Private injuries & revenges no punishments:

From the definition of punishment, I infer, first, that neither private revenges, nor injuries of private men, can properly be styled punishment; because they proceed not from public authority.

Nor denial of preferment:

Secondly, that to be neglected, and unpreferred by the public favour, is not a punishment; because no new evil is thereby on any man inflicted; he is only left in the estate he was in before.

Nor pain inflicted without public hearing;

Thirdly, that the evil inflicted by public authority, without precedent public condemnation, is not to be styled by the name of punishment; but of an hostile act; because the fact for which a man is punished, ought first to be judged by public authority, to be a transgression of the law.

Nor pain inflicted by usurped power;

Fourthly, that the evil inflicted by usurped power, and judges without authority from the sovereign, is not punishment; but an act of hostility; because the acts of power usurped, have not for author, the person condemned; and therefore are not acts of public authority.

Nor pain inflicted without respect to the future good.

Fifthly, that all evil which is inflicted without intention, or possibility of disposing the delinquent, or, by his example, other men, to obey the laws, is not punishment; but an act of hostility: because without such an end, no hurt done is contained under that name.