Encyclopedia of the Philosophy of Law and Social Philosophy

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Alexy’s Theory of Rules and Principles

  • David DuarteEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-94-007-6730-0_380-2


Notwithstanding a mere classification of norms, in its plainest and initial understanding, Alexy’s distinction between rules and principles is the cornerstone of a systematic comprehension of law, encompassing multiple and diversified assertions that compound what might be qualified as a complete theory of law. As the starting point of the principles theory, Alexy’s contraposition between rules and principles has implications in various legal fields, such as, and just naming a few, the theory of norms, the role of proportionality, the methodological operations in the application of law, or the theory of rights. If one takes into account the new framework created in legal science, its large scope, the amount of research made underneath, and its diffusion, Alexy’s principles theory can even be seen as a paradigm shift in legal science.

The Opening Criterion of Distinction

Alexy’s distinction between rules and principles is based, at a first level, on a criterion regarding the type of satisfaction each kind of norms has when applicable. This means, under this criterion, that principles are norms that can be satisfied in varying degrees, depending on the grade of satisfaction not only on what is possible as a matter of fact, empirically, but also on the limits given by other norms, no matter if other principles or rules (Alexy 2000: 295). Therefore, and since their applicability is just externally limited by the factual and legal possibilities present, principles are norms that require something to be realized to its greatest extent. They are, thus, and for this reason, optimization requirements (Alexy 2002: 47). Differently, rules are norms that are always satisfied or not. If a rule is validly applicable, it requires exactly what is defined in its content, without any margin of variation while governing a case. From this follows that rules, as Alexy puts it, are fixed points on the field of the factually and legally possible (Alexy 2002: 48).

Among the various classifications legal science uses to distinguish rules from principles (Pino 2016: 72), the distinction in Alexy has to be qualified as a strong one. It is a qualitative distinction, separating rules from principles through their substantial properties or, in other words, on the basis of a distinct categorical nature. Accordingly, it does not express a mere quantitative difference, as it is typical in weak distinctions, frequently supported on different grades of generality purportedly recognized in rules and principles (Verhej et al. 1998: 3). In another hand, Alexy’s distinction may also be qualified as “exclusive,” in the sense that it exhausts all norms conceivable within a legal order; the so-called “Exclusion Theorem” (Bäcker 2011: 57). The exclusive character of the distinction in Alexy implies, consequently, that any norm is necessarily a rule or a principle (Alexy 2000: 295).

Rules and Principles in Conflicts of Norms

The first-level criterion is complemented by other criteria. One of them is the way both kinds of norms have their conflicts solved. In what regards rules, when more than one are applicable but have incompatible legal consequences, only two outcomes are admissible: (i) one exception in recognized in one of the rules; (ii) one of the norms is qualified as invalid (Alexy 2002: 49). Differently, when principles collide, a solution is obtained through balancing: principles with different legal consequences are weighted under the circumstances of the case at hand, and one principle takes priority against the other. This means that when principles collide, no issue of validity is at stake, as no exception is recognized in any norm, given the fact that principles in collision just lead to a relation of precedence (Alexy 2002: 50). Obviously, and in opposition to rules, a principle can take precedence in one case and be defeated in another.

From this follows that, for Alexy, rules and principles have distinct prima facie characters (Alexy 2002: 57). While principles, as optimization requirements, are not definitive commands because their applicability to a case depends on the weighting carried out with the opposite principle, rules are not definitive commands only for the reason they are subject to accommodate exceptions, whenever this follows from an opposite principle, directly or because it supports the incompatible rule (Alexy 2002: 58; Brożek 2012: 209). Taking into account this specific prima facie character of rules, Alexy frequently stated the broader and global opposition of principles as optimization requirements versus rules as definitive commands (Alexy 2014: 512).

Criticism and Refinement

The understanding of principles as optimization requirements faced some relevant criticism (Aarnio 1997: 27; Sieckmann 2011: 29). The main one was that optimization requirements are commands that can be realized or not, which means that they cannot be satisfied in various degrees: on this basis, the concept of optimization requirements would not be suitable to define the difference between rules and principles (Aarnio 1997: 27). This has led to the subsequent distinction between commands to be optimized and commands to optimize: the former describes the ideal ought, which is what has to be transformed in a real ought through optimization, while the latter just refers to something that has to be done, in the sense of a command not to be optimized but fulfilled by optimization (Alexy 2003c: 109). This distinction clarifies, thus, what is meant by optimization requirements. They are commands to be optimized, a refinement that does not justify, however, the abandonment of the initial concept, given the fact that it expresses in an altogether straightforward way the nature of principles (Alexy 2000: 300).

