Peter Suber, Paradox of Self-Amendment, Section 1

Part One
The Paradox of Self-Amendment

Section 1
Introduction: Logical Paradoxes in Law
Peter Suber, Paradox of Self-Amendment Table of Contents

A. Paradoxes that perplex, and paradoxes that prove

Paradoxes are statements that look meaningful, and often true, but that justify contradictory conclusions. Therefore they cause consternation among logicians and mathematicians.[Note 1] Should they cause consternation among lawyers? That is the background question of this essay.

Paradoxes are disturbing because we have difficulty denying them status as meaningful statements subject to the normal rules about contradiction and truth, and we are rarely willing to amend our logical rules merely to accommodate a string of words that twists them up. The easy solution, then, is to classify the string out of the danger zone: if it is meaningless, or non-cognitive (neither true nor false), or otherwise disqualified, then it is not subject to the logical rules about propositions and it cannot disturb.

Conversely, paradoxical statements disturb the more they resist these attempts at convenient reclassification, the more they seem meaningful and cognitive. If we accept the application of normal logical rules to these statements, hypothetically or because the statements cannot be distinguished on relevant grounds from the propositions of science and daily life, then we must face the contradictions among the rules that result. We will then have brought the system of rules itself into question.

The classical example of a logical paradox is the Epimenides or the Liar: "This very statement is false."[Note 2] It looks meaningful and subject to the ordinary rules for assigning truth-values. But if we call it false, then we must infer that it is true, and if we call it true, then we must infer that it is false. This result violates our expectations, and if we are logicians, it violates our rules, for we believe the general proposition that a statement cannot be both true and false in the same sense at the same time with reference to the same moment. We normally follow the rule that contradictory predicates can be true of the same substantive only by equivocation or in succession.[Note 3] Without appeal to such a rule about truth-values and their distribution, and a metastatement to the effect that the Liar's statement is subject to the rule, we could not call the Liar a paradox.

Many object to the self-reference of the Liar's statement as the semantic violation creating the paradox; but the self-reference is avoidable. The paradox may be restated as two statements that refer to one another without either referring to itself.

A. Statement B is true.
B. Statement A is false.

The larger or indirect circularity of reference, and the reflexivity of the other paradoxes, may well be ineliminable. However, the paradox may not be solved by the flat ban on direct self-reference that one may think of first.

Two paradoxes other than the Liar, discovered in this century, have become important types. The first is Russell's paradox. Sets can have other sets as members; while we can speak of the set of all cats and the set of all dogs, we can also speak of the set of all sets of animals. The latter is a set whose members are other sets. Since sets can be members of sets, some sets can be members of themselves —for example, the set of all sets described in this paragraph, or the set of all non-cats. But of course many sets of sets will not be members of themselves, such as the set of all sets of animals. Hence we can speak of "the set of all sets that are not members of themselves". Is the last-named set a member of itself? If it is, then it isn't, and if it isn't, then it is.[Note 4]

The second paradox is usually called Grelling's paradox. If an adjective correctly describes itself, call it autological. For example, "short" and "polysyllabic" are autological because they are, respectively, short and polysyllabic. If an adjective does not correctly describe itself, call it heterological. "Wooden" and "monosyllabic" are heterological. The paradox arises when we ask whether "heterological" is heterological. If it is, then it isn't, and if it isn't, then it is.[Note 5]

These three paradoxes all display a common feature apart from their self-reference or self-applicability. In each case a predicate and its negation are made equally applicable to the appropriate sort of substantive. Let us call the predicate which a paradox turns against itself the "paradoxical predicate". For the Liar, the paradoxical predicate, applicable to statements, is "being true". For Russell's paradox, applicable to sets, it is "being a member of" a certain set. For Grelling's paradox, applicable to adjectives, it is "being heterological". If a statement were formulated for each paradox which affirmed the paradoxical predicate of the appropriate substantive, then that statement would imply its own negation, and its negation would imply the statement.

This characterization is still too general to differentiate Russell's paradox from Grelling's and the Liar. Russell's paradox is a different logical type, as we will see, but it displays this type of paradoxical predication. The mutual implication of the paradoxical statement and its negation, at least for Grelling's paradox and the Liar, gives this type of paradox a structure that makes it a member of an important family of propositions: tautologies or necessary truths, contradictions or necessary falsehoods, and contingencies or statements whose truth value is contingent upon which possible universe is actual. Each may be defined in many ways, but the definitions below highlight their unity and lay the foundation for a distinction between types of paradox.

A tautology may be defined as a statement which is implied by its own negation, but which does not imply its own negation. A tautology is a formal truth, e.g. "Either proposition p is true or it is not true." This proposition is true by virtue of its form alone, without regard to its content or the truth of its component p. A contradiction implies its own negation, but is not implied by its own negation. A contradiction is a formal falsehood, e.g. "Both proposition p and ~p are true." This proposition is false by virtue of its form. A contingency neither implies nor is implied by its own negation. Its truth is not a function of its form, but only of its content, e.g. "Either proposition p is true or proposition q is true." Following this pattern we may define the Grelling and Liar type of paradox as a statement that both implies and is implied by its negation. In the most common notation:

p is a tautology if and only if ~ (p ~ p) · (~ p p)
contradiction    (p ~ p) · ~ (~ p p)
contingency ~ (p ~ p) · ~ (~ p p)
paradox, Liar type    (p ~ p) · (~ p p)

(This notation is to clarify matters for those who already know it; it will not be used in the body of the book.)

But there is another type of paradox. Logically it is closer to a contradiction, by the definitions above, than a paradox. It includes Russell's paradox, but the type is best illustrated by the popular paradox of the Barber. Suppose a town has a barber who shaves all and only those men in the town who do not shave themselves. If the barber is a shaved man who lives in the town, then does the barber shave himself?[Note 6] At first this looks like the type of paradox already described, for the answer is: if he does, then he doesn't (because he does not shave those who shave themselves), and if he doesn't, then he does (because he shaves all those who do not shave themselves).

