[author]Zhao Zerui
[content]
How Is Law Calculated?—Judicial Procedures That Endows Legal Argumentation with Recursivity
Zhao Zerui
Assistant Research Fellow,
China Institute for Socio-Legal Studies, Shanghai Jiao Tong University
Associate Research Fellow, KoGuan School of Law
Abstract: In judicial adjudication, the computation of law is a recursive cycle in which prior legal decisions are adjusted through legal argumentation to resolve present legal problems. Current conceptions of iterative legal computation, centered on the identity of legal concepts and the equivalence of induction and deduction, regard judicial adjudication as the retrieval and extraction of information from a pre-established set of complete legal propositions. However, such an approach faces the dilemmas of formal paradox and circular reasoning. The paradigm of legal argumentation reveals the recursive principle of computation in judicial adjudication, which revolves around creative repetition and the transformation of value differences, and it demonstrates that legal computation is not a mechanical technique of information retrieval but a humanistic art of information generation. In order to endow legal argumentation with recursive computability, judicial procedure, through role allocation and behavioral norms, guides lawyers to explore the potentiality of law via language games, while providing judges with a third dimension of value judgment to integrate diverse value differences. Under the orchestration of judicial procedure, lawyers and judges repetitively articulate differences, jointly staging a dramatized process of the production and resolution of legal problems.
Keywords: Computational Jurisprudence; Legal Argumentation; Judicial Procedure; Consistency of Adjudication; Recursion
“Computational jurisprudence,” which arises from the integration of legal culture and digital technology, has become an important component of the autonomous knowledge system of Chinese jurisprudence in the digital era. Computational jurisprudence in China should not be confined to the innovation of research tools; rather, it must break through the mechanistic framework of traditional “legal mathematics.” By reconstructing the epistemology, methodology, and axiology of legal culture through technological principles, its essence is a project of legal-cultural reconstruction empowered by technology. Its core mission lies in constructing a new paradigm of jurisprudence that both carries the genetic imprint of Chinese legal culture and embodies the characteristics of digital civilization. Therefore, on the one hand, computational jurisprudence in China must provide people with new cognitive models and analytical paradigms for understanding “what law is” and “the computational modes of legal order,” thereby ensuring the independence of computational jurisprudence as a new interdisciplinary field. On the other hand, it must also offer original insights and illuminating approaches to the core issues long debated in existing legal theories, thereby ensuring the continuity of computational jurisprudence as a branch of legal scholarship. Against this backdrop, this paper seeks to renew the conception of legal computation in judicial adjudication by drawing on the underlying computational principles of machine learning algorithms. From the perspective of “communication–procedure” and within the paradigm of “legal argumentation,” it aims to provide an innovative answer to the question of “how law computes.” In so doing, computational jurisprudence may break free from its stereotypical impressions of circular reasoning, mechanization, and axiological monism, and instead shape a humanistic image rooted in democratic culture, flexible and controllable, and inclusive of diverse values.
To this end, this paper will proceed by addressing three interrelated questions, and through them progressively expound the transformation of legal computation models in the contemporary era: First, why is the current mainstream model of iterative legal computation unable to accurately reflect the actual operational patterns of judicial adjudication? Second, how can the recursive computational principles of machine learning algorithms be applied to construct a new computational model that reflects the division of labor and collaborative mechanisms within the legal community, and captures the emergent effects of judicial practice? Third, how does judicial procedure, through reasonable role allocation and behavioral norms, endow legal argumentation with recursive computability?
1. Iterative Legal Computation Based on Conceptual Identity and Its Dilemmas
The current mainstream paradigm of legal computation is an iterative computational conception that borrows from reductionist cybernetic thought. It takes legal concepts as the smallest units of computation, the formal equivalence of induction and deduction as the computational logic, and the order of identity as the computational goal. This conception understands the judicial adjudication process as a mechanized process of retrieving and extracting specific information from a set of legal propositions. The advantage of this iterative computation lies in its capacity to forcibly strip away all subjective uncertainty from the process of legal computation in judicial adjudication by relying on the subject–object dichotomy. It takes the formally structured concepts presented in objective texts as the starting point of computation, simplifies complex and variable case facts into combinations and iterations of basic legal concepts, and thereby eliminates uncertainty and arbitrariness in judicial adjudication. However, in order to ensure the accuracy and predictability of the legal computation process, this conception may overlook the subjective initiative of lawyers and judges, reducing judicial adjudication to an independent, closed system capable of operating autonomously from society, thereby trapping legal computation in the dilemmas of formal paradox and circular reasoning.
1.1 The Iterative Computational Conception Centered on the Apriority of Legal Concepts and the Equivalence of Induction and Deduction
Simplifying complexity within social chaos is a shared goal of both jurisprudence and computational science. Thus, as early as ancient Rome, scholars introduced computational thinking into legal theory in order to overcome uncertainty and arbitrariness in adjudication. Prior to the twentieth century, computational thinking was largely dominated by inductive–deductive logical atomism (manifesting as reductionism in cybernetics), which simplified the complex structures of the world into iterative combinations of basic structures. In such iterative computation, the identity maintained by these basic structures provides the rationale and legitimacy for decomposing complex facts into simple structural additions. For example, when all natural numbers can be decomposed into iterative combinations of 1, and this 1 remains unchanged throughout the iterative process, it becomes possible to calculate all natural numbers with precision and objectivity. In other words, iterative computation depends upon an a priori and clearly defined minimal mapping structure, which must maintain identity throughout the iterative process of combination. When this computational approach was introduced into jurisprudence, it evolved into legal formalism, which relies on a priori legal concepts and inductive–deductive logic that preserves conceptual equivalence to resolve the complexity of legal disputes. Montesquieu’s portrayal of the mechanical, script-reading ideal judge and the metaphor of adjudication as a “vending machine” both embody the broad application of this iterative computational approach in legal culture. Indeed, the rise of this computational approach within jurisprudence was mainly due to its objectivity and determinacy, which help constrain arbitrariness in judicial adjudication, enhance the predictability of legal decisions, and strengthen judicial independence, thereby overcoming the subjectivity and extreme uncertainty for which ancient legal decisions were criticized.
