Proportionality, Comparability, and Parity: A Discussion on the Rationality of Balancing

Abstract This article analyses the rationality of the principle of proportionality as a justificatory method for solving cases involving conflicts of constitutional principles. It addresses the “problem of comparability”: a set of arguments claiming that proportionalists fail to understand what happens when constitutional principles collide. The problem of comparability suggests that balancing cannot be done if some conflicts of constitutional principles are, in reality, cases of noncomparability, incommensurability, incomparability, or vagueness. In this article, I challenge the views of both proportionalists and their skeptics. Against the skeptics, I argue that proportionality can survive the challenge posed by the problem of comparability. Against the proportionalists, I submit that proportionality cannot be understood as a system of tradeoffs between degrees of satisfaction of principles. If comparison among constitutional principles is to be rational, we need a different approach to normativity—one that allows for the possibility of parity.


I. Introduction
Hard choices in law are rife and it matters how we deal with them.It matters especially if a non-skeptical approach to adjudication is taken, 1 and one, thus, is to understand legal reasoning as a special exercise of practical reason. 2 In this article, I address specifically hard choices concerning conflicting constitutional principles, 3 as the solutions to these are often particularly critical for our societies.Any time we hear about a controversial decision about a "conflict of rights" or "conflict of legitimate constitutional principles," we immediately become interested in the set of arguments justifying a court's decision.And we normally find ourselves in reasonable disagreement with others about the "correct" answer-or whether there is any answer, to begin with.
The predominant method for solving these kinds of cases is proportionality. 4As it is commonly understood, 5 the principle of proportionality conveys the idea that decisions in cases concerning conflicts of constitutional principles can be justified by balancing them.The justifying function of proportionality as balancing within the field of constitutional adjudication seems to be accepted in a wide range of scholarly consensus and judicial practice around the world. 6I take this to be an occasion for addressing the merits of such belief.This article is about the soundness of proportionality as a justificatory method for rights and principles adjudication.
The idea of balancing has intuitive force: if two constitutional principles conflict in a particular case, one may well think that what rationality requires is establishing their "weights" and deciding accordingly.If principle A has more importance than principle B in a particular case, it is justified to give preference to principle A. What could be wrong with this intuition?For some theorists, the problem is that it assumes that constitutional principles are easily "comparable"-that is, that one can always rationally determine how these evaluatively compare with each other.But that, they insist, is not the case when constitutional principles collide: how can you conclusively determine, for instance, that the right to life is more important than individual autonomy?For them, cases involving conflicts of principles are complex because comparing qualitatively different values is either impossible or at least deeply problematic.And any attempt to balance these values in practical situations would constitute a sort of "rationalization": trying to provide a rational explanation for a decision made on non-rational grounds. 7he "problem of comparability," as I term it, poses an important theoretical objection to proportionality.In its different versions, it can be taken to hold that the proportionalists are getting wrong what happens when constitutional principles collide, by believing they can easily compare conflicting constitutional principles when there is really no rational way of doing so.For the skeptics of proportionality, approaching conflicts of principles requires acknowledging that these are noncomparable, incommensurable, incomparable, or vague.The practical implications of this skepticism can 4 See Mathias Klatt, Balancing Rights and Interests: Reconstructing the Asymmetry Thesis 41 OJLS 321, 321 (2021).Earlier in history, a classical notion of proportionality would not mean balancing, but a "proper relation between aims and means."See THOMAS AQUINAS, SUMMA THEOLOGIAE, II-II, q. 64, a.7c.This sense of proportionality is no longer predominant.For a critical study on the shift of views  potentially be problematic for decision-makers and lawyers: either we tolerate the structural pathologies of proportionality because we do not have a better alternative 8 or we get rid of it and admit that the best judges can do in hard cases is relying on legislative virtue. 9But it seems to me that each of the options gives up too quickly on law's rationality.The former preserves the legal concept not because it is reasonable but because its unreasonableness is expedient, while the latter implies giving up on the judges' ability to rationally justify hard choices. 10Is there any way to avoid giving up the intuitive force offered by proportionality while properly facing the problem of comparability?
In this article, I propose an answer that challenges the views of both proportionalists and their skeptics.Against the skeptics, I defend the view that comparative judgments among conflicting constitutional principles are possible even in the hardest situations.Therefore, proportionality may survive the challenges posed by the problem of comparability.But against the proportionalists, I hold the view that balancing cannot be understood as a system of tradeoffs between degrees of satisfaction or nonsatisfaction of constitutional principles, for this commits us to a view of comparability that is unsound in evaluative settings.If comparison among principles is to be rational, we need a different view of comparability, one that does not unreflectively assume-like proportionalists and their skeptics do-that evaluative or normative comparisons proceed just like nonevaluative or merely quantitative comparisons, that is, in terms of a more, less, or equal amount of some attribute.Proportionality may survive the problem of comparability, but only to the extent that it leaves certain assumptions about normativity and comparability behind.To that end, I will follow Ruth Chang's comparativist account of practical reason and shall suggest that proportionalists should be able to adopt an approach to normativity that accounts for the existence of a fourth sui generis comparative relation: parity.The proposal is not novel.This article contributes to the existing body of work by scholars who have attempted to introduce the concept of parity into the discussion on the merits of balancing in rights adjudication-some to defend its rationality, 11 others to challenge it.12I will also take this opportunity to offer my views on the implications of parity for the concept of proportionality.
I develop my arguments in the following way.First, I provide an account of the notion of proportionality, putting particular emphasis on the stage of balancing.Second, I explain the "problem of comparability" and analyze whether the different kinds of arguments against proportionality are justified.Third, I defend that "parity" can explain better what happens in cases involving conflicting constitutional principles.Finally, I discuss the implications of parity for the concept of proportionality.Perhaps a reasonable alternative may be reformulating the dominant understanding of proportionality in a way it does not involve balancing.But then the problem is proposing an account as appealing as the one that is currently held and would have to be abandoned as a result.Martin Luteran proposes an alternative in supra note 5.But it is too early to make a call on this.Before looking for alternatives to proportionality, we need to examine whether it is a sound method for decision-making.

II. The Principle of Proportionality
This section deals with the idea of proportionality.I will not discuss all the accounts of proportionality that there are, or give a detailed account of the great variety of nuanced differences we can find among its proponents; rather, I will concentrate on Robert Alexy's account of proportionality.I do this for two specific reasons: first, because Alexy's is one of the most influential accounts of the principle of proportionality as it is understood today; 13 and second, because Alexy's approach to balancing is paradigmatic to the kinds of accounts some critics claim would not survive the problem of comparability (proportionality as "maximization"). 14Bear in mind, though, that the analysis of the proportionality principle presented here need not be understood as an analysis or critique of Alexy and no one else; this article is concerned with any theory purporting to hold what I think is the core idea of the Alexyan approach to balancing: that conflicts of constitutional principles are solved, at the end of the day, through judgments about tradeoffs in their degrees of satisfaction or non-satisfaction. 15n what follows, I address Alexy's account of proportionality, which is just a part of his broader theory of constitutional rights (TCR). 16For that reason, let us first point out the core ideas of the TCR, which provides a basic framework for explaining how to understand the principle of proportionality in general, and balancing in particular.

A. A Theory of Constitutional Rights
According to the TCR, fundamental rights and collective interests are principles 17that is, normative statements expressing permissions, commands, or prohibitions. 1813 See BARAK, supra note 6, at 5. See also Mathias Kumm, Constitutional Rights as Principles: On the Structure and Domain of Constitutional Justice, 2 INTERNATIONAL JOURNAL OF CONSTITUTIONAL LAW 574, 596 (2004); MÖLLER, supra note 6, at 15; Jamal Greene, Rights as Trumps?132 HARV.L. REV. 28, 61 (2018).14 See, URBINA, supra note 9, at 125; Caviedes, supra note 12, at 168.Urbina identifies two distinct approaches to proportionality.The first group, exemplified by Alexy's view, considers proportionality as a form of "maximization."Alternatively, there is another group that views proportionality as "unconstrained moral reasoning."Urbina argues that the "incommensurability objection," as he calls it, affects the accounts belonging to the first group.It is worth noting, though, that I will delve further into the discussion of the second group in Section V. 15 This core idea is, I believe, present in the works of BEATTY, supra note 6, at 164; Julian Rivers, Proportionality and Variable Intensity of Review, 65 CAMB.L.J. 174, 200-201 (2006); BARAK, supra note 6, at 340 ff.In addition, note that I am assuming that the principle of proportionality does not merely recommend comparison among abstract values.For it is apparent that balancing, in addition to the abstract weight the principles at issue have, also takes into account the concrete situations in which principles are interfered with.See Da Silva, supra note 11, at 286; Klatt supra note 4, at 333-336.For Alexy, principles differ structurally from rules. 19Principles are not just "general rules."Rules are definitive requirements that are applied by subsumption: if a rule applies, the normative consequence must follow.Principles, on the other hand, are optimization requirements-that is, norms requiring something to be realized to the greatest extent possible, given the factual and legal possibilities at hand. 20he distinction becomes more apparent when we look at conflicts of rules and conflicts of principles.Two rules providing for inconsistent legal ought-judgments cannot be applicable at the same time.A conflict of rules is thus solved either by formulating an exception to one of them or by declaring at least one invalid. 21Conflicts of principles do not operate in that way.If two principles provide inconsistent normative requirements, one of them must be "outweighed."Here it should not be considered that the principle outweighed is therefore invalid, nor that an exception has been built into it.In other circumstances, the question of precedence may have to be reversed. 22n every conflict of principles, then, a "conditional relation of precedence" must be established, as one principle is to take priority in the particular case.Consider, for instance, the statement of the German Federal Court in a case involving the right to life and bodily integrity, on the one hand, and the principle promoting the proper functioning of the judicial system, on the other.
[I]f there is a proximate continuing danger that, by continuing a criminal trial, the accused person will lose his life or suffer serious injury to his health, then continuing the legal process breaches his constitutional right under article 2 (2)(1) Basic Law. 23 this case, the ruling of the German Federal Court established a relation of precedence where, for the tribunal, it was justified to prefer the right to life and bodily integrity (P i ) over the principle requiring a proper functioning of the criminal system (P j ) in the particular circumstances where a person's life would be in danger should the trial continue.
But now the question is: How are we to determine such relations of precedence?How is "balancing" to be done?As it turns out, the justification of a relation of precedence between principles is a matter of applying the principle of proportionality.

