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?1See  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”T4wBa5he”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps Ronald Dworkin}, {\\scaps A Matter of Principle} 119\\uc0\\u8211{}20 (1985).}”,”plainCitation”:”Ronald Dworkin, A Matter of Principle 119–20 (1985).”},”citationItems”:[{“id”:83,”uris”:[“http://zotero.org/users/1960061/items/3W5WACXI”],”uri”:[“http://zotero.org/users/1960061/items/3W5WACXI”],”itemData”:{“type”:”book”,”title”:”A Matter of Principle”,”publisher”:”Harvard University Press”,”number-of-pages”:”438″,”source”:”Google Books”,”abstract”:”This is a book about fundamental theoretical issues of political philosophy and jurisprudence. In his familiar forceful and incisive style Professor Dworkin guides the reader through a re-examination of some perennial moral, philosophical, and legal dilemmas.”,”ISBN”:”978-0-19-825574-1″,”language”:”en”,”author”:[{“family”:”Dworkin”,”given”:”Ronald”}],”issued”:{“raw”:”1985″}},”locator”:”119-20″}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Ronald Dworkin, A Matter of Principle 119–20 (1985).. 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?”2Cf. generally  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”n6gGjnVq”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps Wesley Newcomb Hohfeld}, {\\scaps Fundamental Legal Conceptions as Applied in Judicial Reasoning: And Other Legal Essays} (1920).}”,”plainCitation”:”Wesley Newcomb Hohfeld, Fundamental Legal Conceptions as Applied in Judicial Reasoning: And Other Legal Essays (1920).”},”citationItems”:[{“id”:1189,”uris”:[“http://zotero.org/users/1960061/items/VS5CFJ4U”],”uri”:[“http://zotero.org/users/1960061/items/VS5CFJ4U”],”itemData”:{“type”:”book”,”title”:”Fundamental Legal Conceptions as Applied in Judicial Reasoning: And Other Legal Essays”,”publisher”:”Yale University Press”,”number-of-pages”:”126″,”source”:”Google Books”,”shortTitle”:”Fundamental Legal Conceptions as Applied in Judicial Reasoning”,”language”:”en”,”author”:[{“family”:”Hohfeld”,”given”:”Wesley Newcomb”}],”issued”:{“raw”:”1920″}}}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Wesley Newcomb Hohfeld, Fundamental Legal Conceptions as Applied in Judicial Reasoning: And Other Legal Essays (Walter Wheeler Cook ed., 1919).. 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.3See Marc O. DeGirolami, The Tragedy of Religious Freedom 223 n.24 (2013) (citing  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”M3daLlGl”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps Timothy Andrew Orville Endicott}, {\\scaps Vagueness in Law} 72 (2000) (\\uc0\\u8220{}It is a consistent feature of legal systems that legal institutions treat legal standards as if their application were bivalent.\\uc0\\u8221{}).}”,”plainCitation”:”Timothy Andrew Orville Endicott, Vagueness in Law 72 (2000) (“It is a consistent feature of legal systems that legal institutions treat legal standards as if their application were bivalent.”).”},”citationItems”:[{“id”:479,”uris”:[“http://zotero.org/users/1960061/items/CMDNIWWV”],”uri”:[“http://zotero.org/users/1960061/items/CMDNIWWV”],”itemData”:{“type”:”book”,”title”:”Vagueness in Law”,”publisher”:”Oxford University Press”,”number-of-pages”:”230″,”source”:”Google Books”,”abstract”:”Vagueness in law leads to indeterminacies in legal rights and obligations in many cases. The book defends that claim and explains its implications for legal theory. Vague language is the book’s focus, but vagueness is not merely a linguistic feature of law. Law is necessarily vague. That fact seems to threaten the coherence of the ideal of the rule of law. The book defends a new, coherent articulation of that ideal.”,”ISBN”:”978-0-19-826840-6″,”language”:”en”,”author”:[{“family”:”Endicott”,”given”:”Timothy Andrew Orville”}],”issued”:{“raw”:”2000″}},”locator”:”72″,”suffix”:”(\”It is a consistent feature of legal systems that legal institutions treat legal standards as if their application were bivalent.\”)”}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Timothy Andrew Orville Endicott, Vagueness in Law 72 (2000) (“It is a consistent feature of legal systems that legal institutions treat legal standards as if their application were bivalent.”)).. 