On the basis of this computational thinking, legal computability must rest on three formal premises: First, the principle of identity in legal computation, meaning that computable content in law is confined to objective symbolic expressions detached from subjectivity—that is, the legal concepts in legal texts and their formal structural relations. At this stage, legal concepts and social facts are understood to have an a priori, unique, and determinate semantic mapping, which remains identical throughout the inductive–deductive process of legal interpretation. Second, the principle of non-contradiction in legal computation, meaning that the results of legal computation must be uniquely correct; contradictory interpretations of legal concepts or legal propositions under the same time and relational conditions cannot both be true. Accordingly, the legal computation of any dispute must yield a uniquely correct and exclusive solution. Third, the principle of the excluded middle in legal computation, meaning that under the same time and relational conditions, a legal proposition must be either true or false, with no third possibility. This is reflected in the rule that judges may not refuse to adjudicate, and must rule on whether a party’s claim is lawful or unlawful—what is also called law’s binary code.
Thus, once introduced into legal culture, iterative computational thinking gave rise to a research paradigm based on a priori legal concepts as basic computational units and inductive–deductive equivalence as the fundamental logic of computation. For example, conceptual jurisprudence posits that “positive law is derived through formal logic from the highest axiomatic concepts.” Although this form of conceptual jurisprudence was subject to widespread criticism in the nineteenth century for its rigidity and inflexibility, its extension into the syllogistic model continues to exert influence today. In the syllogistic model of legal reasoning grounded in iterative computation, the legal computation of judicial adjudication is constructed as inductive–deductive logic composed of three elements: the major premise (legal provisions), the minor premise (case facts), and the conclusion (judgment). Within this model, legal computation in adjudication is detached from subjective value judgments by means of the subject–object dichotomy, relying instead on objective basic concepts and formal logic. Judicial adjudication is conceived as the input–output process of a vast repository of legal propositions: legislators and scholars, by clearly defining legal concepts and applying inductive–deductive logic, construct a sufficiently large set of legal propositions as the input end of the judicial system; lawyers and judges, by employing the syllogistic model and fact-finding, extract from this set the unique and determinate legal decision, which serves as the output of the judicial system. Hohfeld’s analysis of legal concepts exemplifies this mode of legal computation. He sought to define with precision four pairs of fundamental legal concepts as the least common denominators of all legal propositions. By clearly defining and identifying these four pairs, he decomposed all legal propositions into iterative combinations of them through inductive–deductive formal logic. The legitimacy and precision of such conceptual legal computation derives precisely from these a priori legal concepts and the identity they maintain throughout the iterative process.
This mode of legal computation gives rise to two basic paths for constructing legal-proposition databases: the top-down path of rule-based reasoning models, i.e., expert legal systems; and the bottom-up path of data-based reasoning models, i.e., legal big data systems. The top-down rule-based reasoning path can be understood as a digital attempt at legal conceptual analysis: by using clearly defined legal concepts, all legal rules and their referenced propositions are converted into formal symbols. Case facts are then symbolized and computed via syllogistic induction and deduction, leading to the derivation of uniquely correct legal decisions. In this path, both the basic legal concepts that constitute legal rules and their semantic mappings with facts must be pre-set and input manually, such that judicial practice only needs to provide the facts in order to precisely retrieve the unique legal decision corresponding to the database of legal propositions. The bottom-up data-based reasoning path, on the other hand, draws on statistical theory, relying on large volumes of existing judgment data and empirical research to fit conceptual structures and their semantic mappings with facts that best match the observed distribution of data. These then serve as the a priori, concretely defined set of legal propositions, thereby constructing the set of legal propositions through data learning rather than manual input.
Although these two paths may be regarded as different attempts at computational jurisprudence—one by doctrinal legal science (seeking top-down what the set of legal propositions ought to be), the other by empirical legal research (seeking bottom-up what the set of legal propositions actually is)—both remain reliant on iterative computational thinking based on a priori legal concepts and inductive–deductive equivalence. They merely differ in their views on how to identify the a priori legal concepts and their semantic mappings with facts. Some scholars equate this iterative computational approach with computational thinking, and equate iterative legal computation with computational jurisprudence, identifying its features as “abstraction” (i.e., the search for basic legal concepts and their definitions as minimal units of computation) and “automation” (i.e., iterative addition detached from intersubjectivity, based on the equivalence of induction and deduction). However, such discussions cannot explain why computational jurisprudence must now be promoted when computational thinking has already long been embedded in doctrinal legal science and empirical legal research. Is computational jurisprudence merely an upgraded tool for existing legal computation approaches, or simply a generic term for applying the latest computational tools to traditional legal studies? More importantly, computational jurisprudence premised on this iterative computational approach risks conveying the mechanistic impression that the legal community can be completely replaced by digital technology. It struggles to embody the humanistic dimensions of computational jurisprudence, and fails to respond to the extensive criticisms of iterative computational thinking that currently permeate legal scholarship.