But there is an important difference between the Barber and the Liar paradoxes. We can evade all contradiction in the Barber paradox by concluding that the Barber does not exist as defined. The contradictions only arise from the assumption that such a peculiar Barber exists, and we are free to reject that assumption; in fact, the contradictions give us good reason to do so. There is no comparable assumption in the Liar paradox to reject, unless it is the belief that the words say what they seem to say. "Liar-type" paradoxes, then, make contradiction uncomfortably unavoidable; they demand radical remedies. "Barber-type" paradoxes, by contrast, simply prove that something cannot exist as defined on pain of contradiction. They are "paradoxical" only at first sight; in the last analysis they are proofs.

By the definitions set out above, the statement that the Barber shaves himself is a Liar-type paradox; but the statement that the Barber exists is a contradiction, not a paradox. If we knew nothing about the Barber but his ambition to shave all and only those men in a certain town who did not shave themselves, then we could avoid contradiction by concluding that the Barber was a woman, or did not live in the town, or both. But if the Barber is a man in the town, then we can avoid contradiction only by denying his existence. This turns out to be a perfectly consistent proposition, even if we are surprised at the necessity of concluding it.

Russell's paradox is of the Barber-type, not the Liar-type. It proves with finality that there is no such thing as a set of all sets that are not members of themselves. Russell's paradox is more difficult to cope with than the Barber, not because it is logically different, but because so much has depended on the broad notion of a set as any collection of any elements, and so little has depended on belief in the Barber. This shows how a paradox may require revision of our most fundamental concepts.

Paradoxes of the Barber-type are often called "veridical" paradoxes because they establish the truth of a proposition. The opposite of a veridical paradox is not the Liar-type but a "falsidical" paradox which attempts to prove the falsehood of a proposition by deriving contradictions from its affirmation. Zeno's paradoxes of motion are the most common examples. They purport to prove that motion is impossible, or to falsify the belief that motion is real.[Note 7] Some writers distinguish both veridical and falsidical paradoxes from the Liar-type paradoxes by calling the latter "antinomies". However, that usage has not been widely adopted. I shall call the Liar-type paradoxes simply "paradoxes" or "genuine paradoxes" for emphasis, and the Barber-type, "veridical paradoxes", but only when the context requires that the two types be distinguished.

If the Liar-type or genuine paradox is symbolized,

(p ~ p) · (~ p p)

(p implies its negation and its negation implies p), then the veridical paradox should be symbolized,

{[p (q · ~ q)] ~ p} · ~ (~ p p)

(the fact that p implies a contradiction implies that p is false, and it is not the case that the falsehood of p implies the truth of p). But this formulation is equivalent to the definition of a contradiction. Hence, acceptable alternate terminology is that the Liar-type statement is a genuine paradox, and the Barber-type statement is merely a contradiction, although a surprising one, and one which leads to genuine paradox if affirmed.[Note 8]

B. "Solving" paradoxes in logic and law

To "solve" a paradox is to restate our rules about truth-values or about meaning or about some other feature of the proposition or its interpretation so that the statement no longer violates them, and so that the body of truths determined by the rules is saved, as far as possible, from loss of meaning or truth. These attempts break into two broad classes: those that attack the statement as ill-formed, and those that attack the system of rules as inconsistent or incomplete. The former allows the rules of the system to defend themselves by "overruling" the attempt by the paradox to fit comfortably into the system. The latter allows the statement to stand as a meaningful utterance, and uses it as leverage against the system of rules that cannot, so far, accommodate it.[Note 9] The former attempt often seems to beg the question by presupposing the adequacy of the logical system under suspicion. The latter attempt often risks the destruction, or the relegation to logical limbo, of too much meaningful language, useful rules, and good logic and mathematics.

Laws may be paradoxical within the system of legal rules just as propositions may be paradoxical within the system of logical rules. (What is more interesting is that laws as propositions may be paradoxical within the system of logical rules and yet not be paradoxical within the system of legal rules, as we shall see.) The methods of coping with legal paradoxes fall into the same two broad categories. On the one hand, a paradoxical law may be attacked as void or, on the other, the legal rules about validity, authority, and permissibility may be attacked as inconsistent, incomplete, or inadequate in some other way.

In logic we may reformulate and tighten the statement of rules to eliminate vagueness and inconsistency, and we may add others that seem to solve the problem. But in law only certain officials have these powers. In logic we may decide that the paradoxical statement is to blame, and brand it meaningless or non-cognitive, but in law the power to nullify the paradoxical law is vested only in some, although all have the power to disregard the paradox. A logical solution to a logical paradox in law, therefore, will only rarely be a legal solution too. For the same reason, a logical solution that would brand a paradoxical law as a nullity lacks the requisite legal authority to nullify, and therefore we must admit that a law can persist in legal validity despite paradox or contradiction. This conclusion cannot be winked away by the a priori application of logic to law, nor can its importance to legal reasoning be overestimated.

Many jurists, especially those inclined to respect formal logic, deny that any actual contradictions exist in law. Apparent contradictions, they hold, may always in principle be ironed out by the canons of interpretation and priority; that is, in any apparent conflict of legal rules, all but one rule is either inapplicable to the given circumstances or invalid.[Note 10] One canon of interpretation requires that inconsistent rules be read in a way that reconciles them, and if they are irreconcilable, to favor the recent law at the expense of the older one (see Section 16). David Daube complains that we know very little about impossible and contradictory laws because they are typically "construed away" by judges armed with the canons of interpretation and a fear of contradiction.[Note 11] I will regard the question of the existence of actual (as opposed to apparent) contradictions in law as an empirical question, if only for methodological purposes, to allow me to look for them. To assume a priori that they cannot exist would close many doors of my inquiry and, I believe, distort our understanding of law.

