Section 4
The Denial of Self-Application
Peter Suber, Paradox of Self-Amendment Table of Contents

A. Four ways to avoid self-amendment

Let us explore the horn of the dilemma that denies that the secondary rules of change are self-applicable. If we concede that at least some actual secondary rules of change are mutable by legal procedures, then it seems that we must choose among four possible options.

(1) Secondary rules affect only primary rules, and tertiary rules do not exist. This is Hart's unsupplemented view,[Note 1] and it clearly fails to face the problem or explain the phenomenon of the lawful mutability of the rules of change. It makes self-amendment impermissible because it is ultra vires or beyond the authority of valid law, not impermissible because it is a contradiction or paradox.

(2) Secondary rules may affect other secondary rules, but not themselves. A given subset of all the secondary rules of change may affect others not in that subset. For example, rules of change could affect rules of recognition and adjudication without forcing us up to the tertiary level. Within the sphere of rules of change, a constitutional rule of change could affect a legislative rule of change, for example, without forcing us up to the tertiary level. This requires us to separate the legal and logical hierarchies, and allow many legal levels to exist at the same logical level. This is clearly a coherent theory, and Hart says nothing to exclude it.[Note 2] The problem of self-amendment, however, is not fully addressed until we ask how legal change may occur within the same, especially within the supreme, legal level. How can constitutional rules of change be changed without self-application or revolution?

Suppose an AC has several sections, A, B, and C, each laying out a method of amendment, e.g. by ratification by three-fourths of the states, by convention, and by popular referendum. Then rule A could be used to amend B and C; rule B could be used to amend A and C, and rule C could be used to amend A and B. We may call this the "see-saw" method of amendment (explored more fully in Section 13).

But it is possible that the individual rules of change at a given legal level will not suffice to amend. Rule A may govern part of the amendment procedure, rule B another part, and C another part, such that no single rule could authorize any change. For example, rule A could regulate methods of proposal, rule B could lay out the methods of ratification, and rule C could specify the official who certifies that ratification has occurred. Under these assumptions change could occur either through the entire set only, or through some subset.

(i) If all the rules in the set are necessary to change any member, then none could be changed without self-application or the use of tertiaries. However, change through addition could still occur. The entire set a given time could permit the enactment of a new rule of change, even at the same level as the old. The addition of a new rule will necessarily allow a subset, namely, the original set, to suffice to amend. The addition of many new rules will allow more than one subset to suffice to amend, which leads to the next alternative.

(ii) If a subset may suffice to change rules not in the subset, and especially if more than one subset may suffice, then a subset including rules A and B but not C could authorize the amendment or repeal of rule C.

Again, the entire set at a given time could permit the enactment of a new rule of change. The set of A and B could enact C; then the set of A and new-C could amend B; then the set of new-B and new-C could amend A, and so on. Now we face the tantalizing possibility that this could continue until the content of the entire set matched the sovereign's desire, or that any content whatsoever could be attained from every initial set. If so, then this theory could accommodate all possible changes of rules of change without resort to self-application or the postulate of immutable rules (see Section 13). But such a see-saw of legislation or amendment does not reflect the actual practice of any sovereign, and hence has little power to explain, even if it could permit, the lawful change of existing rules of change.

If certain "key" secondary rules of change were enacted in duplicate, then some of the problems with the see-saw method could be solved. The rule to be changed could always be found outside a subset containing rules sufficient to amend it, for that subset could contain "copies" of every rule of change in the system. However, if a duplicated rule were the one to be changed, and if duplication were needed to permit its amendment (it could be changed only with the assistance of its own "copy"), then at least one copy would remain unchanged after any act of amendment or repeal. This would not frustrate the purpose of amendment so much as it would that of repeal. But even in case of ordinary amendment, the duplication of rules would make it difficult or impossible to change all the tokens of the type, and to avoid the apparent paradox of a new rule denying (at least one copy of) a valid and binding rule in the process of coming into being. Nor of course can duplication, any more than the see-saw, explain how actual legal systems structure the change of their rules of change. It is not from ignorance of logical nicety that legislators do not enact rules in duplicate.