1.2 Critiques of Iterative Legal Computation
Whether in the top-down legal expert system or the bottom-up legal big data system, iterative legal computation, in its pursuit of objectivity and precision, must rely on clear, a priori conceptual definitions and inductive–deductive logic that preserves those definitions unchanged during the computation process. In this way, it constructs a legal computational paradigm that takes conceptual identity as its basis of legitimacy. As the myth of the rule of law, the principle of treating like cases alike in its strictest sense—“identical cases, identical judgments”—is regarded as the ideal scenario that iterative legal computation seeks to realize. However, while iterative legal computation, by constructing an order of conceptual identity, effectively enhances the predictability and objectivity of judicial adjudication, it deprives adjudication of subjective initiative and humanistic concern. It may even lead legal culture toward a purely technical science devoid of value judgment, thereby neglecting that law is not merely a social control technology but also a humanistic art that reflects human concern and the aesthetic order of justice. To equate this narrow, mechanical form of iterative legal computation with computational jurisprudence fosters the unrealistic illusion that intelligent justice can operate independently and in isolation from intersubjectivity, which in turn gives rise to pseudo-questions such as whether artificial intelligence will replace judges and lawyers, or whether code can completely replace law. Scholars have already raised doubts and criticisms of this approach from multiple perspectives.
The first critique centers on the problem of incompleteness. If law were an uncontradicted and consistent conceptual computational system, it would inevitably contain blind spots. This means that for law to achieve fully complete, objective, and accurate judicial computation through inductive–deductive maintenance of conceptual identity, it must necessarily confront cases where no judgment can be rendered. This stands in contradiction to the principle that law must not refuse adjudication, which requires the completeness of law. In reality, however, in order to satisfy the completeness requirement under this principle, judges are often forced to adjudicate in contexts of statutory gaps and interpretive conflicts.
The second critique concerns the problem of circular reasoning. Specifically, iterative legal computation assumes that the relationship between the general propositions that constitute the major premise (the set of legal propositions) and the singular propositions of the minor premise (the facts of the case) is one of identity—moving from universal propositions to singular ones, from general propositions to particularized ones. That is, legal norms are recognized as universal propositions ∧x（Mx→Fx） composed of multiple singular propositions in conjunction: ∧x（Mx→Fx）=（Ma→Fa）∩（Mb→Fb）……（Mn→Fn）. The singular propositions of case facts are already contained within the universal propositions of the legal set. Thus, iterative legal computation is merely an information retrieval technique extracting pre-existing specific propositions from a priori, complete legal databases; it generates no new legal content. By treating the very legal proposition set that already contains the decision as the justification for that decision, it amounts to circular reasoning. This circularity disguises the process of moving from legal rules to case facts and then to legal decisions as a mere technique of formal inference, thereby neglecting the inescapable issue of value judgment in judicial processes. In this way, legal computation is naively assumed to be a mechanical process of searching and retrieving predetermined information from an a priori database of legal propositions.
The third critique mainly targets the inability of iterative legal computation to reflect uncertainty and creativity in adjudication. Within iterative legal computation grounded in conceptual identity, the formal logic whereby judges extract legal decisions objectively and precisely from a priori legal propositions based on case facts strictly excludes uncertainty and creativity. Its principles of identity, non-contradiction, and the excluded middle mean that an a priori, complete set of legal propositions has already pre-specified a unique correct solution for any case. Yet empirical studies have revealed the inherent uncertainties faced in judicial processes, as well as the creative demands of adjudication. Judicial perception of objective rights is often tinged with subjective opinion, and the pursuit of certainty by judges is frequently futile. Uncertainty is inevitable; judicial process is not discovery but creation. The reality that legal actors are reluctant to admit but must accept is that judges, in the judicial process, create law.
1.3 Reflections on Analogy and Incomputability
As the disjunction between iterative legal computation and judicial practice has become increasingly evident, more scholars have criticized this a priori, deterministic, reductionist paradigm of conceptual computation from different angles. Yet owing to a limited understanding of computational theory, the responses prompted by these critiques easily fall into a dilemma: either appealing to analogy to soften the rigidity and mechanization of identity while failing to substantively transcend the limits of iterative computation, or adopting the extreme view that the creativity, uncertainty, and subjectivity of judicial practice are fundamentally opposed to computational thinking, thereby concluding that law is inherently incomputable.
The strategy of understanding treating like cases alike as “similar cases, similar judgments” rather than “identical cases, identical judgments” still fails to escape the paradigm of iterative legal computation. Case–judgment similarity remains interpreted as an embodiment of legal generality. Adjudication is still conceived as a mechanical process of retrieving specific legal decisions from general legal propositions, leaving the problem of circular reasoning unresolved. This response merely shifts the mapping rules of iterative legal computation from strict one-to-one correspondences to one-to-many correspondences, rendering the connotations of legal concepts more flexible through analogy. Thus, unlike “identical cases, identical judgments,” which emphasizes that legal concepts can maintain definitional identity and decision equivalence through inductive–deductive reasoning across different factual situations, “similar cases, similar judgments” acknowledges that adjudication inevitably extends the connotations of legal concepts. Yet the computation of law must still pursue conceptual identity and decision equivalence. Typified reasoning here is treated only as a necessary supplement to iterative legal computation in coping with judicial uncertainty.
In addition, there are extreme responses that regard law as incomputable. For example, American legal realists such as Frank and Llewellyn exposed the variability and subjectivity of law in judicial practice, thereby developing critical jurisprudence grounded in the stance of uncovering subjective value judgments hidden beneath the veil of objective, precise conceptual computation. Other scholars have turned to empirical sciences or economics to supplement explanations of legal uncertainty and subjectivity that iterative computation cannot encompass. Yet they often face the dilemma of either abandoning iterative computation entirely and collapsing into radical indeterminacy, or retaining iterative thinking while failing to resolve its inherent contradictions. This cognitive limitation—treating uncertainty and subjectivity as fundamentally opposed to computational thinking—pushes critiques of iterative legal computation toward the conclusion that law is incomputable. The very thesis that the introduction of computational theory into legal culture sought to overcome—“the radical indeterminacy of law”—thus resurfaces. The ideal of legal computation, transforming plural value conflicts and social chaos into a unified order of fairness and justice, urgently requires a paradigm shift. The fact that law is unsuited to iterative computation does not mean that law is incomputable. Extending critiques of iterative computation to computational jurisprudence as a whole, thereby excluding computational theory and mathematical logic from the horizon of legal studies, amounts to an overcorrection.