Exploration of the possible roots of a paradox may well disclose deeply hidden flaws in our rules of meaning, which make the paradoxical statement seem meaningful, in our rules about truth-values, which seem to support both the truth and the falsehood of the paradoxical statement, or in our legal rules of validity, which give the paradoxical law apparent or actual validity. Or else it may simply stir endless controversy. As Quine put it,[Note 12]

The argument that sustains a paradox may expose the absurdity of a buried premiss or of some preconception previously reckoned as central to physical theory, to mathematics or to the thinking process. Catastrophe may lurk, therefore, in the most innocent-seeming paradox. More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.

Russell's paradox, for example, showed catastrophe in the form of contradiction lurking in the basic assumptions of set theory, namely, that a set could comprise any collection of any elements or that the same rules applied to all sets of any elements. Doing without either of these broad propositions is difficult and inelegant, but preferable to living with paradox or doing without a consistent set theory.

By far the majority of attempts to solve the logical paradoxes have been undertaken by logicians and mathematicians. One cause and one effect of this concentration is that paradoxes have been studied in comparative isolation as statements that must be contended with, simply qua strings of words which our formal and linguistic rules apparently permit, or qua postulates which common sense finds unobjectionable. The appearance of paradoxes has only rarely been studied in particular domains of human thought where purposes other than clarity and consistency are foremost. This essay is a contribution to the study of logical paradoxes in law.

Statements that are logically similar to the paradoxes studied by logicians, and arguments that are logically similar to those that support paradoxes in mathematics, appear in law. Sometimes they appear to judges, sometimes to legislators, and sometimes to advocates (see Section 20). The tangled complexes of rules, the lack of clarity on the ultimate presuppositions of the system, self-reference and circular inferences, and other conditions that give rise to the various paradoxes frequently occur in legal cases, contracts, statutes, constitutions, and legal systems. The phenomenon of logical paradoxes in law is especially intriguing because lawyers do not share the logical scruples of logicians,[Note 13] do not share the luxury or freedom to let controversies run on endlessly, and yet are constrained by their professionalism and by their roles (judge, legislator, advocate, counselor, administrator) to be principled in all their decisions.

Logicians need not ever decide the correct approach to the paradoxes, whereas lawyers must not only resolve them when they arise, but must frequently do so under a deadline. Logicians who have decided, to their own satisfaction, the correct approach to a paradox may nevertheless be grateful for the indefinite time and unhurried pace allowed to their predecessors, and the relative freedom from risk if they are wrong. Lawyers under a deadline are normally unaware of the centuries of prior thought on the paradoxes, and in any case might find it irrelevant to the "condensation" of the paradox in their particular case and to the inherited rules under which they must devise a solution.

Logicians would never be satisfied with a provisional solution to a paradox, or with a solution that admittedly took no heed of the consequences for the ultimate principles of the logical system under scrutiny. For logicians are scientists who are studying the very integrity of their systems. If that integrity is in doubt, then a "solution" that preserved the doubt or hid it behind a practical indifference to theory would be the very opposite of a scientific solution. Scientists are fortunate, then, that their calling requires only honest study and progress, not solution. Even negative results and the discovery of new difficulties are praiseworthy in science. Logicians feel urgency to solve the paradoxes because their systems will be less certain than they might be until a solution can free them from suspicion; but no judge will order them to prepare for final arguments within a month. Logicians would wait one thousand years for an adequate solution to the Liar paradox, and would reconstruct their systems from the ground up if necessary. In fact, they have already waited twice as long and have reconstructed their systems radically more than once. Lawyers cannot wait, must employ provisional solutions, and lack the same freedom to reconstruct their systems.

Lawyers work within legal systems, and strive to preserve them, in a way very different from the way in which logicians work with and strive to preserve logical systems. I will argue that a legal system is distinct from a logical system (inter alia) in its capacity to tolerate contradiction.[Note 14] The difference between the two systems, and the results of conceiving a system of legal rules naively to be a logical-type system of logical-type rules, are central topics of this essay.

Lawyers and logicians also have different concepts of solving a problem. If a paradox is compared to a maze, lawyers would be content to "solve" it by knocking down the walls, if the relevant rules did not forbid it. To a logician that would be the height of intellectual dishonesty and self-deception. But we should not think the lawyer's "solution" a cheat, for it is a solution to a different problem; or, from another perspective, it is an optimum defined by different constraints. To the logician the problem is abstract and epistemic: how to understand the paradoxical statement and the system of rules pertaining to meaning and truth in order to bring the two into harmony at the least cost to the bodies of logical and mathematical science. To the lawyer the problem is concrete and practical: how best to adjudicate the interests of these two parties, or how best to achieve a certain state of affairs, in accordance with an inherited system of rules that contains material content, vagueness, ambiguity, and inconsistency.

In what follows I will consider legal solutions to logical paradoxes in part under criteria suited to legal decisions and actions. Legal solutions are not, to me, dishonest or unscientific merely for emphasizing the practical. They may well be illogical in the technical sense. But if they are, then they may merely reveal and symptomize differences between logical and legal systems, not necessarily defects in the latter.

I will resist the temptation to view formal logic as the rightful legislator for all spheres of intellectual labor.[Note 15] Modern formal logic is more like a game, whose rules should not be allowed to govern non-playing behavior, than a sovereign that can establish its own efficacy by punishing dissenters. That legal rules may be bad logic and good jurisprudence at the same time is yet to be established, of course, but I will at least allow myself to proceed as if that conclusion were not foreclosed a priori.

