How "Rights" Are Like Superscript -1

Legal language, like mathematical language, often gives multiple definitions to similar terms.

|The Volokh Conspiracy |

My post about how the word "right" in American legal usage often includes the entitlements of government as well as of individuals drew this comment:

"But as a matter of American legal language…".

There you go again, conflating law with reality. Legal consensus makes governmental 'rights' no more real than it would make defining 'pi' as 'three' be three, even though that could be enforced by the state with further (not unfamiliar) grotesqueries to compensate for its being unreal.

And this led me to think of this example from math: What does x-1 mean to you? Generally speaking, it would mean 1/x. But what does sin-1 x mean to you, if you know your trigonometry? It wouldn't be 1/sin x, but rather arcsin x, which is to say the inverse of the sin function: sin (arcsin x) = x. More broadly, f-1 (x) is the inverse of the function x, so that f (f-1 (x)) = x. And even if you don't feel at a home with trig and with functions, the point is simple: In math, as in law, we sometimes use the equivalent of homonyms— two different (but often related) concepts that are represented using the same symbols.

The same is true, of course, with computer programming language, where the same symbol can mean quite different things depending on context (usually depending on the data types of the items being operated on)—in the same language, + might mean addition when used with numbers but concatenation when used with text, or date addition when used with a date and an integer. (This is sometimes labeled "overloading" the operator, not in a pejorative sense but just in the sense that the operator has multiple meanings.)

Think of "right" the same way. In ordinary English, of course, the word means many different things—correct, the opposite of left, to restore to an upright position, and more. In law, it usually means a legal entitlement, but of course it has different logical properties and rhetorical qualities when used to refer to different kinds of legal entitlements: entitlements of individuals, groups, or governments, entitlements to get things or to be free from things, entitlements that are seen as stemming from moral principles or entitlements that are seen as stemming from positive law, and more. Just like superscript -1 in math, or the addition symbol in many computer programming languages, it is a sort of homonym, the meaning of which usually has to be understood from context.

Now one could certainly argue that language would be better, because less likely to confuse, if it used fewer homonyms. Perhaps we should stop using sin-1 and instead use arcsin, as many people do, and perhaps we should come up with a new symbol fINV to use for the inverse of a function more generally.

At the same time, language is a grown order, developed over centuries, and there are costs to trying to depart from it or to change it. We often use familiar locutions despite their potential ambiguity, precisely because they are familiar. (For instance, the word "homonym" is somewhat ambiguous—it could refer to two words with the same spelling but different meanings, which are sometimes labeled "homographs," or two words with the same pronunciation but different meanings, which are sometimes labeled "homophones," but I use "homonym" because it's more familiar and because I assume you identified the proper meaning from context.) We certainly don't say that sin-1 is "wrong" in the sense of arcsin just because the same symbol is used for something else in some other context.

The same is true for "right," as in asking whether courts have "a right of ultimate jurisdiction," or whether Congress has a "right" "to tempt the navigators of enemy-vessels to bring them into the American ports." It's not the same sort of right as your right of free speech, or for that matter as your right not to be defamed by a fellow citizen, or your right to have the civil case you bring tried by a jury. As a matter of "reality," or more precisely as a matter of legal analysis, each of these "rights" has somewhat different qualities. But if the question is whether it's correct to call these "rights," that is a question of legal American English (which is to say of convention among users of legal American English), not logic—just as the question whether sin-1 sometimes means arcsin is a question of mathematical language and thus of convention among mathematicians.

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  1. The problem with this entire discussion is that rights are largely in the eye of the beholder. There is no cosmic legislature that declared what would, and would not, be human rights. And conceptually, one of the issues I have with libertarianism (even though I myself have libertarian instincts) is that it makes these broad, sweeping assumptions about the existence and given-ness of rights without offering any real foundational support. I used to be on an e-list with a couple of libertarians; I repeatedly asked them where do these rights come from, and they never had an answer for me; they just took as a given that you have the right to be left alone. So, I will ask that same question of any libertarians here: Where do these rights magically come from? What principled argument would you make to someone who said, “Oh, it’s interesting that you believe in the right to be left alone; you’re entitled to your opinion that that right exists, just as I’m entitled to my opinion that it’s complete nonsense”?

    1. That’s one of the clever things America did though – attached the concept of rights and liberty to our sense of nationalism: ‘WE hold these truths to be self-evident…’

      Even if rights are entirely subjective, we’ve nationally skipped all that epistemology and can move directly into arguing about their substance.