B. The Principle of Proportionality
Alexy submits that the nature of principles entails the principle of proportionality and vice versa. 24The reason is that proportionality is required to determine whether an 19 For Alexy, both rules and principles are norms providing the basis for legal ought-judgments in terms of permissions, commands, and prohibitions.But the difference is not simply one of degree.TCR, supra note 16, at 46-47, and cf.Joseph Raz, Legal Principles and the Limits of Law, 81 YALE L.J. 823, 838 (1972).

24
TCR, supra note 16, at 66.For Alexy, the proportionality principle is entailed by the nature of rights and principles as a matter of conceptual analysis.However, one may go further and defend that the interference with a principle is justified. 25Proportionality is composed of three subprinciples: 26 suitability, necessity, and proportionality in the narrow sense.These subprinciples determine whether it is justified to satisfy one principle through a measure that, at the same time, interferes with another principle.The subprinciple of suitability requires that the measure is adequate to satisfy the principle it intends to satisfy.It excludes unsuitable alternatives. 27The subprinciple of necessity requires that the end sought "cannot well equally be achieved by the use of other means less burdensome." 28And the principle of proportionality in the narrow sense can be expressed by the "Law of Balancing": "The greater the degree of non-satisfaction of, or detriment to, one principle, the greater must be the importance of satisfying the other." 29Particular attention to the balancing stage will be paid here.
Balancing can be broken down into three stages.Assume a particular balance of two principles, P i and P j .The first stage is a matter of establishing the degree of nonsatisfaction of P i .The second is a matter of establishing the degree of satisfaction of P j .And the third stage is about judging whether the importance of satisfying P j justifies the detriment to P i .The way in which judgments are made in the first two stages is through a triadic scale representing different degrees of intensity.Thus, the degree of nonsatisfaction of P i (I i ) and satisfaction of P j (I j ) can be classified as either light (l), moderate (m), or serious (s). 30Alexy insists that, for these considerations to be rational, they must be capable of being presented in an inferential system.31Hence a "Weight Formula" is proposed.The simplest form is the one in which only I i and I j are considered: (W i , j stands for the concrete weight of the principle whose violation is being examined, which is P i ) In the simple formula, no consideration is given to the abstract weights of the competing principles (W i , W j ), nor the reliability of the empirical and normative epi-  TCR, supra note 16, at 102.Alexy sometimes refers to "degrees of interference with" or "importance in satisfying" constitutional principles.I will here adopt, as Da Silva (supra note 11, at 286), the strategy of using the terminology of "satisfaction" and "non-satisfaction."30 See Robert Alexy, On Balancing and Subsumption.A Structural Comparison, 4 RATIO JURIS 433, 440 ff (2003).To be sure, a triadic scale is not necessary.Balancing would be possible with just two grades of intensity.And one could even think of a scale with more than three grades (with the proviso that the number of grades is not too high, which would make classification too difficult).case for both principles (R i , R j ). 32If so, a complete form of the Weight Formula can be represented as follows: Graduation in abstract weights and empirical assumptions can certainly be of importance when analyzing concrete cases.For the sake of expositive simplicity, however, let us assume that constitutional principles' abstract weights will generally (at least in hard proportionality cases, which is our subject) be "the same" 33 and that we can dispense with the issue of epistemic uncertainty about the empirical and normative impacts of a given measure.Let us work only with concrete degrees of interference and importance and see how, according to Alexy's account, they determine relations of precedence.
Here we can work just with the classification (l)-(m)-(s), or attempt, like Alexy, a numerical representation by assigning numbers to each category, where: In the third stage of balancing, we should answer the question of whether or not the degree of non-satisfaction of P i is justified by the satisfaction of P j .If we follow the Law of Balancing, we will have three possible bundles of situations.First, we have a bundle of possible situations in which P i takes precedence over P j .The relation of precedence is numerically represented by a number higher than 1.
Second, there would also be a bundle of cases where P j takes precedence over P i .Here the relation of precedence can be represented by a number lower than 1.
32 Id. at 177.Alexy would even suggest that a "refined" formula would distinguish between empirical and normative assumptions.On the implications and characteristics of R-variables in Alexy's theory, which include their own triadic scale of "reliability": see Id. at 178-179, 185-188.

Legal Theory
Finally, there would be three possible situations of "stalemate," where there is no relation of precedence.
For Alexy, in stalemate situations balancing determines no result.It is permitted either to perform the measure in question (interference with P i ) or not. 34The legislator would enjoy the freedom to "decide as he [sic] wishes." 35Of course, decisions made by means of balancing can always be contested, although Alexy reminds us that contestability does not imply irrationality. 36Balancing can be considered a rational device if it is possible to justify the kind of judgments the Law of Balancing requires.The problem of comparability, in that sense, represents a problem for the rationality of balancing insofar as it challenges the idea that comparative judgments among fundamental principles are possible.

III. The Problem of Comparability
Proportionality may give rise to many theoretical problems, 37  it is misleading and potentially confusing.Incommensurability is a very specific phenomenon, and undue generalization to make it mean more than it does is not helpful.I therefore start this section by making what I hope to be a conceptual clarification of what comparability is and the very different circumstances in which comparison between two items is problematic. 38This exercise shall allow me to identify four specific problems of comparability.As it turns out, each of these subproblems can be thought of as a challenge to the idea that proportionality is the right way to think about hard cases involving conflicting constitutional principles.I shall examine whether-and to what extent-any of them succeeds in doing so.
Two items are comparable if, and only if, there is a positive comparative relation that holds between them.A standard-"traditional"-view of normativity assumes that the scope of possible positive comparative relations is exhausted by the trichotomy "better than," "worse than," and "equally good." 39Later in this article, I will argue that normativity need not be understood in terms of that trichotomy.But the challenge to the standard view can wait.The trichotomy, for now, may help to understand comparability by showing that relations of the kind "better than," "worse than," and "equally good" are positive relations because they describe what the relation between two items is, as opposed to what their relation is not.Note, in addition, that comparisons make sense only if they proceed with respect to a covering consideration.It makes no sense to say "X is better than Y, period." 40There must be some consideration in respect of which the items at stake are compared.We can compare, for instance, two nonevaluative items, such as apples and oranges, with respect to their "sweetness" or "nutritional value." 41Or we can compare evaluative items, such as two governmental policies, with respect to their "justice" or "expediency." 42here are different ways in which making comparisons between two or more items can be problematic.Perhaps you may think that comparing two items is problematic because the covering consideration does not "cover" them both.In such a case, we can say that such two items are "noncomparable." 43  By "nonevaluative items," I do not refer to items that cannot be evaluated in any way.One can sensibly say that there are "good apples" or "bad oranges" based on a relevant covering consideration.My use of the term "nonevaluative" refers to a much simpler idea of items being not morally significant in a choice situation.Perhaps there is a possibility that there are some choice situations in which deciding between apples and oranges may have great moral significance (think, for instance, of the story of Adam and Eve).But choice about which fruit to consume will not ordinarily have moral significance for ordinary people's lives.respect to which principles can be compared.They may assert that without a covering consideration, no comparison between constitutional principles can proceed.Call this the "problem of noncomparability." In a different sense, you may think the comparison between two items is problematic because they cannot be measured by a common scale of units of value.Some skeptics of proportionality thus raise what we can properly call the "problem of incommensurability."Proportionality, incommensurabilists maintain, gets conflicts of principles wrong because it assumes that principles can be measured by a single scale of units of value.But such an assumption would be an unhelpful analogy.In their view, the problem is that constitutional principles are incommensurable.One cannot pretend to determine, for instance, how many units of "constitutional importance" are needed for the right to privacy to be more important than the right to freedom of speech.
Another possibility is thinking that the comparison between two items in relation to a covering consideration is problematic because it is impossible.Here we can distinguish two kinds of impossibility.You may think, first, that two items cannot be compared because, in relation to a covering consideration, it is false that any possible positive comparative relation holds. 44The failure of comparison here is thus determinate.The problem, properly speaking, is one of "incomparability."Some critics of proportionality will thus assert that cases of conflicting principles are cases of incomparability.Proportionality is the wrong way to approach these cases because it assumes that one can determine that constitutional principles can be compared in terms of the trichotomy "better than," "worse than," and "equally good."For the incomparabilist, fundamental principles are so intrinsically irreducible that comparison between them must determinately fail.
But one may also think that comparison is impossible not because it is false that a positive relation holds between two items, but rather because it is neither true nor false that such comparative statements can be made. 45The failure of comparison here is indeterminate and the source of such indeterminacy, the "indeterminist" submits, is some sort of vagueness. 46The comparison between two given items would fail to hold either because the comparative predicates "better than," "worse than," and "equally good" are vague and unable to be applied in borderline cases.The proponents of the "problem of vagueness" may assert that proportionality gets conflicts of principles wrong because it pretends one can always determine a comparative relation among constitutional principles when such cases are really instances of vagueness.
In what follows, I will analyze whether these problems of comparability are capable of undermining the notion of proportionality.As it turns out, the noncomparability and incommensurability objections present no serious challenge to proportionality.Incomparability and vagueness, however, need to be taken more seriously.While I conclude that proportionality may survive these last challenges, it cannot do so untouched.