C. Self-amendment

I have chosen one veridical paradox on which to concentrate. A more complete study of logical paradoxes as they arise in law must wait (but see Section 20). The paradox on which I will focus arises from the question whether the clause of a constitution that authorizes amendments may authorize its own amendment or repeal. May a rule that permits the change of other rules also permit its own change, especially its irrevocable change into a form inconsistent with its original form?[Note 16] This paradox does not have a strict counterpart in logic, for it pertains to changing the rules of the system by means of a rule within the system. In logic rules are traditionally thought to be either immutable and eternal, or arbitrary postulates changeable by logicians but not by the postulates' own authority.[Note 17]

This paradox is a form of another paradox, however, which has been studied more frequently —the paradox of omnipotence. In law the paradox is of parliamentary, legislative, or sovereign omnipotence: the power to make any law at any time.[Note 18] In philosophy generally the paradox is of divine or admittedly hypothetical omnipotence: the power to do any act at any time.[Note 19] If an entity has the power to make any law or do any act at any time, then can it limit its own power to act or make law? If it can, then it can't, and if it can't, then it can. If it can do any act at any time, then it can limit or destroy itself, because that is an act; but it cannot do so, because doing it means it cannot and could not do any act at any time. In the legal version we can say that either there is a law that the sovereign cannot make or a law that it cannot repeal.[Note 20]

Like the Barber, if we allow the postulate of the deity or sovereign's existence to stand unchallenged, then it leads to genuine paradox. But also like the Barber, the postulate of its existence implies the affirmation and the negation of a paradoxical predicate (here "can limit its power irrevocably"). Because the postulate implies a contradiction, it is false, and because its falsehood does not also imply its truth, it is not paradoxical like the Liar and we may call it false with finality. Advocates of (non-hypothetical) omnipotent sovereigns or deities may propose distinctions, particularly as to the duration and self-applicability of power, that may save recognizable versions of their entities from paradox, and these will be examined (see Sections 10 and 11). But at first the legally omnipotent sovereign and metaphysically omnipotent deity appear to suffer the fate of the Barber: they cannot exist as defined.

On this view, a clause that authorizes its own amendment or that actually limits itself by self-amendment is, perhaps surprisingly, a contradiction. On this view amendment clauses are immutable except by illegal or extra-legal means such as revolution. Alternately, any act of self-amendment considered valid by legal authorities is simply a case of "peaceful revolution".

The existence of the contradictory Barber was overruled by logical and metaphysical rules that we are not inclined to bend or qualify just to save the Barber. The Barber is doomed because we believe that what is contradictory is impossible or cannot exist. We may be wrong about this, but it is a strongly held belief that we will not give up just to open the doors of existence to the Barber and his ilk. Is the omnipotent sovereign or self-applicable amendment clause doomed by an equally inflexible judge? Do we believe law is governed by the principle of non-contradiction as ardently as we believe that material reality is governed by it? If we are inclined to this belief, we should at least not be dogmatic about it, because we know that judges and legislators often make vague, inconsistent, and incompetent judgments, which are authoritative for all that. Knowing this, might we give up the belief in order to explain the legality of a commonplace procedure accepted in all jurisdictions where it has been tried?

The paradox of self-amendment is one of the best forms in which to study the paradox of omnipotence. The idea of a sovereign or a deity is vague and requires much preliminary specification before the contours of the problem can come into relief. But the preliminary specification often imports unnoticed theoretical commitments. By contrast, an amendment clause is as plain as language (however much that may be), and open to inspection. Moreover, self-amendment is not hypothetical, as questions about deities and sovereigns are for many, and it is not metaphysical, as deities and sovereigns tend to become. To say that an amendment clause applies to the whole constitution of which it is a part, therefore including itself, is intelligible and even plausible. The self-application of an amendment clause leads to a result, an amended clause, that is easily grasped in a way that the state of diminished divine power is not.

Finally, if we like we can try to amend an amendment clause and see what happens (see Appendices 2 and 3). If we limit or repeal the clause through self-amendment, and no court invalidates the act, then we must confront the important fact that the laboratory test apparently showed what is legal without touching the question what is logical.

Even if we ignore the possible self-application of an amendment clause one could argue that there is a paradox in the very presence of an amendment clause in a constitution. It also resembles the paradox of omnipotence. The constitution is apparently supreme and unlimited by higher law, qualities that also define omnipotence. Changing the constitution from below, say, by statute, is impermissible because a derivative authority cannot alter the supreme authority.[Note 21] But can the constitution change itself, or use its supremacy against itself? A change in the text may well be regarded as a limitation or infringement on its supremacy, for each clause partakes of the supremacy of the whole, and is thereby immunized from displacement from the apex of authority, at least by lower level rules.[Note 22] Any significant change will permit what was once forbidden or forbid what was once permitted. In that sense any significant change will contradict earlier rules, perhaps including the rules that authorized the change (see Section 12.C). But if so, then the amendment clause, even applied irreflexively (to provisions other than itself), is a paradoxical authorization of self-limitation. The self-applicability which alone could justify the result may nevertheless, like that of the omnipotent legislature or deity, be a surprising contradiction.

If we are acquainted with the history of inconsistent cases on constitutional law, then we may tend to think of constitutions as non-paradoxical despite inconsistent rules or interpretations. We may even think of constitutions as so fundamental (or supreme) that they undercut (or transcend) the "requirement" to be non-paradoxical and the damage that ordinarily arises from contradiction. This may be so, on the evidence that legal systems, when they assert a contradiction, do not "crash" like computer programs, or blip out of existence like Barbers caught in contradiction. But it does not explain the curious capacity to absorb and tolerate contradiction.