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”?4See Ashcroft v. Iqbal, 556 U.S. 662, 678–79 (2009). Is this statute’s effect on private speech limited to the punishment of “fighting words”?5Compare, e.g., R.A.V. v. City of St. Paul, 505 U.S. 377, 386–96 (1992), with Snyder v. Phelps, 131 S. Ct. 1207, 1215 n.3 (2011); id. at 1226–27 (Alito, J., dissenting). Would this liquidated damages award constitute a “penalty”?6U.C.C. § 2-718 (Am. Law Inst. & Unif. Law Comm’n 2017); see, e.g., Equitable Lumber Corp. v. IPA Land Dev. Corp., 344 N.E.2d 391, 397 (N.Y. 1976). 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,7See generally Pablo. E. Navarro & Jorge L. Rodríguez, Deontic Logic and Legal Systems (2014). 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.”8H. L. A. Hart, The Concept of Law 123 (2d ed. 1994); see also id. at 124 (“[T]he law must predominantly, but by no means exclusively, refer to classes of person, and to classes of acts, things, and circumstances; and its successful operation over vast areas of social life depends on a widely diffused capacity to recognize particular acts, things, and circumstances as instances of the general classifications which the law makes.”) (emphasis in original). 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.9See generally  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”wttgDoBT”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps Frederick Schauer}, {\\scaps Playing by the Rules: A Philosophical Examination of Rule-Based Decision-Making in Law and in Life} 18\\uc0\\u8211{}21 (1991).}”,”plainCitation”:”Frederick Schauer, Playing by the Rules: A Philosophical Examination of Rule-Based Decision-Making in Law and in Life 18–21 (1991).”},”citationItems”:[{“id”:1103,”uris”:[“http://zotero.org/users/1960061/items/U2MM46QD”],”uri”:[“http://zotero.org/users/1960061/items/U2MM46QD”],”itemData”:{“type”:”book”,”title”:”Playing by the Rules: A Philosophical Examination of Rule-Based Decision-Making in Law and in Life”,”publisher”:”Clarendon Press”,”number-of-pages”:”278″,”source”:”Google Books”,”abstract”:”This is a philosophical but non-technical analysis of the very idea of a rule. Although focused somewhat on the role of rules in the legal system, it is also relevant to the place of rules in morality, religion, etiquette, games, language, and family governance. In both explaining the idea of a rule and making the case for taking rules seriously, the book is a departure both in scope and in perspective from anything that now exists.”,”ISBN”:”978-0-19-101874-9″,”shortTitle”:”Playing by the Rules”,”language”:”en”,”author”:[{“family”:”Schauer”,”given”:”Frederick”}],”issued”:{“raw”:”1991-08-22″}},”locator”:”18-21″}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Frederick Schauer, Playing by the Rules: A Philosophical Examination of Rule-Based Decision-Making in Law and in Life 18–21 (1991).. Indeed, all analysis—legal or otherwise—may at some level be reduced to this type of “lumping” and “splitting.”10 ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”b1tSepuJ”,”properties”:{“formattedCitation”:”{\\rtf Bradley C. Karkkainen, {\\i{}New Governance in Legal Thought and in the World: Some Splitting as Antidote to Overzealous Lumping}, 89 {\\scaps Minn. L. Rev.} 471, 479 (2004) (\\uc0\\u8220{}It  is  sometimes  said  that  the  two  most  basic  intellectual moves  are  \\uc0\\u8221{}lumping\\uc0\\u8220{}  and  \\uc0\\u8221{}splitting\\uc0\\u8220{}-that  is,  finding  relevant common  characteristics  that  allow  us  intelligently  and  usefully to  group  apparently  distinct  phenomena  into  a  single  category [\\uc0\\u8221{}lumping\\uc0\\u8220{}],  3 3  and  finding  relevant  distinguishing  characteris- tics  that  allow  us  intelligently  and  usefully  to  separate  other- wise  similar  phenomena  into  distinct  classes  [\\uc0\\u8221{}splitting\\uc0\\u8220{}].\\uc0\\u8221{}).