This essay originally appeared in the Stanford Literature Review, 7, 1-2 (Spring-Fall 1990) 53-78. Copyright © 1990, Peter Suber.

This summary of my book, The Paradox of Self-Amendment: A Study of Logic, Law, Omnipotence, and Change (Peter Lang Publishing, 1990) was delivered at the conference, "Paradoxes of Self-Reference in the Humanities, Law, and the Social Sciences," May 20-21, 1988, at Stanford University. The book was unpublished at the time of the presentation.

The Paradox of Self-Amendment
in American Constitutional Law
Peter Suber, Philosophy Department, Earlham College


Logical paradoxes in the strict sense produce statements like those of the Liar ("This very statement is false") that are false if true, and true if false. They resist rational solution or at least divide logicians for centuries of apparently irreconcilable wrangling. What happens when similar paradoxes arise in law?

That is a difficult question, but part of the answer is that paradoxes come and go without much notice and are dealt with without much ado. This fact makes the question important as well as difficult. How law copes with paradox sheds light on the nature of legal reasoning and rationality, the nature of legal practicality, and the sense in which law can be reasonable, even 'wise', while being illogical in the technical sense.

In my book I select one principal paradox —the paradox of self-amendment— and explore its variations in several Anglo-American jurisdictions and contexts, but mostly in American constitutional law. If a constitution has an amendment clause (a provision describing or prescribing how to amend that constitution), then can that clause be used to amend itself? Is self-amendment paradoxical? If it is paradoxical, can it be lawful? If it is lawful, can the logic of law be logical?

Some legal rules govern the change of other legal rules. But even these "rules of change" (as Hart calls them) are changeable, usually by higher level rules of change. When a rule of change is supreme within its own system (as a constitutional rule of change probably is), then it is changeable, if at all, only under its very own authority. The paradox of self-amendment arises when a rule is used as the authority for its own amendment. It is sharper when the rule of change is supreme, sharper still when it is changed into a form that is inconsistent with its original form, and sharpest of all when the change purports to be irrevocable.

Earlier thinkers have charged that self-amendment in this sense can be reduced to a formal self-contradiction. After considering some obvious and not so obvious objections and subterfuges, I conclude that this is correct. Moreover, these thinkers have decided on the whole that this contradiction invalidates self-amendment. Their principle seems to be that what is logically impossible must be legally impossible or impermissible as well. I conclude that this is factually incorrect and philosophically arrogant.

One task the earlier scholars have not undertaken is to show that self-amendment has frequently occurred despite all contradiction and paradox. It is commonplace and, in fact, not even controversial in legal circles. By all legal tests of lawfulness self-amendment in the most illogical sense is lawful. It has occurred in 47 of the 50 states, and in every nation with a written constitution that I have investigated (though I have made no systematic study of the latter). Nor have past explorers of these themes asked what law, legal change, legal reasoning, and legal rules are such that they can tolerate this kind of contradiction while forbidding many other things in the name of consistency.

The principle that what is logically impossible must be legally impossible may be philosophically arrogant and ignorant of legal history, but it is not a simple mistake. It is a new variation on the theme of natural law. Instead of finding that human law depends for its validity on an eternal moral law, this version makes it depend on an eternal logical law. One of the most persistent and persuasive objections to traditional natural law theory applies to the new version as well. If human law can be immoral without ceasing to be law, it seems it can be illogical without ceasing to be law. Law has its own tests of what is law, and those tests validate much that is immoral and illogical. To decide that a transcendent moral or logical test supersedes the legal tests, and can invalidate what is otherwise law and validate or enact what is otherwise utterly tacit, is to transform law into morality or logic and unduly diminish its historical and social dimensions.

The Paradox of Self-Amendment is in part a critique of the new natural law theory that replaces the traditional moral test with a logical one. I conclude that logic does not legislate a priori for law any more than morality does, although of course law remains perpetually open to logical and moral criticism. To know what is lawful in a society, we must look only at legal evidence and social phenomena. We need not consult any moral or logical doctrine, even any that happen to be affirmed unanimously, except to the extent that they influence the concrete legal practices of that society. In particular, I conclude that self-amendment may well be a strict self-contradiction, but the evidence of legal history suffices to show its lawfulness even in the face of logical doctrine. I argue that the only alegal source of legal authority is social fact, not normative principles from morality or logic.

The paradox of legal self-amendment is just one variation of the broader paradox of omnipotence: how can a power supposed to be omnipotent irrevocably limit itself? The paradox of omnipotence is usually applied to deities, but it can also be applied to omnipotent legal powers, such as the power to amend a constitution. The U.S. federal amending power can change every rule in the American legal system, with the arguable exception of itself. (It has amended itself by "sunset clause" but not by formal amendment; more than one defeated proposal would have required strict self-amendment; these instances and others are fully discussed in the book.) In that sense it is legally omnipotent within the American system and its use to limit the amending power irrevocably is logically equivalent to the problem of an omnipotent deity using its power to make a stone it could not lift or irrevocably to assume a mortal form.

The paradox of omnipotence is generally thought to be of the "Barber-type". The paradox of the Barber is that of a shaved man who shaves all and only those in his own town who do not shave themselves. Who shaves the Barber? If he shaves himself, then he does not (because he does not shave those who shave themselves); and if he does not shave himself, then he does (because he shaves all those who do not shave themselves). At first this looks like the paradox of the Liar: "This very statement is false." If it is false, then it is true; and if it is true, then it is false.

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.

The paradox of omnipotence seems to be of the "Barber-type", for it seems to prove that omnipotence cannot exist as naively conceived. If we naively assume that a deity can do any act at any time, then can she make a stone so heavy that she cannot lift it? If she can, then she is not omnipotent in the naive sense (for there is something she cannot lift); and if she cannot, then she is not omnipotent in the naive sense (for there is something she cannot create). The natural solution is to deny that there is a deity with the power to do any act at any time.

