The argument is based on the claim that there is an important relation between moral rights and moral duties.  As Goldman puts it: “Since having rights generally entails having duties to honor the same rights of others, it is plausible that when these duties are not fulfilled, the rights cease to exist” (1979: 43).  And once the rights cease to exist, it becomes permissible to treat people in ways that it would otherwise be impermissible to treat them: “by violating the rights of others in their criminal activities, [offenders] have lost or forfeited their legitimate demands that others honor all their formerly held rights” (1979: 43).  Goldman emphasizes that violating one right does not mean you lose all of your rights.  You lose the right that you violated or an equivalent: “if we ask which rights are forfeited in violating rights of others, it is plausible to answer just those rights that one violates (or an equivalent set).  One continues to enjoy rights only as long as one respects those rights in others: violation constitutes forfeiture” (1979: 45).

One way to reconstruct the argument is as follows:

P1: if P has a right to X, then P has a corresponding duty not to violate other people’s right to X
C1: if P does not fulfill his duty not to violate other people’s right to X, then P forfeits his right to X (or an equivalent set of
        rights)
P2: an offender fails to fulfill his duty not to violate someone’s rights
C2: an offender forfeits his own possession of the right he violates in his victim (or forfeits an equivalent set of rights)
P3: if P forfeits a certain right, then it is morally permissible for the state to punish him by harming in ways that would otherwise
        violate that right
C3: it is morally permissible for the state to punish an offender by harming the offender in ways that would otherwise violate the
        right that the offender violated in his victim (or an equivalent set of rights)

Note that the conclusion establishes both the permissibility of punishment and a kind of proportionality limit on the amount of punishment that is permissible.