A. Noncomparability and Incommensurability
A noncomparability argument might be found in one of Webber's critiques of proportionality.He argues that, under the proportionality principle, "judges and scholars rarely identify a common criterion" for evaluating the weights of competing principles. 47In Webber's view, proportionality does not direct judges to "identify a value that ought to be maximized," but to optimize both principles in their own terms, without a common standard to measure their relative weights. 48For him, the problem of proportionality as a method is that "no common measure is appealed to, and, thus, no comparison of the intensity of the interference of one principle with another is possible." 49ut this is not a powerful critique.Alexy would reply that there is indeed a "common perspective," a covering consideration, under which comparisons are made: "constitutional importance." 50Of course, one might debate whether "constitutional importance" is a covering consideration or just a placeholder for whatever matters. 51e that as it may, it seems that the proportionalist would rapidly clarify that proportionality does consider a covering consideration that covers principles in conflict, whether or not one succeeds in determining it accurately.After all, the specification of a covering consideration need not be obvious.Substantive disagreement about what "constitutional importance" means is possible.Some candidates for covering considerations in hard choices concerning principles may refer to "justice," "common good," "fairness," "respect for human dignity," "utility," and so on.There is even the possibility that a more comprehensive value (perhaps nameless) that accounts for the comparability among the candidates can serve as the covering consideration in hard choices. 52hat about the incommensurability problem? 53Does proportionality assume commensurability?Yes, it does.Recall that Alexy proposes that the degrees of interference with, and satisfaction of, constitutional principles can be measured through the "triadic scale": light-medium-serious.These degrees of intensity are assigned numbers in his Weight Formula,54 such that the triadic light-medium-serious can 47 WEBBER, supra note 6, at 91.This might be in line with Webber's concern on Alexy's reply to the non-comparability problem.Webber seems to submit that the "point of view of the constitution" is not a helpful covering consideration.

53
A critique of proportionality along these lines has been advanced by Tsakyrakis, supra note 37, at 472-474.Timothy Endicott seems to raise this doubt as well.See Endicott, supra note 8, at 311, 316, where he states that, "The immeasurability of an interest or set of interests entails incommensurability between that interest and conflicting interests."By "immeasurability," he means the property that something that cannot be quantified has: cf.TIMOTHY ENDICOTT, VAGUENESS IN LAW 46 (2000).be represented numerically by the numbers 1, 2, and 4 respectively.Proportionality, in this sense, proposes an underlying cardinal measure in terms of which degrees can be grouped as "light," "medium," and "serious." 55he next issue is whether proportionality recommends commensurating what is, in fact, incommensurable.Here it seems that Alexy would concede that degrees of satisfaction and non-satisfaction of constitutional principles need not be understood in precise magnitudes.Indeed, he accepts that instead of a triadic scale, one could think in simpler or more complex scales of measurement. 56His insistence on a triadic scale, arbitrary as it is, does not seem to hold that constitutional principles are commensurable in the sense that they can be compared cardinally in a precise manner.I think he would accept that these magnitudes are imprecise in nature-that is, incommensurable.
But before adjudicating a win to the incommensurabilist, we need to ask whether incommensurability has the practical effect its proponents think it has.Indeed, it is often thought that incommensurability precludes rational choice between items.This would amount to the claim that incommensurability entails either incomparability or vagueness.But that is not the case.Incommensurability means only that precise cardinal comparison is not possible.But cardinal comparison between incommensurables can also be imprecise. 57Perhaps a triadic scale purporting to represent degrees of satisfaction and non-satisfaction of constitutional principles is an unhelpful metaphor, but Alexy is right to defend the intuition that principles can be interfered with, or satisfied in, greater or lesser degrees. 58We can certainly perceive that some restrictions are greater than others.We make no equivocation in judging, for instance, that imprisonment is a greater interference with personal liberty than a curfew.If differences in magnitude exist, the fact that such differences cannot be measured precisely does not preclude the possibility of comparison, for those imprecise 55 Klatt, in supra note 4, at 336, confusingly suggests that the triadic scale is not a cardinal scale but rather an ordinal one, and that this would, in his view, make proportionality compatible with Chang's comparativism.This is mistaken, as the triadic scale presupposes cardinality (albeit imprecise) and does not necessarily reject a trichotomous view of normativity.I shall properly develop this point later in Section IV.  (1990); Da Silva, supra note 11, at 283; Klatt, supra note 4, at 336.We need not, however, commit ourselves to such a view.Precise cardinality and mere ordinality are two extremes by which comparative structures can be thought of.Mere ordinal comparisons are indifferent as to whether differences between items are big or small.Precise cardinality, on the other end of the spectrum, requires us to be able to determine exactly how big or small differences are.But there is conceptual space for a middle course, and perhaps most of the comparisons we make are of this type.If items are imprecisely cardinally comparable, there is some magnitude to their difference.We may not be able to measure it in terms of a yardstick-type unit.But we may have more to say about it than just thinking of an ordered ranking.We may be able to tell whether those differences are big or small, or indeed if some item is somewhat better or much better than another.Imprecise cardinal differences are conceptually possible and widespread.For authoritative treatment on the concept of imprecise cardinality.See JAMES GRIFFIN, WELL-BEING 81, 96-98, 104 (1988); CHANG, supra note 40, at 28-33; DEREK PARFIT, ON WHAT MATTERS, VOL 2 556ff (2011).cardinal differences can be great enough to allow for the possibility of determinate comparative relations.
To make sense of this idea, imagine being asked to compare the size of two clouds that seem to be at the same altitude "just by looking at them." 59 Assume now that the two clouds, due to constant and changing winds, move across the sky changing their shape, sometimes stretching and sometimes shrinking.Just by looking at them, it may not be possible to say for sure exactly how big these clouds are, whether reference to their size is made in terms of the area of the sky covered or the volume they occupy in the atmosphere.But let us further suppose one of the clouds is clearly bigger than the other despite how much their corresponding shapes vary over time.We may not be able to say, just by looking at them, precisely how much the difference is in terms of a scale of measurement.But we nevertheless could describe the difference effectively.One may, for instance, come up with a ratio approximating the magnitude of the difference: "The big cloud is between 50 and 70 percent bigger than the small cloud."Or one may instead appeal to other entities to describe the difference: "The big cloud is as big as a regular house, while the small cloud is, at best, as big as a regular car."Be that as it may, it makes sense to assert a comparative relation: the "house cloud" is bigger than the "car cloud."As it turns out, the fact that we cannot precisely measure the items in comparison does not entail incomparability because we can tell whether one is bigger than the other.Similarly, such imprecision does not entail vagueness or indeterminacy either, because this is not a borderline case where the clouds are so similar in size that sometimes, say, the cloud on the right seems bigger and sometimes the one on the left does.
Alexy's account of proportionality should thus be understood as a method for the imprecise comparison of principles.His triadic scale would play the role of a theoretical device, justified on practical grounds, to represent magnitudes as if they were precise. 60The proportionalist may concede, therefore, that degrees of satisfaction and non-satisfaction of constitutional principles are imprecisely cardinally comparable in reality.But she would nonetheless propose that proportionality's proposed measurement scale, whether simpler or more complex than a triadic scale, is a theoretical attempt to approximate the value of these magnitudes.This might, I believe, serve as a sufficient response to the incommensurabilist.

59
For the sake of the example, I shall ask the reader to imagine one is asked to measure the size of two objects "just by looking at them."I do this to avoid a potential objection to the example.In reality, clouds are commensurable.We can tell for sure how big or small they are in terms of a scale of measurement.We can measure them in terms of the area they cover, the volume they occupy and even the number of water particles of which they are composed.However, by stipulating that we are asked to compare them "just by looking at them," I aim to set a scenario where the measurements can only be imprecise.While this stipulation does not eliminate the fact that clouds are in fact commensurable, I hope it can have enough metaphorical value to make sense of the idea of how comparisons between incommensurable items is nevertheless possible.After all, I am not concerned here with comparison of physical entities, but rather normative ones.60 Alexy seems to maintain this view when, in the TCR, supra note 16, at 99, he claims that, in respect to the idea of balancing, "one cannot produce a firm answer on the basis of reliable quantification; rather the outcome-however it is determined-can only be illustrated numerically."It seems, then, that Alexy accepts that his model cannot be a model of precise cardinality.Although he confusingly insists, later on, on proposing a cardinal triadic scale that renders precise magnitudes.
It therefore seems that the noncomparability and incommensurability arguments do not pose a serious challenge to Alexy's account of proportionality.At best, the incommensurability objection may push the proportionalist to admit that the triadic scale for the measurement of degrees of satisfaction and non-satisfaction of constitutional principles is an artificial device for the measurement of imprecise evaluative differences.But none of these problems warrants the claim that comparability in hard cases is precluded.