Similarly, can a constitutional amending clause amend itself —especially, can it do so when it is the only authority for the amendment, when it is the supreme rule of change in that legal system, when the new version of the clause would be inconsistent with the original, when the amendment would diminish the amending power, and when the amendment purports to be irrevocable? If we regard constitutional amending clauses as legally omnipotent, on the evidence that they are the supreme rules of change in their respective systems, then we have replicated the theological version of the paradox of omnipotence. If we do not regard amending clauses as legally omnipotent, then some irrevocable limitation on their power must exist; for if all limitations on it were revocable, some power would be legally omnipotent. There is no contradiction in assuming that such irrevocable limitations exist; there is only the difficulty of turning to legal history and finding them. If we insist they are there even though we cannot find them, then no legal power is omnipotent. That is a logically permissible solution. If we think those irrevocable limitations are not there, or if we suppose they are not for the purposes of inquiry, then we thereby posit at least one omnipotent legal power. If we do so, we have clearly lost the right to reject the assumption of omnipotence —as we reject the assumption of the Barber's existence— when we derive a contradiction from it. So we must apparently choose between the universal absence of legal omnipotence and the permissible presence of legal contradiction. In general I take the latter path, even though it implies that the paradox of self-amendment is more "Liar-like" than "Barber-like".

Because the paradox of self-amendment does not arise in strong forms for changes that are revocable, it raises the question what an irrevocable legal enactment could be. Hence the book also explores the important question of democratic theory whether any legal rules can be strictly immutable or beyond history. Several jurisdictions within the United States and the former colonies of the British Empire have tried to safeguard democracy or some vision of fair procedure by rules that are self-entrenched. A rule is self-entrenched when it is made immutable to legal change by a declaration within the rule itself that says, in effect, "this rule may not be changed". My solution to the paradox of self-amendment shows why such rules are mutable anyway, or why their change is typically approved by courts. It also explains our sense that truly immutable rules would be undemocratic constraints on liberty, even if their content proposed to protect democratic institutions. I argue that self-entrenchment fails, not because it is an undemocratic means to a (contingently) democratic end, but because the nature of law cannot abide it. But I also argue that the nature of law is not independent of the values of the people who make, respect, and disrespect law.

If legal rules that authorize change can be used to change themselves, then we have paradox and contradiction; but if they cannot be used to change themselves (and if there is no higher rule that could authorize their change), then we have immutable rules. Paradox and immutability should create an uncomfortable dilemma for jurists and citizens in western legal systems. It appears that we must give up either a central element of legal rationality or a central element of democratic theory.

I argue both descriptively and normatively that law can tolerate paradox but cannot tolerate immutability. Accepting this conclusion does not compromise a properly nuanced vision of legal rationality, although it does undermine the simplistic vision, sometimes conveyed (if not taught) in law schools, that law is a formal system cursed with content, the civil correlate of mathematics.

Finally, the demonstration that self-amendment is lawful in the Anglo-American legal tradition disproves a common theory of legal change: the theory that all valid change of law must be authorized by prior, higher legal rules. Aside from denying the possibility of what is actual (namely, self-amendment), this view has other absurd consequences. It implies that no new legal system could get started. None could break off from another lawfully, and all that broke off unlawfully would be eternally barred from becoming lawful themselves. Any regime to be called lawful would require an infinite genealogy. Because we want to say that there are some lawful regimes, we must be able to explain how they could get started without at the same time making them mere creatures of prior regimes in an infinite series. Some power, be it contract or revolution or some other, must make law ex proprio vigore or from its own strength. Only a theory of permissible self-amendment can explain this fully, and therefore only such a theory can explain legality per se.

Summary of Analysis and Argument

Because the paradox of self-amendment is a special case of the paradox of omnipotence, our central question is whether a deity or amending clause can irrevocably limit its own power. Logicians using strictly logical criteria offer the answer that such self-limitation is self-contradictory; the same answer is given by lawyers and theologians who adopt the same criteria for their disciplines. No one using strictly legal criteria has ever said that self-limitation is self-contradictory, let alone impermissible.

The logical or formalist view of law as it affects this problem I am calling the inference model of legal change and validity. Under the inference model, legal change is modeled by deductive inference. The authorizing rule of change (for example, the old amending clause) is one premise, the fact of enactment under appropriate procedures is another, and the validity of the new rule (new amending clause) is the conclusion.

Rule: If act A is done, then rule B is valid.
Fact: Act A is done.
Conclusion: Therefore, rule B is valid.

The theory is that the amendment is lawful if and only if this deduction is valid and its premises are true.

Self-amendment occurs when the rule affirmed in the conclusion is meant to replace the rule in the premises, that is, when the rule in the premises refers to itself. Irrevocable self-limitation is self-contradictory on this model because it requires an inconsistency between a premise and the conclusion of the inference, when the premises themselves are (or may as well be) internally consistent. The rule of change that sits in the premise is inconsistent with its own invalidation, or with the exclusive validity of its successor asserted in the conclusion. Alf Ross, the Danish logician and jurist, has argued in detail[Note 1] that by formal criteria such an inference must be invalid. The esteem of Alf Ross in both logic and law forces us to take his arguments and his model of legal change seriously. Under the inference model what is logically impossible is therefore legally impermissible, and any legal practice or tradition to the contrary is itself a violation of both law and logic.

There may well be a way to dissolve the paradox for the inference model or for formal logical generally. (We will see that satisfying the inference model is more difficult than satisfying formal logic alone.) Such dissolution would take the form of removing the inconsistency between premise and conclusion in the inference that models the act of amendment. A dissolution of this kind would satisfy logicians even if it had absurd consequences for law; but I have not been able to find even one dissolution of this kind that stands up to analysis.[Note 2]

We may distinguish two ways to "solve" a paradox. Paradoxes are not mere contradictions; they are not nearly as tame. But their bite lies in their way of making contradiction appear inescapable. One kind of solution is to escape contradiction by a path not seen by others. I am calling this a dissolution of the paradox. Another way is to explain why contradiction is harmless, whether it is escapable or not. This will only work when contradiction is harmless, of course, and most of the argument in support of solutions of this kind will be spent in that cause. This will not dissolve the paradox, but will excuse and domesticate it, removing its sting and threat. It is much like what lawyers call an "affirmative defense": admit that you did it, but claim some excuse like insanity. The second kind of solution is more radical than it seems and requires, in effect, an insanity defense for law itself.

In short, I argue that no dissolution satisfies the terms of the inference model, but that when we reject that model we find a number of plausible domesticating strategies. First I will consider various attempts to dissolve the paradox.