Why is iterative computation unsuitable for law? Because it is a form of computation that overemphasizes formalism while neglecting context. This results in each instance of adjudication being overlooked and assimilated, failing to recognize that every act of legal computation is shaped by the context of adjudication, which directly alters the formal structure of computational content. Under iterative reasoning, legal computation serves only the function of simplification without generating any new information. It reduces legal formal expression to a mechanical process of precise retrieval from an a priori set of propositions in order to preserve conceptual identity. Yet in reality no such absolutely correct, immutable, and complete a priori set exists, and the identity concepts that iterative legal computation relies upon evolve in meaning as communicative contexts shift. Thus, conceiving adjudication as iterative computation inverts the relationship between “form” and its underlying “context,” seeking to construct a context-independent technique of formal logic. The result is that formal logic, when a priorized, appears clumsy and ineffective in judicial practice. If computational jurisprudence in China seeks to propose a legal computational paradigm consistent with the operational logic of judicial practice, it must simultaneously attend to the formal expression of law and its generative context, and explore their interconnection. Only by incorporating both intersubjective communicative contexts and formalized expressions into the horizon of legal computation, rather than attempting to explore context-independent structural relations between forms, can such a paradigm be realized.
2. Recursive Computation Focusing on Repetition and Difference in Legal Argumentation
In order to overcome the dilemma of iterative computation and to reveal the creativity of specialization and collaboration within the legal community, this paper draws on the logic of recursive computation in machine learning and proposes a paradigm of legal recursive computation that centers on repetition and difference in legal argumentation. If the idea of iterative computation can be summarized as the search for an eternal and unchanging minimal computational unit, resolving complex problems through combinations and iterations that maintain identity, then the idea of recursive computation can be briefly described as using past computational results to creatively repeat and thereby address current complex problems. For ease of understanding, social problems can be metaphorically likened to infinite and distinct locks: the unlocking theory of iterative computation is “to find a master key that can open every lock,” whereas the unlocking theory of recursive computation is “to adjust the key that opened previous locks in order to open the current one.” Legal recursive computation, therefore, is conceived as a model that takes legal argumentation as the basic computational unit, employs creative repetition that integrates diverse value differences as its computational logic, and grounds its legitimacy on constructing a third dimension capable of perceiving the intensity of justice. The advantage of such a computational paradigm lies in its independence from the identity of legal concepts, enabling each legal computation to transform diverse value differences in communicative contexts into legally reproducible formal structures through legal argumentation. In sum, legal recursive computation, based on legal argumentation in judicial adjudication, differentiates chaotic, redundant, and indeterminate value differences among subjects into small, fluid, and unbound regulable differences. These regulable differences are then further differentiated into legally regulated differences integrated within the legal knowledge system, according to subjectively perceived intensities of justice. In this way, an information spiral emerges, constantly invoking and adjusting previous legal decisions to resolve current legal issues.
2.1 Recursive Computation with Legal Argumentation as the Basic Unit
Legal argumentation is the manifestation of communication within the realm of normative order. The iterative approach to reducing complexity is to use explicit, finite, but complete a priori conceptual mapping relations to reduce infinite complexities into combinations and iterations of these finite concepts. In pursuit of absolute objectivity and precision, such computation maintains the identity of conceptual mappings during the process, thereby excluding uncertainty. However, as previously noted, there are two reasons why iterative computation does not suit judicial adjudication: first, the rule of law cannot be accomplished overnight, nor can we expect a select few at a particular time and space to define a priori and complete sets of legal concepts and propositions. Second, law must evolve dynamically in connection with the uncertainties of the social context. The uncertainties arising from intersubjectivity in judicial adjudication are not without value; rather, they may drive legal innovation. Such intersubjective and contextual uncertainties must be encompassed by legal computation, not simply excluded. Therefore, a computational paradigm need not rely on a priori, fixed conceptual mapping relations as a necessary basis. Instead, recursive computation, which absorbs contextual differences of each computation into its formal structure, is more suitable for reflecting the principles of legal computation in judicial adjudication.
The concept of “recursion” originates in non-reductive cybernetics, being closely related to the idea of “feedback,” and later came to signify a form of creative looping. Starting from a simple function, recursion can generate complexity by repeatedly invoking itself. Conversely, recursive computation can be understood as a cyclical form of computation that reduces complexity through creative repetition. In programming, recursion refers to a technique whereby a program calls itself to solve complex problems; it has become an underlying mathematical logic of machine learning algorithms. A classic example is the Fibonacci sequence: f(n) = f(n-2) + f(n-1) = f(n-4) + f(n-3) + f(n-3) + f(n-2) … . Its computational method relies on repeated self-calling to achieve repetition that nevertheless produces innovative spiral expansion. This means recursive computation does not depend on reducing complex phenomena or functions to a priori, unchanging conceptual mappings, but instead addresses complexity by repeatedly invoking previous functions. The function as a computational unit itself changes dynamically with each invocation and thus contains uncertainty. From the perspective of recursive computation, computation is not a finite regression reducing complex phenomena to a priori conceptual mappings, but an infinite creative process of understanding future complexity through existing functional relations. To summarize: “Recursion is characterized by a looping movement of returning to and determining itself. Each movement has contingency, and contingency determines its particularity. We can imagine a spiral in which each circular movement is partially determined by the previous one, while the influence of prior movements continues as both concept and effect.”
