Legal Sets

In this Article, I propose that the practices of legal reasoning and analysis are helpfully understood as being primarily concerned not with rules or propositions, but with sets. This Article develops a formal model of the role of sets in the practices of legal actors in a common-law system defined by a recursive relationship between cases and rules. In doing so, it demonstrates how conceiving of legal doctrines as a universe of discourse comprising (sometimes nested or overlapping) sets of cases can clarify the logical structure that governs marginal cases and help organize the available options for resolving such cases according to their form. While many legal professionals may intuitively navigate this set-theoretic structure, the formal model of that structure has important implications for legal theory. In particular, it (1) generates a useful account of the relationships among rules, standards, and principles; (2) provides a novel set of tools for understanding the nature of precedent; and (3) illuminates an extra-linguistic dimension to the problem of judicial discretion. On the last point, I argue that discretion is not merely a product of the imperfect relationship between abstractions and reality, or between natural language and the world, but that it is instead an emergent property of the structure of legal practice: a structure composed of sets “all the way down.”

INTRODUCTION

Legal reasoning and analysis are helpfully understood as being primarily concerned, not with rules, or even with propositions, but with sets. Set-based logic permeates mature legal systems, and constitutes the deep structure connecting legal authorities to the behavior of legal actors. This is particularly so in common-law systems, where judicial opinions serve a dual function as both backward-looking dispositions and forward-looking authorities, setting up a recursive dynamic that can best be modeled using the logic of sets. In this Article, I will explain that logic and describe some of its implications.

Consider that in many areas of law, our adversarial legal system channels disputes toward binary choices: Does this complaint state a cause of action or not? Is this statute constitutional or not? Is this defendant liable or not? Is this claimed element of damages recoverable or not?1 Even where a rule appears not to be framed in terms of binary outcomes, it is usually trivially easy to re-frame it in such a way: instead of asking “how broad is the plaintiff’s right?” we might instead ask “does the plaintiff have a right to x enforceable against this defendant?”2 Indeed, this is an essential move in legal reasoning: the process of resolving legal disputes often consists of channeling amorphous, complex issues into a series of discrete binary questions winnowed and tested via the adversarial process.3 Reaching one of those binary outcomes typically depends on a judgment as to whether the facts of a case satisfy some test defined by a legal rule: Does this paragraph of the plaintiff’s complaint recite “mere conclusory statements”?4 Is this statute’s effect on private speech limited to the punishment of “fighting words”?5 Would this liquidated damages award constitute a “penalty”?6 And so on.

My first claim is that all such legal rules can be understood to define categories, and the application of those rules to the facts of a particular case thus consists of a determination whether a particular state of the world falls within or without a relevant category. This first claim marks a minor departure from the sentential deontic models common in contemporary law and logic theory,7 but it has clear antecedents in Anglo-American legal theory. H. L. A. Hart noted that “[a]ll rules involve recognizing or classifying particular cases as instances of general terms.”8 Likewise, Frederick Schauer draws heavily on the notion of categories (and the related notion of generalizations) in his philosophical investigation of the nature of rules.9 Indeed, all analysis—legal or otherwise—may at some level be reduced to this type of “lumping” and “splitting.”10 But in this Article, my aim is to formalize the insight into a model of legal analysis, because such formalization yields new insights emergent from the model itself. As one prominent history of formal logic points out:

Formalization is a difficult and tricky business, but it serves a valuable purpose. It reveals structure and function in naked clarity, as does a cutaway working model of a machine. When a system has been formalized, the logical relations between . . . propositions are exposed to view; one is able to see the structural patterns of various “strings” of “meaningless” signs, how they hang together, how they are combined, how they nest in one another, and so on.11

This type of formalization must be distinguished from the typical meaning of the term “formalism” in legal theory. Historically, “formalism” in Anglo-American jurisprudence is (fairly or not) a pejorative term: it is a foil for the distinctively American school of Legal Realism.12 Even in its most sophisticated iterations, this concept of formalism is at bottom a substantive and indeed a normative construct: it distinguishes legitimate bases for judicial decision-making from illegitimate ones, and makes claims regarding the extent to which legal authorities determine adjudicative outcomes or constrain judicial behavior.13 In this Article, I will avoid most of the normative questions that divide realists and anti-realists in an attempt to clarify the formal structure of legal analysis in a model that has explanatory power regardless of one’s substantive views or normative commitments regarding the nature of adjudication. As we will see, however, this purely formal model lends some support to realist claims regarding the indeterminacy of legal rules and to positivist claims regarding the primacy of practice in determining what a society’s law is.

My formal model will be built upon a particular vocabulary. In philosophy, logic, and mathematics, category definition is the province of set theory. Thus, if categories truly do play an important role in legal analysis, the tools of set theory may fruitfully be applied to the relationship between legal rules and particular cases, and ultimately to the relationships among legal rules themselves. For these purposes, a “rule” is any legal directive, formulated at any level of precision or generality, that purports to direct the behavior of actors within a legal system.14 A “case” is any state of the world that has generated a legal dispute subject to evaluation within that legal system. In this way of thinking, what lawyers describe as “application of law to facts” is best understood as evaluation of the set-theoretic concept of membership or belonging: whether a particular state of the world can be situated within a category defined by a relevant legal rule.