Attempted Dissolutions

In standard logic all propositions follow validly from a contradiction. Therefore, one solution might be deliberately to make the premises of the inference that models the change internally inconsistent. This would certainly work to make the new amending clause a logically valid consequence of the old one. But it would work only at the cost of assuring that every amending clause is internally inconsistent. This result is inadmissible for the inference model, which cannot allow inconsistent rules simultaneous validity. However, if one likes, one may consider the total invalidation of the amending clause a solution satisfactory to logic that is repugnant only to law. My aim in the book, though, is to explain how actual change of amending clauses has been possible or how far it is logical. The actuality of self-amendment is a fact that is part of the explanandum (documented in the book). Even a logical dissolution must deal with it, much as approaches to the Liar cannot pretend that the lying words do not exist or cannot form the paradoxical statement.

Most philosophers who have approached the problem (primarily through the theological paradox of omnipotence) evidently hope that a solution can be found that does not violate or require the inapplicability of formal logic. I have not argued for any conclusion that should dampen that hope. I have argued that formal logic has very limited application in law, but not that a dissolution of the legal form of the paradox is strictly impossible by logical criteria. Hence I admit —without the same earnestness of hope— that there may well be a dissolution of the paradox satisfactory to formal logic. All I have argued is (1) that the obvious attempts at such dissolution fail, and (2) that in any case law can dispense with such a dissolution. The second point is the more important thesis, and stands (if it stands at all) even if a satisfactory logical dissolution should be discovered tomorrow.

If self-amendment could be logically impeccable if only the amending clause were suitably reworded and its self-change properly transacted, then a legal system may still ignore those niceties utterly. This remains the case whether the required changes are ingeniously simple or hideously complex. Today lawmakers listen to what logicians say they should do only to the extent that logicians form a weighty voting bloc in their constituencies. There is no reason to think that their lawmaking acts will cease to be valid if the logicians should one day become correct, any more than those lawmaking acts are invalidated when the weightier loyal opposition is correct. Legal validity is a matter of power and social practice, not abstract correctness.

There will be no solution satisfactory to formal logic if formal logic bars all self-reference, say, by incorporating a theory of types, and if self-amendment unavoidably requires self-reference. Ross believes this is the case, but has been followed in this by only one commentator,[Note 3] and then only with reservations. Hart[Note 4] leads the opponents of this view in arguing that at least some self-reference is logically and legally unobjectionable. Raz[Note 5] even argues that self-reference is unnecessary in self-amendment. I argue that examples of self-reference and self-application are legally acceptable, even if meaningless to formal logicians, so long as legislators, voters, and judges find some meaning in them. Moreover, if all self-reference is meaningless, then a self-referential amending clause would be unavailable even for ordinary, irreflexive amendment, which proves far more than Ross intended or desired.

Ross distinguishes between logical and legal contradictions. A logical contradiction exists between any statement and its negation. We don't have to assert one or both of the statements for the contradiction to exist. A legal contradiction exists between any two inconsistent laws that are both valid at the same time. If they are not both valid at the same time, they will be logically but not legally contradictory. Armed with this distinction, Ross has an answer to the most common attempt to dissolve the paradox of self-amendment.

The most common attempt to dissolve the paradox has been to insure, or assume, that the old and new amending clauses are never valid and supreme at the same time. Temporal overlap can be prevented; but preventing it merely avoids a legal contradiction, not a logical contradiction. It keeps inconsistent rules from enjoying simultaneous legal validity, but thereby presupposes their logical inconsistency. That inconsistency invalidates the inference that models self-amendment because a logical contradiction between premise and conclusion (when the premises are internally consistent) suffices to invalidate a deduction. As Ross often puts it, the invalidity consists in our attempt to derive from one norm a second norm inconsistent with the first.

If one rejects the distinction between logical and legal contradiction, but still holds the inference model of legal change, then one is no better situated to overcome the invalidity of the inference that models the self-amendment. If the contradiction is supposed to disappear because the old amending clause loses validity at the moment the new one acquires validity, then in the inference that models this process the assertion of a key premise must be suspended in mid-inference. Even if this successfully removes the inconsistency in the inference that models the self-amendment, it replaces it with a new fallacy. Logicians do not have a name for this fallacy because it cannot be performed in ordinary argument.

In discussing Ross's thesis, John Finnis[Note 6] introduces a term that helps here. If some legal rules are valid only as long as the laws that authorize them are themselves valid, then we can say that those legal rules are routinely validated; if some rules are valid even after their authorities are repealed, then they are "transtemporally" validated. For example if a court holds its jurisdiction under a statute, and if the statute is repealed, then the court loses its jurisdiction (ceases to be a court) if it is routinely validated, but continues unaffected if it is transtemporally validated. In general we will be tempted to recognize cases of transtemporal validation wherever self-amendment seems to occur, e.g. when a new amending clause is validated by the old amending clause that lost its validity when it was superseded by the new one.

I take no position the transtemporal validation of law, although superficial observation suggests that some laws are transtemporally validated and others are not. Ross argues that transtemporal validation is illogical, hence impossible, and yet is logically required for strict self-amendment; hence, he concludes that self-amendment is impossible. His reason for rejecting transtemporal validation is essentially that nothing analogous to it can occur in valid deductions. Inferences in which premises are "dis-asserted" just as the conclusion is asserted violate canons of deduction, including the metaphysical one that premises determine their conclusions instantaneously. Any conclusion whose derivation depends on certain premises is no longer validly concluded if we deny one of those premises, whether we deny it before the inference, after, or (supposing it is even possible) in mid-inference, after affirming the premises and before affirming the conclusion.

In this sense the inference model of legal change requires that all validation be routine, or that none be transtemporal. Accordingly, if we reject the distinction between logical and legal contradiction, and assume that temporal separation of the old and new amending clauses suffices to avoid the paradox, then we commit ourselves to transtemporal validation and violate the inference model.

In short, temporal separation does not solve the paradox for the inference model whether we recognize the distinction between logical and legal contradiction or not. If we accept the distinction, then temporal separation only prevents legal contradiction; the logical contradiction remains and suffices to invalidate the deduction that models the legal change. If we reject the distinction, then temporal separation will mean that the new amendment clause is valid only by some magical intervention (which is roughly Ross's solution, as we will see) or by transtemporal separation, which violates the inference model.[Note 7]

So if we are to conclude that self-amendment is lawful without ad hoc or magical makeshifts, then we must abandon the inference model or find another way to dissolve the paradox. I now explore both.