B. Incomparability and Vagueness
1. Incomparability Incomparability, recall, occurs if two items cannot be compared because, in relation to a covering consideration, it is false that any possible positive comparative relation holds between them.On a traditional trichotomous view of normativity, incomparability amounts to the claim that, in relation to a covering consideration, the traditional trichotomy fails to hold between two items.The intuition that the trichotomy sometimes fails to hold can be explained by the "Small Improvement Argument."According to this argument, "a small improvement in one of two items, neither of which is better, does not necessarily make the improved item better." 61It may be possible that, for two items A and B, if (i) A is neither better nor worse than B with respect to V, (ii) A+ is better than A with respect to V, but (iii) A+ is not better than B with respect to V, then it follows that A and B are not related by any of the traditional trichotomous relations.
An argument against proportionality based on incomparability is submitted, I believe, by Urbina in his A Critique of Proportionality.He presents his objection in the following terms: [T]wo things are incommensurable with respect to X when X is not a property by which they can be compared quantitatively, that is, X is not a property by which it can be judged that one of the things is (overall, net) more or less X than, or just as X as, the other-whether or not there is a unit of measurement that can express X. 62 Urbina's definition implies incomparability because it expresses a failure of comparison between two items with respect to a covering consideration "X." 63 Notice that he restricts the conceptual scope of comparability to the trichotomy "better than," "worse than," and "equally good."In Urbina's opinion, alternatives are incomparable when, according to the relevant criteria for assessment, they rank differently according to their attributes. 64One is "better" in some relevant respects and the other is 61 See Chang, supra note 39, at 667-668.

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URBINA, supra note 9, at 40. 63 Urbina's objection is not strictly speaking one of incommensurability.By saying that incommensurability is such "whether or not there is a unit of measurement that can express X," he seems to admit that a failure of comparison can occur regardless of whether or not two items are precisely cardinally comparable -that is, commensurable.better in some other relevant respects, but neither seems to be at least as good as the other overall-that is, in all the relevant respects. 65In other words, two items are incomparable if there is a failure of the traditional trichotomous relations to hold.
Under such an understanding of incomparability, Urbina submits that the proportionality principle unduly recommends comparing the incomparable.Hard choices of conflicting principles would be cases of incomparability insofar as one alternative is better in one relevant respect (it optimizes P i ), the other is relevant in another relevant respect (it optimizes P j ), yet neither seems to be at least as good as the other overall (neither optimizes in a higher degree, say, "(P i , j )" because P i and P j are irreducible to each other.By irreducible, Urbina means that P i and P j are values of a different kind, thus assuming that no improvement in P i can compensate ("replace," "substitute," "count as") an improvement of P j : "a house does not become pretty by being very big, or big by being very pretty." 66For Urbina, the problem is that there is "no unifying property" that captures all the relevant respects for determining how P i and P j compare in relation to each other.He submits that "(P i , P j )," which could also be understood as "what is optimal," is not such a property.
Notice that Urbina's critique is a mixed one.The core of his argument is one of incomparability, as it expresses a failure of the trichotomy to hold in relation to a covering consideration.But the argument also includes an appeal to the notion of noncomparability, as it contends that there is no unifying property, that is, a covering consideration, that captures all the relevant respects to determine how constitutional principles compare with each other.With respect to the problem of noncomparability, one can simply recall what has been said above.The problem of noncomparability can be responded to by asserting that what matters in cases involving conflicting constitutional rights cases can be specified.Urbina may be right in protesting that doing "what is optimal" is not a useful covering consideration for comparison.He would be wrong, though, to suppose that a more comprehensive consideration-perhaps nameless-does not exist.
But what about the incomparability argument?Is it really the case that hard choices involving constitutional principles are cases in which the alternatives are incomparable?It seems to me that the problem lies, strictly speaking, in assuming the trichotomy thesis.If Urbina's argument is based on the idea that the irreducibility of conflicting principles yields a failure of the trichotomy to hold, we can object to his critique by showing that (1) the irreducibility of constitutional principles does not entail the failure of the trichotomy to hold, and (2) the failure of the trichotomy to hold does not entail incomparability.
The claim that the failure of the trichotomy to hold is not entailed by the irreducibility of alternatives can be supported by what Chang calls "nominal-notable" comparisons. 67How could you compare Mozart and Picasso in terms of creativity if each one is creative in completely different respects?The comparison seems impossible because Mozart and Picasso are both "notable" bearers of very distinct aspects of creativity.But you could certainly say that Mozart is clearly more creative than a nominal painter, "X," who bears similar properties to Picasso but in a much poorer way.If

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X is comparable with Mozart, and he bears the same kind of properties displayed by Picasso, we could then imagine a continuum of small improvements of X.We can assume X+, who is slightly better than X, would still be worse than Mozart, and therefore comparable.Chang's "chaining argument" rests on the intuition that small evaluative differences do not trigger incomparability where before there was comparability. 68If we continue our chain of small improvements of X, assuming these small differences will not trigger incomparability, we will reach a point in which X-Prime would be both as creative as Picasso and still comparable to Mozart.Could we then keep thinking that Mozart and Picasso are incomparable? 69Consider Alexy's example of what I would call a "nominal-notable" proportionality case.In 1997, the German Constitutional Court dealt with the issue concerning the tobacco manufacturers' duty to place health warnings with respect to the danger of their products. 70Two principles were in conflict: on the one hand, the legitimate objective of protecting the health of the population; and on the other, the freedom "to pursue one's profession."This is a case in which one may reasonably judge that the non-satisfaction of the tobacco manufacturer's right was "nominal" and the importance of protecting the population from health risks was "notable."Sure, tobacco manufacturers were bound to modify the packaging of their products and introduce a warning about the risk of consuming them.But their product was certainly not banned, which could have been a rather "serious" interference.On the other hand, it seems that taking positive action towards protecting the health of the population was indeed of great-"notable," "serious"-importance, and therefore one would not be mistaken in judging that, overall, the duty to place health warning was justified in virtue of the importance of the legitimate objective pursued and despite certain interference with the tobacco manufacturers' right to freely conduct their business.Now, freedom and health may well be very different values, and perhaps, as Urbina suggests, these are irreducible to each other.But if the Tobacco case makes intuitive sense, and we can accept that nominal interferences with one principle can be justified in virtue of the notable importance of pursuing another one, then Urbina would have to explain where the difference is between nominal-notable and notable-notable cases of proportionality.If comparison-that is, judgments of proportionality-seems adequate in the former, why assert incomparability in the latter?
Urbina has provided a response to a notion similar to that of nominal-notable comparisons.In his A Critique of Proportionality, he addressed what he called "Large/Small trade-offs,"71 which are defined as "cases where our common-sense 68 Chang, supra note 39, at 673-679.See also Caviedes, supra note 12, at 181-185.69 As Chang points out, one possible objection to the chaining argument is that it seems to have the form of a sorites.But such an objection would be well grounded if the failure of the trichotomy to hold was the result of the semantic indeterminacy of the comparative predicates.But showing that hard cases are not vague cases will be dealt with later.
intuition is that we decide by commensurating what incommensurability theorists would consider incommensurable values or principles." 72For Urbina, the problem with these cases is that their appeal is grounded solely on intuition.And mere intuitive appeal would not be a sufficient reason to abandon an argument against proportionality unless there is a positive account of how moral reasoning proceeds, in his words, by "commensurating what incommensurability theorists regard as incommensurable." 73t is important to note that Urbina's argument concerning large/small tradeoffs attacks the idea of commensurability, not comparability.Indeed, Urbina accepts that it is not his claim that it is impossible or unreasonable to choose between incommensurables. 74Instead, he argues that we cannot commensurate principles by quantifying degrees of satisfaction and non-satisfaction. 75To the extent that his argument deals with the idea of incommensurability, I believe recalling what has been discussed in the previous subsection would suffice as a response.However, the core of Urbina's concern is to be taken seriously.For even if his argument against large/small tradeoffs is not a claim of incomparability, it is true that more than mere intuitive appeal is needed to validate the comparability of irreducible principles.In that regard, notice that Chang's chaining argument appeals to more than mere intuition.Chang's account provides a positive account appealing to intuition, logic, and argumentation to assert that small differences linking nominal-notable and notable-notable cases are not sufficient to trigger incomparability where before there was comparability.Hence, I would submit that it is the incomparabilist who would bear the argumentative burden of counter-arguing the chaining argument.A potential objection is claiming vagueness, but I shall deal with this problem in the next sub-section.
Another potential argument from the critics of balancing could be that, in nominal-notable proportionality cases, choosing between incomparable items is possible because one of them is an "absolute consideration," such as, for instance, the absolute prohibition of torture.Some, like Waldron, view these cases as examples of "weak incommensurability," 76 while others, like Endicott, argue that they are instead cases of "radical incommensurability." 77But notice that weak (or radical) incommensurability implies lexical superiority of the conclusive consideration over any competing consideration.If the prohibition of torture is an absolute 72 URBINA, supra note 9, at 64.

