There are at least three different kinds of justification generally given for intentionally punishing those who have done something wrong. One is partly utilitarian in nature and it is the grounds usually discussed in the media when issues such as the justification of capital punishment are raised. Yet it is less relevant to capital punishment than the other two. 

Of course, these justifications are about punishing the guilty, not about mistakenly punishing the innocent, even if they are thought to be guilty and even if, in a court of law, they are mistakenly convicted of a crime they did not actually commit. It also presupposes that the act for which one is being punished actually is something that is wrong to do, and is not something that is merely unreasonably illegal or mistakenly unjustifiably and unreasonably regarded as wrong.

1) Punishment as a deterrent of future wrong-doing, where it works to deter either the same person's repeating the offense (or intentionally doing other wrong things) or where it serves to deter others from committing the same (or other) offense. Punishment sometimes is a deterrent, and as long as it is not a draconian or unfairly harsh deterrent, or is not otherwise inappropriate for the crime or misbehavior, particularly when less harsh or more appropriate deterrents are reasonably available, and as long as the conditions above are met -- that the person actually is guilty and the act for which they are being punished actually is wrong -- and as long as there are no mitigating circumstances and where mercy is not justified, it may be employed.  In other words, if there are two possible punishments that are both appropriate on other grounds, the one that will also serve as a deterrent is better than the one that won't.  Where punishment does not serve as a deterrent, this attempt at justifying it does not apply and cannot be grounds for administering punishment.  It is an empirical question whether punishment in a given instance acts to deter crime or not. Yet it seems that a person who has done something seriously wrong should still be punished, even if it "does no good" for deterring his or anyone else's future bad behavior.

The fact that punishment has to be appropriate and not unfairly harsh also shows that deterrence alone cannot be the justification of punishment, since probably making jaywalking or skipping school a capital offense punishable by death would deter jaywalking and playing hooky. But even if they did prevent those wrong acts, they are too harsh a penalty for the severity (or lack of severity) of those wrongs. This is perhaps the underlying idea behind the clause in the United States Constitution prohibiting cruel and unusual punishments.  So again, even if cruel and unusual punishments would be effective deterrents, they cannot be (in this case, legally) justified.

On the opposite side, if deterrence were the purpose of punishment, then punishments could be rewards, not just inflicted penalties.  Suppose, for example, that we publicly reported the infliction of penalties, such as prison terms or execution, for people convicted of serious offenses, but that in actual fact, we secretly instead put them up in plush circumstances with a generous stipend that allowed them to live in luxury -- with the proviso all that would come to an abrupt halt if they perpetrated any other crimes. Or suppose we banished criminals to a plush, paradise island where they could live out their lives quite comfortably but away from civilization, so they could do no harm to any innocent people. Presumably that might work to deter or prevent crime, but surely it is wrong because it is not punitive in some way that should fit the crime they have already committed. There seems to be something about punishment that is not (just) about preventing future crime, but inflicting a penalty on someone for having committed one already.  Thus it seems that a proper punishment should inflict a penalty on the perpetrators of wrongdoing, but one that is appropriate in a Goldilocks way of being neither too harsh nor not harsh enough (i.e., not a "mere slap on the wrist" if even that much).

There are two different ways in which punishment may serve as a deterrent when it does.  Some punishments act as a deterrent by making someone fear the consequences of (being caught) committing an act, but there are punishments meant more to impress upon perpetrators the seriousness of their act than to behavioristically scare them into not doing it again.  A loved one's showing profound disappointment or anger may do more to get a person to see that what he did was seriously wrong and ought not to be repeated than it will to make him simply eschew repeating the act out of fear of an unpleasant response.  Similarly with a strong condemnation by a judge prior to sentencing a convicted person, or as his rationale for the sentence being handing down.  One can consider these things to be part of the punishment or as something separate from the punishment.  In some cases, they will be intended for one purpose more than another, the difference being whether the punishment is meant to make the person suffer or to get the person to take seriously his crime or misbehavior and truly understand and realize that what s/he did was wrong.  When punishment is intended to "teach you a lesson," the lesson may be either or both that the behavior was wrong (and thus is something no person who truly wants to be good will do) or that it will not be tolerated without unpleasant consequences for the perpetrator whether s/he cares about what is right or wrong, good or bad, or not.

2) Punishment as being earned and deserved as a fitting consequence to doing evil. In this case, punishment has the same sort of justification as a reward or as thanks and appreciation. It is something that is deserved, based on the actions of the person who has done the deed. It is not "forward-looking" as are utilitarian considerations, trying to find out what will do the most good in the future, but it is "backward looking" in the sense of trying to find out who has done good, or harm, in the past, and acting in some way to fittingly treat those who have done either. 

Suppose, in a simple case, someone steals money from another. Surely s/he should at least have to pay back the money. But I believe s/he should also have to pay for the aggravation and grief s/he has caused as well. Perhaps s/he should have to pay for the efforts of the police to apprehend him/her, perhaps the cost of any incarceration if incarceration is necessary. 

In a crime where what is taken cannot be repaid - murder, rape, battery, irreplaceable property loss, defamation of character, etc. - it seems that something else must be paid besides money, whether it is by the labor of the guilty party or by some other means to try to make up in some way for what was lost.

Yet something seems wrong in both cases, for we would not let someone pay first to commit a terrible act, such as murder or rape.  And I doubt we would say it is okay for someone to steal our car and lead police on a wild chase, etc. if he paid the costs of doing so ahead of time.  So the notion of paying for a crime or wrongdoing does not seem to fit; it is not like paying for a candy bar that one then has the right to take and eat or paying inside the store for $30 worth of gas one then has the right to pump into one's tank.