Two non-formalist models of legal authority and change important to this problem are the acceptance and procedural models. (1) The acceptance model makes the validity of new rules depend on a complex of social practice, or on rules that in turn depend on a complex of social practice. This social practice is the joint, cumulative result of the ordinary activity of the people and the officials of the system in recognizing, respecting, and disrespecting law. (2) The procedural model holds that, if a procedure (such as that to amend the constitution) is independently known to be lawful, then the products of its correct application are lawful even there is an inconsistency between them and some defining condition of the procedure itself. I will appeal to these two models shortly.

A very attractive attempt to dissolve the paradox appeals to a distinction derived from J.L. Mackie and stated by Hart.[Note 8] The distinction is between continuing and self-embracing omnipotence. Continuing omnipotence is the power to perform any act compatible with the continuation of this power, i.e. any act except irrevocable self-limitation. Self-embracing omnipotence is the power to perform any act as one's first act, including irrevocable self-limitation. One's subsequent acts are limited only by the consequences of one's earlier acts for the extent of one's power. Recalling the naive notion of omnipotence as the power to do any act at any time, this distinction separates "any act" from "any time" in order to make two coherent species of omnipotence from one incoherent one. Continuing omnipotence cannot perform every possible act; preeminently, it cannot "discontinue" itself by some abdication or diminution that cannot be undone. But of those acts that it can perform, it can perform any of them at any time. Self-embracing omnipotence can perform literally any act, but not necessarily at any time. If one act is irrevocable self-limitation, then thereafter the range of power is limited accordingly.

Using this distinction, it is natural and compelling to argue that irrevocable self-limitation is either (1) non-paradoxically impossible, because the amending clause has continuing omnipotence, or (2) non-paradoxically possible, because the amending clause has self-embracing ominpotence.

The distinction is not only a plausible solution to the paradox, but it is rooted in real-world legal disputes. Every few years someone in England's Labor Party proposes to abolish the House of Lords. Apart from the politics of the question, many have argued that it is legally impossible to abolish the House of Lords.[Note 9] It is one House of Parliament and would therefore have to assent to its own abolition. If the abolition is to be irrevocable, then we are asking whether the House of Lords can limit itself irrevocably. One short answer is that it can, if it has self-embracing omnipotence, and cannot, if it has continuing omnipotence. Similarly, on this line of argument, an amending clause could limit its power irrevocably if it had self-embracing omnipotence, and could not if it had continuing omnipotence.[Note 10]

Unfortunately, this approach fails under both the acceptance and inference models, although it may succeed under the procedural model. First note that a power of self-embracing omnipotence is a very different thing when exercised by a deity and when embodied in constitutional amending clause. If a deity has such a power then it magically creates what it wills; but if an amending clause has such a power, it only authorizes what others will. Keeping this distinction mind we can see why irrevocable self-limitation is not made any less self-contradictory by self-embracing omnipotence. The theory of self-embracing omnipotence offers no reason to believe that the inconsistency in self-amendment has been eliminated other than the reason that the inconsistent outcome is authorized. It is natural to believe that what is specifically authorized by a rule cannot be inconsistent with that rule; but this principle is ultimately untenable. I call it the authorization fallacy.

The fallacy in the authorization fallacy is to confuse legal and logical validity, and to assume that what is legally permitted must be logically unobjectionable —i.e. free of contradiction. But if a norm specifically authorized the derivation of a norm inconsistent with itself, and if it was used to derive such a norm, then clearly (ex hypothesi), despite the authorization, an inconsistency would exist. Authorization, then, does not remove inconsistencies between the authority and the authorized, but only legitimates them.

Under the inference model authorization does not dissolve the paradox. If we accept the distinction between logical and legal contradiction, then authorization goes only to the legal contradiction. If we reject that distinction, then a rule authorizing its own replacement violates transtemporal validation.

The acceptance model cannot accept the theory of self-embracing omnipotence either, but for a very different kind of reason. Under the acceptance model, any self-limitation of an amending clause that purports to be irrevocable can be repealed if future generations decide (as our generation has decided) that no generation can bind its successors irrevocably. If the people and officials accept a view of law that permits the repeal of "immutable" limitations on the amending clause, then the amending clause may be restored to its unlimited power. This means that amending clauses have a kind of continuing omnipotence. Even if they also have a kind of self-embracing omnipotence, it is not the kind that permits irrevocable self-limitation.

Ross's attempted solution was to derive the new amending clause from a tacit rule superior to the constitution that authorizes exactly such changes. I argue that such a rule is ad hoc, fictitious, and unnecessary for the amendment of the amending clause, although a judge who could find no other way to permit a changed amending clause to become valid could permissibly appeal to, or invent, such a rule. Ross's tacit rule, however, does not dissolve the paradox even if taken at face value. It neither removes the contradiction of self-amendment, nor admits and domesticates it. His strategy avoids self-amendment altogether and finds another way to effect the change of constitutional amending clauses. The price for making strict self-amendment unnecessary is to admit an absolutely immutable rule —namely, the tacit, transcendent rule authorizing the change of the amending clause. For Ross all rules of change but one can be changed; the exception, by the nature of the case, is the supreme rule of change. Because self-amendment is ruled out, it cannot change itself and, because it is supreme, there is no higher rule to authorize its change. This particular result reflects his more general principle that all legal change is authorized only by prior, higher rules of change. This principle in effect incorporates a theory of types into his version of the inference model.

The requirements of the inference model that every new rule be authorized by a prior, superior rule, and that none be self-authorized, quickly proves its inapplicability to real legal systems. This view entails that no legal rule could validly come into being without an infinite genealogy. The inference model cannot explain the legal origins of any legal system, or permit any revolutionary regime to become lawful. It could explain and permit these things if it allowed an exceptional self-authorizing rule, an unauthorized rule, or validation by subsequent acquiescence; but each of these is inadmissible under its rigorous, irreflexive formal criteria. Therefore, proponents of the inference model may not, as Ross and others have tried to do, explain self-amendment as peaceful revolution. They can explain the breach that makes revolution, but not the reestablishment of legality after the breach. They cannot even explain how the regime before the breach was lawful.