Within this framework, the concept of ideal ought became central for the distinction between rules and principles, since a correspondence between rules and real ought, on one hand, and principles and ideal ought, on another, is carried out (Alexy 2011: 16). Thus, if “Op” is a rule, then it expresses a real ought, where to do “p” is required without taking into account other opposite duties or rights; it is a definitive ought. However, if “Op” is a principle, then it expresses an ideal ought, which is an obligation not yet relativized to the actual and legal possibilities (Wang 2010: 47). It is a prima facie (or pro-tanto) ought, more appropriately represented as “O Opt Op,” where both optimization and the object of optimization are mandatory. As an ideal ought (Oi), a principle is only a reason for that definitive ought, but it implies a definitive ought when entering into a normative conflict (Alexy 2011: 24). This logical structure of principles has been also criticized, namely, on the logical function of the element “Opt” (Alonso 2016: 54).

Subsumption and Balancing

The distinction between rules and principles has a major projection on the operations performed to apply law to a case: while rules define the consequence for a case through subsumption, a deductive scheme, principles are applied by balancing, an arithmetic scheme (Alexy 2003a: 433). Despite the fact that Alexy presents these operations, subsumption and balancing, respectively, connected to the opposition between rules and principles, it is important to notice that they are not mutually exclusive: even though submitted to balancing, principles have their application also dependent on a previous subsumption (Alexy 2011: 22).

Balancing is the specific basic operation for the application of principles. Regardless, now, how principles are precisely weighted in a concrete case, structurally balancing will always confront two or more principles with incompatible legal consequences under some factual circumstances. Given that there is no absolute precedence between principles, at least at the same formal level, balancing is exactly the operation used to define the conditional prevalence of a principle against another in a strictly circumstantial basis (Alexy 2002: 52). Hence, balancing will have the outcome of a legal consequence, the one assigned to the prevalent principle, for the facts that have called the collision between the principles at hand. This is what Alexy has designated as the “Law of Competing Principles,” formulated as follows: “the circumstances under which one principle takes precedence over another constitute the conditions of a rule which has the same legal consequences as the principle taking precedence” (Alexy 2002: 54). Thus:
  1. 1.

    If P1 takes precedence over P2 in circumstances C and if P1 has the legal consequence Q, then the rule of the case is C→Q.


Balancing and Proportionality

Based on principles as optimization requirements, the theory of principles establishes a close connection between principles and proportionality. In Alexy’s own terms, principle theory implies the principle of proportionality, and the principle of proportionality implies principle theory (Alexy 2000: 297). The consequential meaning of this is that principles are applied under the normative framework given by the three subprinciples of proportionality: (i) appropriateness, (ii) necessity, and (iii) proportionality in a narrow sense. The first and the second regard the factual possibilities of a principle as an optimization requirement, expressing a Pareto-optimality. Thus, if a measure M interferes with P1 in order to achieve the satisfaction of P2, but it is not suitable for that purpose, then the optimization of P1 and P2 imposes that M has to be abandoned: it is not appropriate. A similar scheme is valid for necessity. If measure M1 interferes with P1 in order to achieve the satisfaction of P2, but there is another measure M2, that satisfies P2 similarly with a lower interference in P1, then the optimization of P1 and P2 requires the adoption of M2 instead of M1. As these examples show, proportionality, in these two subprinciples, is only attending the empirical possibilities of satisfying P2. With proportionality in a narrow sense, limitations at the level of the legal possibilities are established. Since the fulfillment of a principle is detrimental to the satisfaction of another, proportionality in a narrow sense defines the legal limits of interference. And this is done through the “Law of Balancing,” more precisely the “Substantive Law of Balancing,” which reads as follows: “the more the interference in one principle, the more important the realization of the other principle” (Alexy 2000: 298).

Proportionality in a narrow sense does not only encompass a command on the limits of the substantive reasons underlying the interference. It also entails the “Epistemic Law of Balancing,” known as the second law of balancing, regarding directly the conditions of empirical certainty that have to be present in a decision of precedence between principles: it reads “the more intensive an interference in a constitutional right is, the greater must be the certainty of its underlying premises” (Alexy 2002: 419). Thus, this means that proportionality in a narrow sense also covers the empirical basis of the situation that called the collision. It defines, hence, the limits of admissible uncertainty, reproaching interferences not sustained in consistent grades of empirical certainty.

The Structure of Balancing and the “Weight Formula”

Proportionality in the narrow sense, particularly the “Substantive Law of Balancing,” shows that balancing can be divided in three stages: (i) establishing the degree of non-satisfaction of a principle, (ii) establishing the degree of satisfaction of the competing principle, and (iii) assessing the question of whether the importance of satisfaction the competing principle justifies the non-satisfaction of the first (Alexy 2003b: 136). Given that the first two stages imply the measurement of intensities regarding the satisfaction and the non-satisfaction of the competing principles, a scale is created to express the levels of interference: light, moderate, and serious (Alexy 2003a: 440). Moreover, and adopting of a geometric sequence of 20, 21, and 22 (1, 2, and 4) to each levels of the scale, from this follows that a formula can be conceived for the purpose of evaluating the concrete weight of a principle in a specific case: W i,j = I i /I j , where I i stands for the value of the intensity of interference with a principle P i and I j stands for the value of importance of the competing principle P j , assigned with the hypothetical intensity of interference caused by omitting the interference in P i . (Alexy 2003a: 444). Being this the most elementary version of what was called the “Weight Formula,” it works as a tool to define the outcome of balancing: if the quotient is higher than 1, it means that P i takes precedence to P j ; if the quotient is lower than 1, it means the opposite. If, however, it leads to a stalemate, then the competing principles are a scenario of structural discretion.