(3) The third method for amending rules of change without self-application may be called the "theory of types" method.[Note 3] The initial set of secondary rules of change, or the entire set at a given time, could be identified (say) by numbers in the 100s. They may be used to enact, alter, and repeal secondaries in the 200s and above; those in the 200s may only affect those in the 300s and above, and so on. No rule of change may affect rules within its own "century" of numbers (on its own level), or with any lower number (at a higher level). This theory leaves the initial set immutable, but shows how all other secondary rules may be changed without self-application. The immutable set in the 100s might contain only one member, to mitigate the offense of immutability. However, to make it empty would not eliminate immutability altogether; on the contrary, it would prevent the enactment of any rules of change, which would make every rule in the system immutable. Nor could the immutable set in the 100s be shrunk after the system got started; all its members are immune to amendment from one another and from subsequently enacted rules.

An infinite regress of rules of change need not develop and could not: rules would be added to the top end of the series as needed by society. Because they would be added seriatim in real time, they could not surpass a finite number.

A real legal system which evolved without regard to such a theory of types or levels might nevertheless be explained by it. If the system had a core of apparently immutable secondary rules, of which some might be tacit, then they could be identified as the initial set, which designates logical not temporal priority. Subsequent enactments could be placed on the scale in their logical positions, assuming the theory is sufficiently faithful to the phenomenon to have made positions for all. Mutability is authorized only by logically prior (lower numbered) rules of logically posterior (higher numbered) rules. If in a real system logically prior rules are enacted later in time than logically posterior rules, then we may say that the logically posterior (temporally prior) rules were either enacted unlawfully or enacted on the basis of logically and temporally prior, but unwritten, rules.

Enacting the logically prior, temporally posterior rules might be considered (i) the enactment of explicit forms of previously tacit or unwritten rules that once authorized existing logically posterior rules, or (ii) the enactment of explicit rules that simultaneously made explicit and amended previously tacit rules.

If a logically posterior rule seemed to authorize the amendment of a logically prior rule, or (which is the same thing) if two rules seemed to authorize one another's amendment, then we could say that the system is violating its own rules, perhaps in ignorance or with a winked eye, or that the system actually uses circular applications. In the latter case the system could not be explained by the theory of types. To insist on the former case without more evidence, then, just to make the theory of types applicable, would prematurely deny the possibility of circular applicability and direct self-application, which might be made intelligible by another theory, explanatory of actual legal phenomena, and free of illegality if not of paradox (see Sections 5, 10, 11, and 21).

(4) The initial set of secondary rules of change, or the entire set at a given time, could be taken as the explanandum, not the explanans. How could it have come into existence by legal means? If self-applicability is to be avoided, then the present set could only have been enacted by means of a logically (and temporally) prior set —tertiary rules— which was not itself self-applicable and so requires an even prior set —quaternary rules— to explain itself, and so on. But once such antecedents are postulated, the same mechanism by which they produced the present set could be used to produce a successor to the present set and all future successors.

This fourth theory really does require an infinite regress. Rules are not added by legislators seriatim in real time as needed; they are added by logicians en masse at the speed of inference in order to explain or justify the present set. The infinity required here is not the infinity of Dworkin or Birmingham,[Note 4] but a temporal and logical series of antecedent authorities that, by its nature, cannot end or, more properly, has no beginning. Hence, it requires that every rule of change in the present that is deemed to be lawful have an infinite genealogy. Even if legal systems contain an infinity of rules, none has an infinite genealogy, if only because the world itself has no infinite genealogy. Hence, either this theory is false or no legal system with rules of change has an adequate explanation for its own lawful existence (see Appendix 1.D).

Either way, this theory shows us that any theory that we accept as explaining how the present rules of change may be amended must also explain how they came into being.