      1. Would you say that there are group rights, or are they all individual?

        1. IMO in the way popular parlance means, they are all individual, though individuals may elect to exercise some in concert.
          But YMMV on that point.

          1. I agree. I was asking because the identity politics that the left has long practiced (that would entail group rights if we embraced it, and to an extent we do) is also now coming from the alt-right, which seeks group rights for their preferred group(s) now as well. I don’t think there is an easy answer either. Thanks.

            1. Identity politics is not new. It’s as old as tribalism itself. Here’s an example that predates the modern left, however defined:

              “That in this free government all white men are and of right ought to be entitled to equal civil and political rights; that the servitude of the African race, as existing in these States, is mutually beneficial to both bond and free, and is abundantly authorized and justified by the experience of mankind, and the revealed will of the Almighty Creator, as recognized by all Christian nations; while the destruction of the existing relations between the two races, as advocated by our sectional enemies, would bring inevitable calamities upon both and desolation upon the fifteen slave-holding States.”

              1. Yes…and? That declaration from the CSA is what I am referring too as coming into vogue again from the farthest wing of the alt-right, the idea of identitarian “group rights” for whites. This idea of group rights, for any race, doesn’t sound as crazy as it once did because of the “group rights” that naturally flow from the identity politics of the left. But Sarcasto’s answer is still better, in that it’s better to think of rights as individualistic, though they may be used collectively.

                1. How did that “identity politics of the left” stuff survive white nationalism; the Muslim ban (and expansive promotion of Christianity); ‘we must have fewer brown people, and no Muslims, but more . . . Norway” and ‘the judge is a Mexican (from Indiana), so . . .’?

                  1. Your comment is stream of consciousness and contains no actual argument as far as I can tell. To comment on what I ballpark that you’re getting at, I’d say that rumors of the demise of identity politics on the left are greatly exaggerated, and on the right they are becoming more mainstream. What’s you’re point?

                    After years of lurking at the VC at the WAPO and being unable to comment because I didn’t want to pay to register, my first give and take with the (in)famous VC commentator Rev. Kirkland is a bit of let down.

      2. But “self-evident” implies that they are NOT entirely subjective.

  2. If I can ask more informed people than myself, is the filibuster unconstitutional? Each house of Congress is permitted to create their own rules, but the Vice President is entitled to vote in a tie. By crafting a rule that prohibits a tie, the Senate it looks like they voided a constitutional provision.

    US Constitution, Article I, Section 3, Clause 4
    The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.

    US Constitution, Article I, Section 5, Clause 2
    Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behavior, and, with the Concurrence of two thirds, expel a member.

    1. If there is no tie, then no problem. The constitution even has things built-in that require a 2/3 vote, so the concept of votes that are other-than-50% was not foreign.

      The founders understood traditional parliamentary law (which comes from British parliament); Jefferson even wrote a manual of parliamentary law that was the basis for the house or senate rules (I forget which). Various things in parliamentary law require supermajorities. One is overriding members’ right to speak before the final vote is taken. Since the senate has no limit on speaking, that gave rise to fillibusters that required (originally) a 2/3 vote to end. The house always had strict speaking limits, so the minority party can slow things down, but not halt them – but to override (end) that slowdown of every house member taking their maximum speaking time for a bill still takes a 2/3 vote in the house.

      IANAL, but I’d say ultimately these are “procedural” votes, not substantive votes, so they are entirely within the power of the house/senate to have in their rules. The house or senate could not make a rule that requires a 2/3 vote to pass a law – and, as the so-called nuclear option showed, a majority could, if they really wanted to, eliminate any supermajority procedural vote required.

      1. Sorry, by “substantive votes”, I mean final votes on a law.

      2. One more quick point: I think the senate appointments decision Noel Canning is pretty close to on-point for your question – as long as the senate can show any ability to transact business (however unlikely) then its own decision on when they are in and out of session control (and the president’s arguments to the contrary got exactly zero votes in SCOTUS).

        Why is that on-point? A filibuster (or cloture) technically just continues debate. It is not a yes/no vote on the actual bill. In practice it might kill a bill (or appointment), but it does not legally kill the bill – there is always a chance (however small) it can make it back for a full vote – thus the similarity to Noel Canning. That happens all the time with procedural votes. While a little different, I’ll bet dozens of politically sensitive legislature issues in your state (whether red or blue or mixed) die every year starting with a “Mr/Ms speaker/president, I move we refer this bill to committee”. Most of the time, referring a bill back to a committee is a quiet way to send a bill someplace to die without ever having to take a real yes/no vote.