Despite the relevancy of intensities of interference for the definition of conditional precedence, balancing also convenes the evaluation of the abstract weight of principles (Alexy 2003a: 446). This comes from the assessment that principles can have different weights – or importance – independently from the concrete case that provokes the balancing, with, for this specific reason, possible autonomous consequences on the balancing outcome (Pulido 2006: 108). With the addition of a third pair of variables representing the “Epistemic Law of Balancing,” the most common version of the “Weight Formula” is obtained: besides intensity of interference and the abstract weight of principles, the equation also includes, even though with a distinct scale of values (20, 2−1 and 2−2 or 1, 1/2, and 1/4), the variables representing the reliability of the empirical assumptions underlying the case (Alexy 2003a: 447). In this version, the formula is

\( {W}_{i. j}=\frac{W_i{I}_i{R}_i}{W_j{I}_j{R}_j} \)

This is what might be called as the “normal” version of the “Weight Formula.” This version, however, was also a starting point for other versions. Particularly: (i) the “Extended Formula” and (ii) the “Refined Complete Weight Formula.” The “Extended Formula” has the same structure and encompasses the same variables; however, it accommodates the cases in which there is more than one principle in one or both sides of the conflict. Thus, and just in addition, the divisor and the dividend can have other sets of the same variables assigned to other principles (Alexy 2002: 409).

The “Refined Complete Weight Formula” expresses a change of broader scope. It describes the idea that the second law of balancing is not only about empirical premises but on the underlying epistemic premises, entailing both normative and empirical assessments: the reliability of knowledge on the applicable law and on the empirical assumptions. For this reason, the reliability variable has to be divided in two: Re for empirical reliability and Rn for normative reliability. With this extension on the epistemic side of the formula, a significant consequence occurs: the effective impact of lower than 1 epistemic values on the substantive values. As Alexy states (Alexy 2014: 515), with the lowest epistemic values in both the empirical and the normative variables, the substantive values are decreased into a point that even the strongest substantive reasons for interference are undermined. Accordingly, this version of the formula gives a much better expression to the “Epistemic Law of Balancing.” In this version, the formula is

\( {W}_{i. j}\frac{W_i{I}_i{R}_i^s{R}_i^n}{W_j{I}_j{R}_j^e{R}_j^n} \)

It is almost endless the list of criticism that has been made to the “Weight Formula,” targeting different aspects such as the concept of abstract weight or importance of a principle (Sieckmann 2010: 113) or the indeterminacy regarding how are the scales of measurement to be determined (Jestaedt 2012: 164). However, a simple point regarding its “function” has to be stressed. The “Weight Formula” is not an equation purported to give a mathematical response to a conflict of norms that leads to a balancing; it is a scheme for the application of law, and, as such, it is just a rationalization of an evaluative intellectual process (Pulido 2006: 109; Klatt 2012: 9). In Alexy’s own words, it is nothing other than a mathematical reconstruction of the Law of Balancing with its link to a second Law of Balancing that refers to epistemic certainty (Alexy 2016: 67).


Alexy’s distinction between rules and principles is the first step of a complete theory of law, entailing a comprehensive understanding of its structure, regarding not only the differences between those two kinds of norms, but also what Alexy calls the two basic operations in the application of law: subsumption and balancing. If the former was already consolidated in the legal field, it is on the later that Alexy’s principles theory gives a totally new insight: balancing is the basic operation to be carried out when principles have to be applied, structured as a process for the solution of a collision of two or more norms that are optimization requirements. As the basic operation for the application of principles, balancing convokes proportionality, and proportionality, specifically with its “substantive” and the “empirical” laws, establishes the conditions for the outcome: a deontic solution given by the prevailing principle and the relevant facts of the case. It is within and for the process of balancing that Alexy formulates the “Weight Formula,” an equation that aims to rationalize balancing, where the collision is solved by assigning numerical values, in specific scales, to three variables: intensity of interference, abstract weight of the principles at hand, and reliability of the empirical assumptions. This version of the formula has been changed into a new one, designated “Refined Complete Weight Formula,” in order to accommodate the idea that the second law of balancing is about the underlying epistemic premises of balancing, entailing both normative and empirical assessments.



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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.University of LisbonLisbonPortugal

Section editors and affiliations

  • Miodrag Jovanovic
    • 1
  1. 1.Department of Theory, Sociology and Philosophy of LawUniversity of Belgrade, Faculty of LawBelgradeSerbia