B. Weaknesses of these four methods

All four of these theories are weak, although only Hart's unsupplemented view fails to provide any answer at all. The others are all empirically implausible as explanations of how actual legal systems accomplish the legal change of their rules of change. The fourth theory saves itself only by denying the adequacy of the legal justification of all actual secondary rules of change, and hence of all legal systems built on them. Such a recourse is implausible but, more important, by denying that any application of any rules (including rules of change) is lawful, it begs the question of how the lawful change of secondary rules of change is actually brought about. The see-saw method employs a mechanism which no one would seriously propose has been used by actual sovereigns in most of the attested cases of amendment to rules of change (see e.g. Appendix 2). Whether it depends for its success upon a particular plurality of rules of change which few, if any, actual ACs already possess, or whether it may be made to work for most, if not all, ACs, will be explored in Section 13. But even if it may be made to work for all ACs, it can explain the historical acts of self-amendment of comparatively few ACs. The theory of types method requires immutable rules which, if nothing else, begs the question of the mutability of the rules of change (see Section 3.B).

Hart does not offer reasons for skirting the question of self-applicability of the rules of change, not even the fear of paradox. He does not even explicitly say he denies self- applicability, but displays his rejection of it in his language.[Note 5]

[T]hese secondary rules are all concerned with the primary rules themselves. They specify ways in which the primary rules may be conclusively ascertained, introduced, eliminated, varied, and the fact of their violation conclusively determined.

Nor does Hart explicitly deny the necessity of tertiary rules, or ever use the term. But his repeated expressions of the adequacy of his theory, without tertiary rules, imply denial. Hart did not apparently see the mutability, justiciability, or recognizability of the secondary rules as a problem.

Because these four theories are unsatisfactory at least as explanations of the actual change of rules of change, let us examine the other horn of the dilemma, the horn affirming self- applicability in the face of apparent paradox. We shall reserve the right to return to these theories if the other horn turns out to be just as implausible.


1. H.L.A. Hart, The Concept of Law, Oxford University Press, 1961, pp. 79, 92. [Resume]

2. See the discussion of Gerber and Benditt, cited in Section 3, note 12. [Resume]

3. The theory of types was devised by Bertrand Russell in order to avoid the contradiction in his paradox of set theory (see text at Section 1, note 4) while preserving as much set theory and mathematics as possible. Basically the theory forbids self-reference and structures a substitute method of reference that preserves most of what had formerly been accomplished through self-reference. For example, under the theory it is simply meaningless to speak of a set being or not being a member of itself, although it is quite meaningful and important to speak of a set of level or type 2 being or not being a member of a set of type 1. Under the theory of types there simply is no set corresponding to the description, "the set of all sets that are not members of themselves". The theory adds one more grammatical or syntactic rule to the group of rules that determine whether a string of symbols is well-formed: no reference may refer to any entity of the same or higher level or type. The hierarchy of levels is admittedly metaphysical and cumbersome, but it does eliminate self-reference and most (but not all) the need for self-reference. See Bertrand Russell, "Mathematical Logic as Based on the Theory of Types," American Journal of Mathematics, 30 (1908) 222-62, reprinted in an anthology of Russell's essays, Logic and Knowledge, ed. Robert C. Marsh, Capricorn Books, 1956, pp. 57-102.

The hierarchy of types postulated by Russell should be distinguished from both the logical (Hartian) hierarchy of primary, secondary, tertiary...rules, and the legal hierarchy rising through adjudications and statutes to constitutional rules. I am not denying the legal hierarchy in any sense (but see Section 21.C). I am assuming the Hartian hierarchy throughout the essay to test its coherence and extend it. And I am merely considering the Russellian hierarchy in this part of this section. Hart's hierarchy may well be a two-tier theory of types, but to call it that without inquiry begs the question at issue, whether secondary rules may apply to themselves. No rules in a proper theory of types could apply to themselves. [Resume]

4. See Section 3, notes 13 and 14. [Resume]

5. Hart, op. cit. at 92, emphases added; see also 79, 93, 94, 97, and 108. [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.