasional victim's relation content, does not really have such a price. It's simply not worth it. Indeed, for most extreme cases self-defence is already accounted for, armed police will shoot to kill and so will our armed forces and intelligence agents authorised to do so (but again, not without review and oversight); which I feel, in combination with life sentencing, mental health provisions in law and its review, is better than the capital punishment systems being adopted at large.