  3. While I enjoyed the blog post about legal language and mathematics and learned something, it is oblique to the commentator’s point.

    We assert that there are things things called natural rights, that belong to us as individuals, and that they are protected by governments and not granted by them. But in reality, we only have the rights that the government is willing to give to us (even if it is a democratic government where the “we” is part of the governing system). The same is true, by extension, of what one stronger government gives as a right to another weaker government. So, that being said, the commentator’s point that we should not confuse rights with reality stands. Rights have no independent existence, and rights that are granted without reality, are not rights. The Soviet Constitution was full of positive rights about housing and pay, but they were nothing but words on paper. We’ve had this debate back and forth in America with the 14th Amendment making rights “real” or not, many times.

    Law professors and the legal profession enjoy the psychological payoffs of thinking that what they do matters, and what they do matters to an extent when no on is really paying attention about cases involving standards for truck mudflaps in interstate commerce (just an example), but ultimately courts are secondary to what the elected branches of the government are doing, and moving backwards a step of causality, what the culture itself, is doing.

  4. Speaking of arcs and tangents and rights…

    In the public policy scenarios below, is Gus equal to Bob? That is, do both Gus and Bob truly enjoy the same privileges and do both Gus and Bob have the same power to possess earned income? More importantly, are both Bob and Gus equal to both Alice and Jessebele? Or do all of the policy scenarios create a caste system? Do the “colors” (races, sexes, etc) of Bob, Gus, Alice, and Jessebele make any difference?


    Both Bob and Gus are contractual employees of Fred. Bob voluntarily hires Intercessor Alice to bargain with Fred.

    One of the following happen:
    (1) Both Bob and Gus benefit as a result of intercession.
    (2) Bob, but not Gus, benefits as a result of intercession.
    (3) Gus, but not Bob, benefits as a result of intercession.
    (4) Neither Gus nor Bob benefits as a result of intercession.

    And then one (or more) of the following happen:
    (A) Alice demands that Gus pay Alice.
    (B) Bob demands that Gus pay Alice.
    (C) Bob demands that Gus pay Bob .
    (D) Fred demands that Gus pay Alice.
    (E) Fred demands that Gus pay Bob.
    (F) The Sovereign demands that Gus pay Alice.
    (G) The Sovereign demands that Gus pay Bob.

    Gus compliantly makes the payment demanded. Gus voluntarily hires Intercessor Jessebele, Intercessor Alice’s equal in all respects, to bargain with Fred. Repeat (A)-(G), changing the names as appropriate .

    1. I feel that this is a truncated version of a larger piece, and I’m not fully getting it here. Is there a link to the original?

  5. I am pretty sure that by bring math and computer science into the analogy, you lost 99.975% of your readers. The rest were already convinced.

    Remember: Theorem #1 of a Math Texts is that readership is inversely proportional to the number of equations.

  6. Is it correct that I write that (right)^-1 is wrong or is literally 1 over right being correct? This post has me confused.

    1. like all languages, meaning depends on context. (right)^-1 could be wrong, or could mean the reciprocal of right.

      The key question is: if the reciprocal of right is wrong, do you want to be right?

  7. “When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean?neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master?that’s all.”

    I think it’s a communication challenge.

    I’ve found three words whose varying definitions can quickly delineate political perspectives:
    * right
    * fair
    * insurance

    For all three, the left-wing, communitarian definition is generally, “something the government pays for”.

    Libertarians generally can’t come up with a common definition for fair.

    Conservatives stick with the right=inalienable, and insurance=protection against rare occurences.

    1. Interesting, but I’m not sure this is quite correct. Just to start with one item, can you elaborate a bit on the “Conservatives stick with the right=inalienable” claim? (Part of the problem is that it isn’t completely clear to me what “inalienable” means in this context.)

      1. Referring to the Declaration of Independence, “endowed by their Creator with certain unalienable Rights”. (oops, I always get the un/in prefix wrong, not sure it makes a difference)

        My main point is that people in different parts of the political spectrum have such distinctly different definitions that communication is almost impossible.

