Academic journal article Fordham Urban Law Journal

Applying the Laws of Logic to the Logic of Law

Academic journal article Fordham Urban Law Journal

Applying the Laws of Logic to the Logic of Law

Article excerpt


Consistency is a necessary condition of a just legal system, without which arbitrariness, unequal treatment, unpredictability, and, ultimately, injustice must result. "The truth," remarked Justice Holmes, "is that the law is always approaching, and never reaching, consistency." (1) But beyond meager intuition, or bare observation, is it possible to rigorously examine internal logical consistency (2)--mutual compatibility among legal deductions--in the rule of law? (3)

Kurt Godel, in a 1931 publication of a German scientific periodical, disproved the then-common assumption that each area of mathematics can be sufficiently axiomatized as to enable the development of an "endless totality of true propositions" about a given area of inquiry. (4) Specifically, he proved that any formal logical system (a concept that I shall more clearly explain below) that entails sufficient means as to support elementary arithmetic (5) is necessarily subject to the inherent characteristic of incompleteness: arithmetical propositions which can be neither proved nor disproved within the system. (6) Impliedly, every such system necessarily inheres either incompleteness or inconsistency. Further, Godel proved the impossibility of establishing "internal logical consistency of a very large class of deductive systems ... unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves." (7)

If applicable to law (a significant contingency indeed), Godel's proof indicates unavoidable judicial susceptibility to inconsistency, since abstinence from adjudication of formally undecidable cases is impractical. Thus, application of Godel's Incompleteness Theorem to the legal context would establish a priori limitations on the capacity for consistency to exist within the law, as well as on the faculty to establish internal logical consistency within the law.

Perhaps more importantly, the law itself manifests plausible limitations on its capacity to realize formal consistency, or to be examined with respect to its consistency. Specifically, formalizing a logical axiomatic legal system--a requirement of rigorously examining internal logical consistency--that retains the fundamental values of justice may prove difficult if not impossible. Further, proving or disproving formal legal consistency may require construction of a legal language sufficiently exact to map, or mirror, meta-legal statements--statements about a formalized legal system--within the legal language itself. Such construction may prove impossible as well.

I begin by discussing the difficulties of proving consistency within a formal system generally. After establishing the importance of a formalized legal model as a prerequisite of rigorous examination of consistency, I investigate issues intrinsic to the current system of law that may prevent formalization of a just legal system as currently conceived. (8) I argue that flexibility inherent in a just legal system (in the sense that judges have the ability to modify, in response to a given case, the presumptions from which that case's outcome will be derived) may foreclose the possibility of legal formalization or any comprehensive model thereof. I conclude, however, that a model whose purpose is the examination of consistency within a system need not necessarily retain the dynamic nature of real-world formalization. Rather, a static model of legal formalization may avoid the complications confronting a comprehensive formalization of law, while retaining the fundamental values critical to examination of consistency within the law.


Suppose the creation of a system in which certain natural laws are presumed true. Further, specified rules are initially established to allow additional laws to be derived from the presumed natural laws, and to allow further additional laws to be derived from other derived laws, and so on. …

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