Applying the laws of logic to the logic of law.

• Author: Bavli, Hillel

INTRODUCTION

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.

1. PROVING CONSISTENCY: A SNAKE EATING ITS OWN TAIL

   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. Let us appropriately call any law that is not a natural law a derived law.

   Suppose that each year many new laws are derived from previously derived laws or from natural laws directly. Can it be shown that after many years beyond the system's creation, and many millions of laws beyond the initial natural laws, a contradiction among the system's laws will not arise? It is certainly invalid to conclude that a contradiction will not or cannot arise from the fact that one has not already arisen. (9)

   It can be shown that internal consistency among the foundational natural laws necessarily implies consistency among further properly derived laws. (10) Thus, to prove impossibility of contradiction among millions of eventual derived laws, one must prove consistency among the relatively few natural laws (assuming proper derivation). The possibility of such a proof--namely, that of consistency among the assumed foundational postulates of a given system--represents the concern of the current section.

   1. The Axiomatic Method

      Pure mathematics can be described as a science of deduction. It is the "subject in which we do not know what we are talking about, or whether what we are saying is true." (11) Its concern is not the truth of the assumed postulates or the deduced conclusions, but only that its deductions follow as necessary logical consequences of its assumptions. (12) The "axiomatic method," discovered by the ancient Greeks, is a system of deriving propositions, or theorems, from accepted postulates known as axioms. (13) In the aforementioned example, natural laws are the system's axioms, and derived laws are the system's theorems. (14) The ancient Greeks utilized the axiomatic method to develop an incredibly complex system of geometry deduced from five simple axioms (e.g., a straight line segment can be drawn joining any two points). (15)

      The Euclidean axioms were presumed to be true statements about space. Thus, insofar as theorems were deduced from such axioms, the possibility of deducing a contradiction escaped consideration. (16) That is, until the discovery of a new, non-Euclidian geometry.

      The Nineteenth Century discovery of different, yet equally valid, systems of geometry, such as elliptical geometry, destroyed the crutch upon which faith in the consistency of Euclidian geometry rested: external truth. (17) How can differing conceptions of a point or line both be true when only a single reality exists? (18) The notion of mathematics as a real-world, rather than an abstract discipline was hereby challenged. (19) Establishing the internal consistency of such systems suddenly took on critical importance.

   2. Solving One Problem by Creating Another

      The task of rigorously establishing the internal consistency of a system--even a simple system--quickly encounters a significant difficulty; specifically, a problematic set of alternative approaches. One approach is to utilize the system's own rules and postulates to establish its consistency. It is difficult, however, to justify reliance on a given system of reasoning to prove consistency in that same system of reasoning. Such analysis is circular and therefore unfounded. "It is like lifting yourself up by your own bootstraps." (20)

      The alternative to grounding a system's own reasoning in itself is to establish grounding in a second system's reasoning. This approach, however, accomplishes little more than shifting the problem to another domain. (21) The proof of one system's consistency becomes reliant upon the consistency of an additional system. A rigorous proof of the former system's consistency would thus require proof of the latter system's consistency, which, in turn, must face the same difficulties plaguing the former system's proof. (22)

      Thus, a dilemma unfolds: proving consistency of a given logical system seemingly requires the illogical reasoning...