}”,”plainCitation”:”Bradley C. Karkkainen, New Governance in Legal Thought and in the World: Some Splitting as Antidote to Overzealous Lumping, 89 Minn. L. Rev. 471, 479 (2004) (“It  is  sometimes  said  that  the  two  most  basic  intellectual moves  are  ”lumping“  and  ”splitting“-that  is,  finding  relevant common  characteristics  that  allow  us  intelligently  and  usefully to  group  apparently  distinct  phenomena  into  a  single  category [”lumping“],  3 3  and  finding  relevant  distinguishing  characteris- tics  that  allow  us  intelligently  and  usefully  to  separate  other- wise  similar  phenomena  into  distinct  classes  [”splitting“].”).”},”citationItems”:[{“id”:2803,”uris”:[“http://zotero.org/users/1960061/items/RX2V2RD4″],”uri”:[“http://zotero.org/users/1960061/items/RX2V2RD4″],”itemData”:{“type”:”article-journal”,”title”:”New governance in legal thought and in the world: some splitting as antidote to overzealous lumping”,”container-title”:”Minn. L. Rev.”,”page”:”471″,”volume”:”89″,”source”:”Google Scholar”,”URL”:”http://heinonline.org/hol-cgi-bin/get_pdf.cgi?handle=hein.journals/mnlr89&section=20″,”shortTitle”:”New governance in legal thought and in the world”,”author”:[{“family”:”Karkkainen”,”given”:”Bradley C.”}],”issued”:{“raw”:”2004″},”accessed”:{“raw”:”2017-07-25T16:44:52Z”}},”locator”:”479″,”suffix”:”(\”It  is  sometimes  said  that  the  two  most  basic  intellectual moves  are  \”lumping\”  and  \”splitting\”-that  is,  finding  relevant common  characteristics  that  allow  us  intelligently  and  usefully to  group  apparently  distinct  phenomena  into  a  single  category (\”lumping\”),  3 3  and  finding  relevant  distinguishing  characteris- tics  that  allow  us  intelligently  and  usefully  to  separate  other- wise  similar  phenomena  into  distinct  classes  (\”splitting\”).\”)”}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Bradley C. Karkkainen, “New Governance” in Legal Thought and in the World: Some Splitting as Antidote to Overzealous Lumping, 89 Minn. L. Rev. 471, 479 (2004) (“It is sometimes said that the two most basic intellectual moves are ‘lumping’ and ‘splitting’—that is, finding relevant common characteristics that allow us intelligently and usefully to group apparently distinct phenomena into a single category (‘lumping’), and finding relevant distinguishing characteristics that allow us intelligently and usefully to separate otherwise similar phenomena into distinct classes (‘splitting’).”).. 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 ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”HZ9z7RpX”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps Ernest Nagel, James R. Newman & Douglas R. Hofstadter}, {\\scaps G del\\uc0\\u8217{}s Proof} (2001), https://books.google.com/books?hl=en&lr=&id=oQsUCgAAQBAJ&oi=fnd&pg=PR3&dq=%22The+Systematic+Codification+of+Formal%22+%22of+the+manuscript,+and+helped+to+clarify+the%22+%22An+Example+of+a+Successful+Absolute+Proof%22+%22The+arithmetization+of%22+%22they+received+from+Professor+John+C.%22+&ots=XsCVfI3Gt7&sig=w_P7IiY_pMO83SWVxawmS0IpDzM (last visited Apr 30, 2016).}”,”plainCitation”:”Ernest Nagel, James R. Newman & Douglas R. Hofstadter, G del’s Proof (2001), https://books.google.com/books?hl=en&lr=&id=oQsUCgAAQBAJ&oi=fnd&pg=PR3&dq=%22The+Systematic+Codification+of+Formal%22+%22of+the+manuscript,+and+helped+to+clarify+the%22+%22An+Example+of+a+Successful+Absolute+Proof%22+%22The+arithmetization+of%22+%22they+received+from+Professor+John+C.%22+&ots=XsCVfI3Gt7&sig=w_P7IiY_pMO83SWVxawmS0IpDzM (last visited Apr 30, 2016).”,”dontUpdate”:true},”citationItems”:[{“id”:1000,”uris”:[“http://zotero.org/users/1960061/items/RRM7FAQQ”],”uri”:[“http://zotero.org/users/1960061/items/RRM7FAQQ”],”itemData”:{“type”:”book”,”title”:”G del’s Proof”,”publisher”:”NYU Press”,”source”:”Google Scholar”,”URL”:”https://books.google.com/books?hl=en&lr=&id=oQsUCgAAQBAJ&oi=fnd&pg=PR3&dq=%22The+Systematic+Codification+of+Formal%22+%22of+the+manuscript,+and+helped+to+clarify+the%22+%22An+Example+of+a+Successful+Absolute+Proof%22+%22The+arithmetization+of%22+%22they+received+from+Professor+John+C.%22+&ots=XsCVfI3Gt7&sig=w_P7IiY_pMO83SWVxawmS0IpDzM”,”author”:[{“family”:”Nagel”,”given”:”Ernest”},{“family”:”Newman”,”given”:”James R.”},{“family”:”Hofstadter”,”given”:”Douglas R.”}],”issued”:{“raw”:”2001″},”accessed”:{“raw”:”2016-04-30T01:37:10Z”}}}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Ernest Nagel & James R. Newman, Gödel’s Proof 26 (Douglas R. Hofstadter ed., rev. ed. 2001)..