        Conservatives generally stick with the definition of “right” that matches the Bill of Rights – restrictions on the government, and base that on the concept that these rights exist outside of government, or at least on the covenant between governed and government.

        The Left-Wing adds on what are sometimes called “positive rights”, e.g., things like a right to jobs, health care, housing, etc. Generally forms of redistribution.

        The conservative definition of “insurance” goes back to the roots of the concept – those thrifty Scottish Ministers banding together to provide against the unlikely but catastrophic events. The Left-Wing views insurance, again, as redistribution, a way for people to have their health care provided by society.

        The word “fair” is where the libertarian-ish cause problems. In general the Left-Wing settle on “equal outcomes”, the Conservatives “equal opportunity”, and the libertarian-ish respond with, “what does it mean to you?”

        The result is that the various factions can attempt to have conversations, but fail completely because the words mean completely different things.

        1. Well, if that’s what you mean, that I’m skeptical about your claim about conservatives.

          1. To begin with, it’s pretty clear that many well-established Bill of Rights provisions are seen as involving rights that arise out of the political institutions of our society, and not on natural rights of the endowed-by-their-Creator kind. Jury trial rights are a classic example — they come from Anglo-American traditions, as well as some judgment about practical utility, but they aren’t natural rights. Yet I’ve rarely seen conservatives deny that they are rights.

          2. Likewise, some provisions that few conservatives would deny are rights are positive rights — entitlements to government action. The right to trial by jury in civil cases, for instance, belongs to plaintiffs, who seek to get governmental enforcement of their underlying legal rights, as well as to defendants, who seek to protect their property from untoward government action. I know of no conservatives who doubt that. My sense is that conservatives also view the rights secured by the Contracts Clause as rights, and those are rights to governmental enforcement.

          3. Beyond that, I haven’t seen conservatives deny that a wide range of subconstitutional legal rights are rights — rights to recover for injuries to person or property, rights to get certain contractually provided benefits, and so on.

        2. My sense is that conservatives, liberals, libertarians, moderates, and pretty much everyone else are perfectly familiar and comfortable with all these broad conventional senses of “rights.” They may disagree about whether particular kinds of constitutional rights (e.g., a right to shelter, a right to a minimum income, etc.) should be recognized, and they may be more skeptical about some (not all) claimed positive rights than liberals are. But this has little to do with their general definition of the word “rights.”

  8. 1) [sin(x)]^-1 = 1/sin(x)–note that the ^-1operator applies to everything within the [ ]’s in the expression to the left of the equal sign. Convention dictates this.

    2) arcsin(x) is indeed the inverse function of sin(x). In the expression “sin^-1 (x)”, the ^-1 applies not to nominally numeric quantity sin(x)–see above, rather to the abstract non-numeric sin function itself.

    Used correctly by convention, both of these are unambiguous. Admittedly, just throwing out sin^-1(x) can be a head-scratcher, especially if context doesn’t help out.

    1. Well done. I was just going to say that but I wasn’t going to say it anything like as clearly.

      1. “Two plus two is four, minus one that’s three quick mafs”. -Big Shaq

  9. In math they try to make statements context-free. So for instance f(sup -1)(f(x)) = x regardless of whether f is a function or a variable whose multiplication is commutative.

  10. Another take on the question of rights: Is Health Care a Human Right?

  11. I don’t think this is actually right, as someone who has a degree in math and law.

    The superscript – symbol is effectively the inverse, and can be thought of in a way as a symbol denoting the “functional” inverse, that is, that the inverse of something acting on that same thing returns an “identity.”

    I.E. x^-1 is in used he the context of algebraeic functions, and means the multiplicative inverse. That is, you can think of x as a simplification of x * 1. This thus means that using the same functionality (multiplication), all x^-1 is really saying is x^-1 * x = 1 (the multiplicative identity function)

    In the same way, f^-1 is again just an inverse in a different context. But instead of an algebraeic functionality we’re talking about a function. So, thus f^-1 (f(x)) = x, the idea being that the inverse of the function acting o the function results in the identity function which is then applied to the variable.

    Again, similaraly sin(arcsin(x)) = x (if you go the other way it gets more confusing, granted, but essentially the product of the sin function acting on the arcsin function is, again the identity.

    In that way, the superscript -1 is always just denoting a type of inverse, and an inverse always returns an identity of some sort when it is acting upon whatever it is the inverse of.

    Forgive my imprecise math language, I haven’t written a proof in 12+ years.

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