Claim rights and liberty rights
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A claim right is a right which entails responsibilities, duties, or obligations on other parties regarding the right-holder. By contrast, a liberty right is a right which does not entail obligations on other parties, but rather only freedom or permission for the right-holder. This distinction originates in American jurist Wesley Newcomb Hohfeld's analysis of "rights" into claims, liberties, powers, and immunities, in his seminal work Fundamental Legal Conceptions, As Applied in Judicial Reasoning and Other Legal Essays.
A person's liberty right to x consists in his permission to do or not do x, while a person's claim right to x consists in an obligation on others to allow or enable him to do x. For example, to assert a liberty right to free speech is to assert that you have permission to speak freely; that is, that you are not doing anything wrong by speaking freely. But that liberty right does not in itself entail that others are obligated to help you communicate the things you wish to say, or even that they would be wrong in preventing you from speaking freely. To say these things would be to assert a claim right to free speech; to assert that others are obliged to refrain (i.e. prohibited) from preventing you from speaking freely (that is, that it would be wrong for them to do so) or even perhaps obliged to aid your efforts at communication (that is, it would be wrong for them to refuse such aid). Conversely, such claim rights do not entail liberty rights; e.g. laws prohibiting vigilante justice (establishing a legal claim right to be free thereof) do not thereby condone or permit all the acts which such violent enforcement might otherwise have prevented.
However, a liberty right can be asserted as the inverse of a claim right: a person has a liberty right permitting him to do something only if there is no other person who has a claim right forbidding him from doing so. This is because the deontic concepts of obligation and permission are De Morgan dual; you are permitted to do all and only things you are not obliged to refrain from.
A world with only liberty rights, without any claim rights, would by definition be a world wherein everything was permitted and no act or omission was prohibited; a world wherein none could rightly claim that they had been wronged or neglected. Conversely, a world with only claim rights and no liberty rights would be a world wherein nothing was merely permitted, but all acts were either obligatory or prohibited. The assertion that people have a claim right to liberty - i.e that people are obliged only to refrain from preventing each other from doing things which are permissible, their liberty rights limited only by the obligation to respect others' liberty - is the central thesis of liberal theories of justice.
The distinction between liberty rights and claim rights should not be confused for the distinction between negative and positive rights. Both liberty and claim rights come in positive and negative varieties: your permission to do something is a positive liberty right, your permission to refrain from something is a negative liberty right, another's obligation to do something for you is a positive claim right, and another's obligation to refrain from doing something to you is a negative claim right. However, the De Morgan dual relationship between claim rights and liberty rights crosses the positive-negative rights distinction: one's positive claims limit others' negative liberties and vice versa (i.e. others are obliged to do something for you if and only if they are not permitted to refrain from doing so); likewise, one's negative claims limit others' positive liberties and vice versa (i.e. others are obliged to refrain from doing something to you if and only if they are not permitted to do so).
The other two terms of Hohfeld's analysis, powers and immunities, refer to second-order rights. Powers are liberty rights regarding the modification of first-order rights, e.g. the U.S. Congress has certain positive powers to modify some of U.S. citizens' legal rights, inasmuch as it can impose or remove legal duties. Immunities, conversely, are claim rights regarding the modification of first-order rights, e.g. U.S. citizens have, per their Constitution, certain negative immunities limiting the positive powers of the U.S. Congress to modify their legal rights. As such, powers and rights are often subsumed within claims and liberties by later authors.