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 ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”LJs2meWQ”,”properties”:{“formattedCitation”:”{\\rtf Compare {\\scaps Brian Z. Tamanaha}, {\\scaps Beyond the Formalist-Realist Divide: The Role of Politics in Judging} (2009); with Brian Leiter, {\\i{}Legal Formalism and Legal Realism: What Is the Issue?}, 16 {\\scaps Leg. Theory} 111 (2010).}”,”plainCitation”:”Compare Brian Z. Tamanaha, Beyond the Formalist-Realist Divide: The Role of Politics in Judging (2009); with Brian Leiter, Legal Formalism and Legal Realism: What Is the Issue?, 16 Leg. Theory 111 (2010).”,”dontUpdate”:true},”citationItems”:[{“id”:805,”uris”:[“http://zotero.org/users/1960061/items/K93AZBWX”],”uri”:[“http://zotero.org/users/1960061/items/K93AZBWX”],”itemData”:{“type”:”book”,”title”:”Beyond the Formalist-Realist Divide: The Role of Politics in Judging”,”publisher”:”Princeton University Press”,”number-of-pages”:”530″,”source”:”Google Books”,”abstract”:”According to conventional wisdom in American legal culture, the 1870s to 1920s was the age of legal formalism, when judges believed that the law was autonomous and logically ordered, and that they mechanically deduced right answers in cases. In the 1920s and 1930s, the story continues, the legal realists discredited this view by demonstrating that the law is marked by gaps and contradictions, arguing that judges construct legal justifications to support desired outcomes. This often-repeated historical account is virtually taken for granted today, and continues to shape understandings about judging. In this groundbreaking book, esteemed legal theorist Brian Tamanaha thoroughly debunks the formalist-realist divide. Drawing from extensive research into the writings of judges and scholars, Tamanaha shows how, over the past century and a half, jurists have regularly expressed a balanced view of judging that acknowledges the limitations of law and of judges, yet recognizes that judges can and do render rule-bound decisions. He reveals how the story about the formalist age was an invention of politically motivated critics of the courts, and how it has led to significant misunderstandings about legal realism. Beyond the Formalist-Realist Divide traces how this false tale has distorted studies of judging by political scientists and debates among legal theorists. Recovering a balanced realism about judging, this book fundamentally rewrites legal history and offers a fresh perspective for theorists, judges, and practitioners of law.”,”ISBN”:”978-1-4008-3198-2″,”shortTitle”:”Beyond the Formalist-Realist Divide”,”language”:”en”,”author”:[{“family”:”Tamanaha”,”given”:”Brian Z.”}],”issued”:{“raw”:”2009-10-26″}},”prefix”:”Compare”},{“id”:530,”uris”:[“http://zotero.org/users/1960061/items/DTEKHDI6″],”uri”:[“http://zotero.org/users/1960061/items/DTEKHDI6″],”itemData”:{“type”:”article-journal”,”title”:”Legal Formalism and Legal Realism: What Is the Issue?”,”container-title”:”Legal Theory”,”page”:”111″,”volume”:”16″,”issue”:”2″,”URL”:”http://chicagounbound.uchicago.edu/journal_articles/2743″,”shortTitle”:”Legal Formalism and Legal Realism”,”author”:[{“family”:”Leiter”,”given”:”Brian”}],”issued”:{“raw”:”2010-01-01″}},”prefix”:”with”}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Compare Brian Z. Tamanaha, Beyond the Formalist-Realist Divide: The Role of Politics in Judging (2010), with Brian Leiter, Legal Formalism and Legal Realism: What Is the Issue?, 16 Legal Theory 111 (2010).. 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.13See generally Leiter, supra note 12. 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.14The word “rule” here is thus not intended to reflect the particular meaning of the word “rule” in contrast with the word “standard” as the dichotomy is typically framed in legal theory; I discuss the relationship between my set-theoretic model and the rules/standards dichotomy (including the dimensions of precision and generality) in Section IV.B, infra.. 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.15See, e.g., Thomas v. United States, 505 F.2d 1282, 1286 (Ct. Cl. 1974) (quoting Haynes v. Eagle-Picher Co., 295 F.2d 761, 763 (10th Cir. 1961)); Duck Creek Tire Serv., Inc. v. Goodyear Corners, L.C., 796 N.W.2d 886, 893 (Iowa 2011); Hartman Ranch Co. v. Associated Oil Co., 73 P.2d 1163, 1168 (1937).. 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.16See infra Section II.A.