If the impossibility of validating any law after revolution or without an infinite genealogy causes actual jurists to despair that they live under an illegal regime (no matter where they live), incurable by any device, then they have found a nook of legal absurdity that makes them irrefutable. And there are scholars with such views.[Note 11]

A lesser attempt to dissolve the paradox by rewording the old and new amending clauses reduces to one of the major types. If the rewording aims at explicit temporal separation, then it reduces to the unsuccessful time-based attempt at dissolution. If the rewording aims at specific authorization of all sorts of self-amendment, then it may put to rest most doubts of the legal permissibility of self-amendment but cannot erase logical inconsistency except through the authorization fallacy.

The see-saw method is one method of changing an amending clause without appeal to a superior rule of change, and without violating or denying the inference model. The see-saw method uses one method of amendment, A, irreflexively to amend another, B, and then B to amend A, back and forth until the desired content is reached. It is not at all clear at first that any desired content can be reached from any initial position. A short-cut to any desired content is possible if an omnipotent rule of change can be added. If the desired content is not to be omnipotent, then the added omnipotent amending clause must be repealed, which raises all the issues of the paradox of self-amendment. In this sense, the see-saw method (which avoids strict self-amendment) can take us anywhere if strict self-amendment is possible. At least self-amendment would be sufficient; whether it is necessary is a difficult question.

Because the see-saw method lets us add rules that we have crafted for the purpose, many self-amendment problems can be solved in ways unavailable to ordinary amending clauses. For example, temporary omnipotent amending clauses added for the shortcut can be reduced to limited power by a sunset clause inserted at the time of enactment. The theory of contingent omnipotence (summarized below) shows one way by which the see-saw method can take us anywere. The same theory will be used to vindicate the legality and coherence of strict self-amendment.

The see-saw method, however, should not be understood to explain how (most) actual amending clauses have been changed, nor as a dissolution of the paradox. Like Ross's tacit, transcendent rule, it allows change of the amending clause without resort to self-amendment, and therefore shows nothing about the logical or legal permissibility of genuine self-amendment. Unlike Ross's solution, it allows rules of the same level to apply to one another. The see-saw method is ultimately impossible under the inference model if the latter is interpreted to incorporate a theory of types.

Insofar as the paradox depends on an inconsistency between premises and conclusion, it may be defined away by adopting a definition of inconsistency that does not cover most instances of self-amendment. This method is not as artificial and evasive as it might at first appear. The definition of negation (hence, inconsistency) for prescriptive statements is disputed by rival deontic logics. And if any formal logic comes close to fitting law, it is deontic logic —the logic of permission and obligation. (It turns out that Alf Ross is a major player in the development of deontic logic and in the controversy about how it should define negation.)

In law there appear to be at least four tests of inconsistency, two of which might apply to self-amendment. The minimal "deontic" test finds inconsistency whenever one rule permits what another forbids or vice versa. The broader "compliance" test finds inconsistency whenever simultaneous compliance with two rules is impossible. The deontic test makes all self-amendment paradoxical except trivial renumbering, rearranging, and rewording designed to leave the substance intact.[Note 12] The compliance test allows some substantial self-amendment to escape contradiction and paradox, but not most irrevocable self-limitation. Neither test, therefore, can explain the substantive cases of self-amendment we find in legal history.

Solutions from the Acceptance Model

Two solutions may be derived from the acceptance model of legal change and validity. One of them doesn't care whether self-amendment is contradictory; if it is, the contradiction can be admitted and excused. The other appears to be a dissolution that eliminates the contradiction from self-amendment, but on closer inspection it eliminates genuine self-amendment along with the contradiction. One shows how genuine self-amendemnt can be lawful whether or not it is self-contradictory; the other shows how amending clauses can be amended lawfully and logically even if the method is not strictly reflexive.

The acceptance model holds that the ultimate rule of recognition is authorized by the acceptance and usage of the people and officials of the system. My exposition of the acceptance model is derived from the work of H.L.A. Hart. For many commentators on Hart, acceptance authorizes only one master rule, which Hart calls the "rule of recognition", which in turn authorizes every other rule of the system. For these commentators the relationship between the rule of recognition and the other rules of the system is strongly hierarchical and formal; if it weren't for the ultimate role of acceptance, this might well be called an inference model. I do not read Hart this way. In particular I read him as asserting that acceptance can operate on particular rules and rulings directly, without being logically or legally funneled through the rule of recognition.

However, to avoid purely exegetical squabbles I am happy to call my view a "modified acceptance theory". The key modification for my purposes is that some rules other than the ultimate rule of recognition are authorized directly by acceptance, and that in principle acceptance can always overrule the rule of recognition to invalidate what the rule authorized or to validate what it did not authorize. The modified theory may be made tidier if the overruling of the rule of recognition is considered ipso facto to amend it. That way the rule of recognition preserves its role as universal arbiter (but not, for Hart, self-arbiter). I have no objection to such a move, provided it does not tidy things beyond recognition, and lead to the denial of the power and legitimacy of acceptance as an alegal source of legal authority to interfere at any time for or against any rule.

Now that the modification is clear, I'd like to call the position the "direct" acceptance theory. I prefer that the name of the position emphasize, not that I differ from some other readers of Hart, but that acceptance works directly on all fronts of a legal system at once, not indirectly through one master rule, and that its authorizing effect arises from social practice rather than inferentially trickling down from the pinnacle of a pyramid of rules. The direct acceptance theory does not deny that there is a hierarchy of rules in which superiors authorize inferiors; it only insists that this is contingent on what is accepted, not necessary for legality as such. It may be that most rules most of the time are authorized by other, higher rules, and that acceptance only rarely "intervenes" to do its work directly. The direct acceptance theory requires only that we admit that this can happen in principle —namely, when it is accepted as happening.

The two solutions provided by the direct acceptence theory are as follows. First, self-amendment may be accepted as valid despite the contradiction inherent in it, which may be conceded to exist. If the contradiction can really be dissolved we need not do so; if it cannot, we need not resort to legal fictions that allow us to act as if it were dissolved. This is possible because acceptance is not bound by any formal logic. If the people and officials in the appropriately complex sense accept self-amendment, despite its contradiction, then their acceptance suffices to make it lawful.