Starting with this foundational notion of the relationship between rules and cases, I construct in this Article a set-theoretic and purely formal model of the structure of legal doctrine in a common-law system. In doing so I will rely primarily (though not exclusively) on examples from my own fields of substantive expertise: property and intellectual property (IP) law. For example: whether or not a transferee of a leasehold interest can be held directly liable to the original lessor for failure to pay rent often depends on whether the transfer falls into the category of “assignment” or the category of “sublease.” Which of those two categories any particular transfer falls into typically depends, in turn, on whether the instrument of transfer provides for the original lessee to retain any part of the leasehold interest.15 This set of rules can be formalized and related to one another in an overarching structure via the set-theoretic relations of membership and inclusion—as I will illustrate below.16

Moreover, as we will see, in common-law systems this structure is recursive: rules inform the disposition of cases, while the dispositions of cases collectively and inductively inform the development of new rules over time. Conceiving of legal doctrine in terms of sets, rather than propositions, allows a clearer understanding of the dynamic process by which the practices of legal actors generate the development and modification of legal directives. And importantly, it reveals how certain dynamics of doctrinal change emerge predictably from the structure of a common-law system, rather than the substance of particular rules.

Formalization along these lines is the key to my second claim: that understanding the structure of legal analysis in terms of set theory reveals subtle formal distinctions in the behavioral strategies legal actors may deploy when faced with underdeterminate (or contradictory) legal rules. Thus, regardless of their substantive conclusions regarding the outcome of a particular case, and indeed independently of the justification for that outcome, legal actors will often have the freedom to implement their conclusion in a number of formally distinguishable ways. Distinctions among these strategies map directly to distinctions between the set-theoretic relations of membership and inclusion. We will see that the choice of a formal strategy has important implications for the claims made by any particular act of advocacy or adjudication against the body of doctrine within which it is situated.

Understanding these formal distinctions in the behavior of legal actors leads to my third claim: that important objects of study and debate in substantive legal theory—such as the distinction between rules and standards, the nature of precedent, and the problem of judicial discretion—are emergent properties of the set-theoretic structure within which legal practice operates: a system of cases categorized according to multiple, overlapping rules. In short, the logical structure of the relationship between rules and cases is what generates some of our most persistent jurisprudential concerns. These concerns are emergent features of the system’s most elementary structures.

To be clear: I intend to demonstrate that a set-theoretic model of the structure of legal systems is extremely useful in understanding those systems, but I do not claim—nor do I in fact believe—that set theory can provide a complete model of legal systems, nor that it can answer all the important questions in legal theory or jurisprudence. I am not here making a claim about what law is in any metaphysical sense, about what makes it law as opposed to something else, about the appropriate scope of precedent, about the appropriate degree of constraint or discretion for judges, about the sources of normative content that do or should guide judicial decision-making, about law’s relation to morality or to tradition, or any similar jurisprudential concern. Nor do I intend to defend any claim about the proper interpretation of legal texts or any other prescriptive theory of adjudication. On the contrary, my point is merely that the logical structure of legal doctrine necessarily implies certain behavioral options in the context of specific legal disputes, and that these options are in fact independent of any such substantive or normative concerns—that they emerge from the structure of the legal system itself.

These claims, though modest, fill an unfortunate gap in the literature. There has never been an effort to formalize legal reasoning along the lines I attempt in this Article. This may be because the theoretical tools I will rely on were just gaining traction in philosophy at roughly the same time the American Legal Realists launched their program against formalist models of adjudication,17 and have since been largely abandoned to mathematicians as twentieth-century philosophy took its linguistic turn.18 Today, the formal logic of legal systems is a subject that is mainly of interest to philosophers,19 to those interested in the possibility of representing legal reasoning using the tools of artificial intelligence,20 and to scholars in the code-based civil law tradition21—with some notable and worthy exceptions.22 But lawyers and common-law legal scholars can benefit from greater attention to the common law as a distinctive logical system.

One important area of application is in legal education. The model developed in this paper will be useful to legal educators and law students, as a guide to the types of analytical moves that are part of “thinking like a lawyer.” For law students who tend to think graphically or spatially, in particular, the familiar graphical representations of sets in Venn diagrams is likely to be a particular aid to understanding and mastery of legal habits of mind such as analogy and distinction, the extraction of rules from cases, and the flexible scope of rule-application. Second, the tools of set theory are also of use to the practicing lawyer, judge, and legal commentator, because they offer a more precise vocabulary for identifying and critiquing poor legal reasoning that the typical rhetorical approach may gloss over. Finally, the set-theoretic model I will develop here has implications for legal theory, where heated substantive debates often leave important formal ambiguities unexamined. Not only can the set-theoretic model help cut through those ambiguities, but it also shows how some of these substantive debates are actually generated by inescapable structural features of law as a social practice.

The Article proceeds in four parts. Part I provides a brief overview of naïve set theory for those unfamiliar with it, introducing the terminology and concepts that will be deployed in the analysis that follows; those who feel comfortable with the concepts and notation systems of set theory and predicate logic may skim or skip this section. Parts II and III build the scaffolding of a set-theoretic model of law. Part II demonstrates how the tools of set theory can be deployed to analyze the relationship between rules and cases, and how legal doctrines can be modeled using set-theoretic concepts. Part III demonstrates more complex interactions of rules and cases, focusing on “hard cases” in which multiple applicable legal rules appear to contradict one another. This Part provides a more thorough description of the strategies legal actors can use to resolve such doctrinal conflicts at various levels of formal structure. Part IV discusses some implications and limitations of a set-theoretic understanding of legal doctrine, including its interaction with other aspects of legal theory.

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* Professor of Law & Faculty Director, Intellectual Property Law Center, St. John’s University School of Law. I am grateful for comments and suggestions from Brian Bix, David Carlson, Marc DeGirolami, Greg Keating, Anita Krishnakumar, Sara Lawsky, Josh Sarnoff, and Larry Solum. All errors are the author’s alone.