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,17For example, when Russell and Whitehead’s Principia Mathematica was first published, Jerome Frank was already a law student on his way to private practice. Compare Frank, Jerome New, Fed. Jud. Ctr., https://‌www.fjc.gov/‌history/‌judges/‌frank-jerome-new [https://‌perma.cc/‌TC5Q-‌PX6K] (last visited Mar. 26, 2019), with Andrew David Irvine, Principia Mathematica, Stan. Encyclopedia Phil., https://‌plato.stanford.edu/‌archives/‌spr2015/‌entries/‌principia-mathematica [https://‌perma.cc/‌ZXN4-‌NA96] (last updated Mar. 10, 2015). By the time the disciplines of philosophy and mathematics had sufficiently digested the Principia to generate Gödel’s breakthroughs, Karl Llewellyn and Max Radin were busy advancing the Realist program from perches on prominent law school faculties. See generally, e.g., Karl N. Llewellyn, A Realistic Jurisprudence—The Next Step, 30 Colum. L. Rev. 431 (1930); Max Radin, Statutory Interpretation, 43 Harv. L. Rev. 863 (1930). Indeed, when Radin refers to the “determinables” and “determinates” of W. E. Johnson’s Logic, in his disquisition on statutory interpretation, he is unwittingly relying on a philosopher whose efforts were soon to be overtaken by the system of Russell and Whitehead (and by the increasingly formal logics that followed), in keeping with the more general movement of the study of logic from philosophers to mathematicians. See id. at 868–70;  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”8snwQwqu”,”properties”:{“formattedCitation”:”{\\rtf {\\scaps John Passmore}, {\\scaps A \nHundred Years of Philosophy} 343\\uc0\\u8211{}45,  394 (2d ed. 1966).}”,”plainCitation”:”John Passmore, A Hundred Years of Philosophy 343–45,  394 (2d ed. 1966).”},”citationItems”:[{“id”:52,”uris”:[“http://zotero.org/users/1960061/items/36CDHC68″],”uri”:[“http://zotero.org/users/1960061/items/36CDHC68″],”itemData”:{“type”:”book”,”title”:”A \nHundred Years of Philosophy”,”publisher”:”Penguin Books”,”number-of-pages”:”644″,”edition”:”2d”,”source”:”Google Books”,”ISBN”:”978-0-14-013671-5″,”note”:”Google-Books-ID: 7ltRAAAAYAAJ”,”language”:”en”,”author”:[{“family”:”Passmore”,”given”:”John”}],”issued”:{“raw”:”1966″}},”locator”:”343-45,  394″}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} John Passmore, A Hundred Years of Philosophy 343–45, 394 (2d ed. 1966).. and have since been largely abandoned to mathematicians as twentieth-century philosophy took its linguistic turn.18The “linguistic turn” is a term of art in intellectual history and refers to an increased focus in post-war analytic (i.e., Anglo-American) philosophy on the philosophy of language, for which first-order symbolic logic of the type relied on in this Article was quickly deemed unsuitable. The term itself was popularized in The Linguistic Turn: Essays in Philosophical Method (Richard M. Rorty ed., 1992). See generally  ADDIN ZOTERO_ITEM CSL_CITATION {“citationID”:”i7XiPxvy”,”properties”:{“formattedCitation”:”{\\rtf P.M.S. Hacker, {\\i{}The Linguistic Turn in Analytic Philosophy}, {\\i{}in} {\\scaps The Oxford Handbook of the History of Analytic Philosophy} 926 (Micahel Beaney ed., 2013).}”,”plainCitation”:”P.M.S. Hacker, The Linguistic Turn in Analytic Philosophy, in The Oxford Handbook of the History of Analytic Philosophy 926 (Micahel Beaney ed., 2013).”