The second solution is that the new amending clause may derive its authority directly from acceptance (or from a rule of recognition amended by acceptance), rather from the old amending clause, even if the procedures of the old amending clause were used to propose and ratify the new one. By shifting the authority for the new amending clause from the old one to direct acceptance, we deny in effect that real self-amendment has occurred. If there is a contradiction in strict self-amendment, this method bypasses it. If we assume that Hart's rule of recognition would not recognize a contradictory procedure, then both these solutions require the modified or direct, as opposed to the "inferential" acceptance theory.

The first method allows genuine self-amendment to occur, and disregards or forgives any contradiction in the process. The second holds in effect that the appearance of self-amendment is illusory, and that while the procedures of the old amending clause may have been followed, the authority of the new amending clause derives from another source.

The first solution works as well with any model of law that can explain legal tolerance for contradiction, such as the procedural model, which I mentioned, and many others that I have not discussed, such as an ideological model that identifies valid law by its content only and never by its form; theological and militaristic models that replace social acceptance with the will of a deity or junta; or the drunken judge model that exonerates all official error and allows it to live the normal life of law —which is not logic but experience.

The second solution works as well with any posited source of authority for the new amending clause other than the old amending clause. Ross's tacit, transcendent rule is in this category, as is any social contract theory of legality. Principles in Ronald Dworkin's sense of the term are also candidates for the source of the authority of the new amending clause; such principles might pertain to popular sovereignty, justice across generations, or deference to the action of a constitutional convention.

The first solution not only forgives the contradiction that Ross found in the inference modeling self-amendment, but also the absurdities (if that is what they are) in transtemporal validation. In strict self-amendment when the new clause is authorized by its predecessor, the authority for the new clause is either repealed by the act of amendment, requiring transtemporal validation, or it persists despite its replacement, introducing a new kind of queerness. Nevertheless, all these difficulties are merely theoretical; lawmakers may heed them or not, as they wish.

The second solution can be made more venerable, if not more plausible, by putting it in more classical terms. Just as a contract's equivalent of an amending clause may be amended by the parties without paradox, because the validity of the new clause derives (on one reading) from their agreement, not from the old clause, so the constitutional amending clause might be changed if the amendment process were interpreted as the act of parties to a contract. The acceptance theory, in fact, is a variant of a classical consent or contract theory of legality. It captures the sense in which the decisions of the people are superior to all law, by their capacity to change and supersede all law, while escaping the less plausible, historical or hypothetical presuppositions of an actual contract theory. The acceptance theory locates legal authority not in a contract but in the less explicit, less rule-like, less reciprocal, more ambiguous, more mutable, and more responsive "instrument" of social practice.

Of the two solutions made possible by the direct acceptance theory I prefer the first —that there really is self-amendment and any contradiction in it is forgivable— although there is no need to choose. I believe that the first solution comes closer to explaining how actual amending clauses actually change, namely, through literal self-amendment, disregarding and plowing under any self-contradiction. The alegal sources of legal authority emphasized in the second solution still exist —they are consequences of the direct acceptance theory. But they need not be invoked to bypass self-amendment when they can be invoked to explain self-amendment. While the self-limitation of deities remains mysterious, we should remember that the self-amendment of constitutional amending clauses is transacted and reviewed by human beings. If they don't notice the contradiction of self-amendment, of if they don't care, then they have made law anyway —as they always do, in their own image. Eternal, formal criteria of consistency have no standing to complain unless their cause is taken up by some human being.

Because self-application of a rule of change is legally permissible, and because it explains self-amendment, I prefer to supplement Hart by making his secondary rules self-applicable. The alternative is that the rules of recognition, change, and adjudication will be unrecognizable, immutable, and injusticiable. He could have made his secondary rules self-applicable without jeopardizing any part of his theory of law. Indeed he would have strengthened both his theory of law in general and his response to the paradox of self-amendment in particular, which depends now on the common but irrelevant objection that the old and new amending clauses could be kept separate in time.[Note 13]

The solution I prefer allows self-amendment despite the self-contradiction of its nature. A chapter of the book explores how acceptance can permit law to absorb, tolerate, or forgive some contradiction, but not to tolerate all contradiction.

The direct acceptance theory of legal change and validity offers, then, two satisfactory solutions to the paradox of self-amendment. Justifying the direct acceptance theory has only been a parenthetical task of this essay. Demonstrating the inadequacy of the inference model has been much closer to the center of concern.[Note 14] Ross's postulate of a tacit, transcendent rule is theoretically coherent, but it is designed to make strict (reflexive) self-amendment unnecessary, which is itself unnecessary, and it clearly fails to explain actual practice.

Whether self-amendment is legally valid depends not on a nation's law or its amending clause, but on its concept of law. Permitting change of the amending clause is compatible with many theories of law, even the inference model, but permitting strict self-amendment of the clause is incompatible with the inference model. Permitting self-amendment in fact is itself permissible in any legal system that currently seems to bar self-amendment (although I know of none of these). One act of accepted self-amendment would overturn the inference model as a descriptive model of law for that system. The proposal and ratification of the reflexive amendment would at the time run afoul of the inference model and perhaps some explicit rules of procedure. But direct acceptance could cure these defects after the fact, contingently, if acceptance shifted in their favor. The reason that violations can be accepted as amendments is that the question of legal permissibility is empirical and turns on what we find when we look. Whether self-amendment is formally contradictory may be a matter for a priori analysis, but that is always distinct from the question of the content of law, even the question of the logical content of law.

Contingent Immutability and Contingent Omnipotence

The direct acceptance theory implies that no rule is absolutely immutable. It follows from this that every amending clause and even acceptance itself have a sort of continuing omnipotence. If a rule purports to be immutable, for example through complete self-entrenchment, then it may be amended or repealed if the requisite acceptance is obtained, which may be called the "transmutation" of the "immutable" rule into a "mutable" rule. The future generations that were to have been bound immutably may decide that they cannot be bound immutably, and repeal what was earlier believed to be an immutable rule. This is true, of course, even if the generation that enacted the "immutable" rule intended to bar all repeal, and reasonably believed that they had used legal devices that made repeal unlawful. This means that rules that purport to be immutable are really mutable, but only contingently. With the requisite acceptance, the people can repeal any law; but it is always a matter of historical contingency whether that requisite acceptance will be summoned. And some "immutable" laws may remain unchanged forever if the acceptance that would support amendment or repeal contingently never arises. So while some rules might contingently remain unchanged forever, none is totally immutable. The possibility of contingent amendment and repeal is permanent.