In judicial adjudication, legal recursive computation revolves around legal argumentation, since legal argumentation incorporates both the formal structures of law and the communicative contexts that determine those structures into the computational scope. The recursive nature of communication is a central entry point in Luhmann’s explanation of how the legal system remains simultaneously closed and open, achieving autopoiesis. As Luhmann notes in the conclusion of A Sociological Theory of Law: “Recursion, as a function that relates the system to its environment and serves to reduce environmental complexity, presupposes the environment. Such recursive operation must confront difference, and system unity can only appear within difference from the environment. In facing difference, the system must be able to observe itself in relation to the other. It thus presents itself within the environment.” Foerster, from a mathematical-logical perspective, systematically argued the proposition that “communication is recursion.” Similarly, several jurists have revealed the recursive nature of communicative acts in judicial adjudication. In Habermas’s discourse theory, for example, “communication” is identified as both the mode of existence of law and the source of legitimacy, breaking the one-directional, linear paradigm of legitimacy and avoiding the dilemma of endlessly searching upward for a priori sources of legitimacy. In Habermas’s words: “The creation of law cannot be regarded as a one-way process from ‘citizens—elections—parliamentary legislation—judicial application.’ The significant intensification of law and social complexity has rendered this schema a cliché.” Likewise, Luhmann’s later thought expressed that the identification of law is the result of communication through specific patterns, a second-order observation achieved through a recursive network (circular inference) of closure. This recursive, looped logic of law ensures that judicial legitimacy has no single predetermined standard. To distinguish this from other forms of communication in society, communication specifically within the legal domain can be collectively termed “legal argumentation.”
In conclusion, recursive computation is a method of reducing complexity by repeatedly invoking and adjusting prior functions. It incorporates uncertainties arising from communicative contexts into the formal system of conceptual content and structure. Based on recursive computation through legal argumentation, the content and structure of legal concepts can be continuously invoked and revised in response to contextual differences in judicial adjudication, thereby achieving the creative repetition of law.
2.2 Repetition in Legal Recursive Computation
To achieve recursive computation in legal argumentation, it is necessary to constantly invoke prior legal decisions to resolve current legal issues, which inevitably leads to the repeated interpretation of legal concepts. However, the key difference between recursive computation and iterative computation lies in the nature of repetition: repetition in recursive computation is not a manifestation of legal generality. It does not deliberately pursue the identity of a priori concepts, but instead highlights the singularity of value judgments in case adjudication. The legal generality pursued by iterative computation—whether conceptual identity or similarity—rests on a tacit assumption: that all cases can ultimately be reduced to a set of basic units, and that these units are interchangeable or substitutable. On this basis, cases with identical or similar units are regarded as the same case, and thus yield identical legal decisions. In contrast, recursive computation based on legal argumentation rejects the existence of fully identical cases, positing instead that each case involves an irreplaceable and non-substitutable communicative context. What links together these singular cases across the flow of time is the act of repetition in legal argumentation. Here, repetition is not a mere formal characteristic of iterative computation, but rather a computational act that modifies or differentiates conceptual content.
In iterative computation, the equal sign is merely a formal marker of legal generality. It indicates that the legal major premise and factual minor premise on the left-hand side of the equation are identical to the legal decision on the right-hand side. This equivalence presumes informational symmetry and, therefore, interchangeability and substitutability. Consequently, iterative computation is devoid of temporal or spatial dimensions, since case practice does not alter the informational content of the law. By contrast, in recursive computation, each equal sign denotes a repeated invocation of law. Much like the equal sign in programming, it functions contextually—subject to judicial adjudication—and may redefine the value content of legal concepts while potentially revising their formal structures. Under this paradigm, the singularities of argumentation in each case are incorporated into the formal structure of legal concepts through creative and repetitive acts of interpretation. This means that what stands before and after the equal sign in recursive computation is neither interchangeable nor substitutable; law itself evolves in the process of computation. Thus, legal computation acquires temporal and spatial dimensions. Repetition in judicial adjudication, then, is not the specialization of legal generality, but a cyclical practice of interpretative invocation and revision. For instance, questions such as what counts as “notice” or whether the exemption conditions of the “red flag rule” are satisfied have been repeatedly but divergently interpreted by courts across different times and places, always in relation to specific case contexts. These judicial practices demonstrate that repetition in legal computation is a creative act of contextualized normative determination, not a mere reduction of cases to interchangeable combinations of a priori concepts.
Accordingly, repetition in legal argumentation is a skill imbued with intersubjectivity and creativity—an artistic act that can never be fully reproduced. In each case of judicial adjudication, the computational content of legal argumentation can only be re-invoked and revised in the future, never wholly substituted or reenacted. This practice of transmitting and revising the values of civilizational order by repeating an unrepeatable event is precisely the hallmark of human art, as seen in festivals and painting. Festivals are celebrated through repetitive acts, yet these repetitions are directed at unique, non-repeatable events; paintings, likewise, attempt to replicate an irretrievable image or scene. Whether in festivals or painting, the singularity embodied in artistic repetition relies on subjective perception to recreate the immersive experience of its contextual background. It cannot be fully anchored by the a priori content of preestablished concepts. This explains why judges and lawyers cannot be likened to vending machines, and why artificial intelligence cannot replace them. Judicial adjudication is a recursive computation propelled by legal argumentation, a humanistic craft of repeating to articulate difference and speaking the same to express the different, not a mechanical technique of formal reasoning that reduces a priori legal concepts to equivalence through inductive and deductive processes.
Having introduced the fundamental principles of recursive computation, we can now reinterpret the principle of treating like cases alike in judicial adjudication. Within recursive computation centered on legal argumentation, this principle is a normative requirement of creative repetition. It prescribes that legal argumentation must employ repeated legal discourse to articulate value claims and judgments that express their differences. Placed in the temporal dimension of adjudication, the principle means that “precedent binds subsequent cases.” This is why some scholars clarify that the principle of treating like cases alike is not a descriptive feature of legal computation, but rather a tautology of “adjudicating according to law”—a normative principle that constrains the conduct of legal argumentation. Such clarifications can only be fully understood when detached from the framework of iterative computation and re-situated within the perspective of recursive computation. Legal recursive computation, relying on prior legal decisions to resolve current legal issues, requires that every act of legal argumentation be performed in a form of interpretation marked by repetition. This ensures that while prior decisions provide grounds for resolving present disputes, present decisions likewise acquire repeatability to serve as bases for future disputes. The principle of treating like cases alike ensures this recursive computability of legal argumentation by normatively requiring judges to “engage in creative repetition within the differentiated context of case adjudication.” It is thus a tautology of “adjudicating according to law.” From the perspective of recursive computation in legal argumentation, the intersubjective differences in case contexts and the normative constraint of repeatability imposed by the principle of treating like cases alike complement each other, jointly guiding the legal community to achieve creative contributions in legal computation.