Or we may think of constitutions as containing an implied limitation on their validity or supremacy: "valid and supreme until amended". This qualification may or may not rid us of the paradox (see Section 10). But in any event it leaves us with qualified supremacy, just as comparable tactics leave us with qualified omnipotence and limited sovereignty. We may be content with finite deities and sovereigns, but a finite amendment power may imply the immutability of some legal rules (see Section 8), a conclusion that is much less palatable.

If the supreme clauses of the constitution may be replaced by a method that it provides for itself, then may an amendment clause replace itself by its own method? The question is logically similar to the question whether an omnipotent being could limit or annihilate itself by its own power. For a power, authority, rule, agent, or entity to displace itself from supremacy, or dethrone itself from omnipotence, is prima facie paradoxical like the Barber. The spectacle of such reflexivities in law creates a dilemma for jurists: to admit that, if the self-amendment is impossible, then the system contains immutable rules (the supreme amendment clause), or that, if self-amendment is allowed, then the system contains contradiction (self-amendment).

D. Logical v. legal approaches to self-amendment

To answer the question of self-amendment a logician might lay out all the relevant rules of the system, attempt to make explicit all principles and procedures taken for granted or within the discretion of judges to employ, and then decide whether self-amendment is permissible under the rules. Permissibility would be a function of consistency and deducibility. If self-amendment is either inconsistent or beyond the scope of available premises, then a logician might sketch out a variety of premises that would allow self-amendment and render it self-consistent, preferably without making other common legal procedures thereby impermissible. If the plethora of available rules and principles is internally inconsistent, or if it authorizes inconsistent outcomes, which will almost always be the case, the logician might try to show which consistent subsets, if any, would permit self-amendment and which have the least destructive impact on the rest of law.

A lawyer would approach the problem differently. Amendment clauses to constitutions already exist and take particular forms. Perhaps a state or nation has already amended its amendment clause, or tried (see Appendix 2). If so, perhaps the amendment's validity was challenged in court. If so, then the holding of that court will be more or less persuasive in one's own jurisdiction according to the complex and partially indefinite rules of stare decisis, the doctrine of precedent. If the lawyer does not care to answer the question for a particular jurisdiction, but wants a general answer, then she is already going beyond the limits of her professional role and seeking an answer more qua philosopher or social theorist than qua lawyer. But a general answer, not tied to a particular jurisdiction, may be obtained by legal research nevertheless. A survey of jurisdictions will reveal the dominant rule, which is a vector of historical decisions, not the conclusion of a deductive or even an inductive argument. The historical decisions may all have been wrong by criteria invoked by logicians and wise citizens, even using the lawyer's own premises, but they are authoritative and to that extent not completely wrong by criteria invoked by lawyers.[Note 23] On the other hand, the lawyer may consult the classic texts of jurisprudence and find there a direct answer, a dominant view, or principles that provide an answer. This authority may be set against the case law, not as higher law, but as a more reasonable view. The lawyer then again acts more like a philosopher, accustomed to free inquiry, than a legal practitioner, accustomed to answering questions within the confines of an inherited system, no matter how silent, inconsistent, or unreasonable it may be on the question. If there is no case law on the question, then the views of the writers may be decisive in guiding a judge who faces the question for the first time. Then philosophers may assume the role of indirect legislators.

I will argue in Section 6 that one of Ross's difficulties lies in mistaking the reasonable view for higher law. If the logical or reasonable solution to the paradox requires appeal to a rule of reason or inference, then it would still not follow that such a rule actually exists tacitly in every legal system. The proposition that logical rules are higher laws binding ordinary laws is challenging and important. However, to make it an a priori assertion is to try to legislate for law from the standpoint of logic, not to try to understand the actual relations between law and logic.

I will take both the philosopher's and lawyer's approach, and ask what is the most reasonable answer and what is the legal answer to the question whether an amendment clause may authorize its own amendment, or whether a legal power may limit or eliminate itself, or whether an authority may authorize its own revision or repeal. The ultimate premises from which answers will be sought will include those of our legal system as well as logical principles. If real amendment clauses are mutable only through contradiction or revolution, then that will not automatically mean that such amendment is legally impossible; for contradiction only defines logical, not legal, impossibility. If legal practice allows self-amendment despite its logical impossibility, then honest scholars will revise their opinions of legal possibility —not to mention their concepts of legal reasoning and legal rationality. Even if logical possibility is an a priori, not an empirical, matter[Note 24] then legal possibility is certainly an empirical matter, to be determined by the proper legal agencies within the world of experience and to be discovered by scholars using empirical methods. A priori reasoning about what is legally possible or permissible begs the question whether and to what extent legal systems are (and ought to be) logical. One may regret the lapse of law from abstract logic, appreciate the equitable flexibility it affords, take satisfaction in the pretensions it punctures, or decry the dangers it makes possible. Here my primary concern is simply to show it.


1. There is an enormous literature on logical paradox; see Peter Suber, "A Bibliography of Works on Reflexivity," in S.J. Bartlett and P. Suber (eds)., Self-Reference: Reflections on Reflexivity, Martinus Nijhoff, 1987, pp. 259-362. For good general discussions of logical paradoxes see Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1970, pp. 60-65; John van Heijenoort, "Logical Paradoxes," in Paul Edwards (ed.), Encyclopedia of Philosophy, Macmillan, 1967, 5:45-51; Susan Haack, Philosophy of Logics, Cambridge University Press, 1978, pp. 135-51; J.L. Mackie, Truth, Probability, and Paradox: Studies in Philosophical Logic, Oxford University Press, 1973, Ch. 6 and the Appendix; T.S. Champlin, Reflexive Paradoxes, Routledge, 1988. [Resume]