},”citationItems”:[{“id”:853,”uris”:[“http://zotero.org/users/1960061/items/MNSR5J3N”],”uri”:[“http://zotero.org/users/1960061/items/MNSR5J3N”],”itemData”:{“type”:”chapter”,”title”:”The Linguistic Turn in Analytic Philosophy”,”container-title”:”The Oxford Handbook of the History of Analytic Philosophy”,”page”:”926″,”URL”:”http://info.sjc.ox.ac.uk/scr/hacker/docs/TheLinguisticTurn.pdf”,”author”:[{“family”:”Hacker”,”given”:”P.M.S.”}],”editor”:[{“family”:”Beaney”,”given”:”Micahel”}],”issued”:{“raw”:”2013″},”accessed”:{“raw”:”2016-08-27T00:25:30Z”}}}],”schema”:”https://github.com/citation-style-language/schema/raw/master/csl-citation.json”} Peter M.S. Hacker, The Linguistic Turn in Analytic Philosophy, in The Oxford Handbook of the History of Analytic Philosophy 926 (Michael Beaney ed., 2013).. Today, the formal logic of legal systems is a subject that is mainly of interest to philosophers,19See generally, e.g., Carlos E. Alchourrón, On Law and Logic, 9 Ratio Juris 331 (1996); Richard Holton, Modeling Legal Rules, in Philosophical Foundations of Language in the Law 165 (Andrei Marmor & Scott Soames eds., 2011). to those interested in the possibility of representing legal reasoning using the tools of artificial intelligence,20See generally, e.g., Carlos E. Alchourrón, Philosophical Foundations of Deontic Logic and the Logic of Defeasible Conditionals, in Deontic Logic in Computer Science: Normative System Specification 43 (John-Jules Ch. Meyer & Roel J. Wieringa eds., 1993); John F. Horty & Trevor J.M. Bench-Capon, A Factor-Based Definition of Precedential Constraint, 20 Artificial Intelligence & L. 181 (2012); Henry Prakken, Logical Tools for Modelling Legal Argument: A Study of Defeasible Reasoning in Law, 64 Studia Logica: An Int’l J. for Symbolic Logic 143 (2000). and to scholars in the code-based civil law tradition21For example, one recent compilation of essays (in English) on legal logic included contributions from twenty-four authors, only six of whom are from common-law countries. The Logic of Legal Requirements: Essays on Defeasibility, at xi (Jordi Ferrer Beltrán & Giovanni Battista Ratti eds., 2012). See also, e.g., Navarro & Rodriguez, supra note 7 (Argentinian authors); Jaap Hage, Law, Logic and Defeasibility, 11 Artificial Intelligence & L. 221 (2003) (Dutch author). —with some notable and worthy exceptions.22See generally Horty & Bench-Capon, supra note 20; Henry Prakken & Giovanni Sartor, Modelling Reasoning with Precedents in a Formal Dialogue Game, in Judicial Applications of Artificial Intelligence 127 (Giovanni Sartor & Karl Branting eds., 1998). One leading American scholar of legal logic, Sara Lawsky, not coincidentally works primarily in the heavily codified field of tax law. See generally, e.g., Sarah B. Lawsky, A Logic for Statutes, 21 Fla. Tax Rev. 60 (2017). Other innovative scholarship has focused on the potential of non-classical logics to illuminate particular aspects of legal reasoning such as fact-finding. See generally, e.g., Kevin M. Clermont, Conjunction of Evidence and Multivalent Logic, in Law and the New Logics 32 (H. Patrick Glenn & Lionel D. Smith eds., 2017); Vern R. Walker, A Default-Logic Paradigm for Legal Fact-Finding, 47 Jurimetrics J. 193 (2007). 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.

* * * * *