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For clarity, here I am assuming that the possibility of rational choice is entailed by comparability.This is because I subscribe an approach to practical reason under which the normative force of rational choice is given by comparative facts.I have no space to defend this claim in depth, but one can consult Chang, supra note 40 for a persuasive defense of what she calls "indirect comparativism."Therefore, even Urbina's claim (URBINA, supra note 9, at 66) that choice between incommensurable values can be possible only relative to a particular life commitment entails precisely the possibility a comparing the importance of these values in relation to that particular life commitment, which would play the role of covering consideration in such scenarios.Endicott, supra note 8, at 322.
Legal Theory consideration, it is precisely because it is comparable with its competitors: it is lexically superior. 78t seems, then, that while failures of the traditional trichotomy to hold are possible, it is not because of the irreducibility of the alternatives at stake.If it seems wrong to assert relations of precedence between two principles pursuing the realization of conflicting constitutional principles, it is not because the principles at stake are incomparable as such.We can find nominal-notable examples in which comparisons between degrees of satisfaction of each principle make sense.
But how do we explain what happens in notable-notable proportionality cases?Perhaps constitutional principles are not per se incomparable.But one could still protest that in hard notable-notable cases of proportionality the trichotomy fails to hold.If one principle is not clearly more important than another in a given situation, it is difficult to maintain that the alternatives must be "equally important."In such a scenario, we could imagine that a small improvement in each alternative would break the "stalemate." 79But legal reasoning is more complex than that.In "stalemate" situations, we cannot assume, with indifference, that each alternative is "evaluatively the same."We can rightly insist that alternatives are indeed evaluatively different.Thus, if alternatives are not incomparable, and yet there is a failure of the trichotomy to hold, we are faced with the following problem: either the alternatives are comparable by virtue of a fourth sui generis relation or the comparative predicates "more important," "less important," and "equally important," when relativized to a covering consideration, are vague.

Vagueness
Some legal theorists submit that a failure of the trichotomy to hold may be a result of vagueness in the comparative predicates.Consider Endicott's example about what "vague incommensurability" is: Colors are vaguely incommensurable because of their complex components.A patch of turquoise may be neither bluer than a patch of navy nor less blue nor equally blue.If the turquoise becomes gradually more green, or if the navy becomes darker, one will eventually be less blue than the other.The two may never be precisely as blue as each other.The property of blueness 78 Lexical superiority is characteristic of what Chang calls "emphatic comparisons", see CHANG, supra note 40, at 113-116.In a similar fashion, the claim that it makes intuitive sense that the notable importance of protecting a principle justifies a nominal non-satisfaction of another could be challenged by the idea that certain category of principles, like rights, have a special force.Tsakyrakis (supra note 37, at 473, 492-493), for instance, would argue that, due to the special force of rights, no satisfaction of a public goal would ever justify the non-satisfaction of an individual right.But this would, in my estimation, be yet another case of an "emphatic comparison," where an alternative is lexically superior to another.As it turns out, subscribing to an understanding of rights as trumps does not entail incomparability.Rather the opposite: it asserts emphatic comparability, as comparing rights and public goals will always render a practical situation where, in relation to what matters, the importance of satisfying a public goal would never be other than "nominal" when compared with an individual right, the importance of which would never be other than "notable."
involves hue and saturation and brightness, and these properties are vaguely incommensurable. 80r Endicott, incomparability is the result of vagueness in the specific comparative predicates.Notice that his account is consistent with the possibility of "clear cases" of comparison.We can think of some pairs of patches in which one is definitely "bluer" than the other, or where one could assert that they are equally blue.But there will be some borderline cases where it will be indeterminate whether a patch is "bluer than" the other.The reason for the indeterminacy, Endicott maintains, is that the comparative term "bluer than" is vague.Some hard proportionality cases would present the same "vague incommensurability."For Endicott, R (Quila) v Secretary of State for the Home Department 81 could serve as an example.In the Quila case, the measure under examination was Rule 277 of the Immigration Rules, which provided that a marriage visa would not be granted unless both partners were at least twenty-one years old.The objective of the rule was to protect young women from family pressure, motivated by the prospect of getting a visa, and thus deter forced marriage.But it was debated whether such a regulation disproportionately affected the right to family life of eighteen-to twenty-year-olds whose marriage was not forced.Two interests were conflicting: on the one hand, the legitimate objective of deterring forced marriage; and on the other, the legitimate interest of young voluntary couples who, for legal reasons, would have to delay the entry of one of the partners to the country.On this case, Endicott asserts: It makes sense to say that the impact of public action on a protected interest is less important than the attainment of the purpose of the action.It may be definitely true or definitely false in some cases, and neither definitely true nor definitely false in others.Some public actions have a detrimental impact on family life that is clearly less important than the pursuit of some public purposes, and some public actions have a detrimental impact on family life that is clearly more important than the pursuit of some public purposes. 82r Endicott, the Quila case exemplifies vague incommensurability insofar as there would be a range of possibilities in which, in some cases, it would be clear that the Immigration Rules are either disproportionate or not, and some others in which it would be indeterminate-that is, neither true nor false-that there is a disproportionate detriment to the right to family life.This is because if the law's impact on family life is significant and the practice of forced married is virtually unaffected, the measure would clearly be disproportionate.Similar certainty would hold if the age requirement for getting a marriage visa would have a great dampening effect on the practice of forced marriages and be practically no detriment to the family life

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of young voluntary couples: the measure would then be clearly proportionate.But the problem, Endicott suggests, is that it is also possible to have cases in which, despite having all the relevant information concerning the actual impact of the law in both the situation of young couples and the practice of forced marriages, there would be no conclusive reason for determining which alternative is preferable to the other.There are some cases in which, despite having all the relevant information concerning the dispute, there would be no conclusive rational grounds for judging that a good effect would be greater than a detrimental impact. 83ndicott's account of vague incommensurability suggests a view of vagueness.Notice that he accepts that there may be easy cases of proportionality-those in which it is clear whether a measure is disproportionate.But he believes there would be hard cases in which the trichotomy fails to hold: it would not be clear that a legitimate policy is more or less importantor even equally importantthan the right to family life of young partners affected by the law.He considers that in such cases there is vagueness, and no rational conclusion concerning how those interests, and the way they are affected, compare with one another regarding "what is proportionate."But is it really the case that hard cases concerning constitutional principles are cases of vagueness?
If hard constitutional cases really are cases of vagueness, we would have to accept that the only practical response available in such cases is arbitrary stipulation.If there are no conclusive reasons for determining how the alternatives relate to each other with respect to what matters in the case, one could just stipulate that one is better. 84hat arbitrary stipulation is the practical response to vagueness, however, comes with a problem for the indeterminist.What Chang calls "resolutional remainder," I believe, can dissuade us from taking it as an adequate explanation of notable-notable proportionality cases.The resolutional remainder suggests that arbitrary stipulation never intrinsically resolves a hard choice situation.If, in a practical problem, one stipulates that A is better than B with respect to V, one has not settled the normative questionthat is, the substantive problem of what normative relation holds between A and B. Vagueness accounts cannot accommodate the resolutional remainder that is left after stipulation takes place.
Here, the indeterminist would reply that, nonetheless, her theory can rely on extrinsic factors to offer a resolution of hard choices in a way that is not arbitrary.It seems to me that Endicott offers such an extrinsic justification of stipulation in proportionality cases: 83 Id. at 321.

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Notice that arbitrary stipulation is recommended by semantic and metaphysical accounts of vagueness.The former recommends semantic stipulation ("if the predicate 'constitutionally more important than X in the case at stake' is vague, let us just sharpen it up to the point in which we can stipulate that it means that family life is constitutionally more important than the policy pursued by the Immigration Rules").The latter recommends metaphysical stipulation ("from the many possible worlds, let us just consider that the world in which family life is constitutionally more important than the policy pursued by the Immigration Rules is the actual world").Perhaps we can say that the judges' power to balance the unbalanceable is not arbitrary (in the pejorative sense of arbitrariness that is relevant to the rule of law), where it is necessary, for good legal purposes, that judges should have that power. 85e extrinsic reason for allowing arbitrary (non)resolution is, for Endicott, that it is often necessary, for good legal reasons, that courts "often pull off such impossible feats." 86But this does not eliminate the resolutional remainder.Indeed, it does not even address the problem.It ignores it because of the belief that solving it is impossible.But the resolutional remainder is a serious issue for law's rationality, for it is intrinsic to the practical choice situation: it will remain even when the problem is artificially settled by virtue of extrinsic factors, such as the legally granted discretion to stipulate the decision.
To understand better what the resolutional remainder is and why it is a problem, we need to distinguish two different kinds of normative problems.The first normative problem concerns two particular constitutional principles in a problem case.Call it the "first-order normative problem."Imagine, like the indeterminist does, that a hard proportionality case is a vague case.You would believe that there is no way to rationally determine which principle is to prevail, for it is indeterminate whether any of the trichotomous relations hold.Such a belief would give rise to a different, "second-order," normative problem: What is to be done in the face of "unresolvable cases?"The second-order normative problem may be very complex, for many courses of action can be taken in the face of unresolvable cases.For example, one could acknowledge the impossibility of solving the first-order normative problem, refrain from stipulating and compensate both parties: "The system owes you an answer, and when it cannot give you an answer you shall be compensated."One may also authorize the parties to settle the dispute by themselves: "The system owes you an answer, when there is an answer.If there is none, the system must at least ensure that the disputing parties can come up with a negotiated solution without having recourse to violence."Or one may, as Endicott submits, stipulate a decision because judges should have the power to resolve the otherwise unresolvable for good legal purposes: "The system owes you an answer, and when there is none, the decisionmaker shall stipulate the matter and make her decision the answer.After all, the function of law's authority is to settle coordination problems."One may argue the merits of any course of action intended to answer the second-order normative problem.One may argue which one fits better the practice of legal systems and is better justified as a practical solution.But whatever one may think is the best solution for the secondorder normative problem, one is to bear in mind that such a solution is a solution for the second-order normative problem, not for the first-order one.This is the idea behind the resolutional remainder: the solution for the second-order normative problem is not a solution for the first-order normative problem.And pretending it can be so is problematic because judges do not justify their decisions by arguing that the case is vague and thus unresolvable.Here I would agree with Dworkin's 85 Endicott, supra note 8, at 325.