2. See Robert L. Martin (ed.), The Paradox of the Liar, Yale University Press, 1970, 2d. ed. 1987, which contains an excellent bibliography, and Recent Essays on Truth and the Liar Paradox, Oxford University Press, 1984; Alexander Rüstow, Der Lügner: Theorie, Geschichte und Auflösung, Erlangen: Teubner Verlag, 1910; and Jon Barwise and John Etchemendy, The Liar: An Essay on Truth and Circular Propositions, Oxford University Press, 1987. The Liar paradox is at least as old as St. Paul's Epistle to Titus (1:12-13), and has been studied for centuries. For its history prior to its treatment under modern logic, which is well covered by Martin's 1970 anthology and bibliography, see Alan Paul Anderson's historical essay in Martin (ed.), op. cit., pp. 1-11; Francesco Bottin, Le Antinomie Semantliche nella Logica Medievale, Padua: Editrice Antenore, 1976; and Paul V. Spade, The Medieval Liar: A Catalogue of the Insolubilia Literature, Toronto: Pontifical Institute of Medieval Studies, 1975. [Resume]

3. By "in succession" I mean in time and after a process of change, as a white piece of paper becomes black after burning. Cf. Immanuel Kant, Critique of Pure Reason, St. Martin's Press, 1968 (original 1781, 1787), B.49; see also B.191 and B.148. [Resume]

4. Bertrand Russell, The Principles of Mathematics, Cambridge University Press, 1910, Chapter 10. [Resume]

5. Kurt Grelling and Leonard Nelson, "Bemerkungen zu den Paradoxien von Russell und Burali-Forti," Abhandlungen der Fries'schen Schule, n.s., 2 (1907-08) 300-334. [Resume]

6. W.V.O. Quine, "Paradox," Scientific American, 206 (1962) 84-96 at p. 84. The paradox was apparently first published in its present form by Bertrand Russell in 1918, although he attributed it to a German whose name he could not recall. An earlier variation occurs in Thomas Aquinas concerning a teacher who teaches all and only those in his town who do not teach themselves. Quaestiones Disputatae de Veritate q.11, a.2 (1256-59). See Pierre H. Conway, "The 'Barber' Paradox," Laval Theologique et Philosophique, 18 (1962) 161-76. [Resume]

7. The distinction between veridical and falsidical paradoxes is spurious under the law of excluded middle in a standard two-valued logic. This is seen by the by the equivalency in such a logic of describing Zeno's purpose to be the falsification of a common belief and the establishment of its uncommon negation. I will not use the distinction, but neither will I rely on the law of excluded middle. For a selection of modern essays on Zeno's paradoxes, and an excellent bibliography, see Wesley C. Salmon (ed.), Zeno's Paradoxes, Bobbs-Merrill, 1970. [Resume]

8. Without the characteristic of leading to genuine paradox if affirmed, it must be admitted that the Barber-type paradox would be an ordinary contradiction distinguished from other contradictions only by the plausibility or apparent innocuousness of the "paradoxical" statement. [Resume]

9. The same alternatives apply to Zeno's paradoxes. We are free to choose between affirming the logic which makes our experience of motion illusory, and affirming the veridicity of our experience, at least of motion, and upsetting the logic that would deny it. Proposed solutions to Zeno's paradoxes have traditionally been variations on the latter theme. The fact that we are apparently free to deny Zeno and revise our logic in the light of experience supports the theories of "mutable" logic cited below in note 17. [Resume]

10. Hans Kelsen originally took this position in his General Theory of Law and the State, Harvard University Press, 1949, pp. 363, 375, and in his Pure Theory of Law, University of California Press, 1967, pp. 74, 328. Kelsen partially abandoned this position in his essay, "Derogation," in Ralph A. Newman (ed.), Essays in Jurisprudence in Honor of Roscoe Pound, Bobbs-Merrill, 1962, pp. 339-55, at pp. 351ff, holding that conflicting norms do not automatically imply repeal, although he also says (p. 351) that such conflict should not be compared to logical contradiction. See H.L.A. Hart, "Kelsen's Doctrine of the Unity of Law," in his Essays in Jurisprudence and Philosophy, Oxford University Press, 1983, pp. 309-42. See also Eduardo Garcia Maynez, "Some Considerations on the Problem of Antinomies in the Law," Archiv für Rechts- und Sozialphilosophie, 49, 1 (1963) 1-14. [Resume]

11. David Daube, "Greek and Roman Reflections on Impossible Laws," Natural Law Forum, 12 (1967) 1-84 at p. 4. [Resume]

12. Quine, op. cit. at p. 84 [Resume]

13. As an example one may cite the 1897 action of the Indiana House of Representatives in voting unanimously to legislate an incorrect value of pi, and to charge non-residents of Indiana a royalty for use of the value. The bill was approved on its first reading in the state Senate, but on the second reading indefinitely postponed. Petr Beckman, A History of Pi, St. Martin's Press, 1971, pp. 174-79.

However, if lawyers often lack the logical scruples of logicians, then as Alf Ross the logician will demonstrate despite Alf Ross the jurist, logicians just as often lack the legal scruples of lawyers. [Resume]

14. See the the remainder of Part One, and Section 21.B.

By "logical system" I will mean a formal, abstract system built on, or around, the principle of non-contradiction, as opposed to dialectical logics which embrace and employ contradiction. Paradigm examples of logical systems are those built by Aristotle, Frege, and by Whitehead and Russell. Philosophers in the dialectical tradition, preeminently Hegel, have developed logical systems that contain and depend on contradiction. None of the primary modern expositions of dialectic in Hegel, Marx, Engels, or Lenin addresses the logical paradoxes specifically. But as a logic that embraces contradiction dialectic should at least appropriate the malevolence of the paradoxes, if not "solve" them. Such an argument is made for modern dialectic by Henri Wald in his Introduction to Dialectical Logic, B.R. Grüner B.V., 1975, pp. 228-30. Ancient or Platonic dialectic differs in many ways from Hegelian and Marxist dialectic. But we know that Zeno of Citium (the founder of Stoicism, not the inventor of the paradoxes of motion) believed that the logical paradoxes could be solved by dialectic. See Plutarch, "On the Contradictions of the Stoics," 8:1034E, Moralia, vol. XIII of the Plutarch series of the Loeb Classical Library, Harvard University Press, 1976. [Resume]