2.3 Difference in Legal Recursive Computation
In 2000, Joseph Raz, in a lecture on normativity, emphasized “the legitimacy of accepting difference.” He noted: “The recognition of the value of ways of life unfamiliar or repugnant to us can enrich our humanity, and protect us from smugness and narrowness. Our knowledge of and respect for the behavior of others also gives us opportunities for change.” Legal recursive computation, centered on legal argumentation, reveals how judicial adjudication transforms chaotic and redundant value pluralism in disputes into the raw material of legal computation. In doing so, legal computation frees itself from a priori concepts and complete sets of propositions, instead grounding itself in the diversity of values. From this standpoint, legal recursive computation reinterprets the essence of legal theory: it is not a set of a priori solutions determined by values, but rather a problematizer that incorporates pluralistic value differences.
From the perspective of recursive computation, normative legal theory is embodied in the argumentative process of “problematization,” whereby the emergence of legal problems is itself an expression of theory. By contrast, actual legal decisions represent the solutions to those problems. Thus, normative legal theory and actual legal decisions manifest themselves within legal argumentation as, respectively, the posing of legal problems and their resolution. Iterative computation confuses the two—normativity and actuality—by conflating problems with solutions. Recursive computation, however, distinguishes them by revealing the process through which differences are transformed, and by demonstrating the principles through which problems are generated and resolved in judicial adjudication.
When disputes enter the realm of legal argumentation, the indeterminate plurality of value differences is differentiated by legal theory into regulable legal problems. This is the process of problem generation. Recursive computation relies on the mutual determinacy among differential elements (legal concepts) to transform pluralistic value differences into regulable legal problems. At this stage, legal theory functions as a universal yet indeterminate problematizer. Althusser’s interpretation of economic theory helps us understand legal theory in this sense: “‘The economy’ has never strictly speaking been given; it refers to a potentiality that is always mediated by its forms of actualization, a differential potential, a ‘topic,’ a problematizer always covered over by its concrete solutions. In short, economic theory is the dialectic of society—it poses to a given society, to its field of totality and its field of problems, the ensemble of its questions. Marx’s famous statement in the Preface to A Contribution to the Critique of Political Economy—that ‘humanity only sets itself such problems as it can solve’—does not mean that problems are mere appearances, nor that they are already resolved. It means instead that the economic conditions of problems determine or produce a method by which problems find their solutions within the framework of social relations.” Similarly, legal theory should be understood not as an a priori solution to all legal problems, but as a problematizer capable of transforming indeterminate plural value conflicts into regulable legal problems. Only in this way can normative legal theory as problematizer be distinguished from actual legal decisions as problem-solvers.
It is important to emphasize that chaotic, pluralistic value claims in disputes cannot be subjectively perceived or judged until they are differentiated by legal theory. For example, without lawyers translating the diverse claims of parties into the language of law based on legal theory, neither judges nor scholars could perform value judgments from a legal perspective. Legal theory here does not function to provide preestablished solutions, but instead transforms plural value differences into legal problems that can be subjectively perceived and judged. As Deleuze explains in his reading of Kant on the nature of theory: “Kant sometimes describes theories as ‘problems without solutions.’ He does not mean that theories are necessarily false problems and thus unsolvable, but rather that real problems are themselves theories, and cannot be eliminated by their solutions, since they are the indispensable conditions of any solution’s existence. The legitimate use of theory must be related to the concepts of the understanding; conversely, concepts of the understanding can only find their sufficient empirical application when they are related to problematic theories—either organized along lines converging toward ideal foci beyond experience, or reflected against the backdrop of a higher horizon encompassing all concepts. These foci and horizons are theories possessing both intrinsic and transcendent qualities—that is, problems as they are.”
In recursive computation centered on legal argumentation, normative legal theory as problematizer and actual legal decisions as solutions maintain an internal connection yet differ in essence. Their internal connection lies in the fact that the content and structure of legal concepts—mutually determinative within theory—change dynamically through legal decisions in adjudication, while also forming the preconditions for future problems. Their essential difference lies in their function: legal theory, to maintain universality beyond specific contexts, must remain indeterminate as a problematizer and serve as a necessary tool of legal computation, responsible for transforming indeterminate plural values into regulable legal problems. Legal decisions, by contrast, are inextricably tied to specific argumentative contexts, requiring subjective perception to deliver contextualized and concrete value judgments.
Thus, when confronted with chaotic and complex plural disputes in social life, legal argumentation employs legal theory to differentiate them into focused, small-scale regulable differences. Through the posing, resolution, and reproduction of legal problems, the differences of argumentative contexts are incorporated into the reproducible formal structures of law. At this stage, recursive computation based on legal argumentation is not a mechanical technique of information retrieval and extraction, but a humanistic craft that achieves spiral expansion of information through the production, resolution, and reproduction of legal problems. The following section will elaborate on how judicial procedures, through the design of role assignments and behavioral rules, guide and constrain this spiral expansion of legal argumentation, that is, how judicial procedure confers recursive computability (hereinafter referred to as “recursivity”) upon legal argumentation. Put differently, the foregoing discussion has explained what recursive computation based on legal argumentation is, and the following will explain how it is realized.