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observation that judges justify their decisions based on the normative force their choice purportedly has. 87s it turns out, the indeterminist may have to think of a further argument if they are to insist on vagueness as an explanation of hard proportionality cases, although I can imagine that some legal theorists may feel satisfied with Endicott's account of hard proportionality cases as cases of vagueness.After all, the legal system does not claim the ability to solve every normative question that may arise, and therefore the legal theorist does not have the burden of offering an account of the intrinsic resolution of hard proportionality cases.Hence, it might be thought that while vagueness may not be the best possible explanation of hard proportionality cases, it is nonetheless good enough.But what if the resolutional remainder is not invincible and hard choices concerning constitutional principles are capable of rational intrinsic resolution?In the following section, I submit that a more satisfactory account of hard proportionality cases can be proposed by adopting Chang's approach to normativity.

IV. Rethinking Hard Choices Concerning Constitutional Principles
Can the principle of proportionality survive the challenges posed by the problem of comparability?Is it true that proportionality rests on mistaken assumptions about how normativity works in evaluative comparisons?We have seen in the last section that there are some cases, which I call nominal-notable proportionality cases, in which it seems conceivable that judgments concerning the degrees of satisfaction or non-satisfaction of irreducible constitutional principles make sense.Proportionality seems to deliver as a method for solving conflicts of principles in "clear" proportionality cases.However, it has also been seen that there are some cases, which I have called notable-notable proportionality cases, in which the trichotomy fails to hold.These are not cases of stalemate-that is, cases of evaluative equality.If that were the case, judges could just flip a coin and decide without rational regret.But it matters what and how judges decide.If the trichotomy fails to hold, how do we explain what happens in some of the hardest proportionality cases?Some theorists suggest that those cases are vague.If so, judges would have no conclusive reason to establish the normative relation and answer which principle should prevail.And in the face of indeterminacy, judges would be authorized, for good legal reasons, to arbitrarily stipulate the choice.We now need to ask ourselves whether we should be satisfied with this explanation and live with the resolutional remainder or attempt a different, more satisfactory one.
In this section, I will join Da Silva, Klatt, and Caviedes in submitting that parity is worth being introduced into the discussion on proportionality.I will defend the following idea: hard proportionality cases-that is, cases where the trichotomous relations fail to hold-are instances of a fourth comparative relation ignored both by proportionalists and their critics: parity (see Section IV.A).If there is room for parity in hard constitutional choices, the principle of proportionality may survive the problem of comparability and retain its status as a method for resolving conflicts of rights.However, since parity requires adopting a different view of normativity, proportionality cannot survive untouched, for it is built on the same assumptions on which its critics rely (see Section IV.B).If my suggestion is correct, I will ultimately discuss some implications for the concept of proportionality resulting from the possibility of parity in Section V.

A. The Possibility of Parity
Part of the argument for believing there is a fourth comparative relation was implicitly presented in the previous section.Indeed, we can take the analysis of the problems of incomparability and vagueness as providing an argument from elimination for the possibility of parity.The argument can be reconstructed in three steps. 88First, it was shown that sometimes the traditional trichotomy fails the hold.This is most likely to be the case in notable-notable cases of proportionality, where it seems wrong to judge that one constitutional principle is more important than the other in a particular case and where assuming evaluative equality would lead to the counter-intuitive implication that whoever is in a position to break the stalemate could simply flip a coin an decide on either option without rational regret.Second, it was shown that the failure of the trichotomy to hold does not entail incomparability.For every notable-notable case of proportionality, one could imagine a nominal-notable case of proportionality where comparison is not problematic.One could then imagine a continuum of small improvements linking the nominal-notable with the notable-notable case.This "chaining argument" rests on the intuition that, in principle, a small improvement does not trigger incomparability where before there was comparability. 89Therefore, if the trichotomy sometimes fails to hold, and at least in some cases that failure does not entail incomparability, there is conceptual room for a fourth positive comparative relation.Chang calls this fourth relation "parity."The third stage of the argument involves showing that a failure of the trichotomy to hold that does not entail incomparability is not just a borderline application of vague predicates.Our assessment of vagueness has shown that we have reason to believe it is not the most satisfactory account of what happens in hard proportionality cases.
But showing how parity can better explain what happens in hard proportionality cases may seem mysterious.That, I believe, is because we have so far been viewing normativity through the lens of the standard trichotomous approach.We have done so because such an approach to normativity is the one assumed by both proportionalists and their critics.If we are to understand parity, we need to leave such a view behind.As it turns out, parity can indeed explain the hardest cases of conflicting constitutional principles.However, the theoretical benefits of parity cannot come without sacrifice.If proportionality is to endure, it must relinquish certain mistaken assumptions that arise from adopting the standard traditional view. 9088 The argument for the possibility of parity is not my own.Here I am relying on Chang's argument as developed in supra note 39. 89 Cf.Id. 673-679.90   In this important sense, I disagree with the assessment provided by Da Silva in supra note 11, as he believes that the principle of proportionality can accommodate without problem the concept of parity.But, as I will argue, such an approach is not right, for the principle of proportionality is just as guilty of adopting the traditional view of normativity as its skeptics.
An alternative view of normativity requires, first, thinking of comparability not in terms of positive evaluative relations but in terms of evaluative differences.If two items have evaluative differences, then they are comparable.If there is no evaluative difference, not even a zero difference, 91 they are incomparable.Thinking in terms of evaluative differences makes the notion of parity more apparent.Chang submits that evaluative differences can be understood in two axes: magnitude (whether a difference between two items is zero or nonzero) and direction (whether a difference is biased or unbiased).The traditional trichotomy can thus be explained in the following terms: • If A is better than B with respect to V, the difference between them is nonzero and biased towards A. • If A is worse than B with respect to V, the difference between them is nonzero and biased towards B. • If A and B are equally good with respect to V, the difference is zero and unbiased.
Parity can be explained, in these terms, as a nonzero-unbiased difference.If A and B are on a par, none is better than the other-that is, the difference is not biased towards either of them, yet the difference is nonzero.But how can this be possible?
The explanation requires leaving what I call the "nonnormative sting" behind.

B. Leaving the Nonnormative Sting Behind
Thinking that evaluative comparisons proceed in terms of items being "better than," "worse than," and "equally good" may be due to the fact that nonnormative comparisons proceed in terms of items being "more than," "less than," and "equal amount of" some relevant attribute. 92Unreflective assimilation of normative comparisons to nonnormative comparisons may unnecessarily restrict the way we understand practical reason in two respects.First, it may restrict the scope of evaluative differences to cardinality.And since nonevaluative comparisons are, generally, capable of precise cardinal measurement, one may further think that therefore evaluative comparisons should generally be able of a precise determination.This is perhaps why theorists such as Alexy think the principle of proportionality needs-for it to be justified-a scale of intensities that allows for numerical representation. 93But the evaluative realm does not work in that way.In my assessment of the problem of incommensurability, it has been suggested that cardinality can also be imprecise.Indeed, I believe Alexy would agree that his triadic scale is an idealization, justified on practical grounds, of a model that attempts to achieve satisfactory approximation to magnitudes that, in substance, are not capable of precise measurement.
On the alternative view of normativity, imprecise cardinality need not be hidden under the mask of commensurability.Indeed, imprecise cardinality explains why unbiased nonzero evaluative differences-that is, parity-are intuitively possible.It should not be shocking to acknowledge that normative considerations and evaluative properties are manifested in the world with some sort of imprecision.Values like beauty, justice, and humility would, when instantiated in the world, manifest themselves inexactly. 94And if this is the case, we have at least one ground to distinguish normative comparisons from nonnormative comparisons.If at least some normative considerations are "born" with such imprecision, we cannot, in principle, take for granted that the trichotomous relations exhaust the scope of comparative analysis.Nonzero unbiased evaluative differences could, in principle, be possible.If so, it might be thought that parity is a sort of "rough" or "imprecise" equality.The difference between items on a par is not zero, but the reason why the difference is nonetheless unbiased is that the options "oscillate" in the "same region" of value.
Parity, however, is not to be understood as reducible to equality, 95 for it has a different logical structure. 96Although parity and equality are both "symmetric" (if A is on a par with B, then B is on a par with A), they differ in that only the former is "irreflexive" (A is never on a par with itself) and "non-transitive" (if A is on a par with B, and B on a par with C, it does not follow that A is on a par with C). 97 If parity is not-just-a sort of imprecise equality, it is right to think of it as a sui generis comparative relation.
The other reason why the nonnormative sting unnecessarily restricts the way we understand practical reason has to do with the structure of normativity.For parity goes even beyond the notion of imprecise quantitative measurement.In more recent work, Chang has made it clear that imprecise cardinality is just a way in which parity might arise.The notion of parity is much deeper, and its explanation is not to be restricted to the domain of cardinality. 98For Chang, what makes parity a distinctive comparative relation is the very structure of normative considerations.The evaluative realm is not manifested only in terms of magnitudes: evaluation is certainly not only a matter of how much value things have.The alternative view here proposed takes into consideration that values have also qualitative manifestations.In her words: Parity holds in the normative realm because qualitative differences in normativity give rise to a fourth basic way in which items can compare.Two qualitatively diverse items that are nevertheless in the same "neighborhood" of value are neither better than one another nor equally good.They are on a par.The idea of unbiased nonzero evaluative differences, or parity, can indeed be explained by appealing to the fact that values present magnitude as well as quality or significance. 100Consider, for instance, love.One may sensibly hold that love is manifested in greater or lesser degrees of intensity.But we can also think that there are manifestations of love that are qualitatively different.Love can be "romantic," "innocent," "obsessive," and so on. 101In that way, parity may arise not as a result of cardinal (whether precise or imprecise) comparisons, but as a result of qualitative differences that cannot be represented by the trichotomy better than, worse than, or equally-even "roughly equally"-good. 102In these cases, we could properly say that two items are on a par because they are "qualitatively different." In light of what has been said so far, parity is not only a possible comparative relation; it potentially is the comparative relation holding in hard cases involving conflicts of constitutional principles.Leaving the nonnormative sting behind might help us to better understand the nature of parity.Now it is time for parity to help us examine the rationality of proportionality.
V. Some Practical Implications of Parity for the Concept of Proportionality If parity is the comparative relation governing some of the hardest proportionality cases, what follows from that fact?The final section of this article discusses what I believe are parity's implications for the concept of proportionality.