If we were to act only on utilitarian -- merely forward-looking-- grounds, we would be compelled to reward or bribe people before they have done good things, and punish people before they have done bad things; but once they have done either good or evil, nothing should be done about it because it is "too late" to affect their past action.  We would only be trying (perhaps unnecessarily) to reinforce their good acts or (perhaps unsuccessfully) to change their bad behaviors.  In fact, one of the utilitarian arguments against capital punishment is that it won't bring back the victim and it often is not a deterrent to others.  But no punishment will bring back a murder victim or restore the former peace of mind of a rape or torture victim, and also it seems odd, at least, to be willing to harm someone to prevent future crime but not harm them for having committed a past crime.  A possible future crime or intention or threat to commit a crime is surely not as bad as having already done it, so why should we be willing to inflict a harm in the form of a deterrent that we would not be willing to inflict in the form of a punishment.  It is like saying to someone who is about to break an expensive item in your store that it will cost him at least $25,000 if he does that, and then after he has done it, saying since it is broken, it is not worth anything any more but you want $50 to pay for the clean up of the mess he caused.  Or it is like saying that it will still cost him $25,000, but not for the one he broke, since it is already broken, but for one he has not broken but might.

Further, on utilitarian, forward-looking, grounds, good people should receive no incentive or attention because they are likely to do good, but people who are about to do great harm should be either coerced or paid not to do it. Incorrigibly good people can be ignored because they need no external incentives to do the right thing.

3) The strongest argument, I believe, is an extension of the view that no one should be allowed to profit from harm s/he has done others. The actual principle I have in mind is: We ought, as much as possible, to increase the amount of deserved good and deserved joy and contentment in the world, and we ought, as much as possible, to reduce the amount, not only of bad or evil, but of undeserved happiness that results from wrongdoing. We ought to be trying to increase, or at least keep, the balance of deserved happiness over ill and evil, and over undeserved happiness caused by wrongdoing. To that end, anyone who is imminently about to cause, or who has already caused, suffering and sorrow, may have, or should have, at least that much joy and contentment taken from him/her. This is the justification for capital punishment in cases where egregious and irreparable harm has been done. There is no reason to allow someone to experience joy of any sort who has robbed an innocent person of the ability to have such joy. This is also the reason, however, where sincere remorse and repentance over one's actions may make capital punishment unnecessary if the offender's life is filled from then on with abject sorrow and penitence about their past wrong actions, and if they have atoned, or made up to humanity, for the evil they have done by creating at least a commensurate amount of good that only they could do.  The reason I qualify "undeserved happiness" with the phrase "caused by wrongdoing" is that we might consider happiness that occurs simply by luck or by nature in some way not to be "deserved" but that does not make it necessarily undeserved.  Or if it does mean it is undeserved, it is still not wrong or undeserved in some problematic way.  For example if someone wins a lottery or survives a plane crash, epidemic, combat mission, or war in which ones friends die through no fault of his or her own, that does not necessarily make one's good fortune wrong to keep.

By this principle, reward for good behavior or good deeds is also explained and supported, since someone who does a good deed, deserves to have good happen to him or her.  Plus insofar as benefits and burdens should be fairly and reasonably distributed, insofar as one causes benefits to occur to others, some should come to him/her as well if possible.  And in the case of irreparable harm done by someone, since they cannot make it up, by adding deserved goodness to the victim, they must pay for it in terms of forfeited undeserved goodness that would happen to them.  In other words they should have to suffer to the extent that they caused a deserving person to suffer in order to not permit the balance of undeserved good to increase over the amount of deserved good or undeserved harm.

It is easy to see that harm may be inflicted on someone who poses a clear and present danger to an innocent person, if such harm is necessary to remove the threat or prevent the evil about to occur, and if the amount or significance of the harm balances (or is proportionate to) the amount of harm/evil to be done. The intended victim is innocent and does not deserve to be harmed; the perpetrator is now no longer innocent, and stopping him will create the most deserved good, or the most good for the most deserving people. Also, this justification allows for stopping a wrong act that is directed at a smaller number of victims by a greater number of perpetrators, which utilitarian considerations by themselves do not, since utilitarianism only takes into account amount of good or harm and the number of people, not whether the distribution is fair or the people deserving.

It would be irrational and ludicrous to hold that such measures to prevent evil are justified but are not justified to punish evil that has already been perpetrated. The threat of imminent perpetration of evil is clearly not as bad as the actual commitment of evil, and so what is justified as a deterrent to evil may also serve as a punishment for it. Surely it cannot be right to take the life of someone who is about to murder an innocent person unless it is also right to take that life after they have murdered an innocent person. You cannot hold that innocent life is important only while it is alive; and you cannot then consistently hold it is impermissible to take an innocent life, but it is acceptable and allowable to have taken one. And you cannot reasonably hold it is acceptable to use whatever proportionate means are necessary to prevent property crimes or other kinds of crimes and evils if you do not hold that such crimes, once they have actually been committed, do not deserve punishment commensurate with both the evil of the crime and/or the harm of the means necessary to prevent it.  If crime and evil are important enough to prevent people from doing, they should be important enough to punish people for having done.  But again, whether in preventing evil or punishing it, the amount of force/harm used is only right insofar as it balances as much as possible the harm/evil that is being threatened or that was done.  That is, it would be wrong to set up a deadly deterrent to protect someone from stealing your lunch from the company refrigerator as much as it would be wrong to execute someone for having stolen it.  But insofar as causing some minor pain or discomfort to prevent the theft is okay, it should be okay to cause it to punish the theft.

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