3. Judicial Procedure as the Regulation of Legal Argumentation for Recursive Computation
“As for what is called procedure, it is nothing more than an institutionalized arrangement for the ideal conditions of argumentation.” Recursive computation centered on legal argumentation requires the constant invocation and revision of prior legal decisions to resolve current legal problems. To realize such recursive computation, judicial procedure must institutionalize the handling of difference and repetition in the generation and resolution of legal problems within argumentation. The guidance and constraints judicial procedure imposes on legal argumentation include: ensuring that the indeterminate differences arising from plural value claims can be reasonably differentiated by legal theory; enabling lawyers, through linguistic games, to produce legal problems that contain determinable differences; and thereby providing the material of computation for judges to adjust prior legal decisions in light of case-specific contexts so as to resolve current legal problems. In institutionalizing the setting of legal argumentation, judicial procedure regulates it by assigning roles and behavioral rules, conceptualizes neutrality as a perceptible third dimension of justice, and aims to guide the repetitive affirmation of differences within plural values through legal interpretation.
3.1 How Judicial Procedure Regulates Lawyers in Uncovering the Potentiality of Law
Law, through legal argumentation regulated by judicial procedure, must constantly transform the indeterminate differences of plural values in cases into determinable differences in legal problems, and further incorporate them into the substantive content of law through creative repetition. This means that the process of legal computation in judicial adjudication is not a specialization of general law but a realization of the potentiality of law. The difference lies in that the former is a matter of retrieving and extracting preestablished information, while the latter is a spiral expansion of information. In recursive computation centered on legal argumentation, lawyers play the role of condensing litigation claims from chaotic and heterogeneous plural value disputes on the basis of legal theory, thereby providing determinable differences for the development of law’s potentiality. Judicial procedure, through strict formal requirements for case initiation, urges lawyers to propose litigation claims that meet procedural standards and that, guided by legal theory, aim at realizing the pluralized values of the parties. Under the procedural requirements, lawyers must continuously communicate with parties, collect evidence, and clarify demands, transforming the miscellaneous indeterminate differences of disputes into litigation claims expressed in legal language and framed within legal theory. Thereafter, through repeated linguistic games between opposing lawyers, the conflicting claims are transformed into clear, specific, determinable legal problems—commonly known as the disputed issues of the case. Iterative computation of law focuses only on the resolution of legal problems—i.e., judicial reasoning in judgments—while neglecting the process by which legal problems are generated. This results in a severe underestimation of the foundational significance of lawyers and their linguistic games for the making of legal decisions.
At this point, it is necessary to clarify the notion of law’s potentiality. In recursive computation centered on legal argumentation, law’s potentiality is distinct from the complete determinacy of case judgments, but it is also not sheer contingency. Rather, it is a controllable indeterminacy constrained by the mutual determination of linguistic elements. To put it another way: the chaotic value claims of disputing parties in society are entirely contingent; but in order for a dispute to be filed and enter the litigation process, lawyers must engage in repeated communication with the parties and, based on the mutually determined concepts and structural relations of legal theory, reorganize and articulate these plural value claims. In this way, wholly contingent disputes are transformed into litigation claims constrained by the structural determination of legal language. When the parties’ claims differ but remain constrained by the mutual determinacy of legal language, linguistic games in legal argumentation can fully utilize the indeterminate differences of case adjudication to uncover the potentiality of law—that is, to raise a series of indeterminate but determinable legal problems. The emphasis on law’s potentiality in legal argumentation signifies that the substantive content of law is not completely predetermined by legal texts. The clear provisions of legal texts provide only the mutual determinacy of linguistic elements—that is, the formal structural relations between legal concepts. Yet this mutual determinacy provided by legal texts can transform the indeterminate value-difference claims of disputes into determinable legal problems—i.e., the disputed issues for judges to decide—only through the guidance of judicial procedure and the linguistic games among lawyers.
3.2 How Judicial Procedure Guides Judges in Producing Consensus from Legal Disputes
If recursive computation of law based on legal argumentation begins from plural values, how does it produce consensus? From this perspective, the mechanism of consensus production in judicial adjudication is not realized by negating differences among plural values, but by affirming them and integrating them into a third dimension. Within the two-dimensional differences generated by the linguistic games of opposing lawyers, judicial procedure requires judges to enter as neutral third parties representing the state, thereby creating an entirely new third dimension—the intensity of justice—in order to affirm determinable differences. Recursive computation centered on legal argumentation understands that the differentiated litigation claims presented by lawyers through legal interpretation are determinable differences each arising from distinct personal value claims. These differences are not matters of right or wrong, but manifestations of law’s potentiality. Judicial procedure, however, requires judges to appear in the role of impartial adjudicators, independent of the parties and representing the state and public interest. In this role, they must affirmatively integrate determinable differences representing plural personal values into a new third dimension aligned with state and public interest. At this point, the mechanism of consensus production in legal argumentation parallels the mechanism by which animal neural networks generate three-dimensional vision. Visual neural networks create a fictive and unforeseeable third dimension—depth perception—by integrating the differentiated two-dimensional images received by two eyes located in different positions, thereby producing a unified, stereoscopic visual effect. This dimension is new and unforeseeable, because it neither depends on nor eliminates any single two-dimensional image, but instead integrates the differences between them to create depth perception. In this way, human beings are able to generate three-dimensional vision based on two differentiated two-dimensional images.