A. Does Parity Undermine the Concept of Proportionality?
Under the proposed approach to normativity, both quantitative and qualitative aspects determine the overall value one item has with respect to what matters in a choice situation.Evaluative differences are sensitive to the different quantitative and qualitative manifestations of the items at stake, and perhaps this is what critics like Urbina are concerned about when criticizing theories like Alexy's.By adopting a traditional trichotomous approach, Alexy's account pays much attention to the quantitative aspect in which principles are manifested in particular cases, and little to no attention to their qualitative manifestations.Such inattention, I believe, makes balancing unconvincing whenever a notable-notable case of proportionality arises.Why assume the "tradeoff" game recommended by the Law of Balancing applies indistinctively in all conflicts of principles?Again, if all that matters is magnitude, we could make a reasonable case for such a method.But the reality is that principles also manifest qualitative aspects.One cannot assume that any light interference with a principle will be justified whenever there is a medium or serious concern about satisfying another principle. 103One could imagine situations in which, though a measure can be classified as a light interference, it nonetheless seems qualitatively more serious than that.Consider a measure imposing an "inheritance tax.Assume the tax rate imposed is minimal.Some may rightfully think of it as a light interference with the right to property.But reasonable disagreement about the quality of such interference can be found.Some may think it "just," others "insulting," and yet others even "trivial."And notice that these judgments can be made without even thinking about the value on the other side of the balance-which we can assume is, in any event, a legitimate concern for the common good.
The qualitative dimension of constitutional principles creates many complexities.I would thus agree with Caviedes that we cannot assume, as Da Silva does, 104 that parity arises only in what Alexy calls "stalemate" situations. 105If we take seriously the qualitative dimension of normativity, we could have parity in cases in which our analysis of the degrees of satisfaction of the principles at issue renders a relation of precedence.This would entail a seemingly paradoxical scenario: we could have situations where (1) according to the law of balancing, the trichotomy holds, yet (2) it also fails to hold, as the constitutional principles involved are considered on a par.
Of course, the paradox is merely apparent, because in (1) only the quantitative aspect of the items at stake is taken into account, whereas in (2) both the quantitative and the qualitative aspects of the items at stake are considered.But if we are ready to accept that the normative world manifests both quantitative and qualitative dimensions, the proportionalist would be faced with a serious mismatch between theory and reality.For balancing would not account for the concept of parity and would thus fail as an account for decision-making.
The proportionalists, if they were to insist on their account, would have to modify their method to incorporate qualitative differences into it. 106This is not necessarily impossible.For one could propose a system capable of determining the value of principles that manifest both quantitative and qualitative aspects. 107Proponents of proportionality may try to come up with such a method, thus replacing the Law of Balancing with an alternative account where, in relation to a covering consideration, one may be able to determine whether one principle is more/less important than, as important as, or on par with, another principle.However, while theoretically plausible, this strategy seems to me practically dispensable for two reasons.First, while it is true that coming up with a theoretical replacement for balancing would make comparisons more precise, it would make decision-making much more complicated, and perhaps unnecessarily so.Once the idea of parity is accepted and its nature understood, one could find and justify the existence of parity without the need for a highly developed and complex method to confirm what sound judgment and practical experience may achieve.And second, modifying the formula would not contribute to determining what course of action is to be taken when two items are on a par, limiting its utility to that of comparative diagnosis.And as I shall explain in the following Here is where I shall respectfully disagree with Caviedes, since he seems to maintain that parity has a similar invalidating effect on proportionality than incomparability.For him, parity renders balancing inapplicable.See Caviedes, supra note 12, at 189-190.107   Chang, for instance, has proposed a componential-organic account for determining the structure of values and the comparisons among bearers of such values that (1) takes into account quantitative and qualitative differences, and (2) does not assume cardinality.See CHANG, supra note 40, at 10-25.
sub-section, the resolution of parity cases takes more than mere comparative diagnosis.
Now, how about abandoning a "maximizing account of proportionality" like Alexy's and relying instead on a more flexible account of proportionality as "unconstrained moral reasoning"?Urbina suggests that what makes these accounts different is that they do not approach balancing as a method for the commensuration of units of value, but simply as an authorization for "balancing reasons." 108Under such a view, proportionality would be an authorization to engage in "open-ended balancing" 109 without the need for a method to calculate normative weight. 110Without an explicit tie to cardinality, one may think, this type of proportionality accounts would be free of the problem of comparability.Moreover, thanks to the flexibility of the appeal to "general practical reasoning," these accounts could also incorporate the concept of parity.
The argument is plausible, but it requires further clarification to support its conclusion.It should be noted that relying on "general practical reasoning" does not inherently exclude a purely cardinal or quantitative view of comparability.Indeed, to the contrary, the strength of the appeal to general practical reasoning lies in its adaptable nature, which allows for the inclusion or exclusion of any approach to comparability.If that is the case, I would not say there are conclusive reasons to believe that appealing to general practical reasoning is sufficient per se to overcome the problem of comparability, for there are many practical theories-perhaps most of them-that adopt the trichotomy thesis and therefore would be ready to assert noncomparability, incommensurability, incomparability, or vagueness before considering the possibility of parity.This is why I would not hastily conclude that proportionality theories understood as "unconstrained moral reasoning" are free from the problem of comparability.To reach that conclusion, an intermediate step is necessary: abandoning a trichotomous approach to normativity and accepting the possibility of parity.As it turns out, it is the concept of parity, rather than an appeal to general practical reasoning, that would equip a theory of balancing with the conceptual tools to face the problem of comparability.
I would thus conclude the following: the principle of proportionality, either as "maximization" or as "unconstrained moral reasoning", may survive the problem of comparability, but only insofar as a trichotomous approach to normativity is ruled out and the possibility of parity is acknowledged.As Greene suggests, "A commitment to proportionality does not suppose a commitment to any particular model or formulation."See Greene supra note 13, at 60. relation between two items.Chang suggests that when two items are on par, it is rationally permissible to choose any of the alternatives.But how is this different from what happens when two items are equally meritorious, incomparable, or vague?When two items are equally good, one could choose whichever option without rational regret. 111hen two items are incomparable, one could choose either option based on personal subjective preferences. 112And if the comparison is vague, one could just estipulate an answer arbitrarily. 113One question becomes imperative: Why does all this matter if, at the end of the day, one can choose either of the alternatives?
It matters a great deal whether hard cases involving conflicting constitutional cases are instances of parity, rather than equality, incomparability, or vagueness.It matters because the practical attitudes and possibilities of the decision-making authority are substantively different.When two items are evaluatively equal, picking whichever option is rationally permissible, as there are sufficient reasons for choosing between two items the evaluative difference of which is zero.As Chang puts it, one might as well just flip a coin without rational regret.When items are incomparable, the scenario is not one of rational comfortability with the choice.Rationality is paralyzed, for there are no reasons to pick between the incomparable alternatives.There are no reasons even for flipping a coin.Rational agency plays no role, and hence a choice can only be made by virtue of arational considerations, like purely subjective emotional or existential preferences.And when the comparison between two items is vague, the practical response of the decision-maker, as stated above, would be that of believing that the first-order choice situation is indeterminate (irresolvable) and thus arbitrary external stipulation would be permissible. 114arity, however, suggests a different practical response.When two items are on par, rationality must actively be exercised.The decision-maker is thus called to exercise their rational agency to provide a resolution.Chang suggests there are two ways in which rational agency can be exercised in parity cases.The first is committing to one of the alternatives.By committing to one of the alternatives, the decision-maker creates a reason for choosing it by willing so.It is an act of self-constitution in which one "puts herself behind" a consideration and thus creates the normativity for choosing. 115Now, I am aware that the possibility of robust creative normative powers can be controversial and may seem mysterious (even radical!) in the context of morality.However, within the context of legal reasoning, it seems to me plausible.In fact, the image of judges flipping coins, or doing what they feel like, or stipulating an answer extrinsic to the choice situation, 116 may seem either disturbing or strange.On the other hand, the more solemn and recurrent idea of judges voting in a court of appeal-literally putting their will behind a proposed ruling for a hard caseseems not only acceptable but also the right thing to do.And it seems justified because decision-makers' normative powers are, at least in the technical legal domain, Legal Theory justified by legal reasons.A second way in which rational agency could be exercised in parity cases is by drifting.One could abstain from the self-constituting action of committing and drift into one alternative in virtue of an existing consideration that counts in its favor. 117Imagine, in that sense, a judge who, faced with a hard choice, drifts into the alternative most coherent, overall, with the legal system because coherence with the existing law is a reasonable consideration in favor of a proposed ruling.
I believe the difference between commitment and drifting offers an important layer of conceptual refinement.A justified choice among alternatives on a par certainly involves making evaluative judgments.Sunstein, in that sense, has submitted that resolution of hard choices requires from the agent an act of self-understanding 118 -that is, an act of constructive interpretation (in the Dworkinian sense) 119 about what the agent values most.But the resolution of hard choices goes beyond acts of understanding and evaluation.Hard cases are hard because it takes will to resolve them.The difference between commitment and drifting expresses, even if poorly, a certain gradation in the way will operates in decision-making: either you create a reason for choosing by committing, or you just drift into one existing consideration and settle the matter without changing yourself.In some practical contexts, perhaps, the difference between a committed decision and a non-committed one might be trivial.But surely such a difference matters in some other contexts.At this point, repeating this article's introductory sentence might be appropriate: hard cases in law are rife, and it matters how we deal with them.