For the same reason any limitation on the amending power, original or self-imposed, is contingently revocable by the amending clause even if the limitation purports to be immutable. If an amending clause is used to overcome its own limitations (to disentrench itself), its success is contingent upon future events —namely, the mustering of sufficient acceptance. These somewhat anomalous powers may be called the contingent omnipotence of the amending clause and the contingent immutability of rules, as opposed to categorical omnipotence of the amending clause and the categorical immutability of rules.

Contingent omnipotence implies contingent, continuing omnipotence. The contingent omnipotence of the amending clause cannot be limited irrevocably, for the subsequent repeal of the limitation is always contingently possible, not categorically impossible. Similarly, acceptance itself cannot irrevocably be deprived of its power to authorize law, under the direct acceptance theory, for any such attempt could be invalidated in the future by a shift of acceptance. If we try to prevent this contingency by law, we must use contingently mutable rules; by the nature of the case, this will fail. So while the people and officials may in a spasm of bad judgment turn their law-authorizing power over to a military or priestly caste, the new regime lasts only as long as the people and officials accept it; acceptance can always revoke its delegated authority and restore its sovereignty.[Note 15]

This view has the merit of allowing continuing omnipotence without any immutable rule to guarantee its continuity. Until now it seemed that the paradox of self-amendment put us in a harsh dilemma in which we had to choose between paradox and immutability. The direct acceptance theory eliminates immutability and softens paradox to a literally acceptable form. The effect is not only theoretically elegant; by eliminating the need for immutable rules, the direct acceptance theorey eliminates an ancient ground for bad faith in recognizing our responsibility for law. No law inherited from our ancestors is immutable; if unjust laws persist, we are responsible for them.

The omnipotence of the amending clause may be abridged by self-imposed limitations, and these limitations may contingently last forever. But because they are always revocable, the amending clause has a kind of continuing omnipotence.[Note 16] Even acceptance is not guaranteed continuity by a categorically immutable rule, for it may contingently fail to overcome its self-imposed limitations or its transfer of powers to another source of authority. However, if its failure to shed its limitations is only contingent, then it is permanently possible that it may contingently succeed at another time.

If contingent omnipotence implies a kind of continuing omnipotence, it equally implies a kind of self-embracing omnipotence. A contingently omnipotent amending clause may limit itself in any way, including ways that appear irrevocable at the time and ways that are contingently unrevoked forever. It may also strip away any limitations, including those that appeared irrevocable when enacted.

This view, then, escapes through the horns of a second dilemma posed by categorical self-embracing and categorical continuing omnipotence. One seems bound to choose between a power that can make immutable rules (self-embracing omnipotence) and one that is defined by immutable rules (continuing omnipotence), or —this is a tongue twister— between a power that is unlimited but limitable immutably and a power that is immutably limited but otherwise illimitable. If one denies omnipotence altogether, then one thereby admits that immutable rules exist to limit power. Contingent omnipotence cannot make categorically immutable rules, and is not defined by categorically immutable rules. Therefore, contingent, continuing omnipotence is also contingent, self-embracing omnipotence. Under this view, there need not be any categorically immutable rules. All the elements of contingently omnipotent rules of change, including their contingent, continuing omnipotence, are subject to self-amendment, and therefore are contingently mutable.

On this view the advantage that no rule is categorically immutable is bought at the price of making every rule of change contingently omnipotent. However, this can be shown to fit the facts of actual practice.[Note 17]

Another virtue of this view is its consonance with democratic theory. The people cannot categorically limit or repeal their right to make law. This limitation is not paternalistic but an expression of the inalienability of their sovereignty. Actually, their sovereignty is contingently alienable, but any alienation of it is contingently revocable as well. This sovereignty, of course, takes the greatly weakened form of acceptance alone. The equal sovereignty of future generations, and their equal right to make and change their laws is contingently inviolate: if we violate the right of future generations to make and change law, then they may restore this right —revoke our limitation on their power— at will. No generation can bind its successors with categorically immutable rules.

The paradox of omnipotence purports to show that no entity can be omnipotent in an unqualified sense. In this way it is like the Barber paradox. For many philosophers the paradox of omnipotence shows that there be no omnipotence as defined in the usual unqualified ways. In one sense I have given the paradox its victory, for contingent omnipotence is far from unqualified omnipotence. But on the other hand, the contradictions that are supposed to preclude the existence of unqualified omnipotence are either eliminated or made acceptable by my solution to the paradox. Self-amendment can be lawful despite acknowledged self-contradiction. Self-limitation can be permanent, contingently, despite the persistence of omnipotence. These consequences are made possible by special properties of law not available to logicians seeking a purely logical solution.

Not surprisingly, there is a theological parallel to my solution available to any theologian willing to assume that an omnipotent deity might perform contradictions. The most distinguished example of such a theologian is Descartes, who wrote to Mersenne in 1634 that[Note 18]

God was as free to make it false that all the radii of a circle are equal as to refrain from creating the world.
and to Mesland in 1644 that
As for the difficulty in conceiving how it was a matter of freedom and indifference to God to make it true that the three angles of a triangle should equal two right angles, or generally that contradictions should not be able to be together, one can easily remove it by considering that the power of God can have no limit....God cannot have been determined to make it true that contradictions cannot be together, and consequently He could have done the contrary.
and to Mersenne in 1630 that

You will be told that if God established these truths He would be able to change them, as a king does his laws; to which it is necessary to reply that this is correct.
Lawyers and theologians who deny that the principle of non-contradiction rules in their realm may accept the paradox with equanimity, as a dialectical logician might and a formal logician never could. I will let theologians speak for themselves from this point, and say only that lawyers who take this path are not simply choosing to play a different game from formal logic; they are arguing that formal logic has limited applicability in their domain, which is a domain of actuality. Formal logic will never accept these arguments as "repeals", but only as "violations", just as the formalist view of self-entrenchment clauses is that all attempted repeals are violations. But self-entrenchment clauses actually fall, and formal logic is the logic of regret, not prevention, or of powerless prescription, not true description, or of abstractly constructed fantasy, not reality.