15. Recognizing the utility of dialectic is only one reason to resist the temptation. [Resume]

16. The discussion of this paradox has been limited to virtually one man. See Alf Ross, "On Self-Reference and a Puzzle in Constitutional Law," Mind, 78 (1969) 1-24; earlier formulations appeared in his On Law and Justice, London, 1958, Section 16, pp. 80-84, his Dansk Statsforfatningsret [Danish Constitutional Law], Copenhagen: Nyt Nordisk Forlag, 2 vols., 1966, Sections 41 and 46, and in his Theorie der Rechtsquellen, F. Deuticke, 1929, Chapter XIV. Danish responses to his earlier formulations, which Ross took into account in his latest formulation, are cited in the Mind article, p. 7.n.1. See also Joseph Raz, "Professor Ross and Some Legal Puzzles," Mind, 81 (1972) 415-21; Norbert Hoerster, "On Alf Ross's Alleged Puzzle in Constitutional Law," Mind, 81 (1972) 422-26; J.M. Finnis, "Revolutions and Continuity of Law," in A.W.B. Simpson (ed.), Oxford Essays in Jurisprudence, Second Series, Oxford University Press, 1973, pp. 44-76 esp. pp. 53ff.

Note that what is often called "Ross's Paradox" is not the paradox of self-amendment, but another paradox discovered by Alf Ross: if "'p' implies 'p or q'", then why do we hesitate to affirm that "'Smith is obligated to do p' implies that 'Smith is obligated to do p or q'"? See Jaakko Hintikka, "The Ross Paradox as Evidence for the Reality of Semantical Games," Monist, 60 (1977) 370-79; Azizah Al-Hibri, "Understanding Ross's Paradox," Southwestern Journal of Philosophy, 10 (1979) 163-70. Ross is as well-known for his contribution to deontic logic as for his jurisprudence. [Resume]

17. Logical rules may be optional even though immutable or eternal. They are optional in the sense that the choice of initial axioms is either arbitrary or arbitrary within limits, and different initial axioms determine different subsequent rules. They are eternal in the sense that, once options are exercised and a system of valid rules is determined, the validity of the rules is not subject to change over time. Another way to make the same point is to say that experience, or events in time, cannot affect the validity of logical rules. A strong statement of this position is to be found in Wittgenstein:

Our fundamental principle is that whenever a question can be decided by logic at all it must be possible to decide it without more ado.

(And if we get into a position where we have to look at the world for an answer to such a problem, that shows that we are on a completely wrong track.)

Ludwig Wittgenstein, Tractatus Logico-Philosophicus, Routledge & Kegan Paul, 1969 (original 1921), 5.551. Note that several philosophers have argued, on the contrary, that experience can conflict with belief, and that we always have a choice whether to change our beliefs in the face of such experience or to change the logic that defines the conflict as a contradiction. This position asserts that logic is mutable, or at least deniable, in light of experience. See W.V.O. Quine, "The Two Dogmas of Empiricism," in his From a Logical Point of View, Harvard University Press, 1953, pp. xiv and 43, and his Methods of Logic, rev. ed. 1959, p. xiv; Israel Scheffler, Science and Subjectivity, Bobbs-Merrill, 1967, pp. 115-16; Paul K. Feyerabend, "How To Be A Good Empiricist," in P.H. Nidditch (ed.), The Philosophy of Science, Oxford University Press, 1968, pp. 12-39; Hilary Putnam, "Is Logic Empirical?" in R.S. Cohen and M.R. Wartofsky (eds.), Boston Studies in the Philosophy of Science, vol. V, Reidel Pub. Co., 1969, p. 216; Susan Haack, Deviant Logic: Some Philosophical Issues, Cambridge University Press, 1974, pp. 25-46. This position has been criticized by Carl R. Kordig, "Some Statements Are Immune to Revision," The New Scholasticism, 56 (1981) 69-76, and Elliott Sober, "Revisability, A Priori Truth, and Evolution," Australasian Journal of Philosophy, 59 (1981). I might add, for the purposes of this discussion, that the "mutability" thesis may be construed to emphasize the optional character of logical rules without denying their (conditional) eternality. In short, the mutability of logical rules, even in the opinion of those who think logical rules are subject to revision in the light of experience, is not the same as the mutability of legal rules. The paradox of self-amendment cannot arise in logic because no logical rules of any logical system authorize one (logician, critic, dissenter, incendiary, subject of anomalous experiences) to change other rules of the system. Even dialectical logics, which contain changing rules, contain no rules that authorize change on the decision of human agents. No logical system tolerates such license in the logician, or tolerates meddling in real time with valid rules according to alogical standards. Legal systems are fundamentally different in this regard, and this difference must be kept in mind to prevent an oversimplified, logicized concept of law. [Resume]

18. The best article on legal omnipotence is Ilmar Tammelo, "The Antinomy of Parliamentary Sovereignty," Archiv für Rechts- und Sozialphilosophie, 44 (1958) 495-513, which is immediately followed by a commentary by Jaakko Hintikka, "Remarks on a Paradox," ibid., 514-16. See also Geoffrey Marshall, Parliamentary Sovereignty and the Commonwealth, Oxford University Press, 1957, passim, and his "Parliamentary Sovereignty and the Language of Constitutional Limitation," Juridical Review, 67 (1955) 62. See also Zelman Cowen, "Parliamentary Sovereignty and the Limits of Legal Change," Australian Law Journal, 26 (1952) 237-40; Sir William Ivor Jennings, The Law and the Constitution, University of London Press, 3d. ed., 1943, pp. 142-45, 148-53; and H.L.A. Hart, The Concept of Law, Oxford University Press, 1961, pp. 64-76, 144-50.