The principle of this third dimension also helps explain why artificial intelligence, skilled in formal reasoning, cannot fully replace the legal profession—or, in other words, why human subjective perception is indispensable to legal computation. In the conception of iterative legal computation focused on concepts, in order to pursue precision and objectivity, the subjectivity of human beings must be replaced by computers that lack subjective perception. However, as numerous scholars have demonstrated, the meaning of law is dynamically evolving, and this dynamic evolution of substantive legal content must be driven by subjective perception within the third dimension. Detached from the argumentative context of cases, the quantitative differences embedded within legal concepts cannot be directly computed or perceived, for such differences are eliminated and equalized through the extension of legal concepts. Yet these quantitative differences—through which legal concepts evolve with changing social needs—can only be subjectively perceived by establishing the third dimension. Whether it concerns the quantitative difference of subjective intent in constituting “intentionality,” the quantitative difference of informational asymmetry in constituting “substantial misunderstanding,” or the quantitative difference of harm to interests in constituting “legitimate defense,” none can be computed purely rationally and objectively, nor can they be directly perceived subjectively. They must instead be placed within the legal argumentation constructed by judicial procedure, where judges, in the third dimension as impartial and independent actors, integrate the determinable differences generated by the linguistic games of lawyers. Only then can these differences be subjectively perceived and computed. As Brouwer noted: “Mathematics is a mental activity; mathematical objects with specific properties exist for an individual only when that individual can construct them in their mind. Language and logic come into play only when attempting to convince others of the existence of such mathematical constructions. The goal of mathematical discourse is to help other interested mathematicians reconstruct the same type of objects in their minds.” This intuitionist view of mathematics, foundational to logic programming, helps us understand why recursive computation centered on legal argumentation must depend upon subjective perception within the third dimension. More precisely, under the subjective perception of the third dimension, judges’ value judgments reasonably distribute the determinable differences in legal argumentation into the extensible interpretations of legal concepts. This recursive process, while dissolving differences and forging consensus, also propels the evolution of law, transforming the judges’ subjective value judgments in case-specific argumentation into repeatable legal common sense. In this way, judicial conscience and legal common sense mutually direct and reflect one another.
In sum, although both lawyers and judges, in order to invoke prior legal decisions to resolve current legal problems, repeatedly interpret legal concepts, judicial procedure distinguishes their roles by institutionalizing role assignments. Lawyers’ legal interpretations involve repeatedly articulating differentiated litigation claims that correspond to personal plural values. Judges’ legal interpretations, by contrast, involve repeating from the dimension of impartiality in order to integrate the determinable differences produced by the linguistic games of lawyers into determinable legal decisions. Through the role assignments and behavioral norms of judicial procedure, legal argumentation becomes a dramatized performance in which lawyers and judges play distinct roles. This argumentative drama, which requires the craft of law as an art, like other forms of human artistic activity, continually incorporates value differences across times and spaces through creative repetition. It thereby consolidates social consensus while recognizing the dynamic transformations of plural values—speaking the same while expressing the different.
4. Conclusion
In order to resolve infinitely complex disputes with finite and concise rules, law must rely on cyclical forms of computation. The iterative computation that has been most commonly introduced into legal research is a mode of cyclical calculation centered on the a priori nature and identity of concepts. It attempts to reduce all case facts to iterative additions of a priori legal concepts, and through the formal logic of induction and deduction, eliminate subjective uncertainty in the computational process. By ensuring the identity of legal concepts during cyclical computation, it renders the generation of legal decisions in judicial adjudication as a process of particularizing general legal propositions, thereby pursuing precision and objectivity in legal computation. However, this approach to iterative legal computation, by entirely excluding intersubjectivity and uncertainty, confronts problems such as formal paradoxes and circular reasoning, and cannot fully reflect the operational principles of judicial adjudication. Moreover, within iterative legal computation, the dynamic demand serving as the starting point of computation and the prerequisite conditions serving as the computational premise are mutually contradictory. If there were indeed an a priori system of legal concepts that maintained identity, then such a conceptual system would necessarily be an absolutely balanced state of symbolic order. In such a case, there would be no need for cyclical computation through repeated interpretations of legal concepts; law would only need to be perfectly formulated and interpreted once, and then implemented by machines. The significance of adjudication would collapse into mere law enforcement, and legal actors such as lawyers, judges, and even jurists would lose their humanistic value, destined in the future to be replaced by more precise and efficient artificial intelligence. Yet, ever since its inception, legal culture has engaged in interpretive practices that, though seemingly repetitive, perpetually initiate cyclical computation. This phenomenon demonstrates that humanity has never, from antiquity to the present, discovered such an a priori system of absolute symbolic order in law. This historical fact proves that the rule of law is not a one-time technical project, but rather a humanistic craft requiring unremitting inheritance and perseverance.
Therefore, in order to understand why law requires human beings to engage in continuous and seemingly repetitive acts of interpretation, contemporary computational jurisprudence must draw on the computational principles of machine learning and propose a new cyclical computational paradigm, thereby providing a more concrete model for understanding legal computation in judicial adjudication. This new paradigm of legal computation is the recursive computation centered on repetition and difference within legal argumentation. Repetition is not identity, and difference is not mere multiplicity. Recursive computation grounded in legal argumentation posits that legal order has always been immersed in chaos, and that the purpose of cyclical computation is to continuously excavate the potentiality of law from chaos and actualize it through the posing, solving, and re-posing of legal questions. This paradigm of computational jurisprudence, rooted in legal culture and refined through the principles of machine learning, is reshaping the image of the rule of law in the digital age. Specifically, recursive computation centered on legal argumentation functions through the allocation of roles and behavioral norms in judicial procedure. Lawyers, drawing on legal theory, extract regulative differences from the disordered and multifaceted value claims arising from disputes, while judges, based on a third dimension of subjective perception, differentiate the value judgments concerning the connotations and formal structures of legal concepts. This recursive mode of computation, revolving around difference and repetition within legal argumentation, reveals that judicial adjudication is a humanistic craft—an interpretive practice of creative repetition that affirms regulative differences within plural individual values—rather than a mechanical technique that negates non-identical value claims by reference to a priori legal concepts or complete sets of legal propositions.
Originally published in Qiúsuǒ (Seeking Truth), Issue 3, 2025. Reprinted with the authorization of the WeChat public account “Qiúsuǒ Magazine.”
Assistant Editor: Yang Ziyue
Executive Editor: Zhao Zerui
Proofreading: Ji Weidong