C. A Challenge Arising Out the Existence of Parity
Finally, if parity cases require active rational agency through the exercise of will, this seems to raise a shift in the diagnosis of the hardness of cases involving conflicting constitutional principles.Once the existence of parity is acknowledged, the difficulty is not so much determining what comparative relation holds between two constitutional principles as manifested in a particular case.Once we allow for the possibility of parity, most hard proportionality cases could be intuitively identified as cases of parity.The hardness of these situations is therefore, volitional, 120 as the decisionmaker now faces the question of whether to commit or to drift when carrying out their duties to settle disputes on behalf of the state.
In light of this situation, a significant challenge arises for the legal system: the problem of "authority allocation."The problem of authority allocation groups different concerns about how legal institutions are to act in the face of choices involving conflicting constitutional principles that are on par.Should judges always commit?Could judges, on certain occasions, just drift?Is the legislature better positioned than the judiciary to commit on behalf of the state?The answer to these questions goes beyond the discussion on the soundness of proportionality.For the problem of authority allocation is not a problem about which method should we use to justify the choice, but about who should have the authority to make hard choices.It requires substantive argument about the best political arrangement to best satisfy constitutional values that, on occasions, may conflict and render hard choice situations.It is, therefore, beyond the scope of this article to answer and discuss in depth whether it is best to leave questions of constitutional adjudication in the presence of parity to the judges or the legislature.Da Silva, for instance, holds that parity is a ground for judicial deference in favor of the legislature. 121But such a conclusion may be hasty.There might be considerations of democracy for leaving the hardest parity cases to the lawmaker, but I wonder if there are no competing reasons of, perhaps, substantive justice to think the courts might also play a role in the state's institutional commitments. 122The problem of authority allocation is a serious one.And perhaps addressing it could best be achieved in a separate study engaging fully with the substance of different political, constitutional, and moral theories.
In any case, the best I can do for now is to sketch a view I have defended in the past 123 and that, although in need of revision, I do not find entirely inconsistent with my current ideas.I would propose a system of mixed-authority allocation where judges and lawmakers would share the authority to commit in different circumstances and according to their institutional roles.Judges, I think, could commit or drift depending on their assessment of a particular case.They could commit when, for instance, what matters is an issue of substantive justice that could not be left to the intricacies and tradeoffs of legislative deliberation.But I also think they should be at least allowed to drift on occasions when they find themselves incapable of committing, perhaps on grounds of humility, to a certain kind of resolution that is far from definitive.That way, if judges were confronted with a case where two constitutional principles were on par, they could drift into a solution they deem sufficiently justified and honor their duty to solve the case at hand, but without having to determine with the authority of precedent that the choice made there is the choice any judge should make. 124In those cases of judicial drifting, the legal system would then defer to the legislative apparatus to discuss the state's stance on certain evaluative conflicts and achieve, preferably and in the absence of judicial commitment, at least a democratic decision. 125I am aware that this hypothetical system may leave many dissatisfied.121 Da Silva, supra note 11, at 300.

122
Here I would agree with Kumm's observation that democratic processes sometimes suffer from pathologies that judicial review might help to address.See Kumm, supra note 109, at 163.And perhaps there are more intelligent ways of arranging the legal systems (even existing ones) to incorporate all the possibilities arising out of the possibility of parity.However, the problem of authority allocation may serve at least to incentivize further scholarly discussion on how legal systems should operate in hard proportionality cases where normative powers need to be exercised to justify rational choice.

VI. Concluding Remarks
In this article, I have tried to analyze the extent to which the principle of proportionality can survive the many challenges posed by the problem of comparability.Proportionality accounts like Alexy's may survive the challenges posed by the problems of noncomparability and incommensurability; however, it struggles much more when notable-notable cases of conflicting principles appear and the traditional trichotomy fails to hold.The tradeoff game recommended by Alexy's account of proportionality would not be able to accommodate the nature of parity, as it adopts a trichotomous approach to normativity that unnecessarily restricts its scope of analysis to cardinality, thus ignoring the possibility of qualitative evaluative differences.The proportionalist, therefore, should be able to leave the "nonnormative sting behind" and try to account for the possibility of parity.This is not impossible, for proportionality may be understood more broadly as encompassing a sort of "open-ended" practical reasoning.
As it turns out, parity comes with conceptual advantages.It allows for rational justification in even the hardest practical scenarios.And it accounts for practical attitudes and powers that judges more likely have when making hard choices.However, the adoption of parity also presents a challenge.The allocation of authority becomes a crucial issue when granting normative powers for making difficult decisions.Political and philosophical discussions surrounding which institution should bear this authority, and whether to commit or drift, take on significant importance.Drawing strong conclusions at this stage would be premature.A mixed system of authority allocations is a cautious suggestion that will need to be explored further. 5

B
. The Relevance of Parity for Decision-Making in Hard Choices Involving Conflicting Constitutional Principles What is the significance of the existence of parity for decision-making?Does it make any difference?It all comes down to how choice is made when parity is the governing 108 Francisco Urbina, A Critique of Proportionality, 57 AM.J. JURIS.49, 66 (2012) 109 Mattias Kumm, The Idea of Socratic Contestation and the Right to Justification: The Point of Rights-Based Proportionality Review, 4 LEHR 141, 146-147, 150; Moshe Cohen-Eliya and Iddo Porat, Proportionality in the Age of Populism, 69 AM.J. COMP.L 449, 454 (2021). 110
around proportionality, see Martin Luteran, The Lost Meaning of Proportionality, in PROPORTIONALITY AND THE RULE OF LAW: RIGHTS, JUSTIFICATION, REASONING 21 (Grant Huscroft et al., eds., 2014).See AHARON BARAK, PROPORTIONALITY: CONSTITUTIONAL RIGHTS AND THEIR LIMITATIONS 457 (2012), asserting that "we now live in the age of proportionality."See also T. Alexander Aleinikoff, Constitutional law in the Age of Balancing 96 YALE L.J. 943, 944 (1987), DAVID BEATTY, THE ULTIMATE RULE OF LAW 163 (2004); KAI MÖLLER, THE GLOBAL MODEL OF CONSTITUTIONAL RIGHTS 99 (2012).And, despite being a critic of proportionality, see GRÉGOIRE WEBBER, THE NEGOTIABLE CONSTITUTION: ON THE LIMITATION OF RIGHTS 71 (2009) 71. 6

stemic assumptions concerning what the measure in question means in the concrete necessity of proportionality for rights adjudication is not only analytical or conceptual, but also normative
or substantive.For a proposal of such a view, see Mathias Klatt, Proportionality and Justification, in CONSTITUTIONALISM JUSTIFIED: REINER FORST IN DISCOURSE 159 (Ester Herlin-Karnell et al., eds., 2016).
It can be challenged, for instance, that proportionality amounts to no more than mere rationalization, or that it unduly avoids moral issues by claiming neutrality.For an example of the former, see FINNIS, supra note 7, at 94; and for an example of the latter, see Stavros Tsakyrakis, Proportionality: An Assault on Human Rights?7 INT.J. CONST.LAW.468, 474 (2009). 37

So, if asked to compare Beethoven's Symphony N°5 and Da Vinci's Mona Lisa with respect to "musical beauty," you may reasonably raise the point that, as "musical beauty" does not cover the Mona Lisa, it and the Symphony N°5 are noncomparable. As it turns out, some skeptics of propor- tionality may deny that proportionality gets right what happens when
Parity, Imprecise Comparability and the Repugnant Conclusion, 82 THEORIA 182, 186-188 (2016).Chang's difference between precise and imprecise cardinality is an illuminating one, for one might think, like Luban and Klatt, that comparisons are either precisely cardinal or ordinal.Cf.David Luban, Incommensurable Values, Rational Choice, and Moral Absolutes 38 CLEV ST L REV 65, 66-7 56See Alexy, supra note 30, at 445. 57 See Ruth Chang,