See the literature that has grown up around the famous English case, London Street Tramways Co. v. London County Council, (1898) A.C. 375, (discussed in Section 15.B below) in which the House of Lords settled a long unsettled question and limited its own power by declaring that it is bound by its own decisions and can, to that extent, decide to limit itself further in the future. The case presents the paradox of omnipotence sharply, but also the paradox of supporting, through the rules of stare decisis, the principle that stare decisis binds the House of Lords. See e.g. Salmond on Jurisprudence, Sweet & Maxwell, 11th ed. by Glanville Williams, 1957, pp. 186-88, 519-24, and the 12th ed. by P.J. Fitzgerald, 1966, pp. 21-28; A.W.B. Simpson, "The Ratio Decidendi of a Case and the Doctrine of Binding Precedent," in A.G. Guest (ed.), Oxford Essays in Jurisprudence, Oxford University Press, 1961, Chapter VI. On the equally paradoxical overruling of London Tramways see e.g. John H. Langbein, "Modern Jurisprudence and the House of Lords: The Passing of London Tramways," Cornell Law Review, 53 (1968) 806-13. [Resume]

19. The leading essay is J.L. Mackie, "Evil and Omnipotence," Mind, 64 (1955) 200-12, revised in his "Omnipotence," Sophia, 1 (1962) 13-25. See also, in roughly chronological order, S.A. Grave, "On Evil and Omnipotence," Mind, 65 (1956) 259-62; I.T. Ramsey, "The Paradox of Omnipotence," Mind, 65 (1956) 263-66; P.M. Farrell, "Evil and Omnipotence," Mind, 67 (1958) 399-403; G.B. Keene, "A Simpler Solution to the Paradox of Omnipotence," Mind, 69 (1960) 74-75; Bernard Mayo, "Mr. Keene on Omnipotence," Mind, 70 (1961) 249-50; G.B. Keene, "Capacity-Limiting Statements," Mind, 70 (1961) 251-52; George I. Mavrodes, "Some Puzzles Concerning Omnipotence," Philosophical Review, 72 (1963) 221-23; Harry G. Frankfurt, "The Logic of Omnipotence," Philosophical Review, 73 (1964) 262-63; Armando F. Bonifacio, "On Capacity Limiting Statements," Mind, 74 (1965) 87-88; G. Schlesinger, "Omnipotence and Evil: An Incoherent Problem," Sophia, 4 (1965) 21-24; Sidney Gendin, "Omnidoing", Sophia, 6 (1967) 17-22; L. Wade Savage, "The Paradox of the Stone," Philosophical Review, 76 (1967) 74-79; James Cargile, "On Omnipotence," Nous, 1 (1967) 201-05; George F. Englebretsen, "The Incompatibility of God's Existence and Omnipotence," Sophia, 10 (1971) 28-31, with comments by David Londey, Barry Miller, and John King-Farlow; P.T. Geach, "Omnipotence," Philosophy, 48 (1973) 7-20; Richard Swineburne, "Omnipotence," American Philosophical Quarterly, 10 (1973) 231-37; Jerome Gellman, "The Paradox of Omnipotence and Perfection," Sophia, 14 (1975) 31-39; Paul Helm, "Omnipotence and Change," Philosophy, 51 (1976) 454-61; Edward J. Khamara, "In Defense of Omnipotence," Philosophical Quarterly, 28 (1978) 215-28; Joshua Hoffman, "Mavrodes on Defining Omnipotence," Philosophical Studies, 35 (1979) 311-13; Bruce R. Reichenback, "Mavrodes On Omnipotence," Philosophical Studies, 37 (1980) 211-14; Gary Rosenkrantz and Joshua Hoffman, "What An Omnipotent Agent Can Do," International Journal for Philosophy of Religion, 11 (1980) 1-19; Loren Meierding, "The Impossibility of Necessary Omnitemporal Omnipotence," International Journal for Philosophy of Religion, 11 (1980) 21-26. [Resume]

20. This formulation of the paradox is probably vulnerable to time-based objections (see Section 10) which a more sophisticated formulation (Section 3) would succeed in evading. [Resume]

21. Exceptions to this principle, especially unusual and possibly non-supreme methods of amending the federal constitution, are considered in Part Two. [Resume]

22. Is a constitutional provision allowing "home rule" of cities at their option an example of paradoxical self-limitation? A city that votes to adopt home rule inverts the pyramid of power, in some respects, but under the authority of the apex. In that it differs from an attempt by the apex completely to sever its sovereignty over a former subject (see Section 20). [Resume]

23. The reflexivity of judicial opinions that base themselves on other judicial opinions and can convert even bad holdings into valid law is carefully discussed by John M. Farago, "Judicial Cybernetics: The Effect of Self-Reference On Dworkin's Rights Thesis," Valparaiso University Law Review, 14 (1980) 371-425. Farago uses the term "self-reference" broadly to cover activities in which referring and reference play no significant part, particularly self-justifying arguments, self-affecting decisions, and what may be called self-reification or the peculiar ontological power of judicial pronouncements to create certain legal realities merely by asserting them to be legal realities. Hence his essay is broader than his title may indicate. [Resume]

24. See note 17, above. [Resume]

This file is one section of the book, The Paradox of Self-Amendment. Return to the Table of Contents.

Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A. Copyright © 1990, Peter Suber.