In summary, the direct acceptance theory implies that rules of law have contingent immutability and that amendment clauses have contingent omnipotence. This frees us from two difficult dilemmas, one that would have us choose between paradox and immutability, and another that would have us choose between continuing and self-embracing omnipotence. It also explains self-amendment without ironing it out into a fictitiously linear operation. By explaining genuinely reflexive self-amendment, it explains how law can be lawful without an infinite genealogy, and hence how legality can exist at all. Finally, it reminds us that we can erect no immutable barriers to evil law, and must recognize no immutable obstacles to good law. Law is firm enough to save us from ourselves, and to thwart us, for only a very short time; the only final check to our worst tendencies and protection of our best achievements is our will.


1. Alf Ross, "On Self-Reference and a Puzzle of Constitutional Law," Mind, 78 (1969) 1-24. [Resume]

2. Ordinary deontic logics —the logics of permission and obligation— do not suffice for reasons that I explain in the book and that will appear briefly below. But I am coming to suspect that a temporally indexed deontic logic, capable of describing permissions and obligations that are enacted, amended, and repealed in time, may be made to work. No one has yet applied a temporally indexed temporal logic to the paradox of self-amendment, but for an idea of what such a logic looks like, see L. Thorne McCarty, "Permissions and Obligations: An Informal Introduction," forthcoming, North-Holland. For reasons that will appear, even if temporally indexed deontic logic dissolves the paradox, it cannot do so for the inference model. [Resume]

3. J.C. Hicks, "The Liar Paradox in Legal Reasoning," Cambridge Law Journal, 29 (1971) 275-91. [Resume]

4. H.L.A. Hart, "Self-Referring Laws," in Festskrift Till—gnad Karl Olivecrona, Stockholm: Kunglia Boktryckeriet, P.A. Norstedt and S—ner, 1964, pp. 307-16; reprinted in his Essays in Jurisprudence and Philosophy, Oxford University Press, pp. 170-78. [Resume]

5. Joseph Raz, "Professor Ross and Some Legal Puzzles," Mind, 81 (1972) 415-21. My book contains a comprehensive bibliography of the discussion of Ross's thesis and of self-reference in law. Meantime for further reading see P. 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. The bibliography is indexed inter alia for the paradox of omnipotence and reflexivity in law. [Resume]

6. John M. Finnis, "Revolutions and Continuity of Law," in A.W.B. Simpson (ed.), Oxford Essays in Jurisprudence, Second Series, Oxford University Press, 1973. [Resume]

7. If a temporally indexed deontic logic that statically described the dynamic change of norms could dissolve the paradox, it would be by rigorously enforcing the requirement of temporal separation. It might succeed at this, but would for that reason violate transtemporal validation. Hence it would not satisfy the inference model. [Resume]

8. J.L. Mackie, "Evil and Omnipotence," Mind, 64 (1955) 200-12; H.L.A. Hart, The Concept of Law, Oxford University Press, 1961. [Resume]

9. For opposing views on the question whether the House of Lords can be abolished, see Peter Mirfield, "Can the House of Lords Lawfully Be Abolished?" Law Quarterly Review, 95 (1979) 36-58, and George Winterton, "Is the House of Lords Immortal?" Law Quarterly Review, 95 (1979) 386-92. Also see Winterton's "The British Grundnorm: Parliamentary Supremacy Reexamined," Law Quarterly Review, 92 (1976) 591-617; O. Hood Phillips, "Self-Limitation by the United Kingdom Parliament," Hastings Constitutional Law Quarterly, 2 (1975) 443-78; and Ilmar Tammelo, "The Antinomy of Parliamentary Sovereignty", Archiv f—r Rechts- und Sozialphilosophie, 44 (1958) 495-513. Geoffrey Marshall argues that the English Parliament formerly had continuing omnipotence but has edged toward self-embracing omnipotence; see his "Parliamentary Sovereignty: A Recent Development," McGill Law Journal, 12 (1966-67) 523-27. [Resume]

10. Shakespeare's King Richard believed he possessed self-embracing omnipotence. Despite the appearance of insurrection, no one could depose him; he could only abdicate. He emphasizes this in his answer to Bolingbroke, who may have thought himself the King's overthrower:

Now mark me, how I will undo myself: [...]
With mine own tears I wash away my balm,
With mine own hands I give away my crown,
With mine own tongue deny my sacred state,
With mine own breath release all duteous oaths.
Richard II, Act IV, scene I, lines 203-210. [Resume]

11. This view is implied by the position taken by Thomas Reed Powell in his essay, "Changing Constitutional Phases," Boston University Law Review, 19 (1939) 509-32. [Resume]

12. This type of self-amendment has actually occurred, most recently in Pennsylvania in 1967, Minnesota in 1974, Hawaii in 1978, and North Dakota in 1979. [Resume]

13. This is essentially the position that Hart takes in the essay cited in footnote 4, above. [Resume]

14. This is true of the larger book as well, although a section of the conclusion of the book addresses the merits of the direct acceptance theory apart from its ability to solve the paradox of self-amendment. [Resume]

15. This is not to say that victims of oppressive laws are to blame for their condition. Any acceptance theory of legal authority and change, when fully worked out, must have a place for (1) ideological acceptance, which lacks the cognitive element of free consent, and (2) involuntary acceptance, which lacks the volitional element. The former is unfeigned, perhaps wholehearted and enthusiastic, but ignorant or delusional, and may contradict the real interests of the acceptors; the latter is qualified and wary, resented acquiescence coerced by violence and the threat of violence. Both are pathological types of acceptance that create pathological legal systems in which legality and legitimacy are both compromised. [Resume]

16. I argue in the book that the repeal of an amending clause makes acceptance of its self-resurrection significantly less likely but not categorically impossible. Such self-resurrection now seems on the verge of happening on Colombia. [Resume]

17. In the book I argue in detail that the elimination of categorically immutable rules does not require one categorically immutable rule to bar others. [Resume]

18. Descartes's letters were first brought into the discussion of the paradox of omnipotence by Harry G. Frankfurt, "The Logic of Omnipotence," Philosophical Review, 73 (1964) 262-63. [Resume]

This essay has been translated into Portuguese by Fernando Borges Araújo, "O Paradoxo da Auto-Revisão no Direito Constitucional," Revista da Faculdade de Direito da Universidade de Lisboa, 31 (1990) 93-128.

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