SL 41 (2001), 3-38
G. Hägele & F. Pukelsheim

Llull´s Writings
on Electoral Systems *

• 1. Introduction
• 2. The sources
• 3. Artifitium electionis personarum (Biblioteca Apostolica Vaticana, Cod. Vat. lat. 9332, f. 11r-12v)
• 4. En qual manera Natana fo eleta a abadessa (Bayerische Staatsbibliothek, Cod. Hisp. 67, f. 32v-34r)
• 5. De arte eleccionis (Sankt Nikolaus-Hospital/Cusanusstift, Cod. Cus. 83, f. 47v-48r)
• 6. Llull´s electoral systems
• • 6.1. The trail to truth
  • 6.2. The bookkeeping of sincerity
  • 6.3. The evaluation of a single pairwise comparison
  • 6.4. The overall evaluation of the election
  • 6.5. A hypothetical convergence of systems
  • 6.6. The reduction of complexity
  • 6.7. The abandonment of saniority
• 7. Discussion
• 8. Résumé
• 9. References

1. Introduction

We present three of Llull´s works dealing with electoral systems, to be cited with the abbreviations given in the parentheses:

• Artifitium electionis personarum (AEP)
• En qual manera Natana fo eleta a abadessa (B24)
• De arte eleccionis (DAE)

B24 is the twenty-fourth chapter of the novel Blaquerna. For a long time this was the only source on Llull´s electoral systems until, in 1937, Honecker [10] rediscovered DAE. In his commentary, Honecker [10, page 307, n. 11a] doubted whether the available bibliographical evidence would imply the existence of yet another work of similar title and content. In 1959, finally, Pérez Martínez [28] announced the discovery of the manuscript of AEP, and later included a description of it in his catalogue of Roman libraries [29, p. 39]. While Pérez Martínez was referenced by Platzeck [30, vol. II, p. 7] and Llinarès [12, p. 177, n. 25], neither of the three authors mentioned Honecker´s discovery of DAE. Conversely, in vol. 17 (1989) of the authoritative Raimundi Lulli Opera Latina [20, p. X] the editors claimed that the Ars electionis is preserved only in a manuscript in Kues, that is DAE, thus neglecting AEP.

The oldest list of Llull´s works is that in his autobiographical Vita coetanea [19, p. 307], reproduced with annotations in Hillgarth [9, p. 341]. The items of interest to us are:

• 86. Ars notandi
• 87. Ars eleccionis
• 88. Alia ars eleccionis

We believe AEP to be identical with number 87 and DAE to coincide with number 88, on the grounds that in Cod. Vat. lat. 9332 the text AEP is immediately preceded by the Ars notandi, and that AEP is longer and more detailed than DAE. DAE comprises about 650 words, while AEP is about 100 words longer. Furthermore, Honecker [10, p. 304, n. 2a] quotes two titles, De arte eligendorum prelatorum magna and De arte electionis compendiosa, from a Renaissance bibliography drawn up by Arias de Loyola. Honecker identifies DAE with the second title. This assignment seems persuasive to us, in that it leaves the first title for AEP, which indeed mentions repeatedly that its electoral system is intended for the election of prelates.

The recent literature, surveyed by Lohr/Bonner [22], concentrates on Llull´s philosophy and does not embrace his electoral systems. Moreover, the electoral systems proposed both by Llull and by Cusanus have long been ignored in the political science literature. This is beginning to change, thanks chiefly to the research of McLean/London [24, 25], Meuthen [27], and Felsenthal/Machover [5], as evidenced in the recent monographs of McLean/Urken [26] and Colomer [3].

Our paper is organized as follows. In Section 2 we comment on the chronological sequence of AEP, B24, and DAE. For a particular work we refer to the number and the date of composition given in A Chronological Catalogue of Ramon Llull´s Works of Bonner [1, vol. II, pp. 1257-1304], quoted as Bo. Sections 3-5 contain transcriptions and translations of the three texts. In Section 6 we describe and compare the electoral systems in these writings. The tract AEP, edited here for the first time, contributes at least three details leading beyond B24 and DAE:

• (1) multiple records are kept during the election session,
• (2) unus punctus is recorded for the winner of each pairwise comparison,
• (3) lots are thrown if candidates are tied with the same number of winning points.

The present paper is accompanied by a web edition at www.uni-augsburg.de/llull/ which offers a linked triple text consisting of facsimile, transcription and translation, for each of the three works. For details of this web edition, see Drton et al. [4]. A narrative on the somewhat accidental circumstances that led to the present research can be found in Pukelsheim [34].

2. The sources

The dating of the first text, AEP (Bo II.A.10: 1274-83), is least certain. Platzeck [30, vol. II, p. 7] places it in the period 1273-5, without providing any detailed evidence. The recent study of Santanach i Suñol [39] reconfirms the period 1274-83 from Bonner´s catalogue. The expert opinion is that, in all probability, Llull completed the novel Blaquerna 1283 in Montpellier, see Soler i Llopart [42]. We shall argue in Section 6 that the electoral systems in AEP and in B24 are distinct from each other. In fact, they are so different that it would seem to us most unlikely that Llull wrote the two works back to back. Therefore we would tend to believe that AEP was written quite some time before 1283.

The AEP is preserved in a single manuscript, Codex Vaticanus latinus 9332 of the Biblioteca Apostolica Vaticana, as reported by Pérez Martínez [28]. The call number 9322 quoted in Pérez Martínez [29, p. 38] is a misprint. The bibliographic information given by Glorieux [8, p. 301], that AEP appears in codex Clm. 9332 in Munich, confuses libraries; at no time did the Bayerische Staatsbibliothek own a codex with this call number.

Cod. Vat. lat. 9332 is an undecorated collection of fascicles in quarto (22.5 x 15 cm), consisting of several distinct parts and totaling 326 folios (paper). It begins with Llull´s Ars notatoria (Bo II.A.2: 1274-6), immediately followed by AEP on folios 11r-12v. Both texts date from the second half of the fifteenth century, and are written by the same hand. The first text starts with a heading giving the title. The second text, AEP, has no heading, instead the title has to be inferred from the colophon: Finis artifitii electionis personarum. The appearance of AEP arouses sympathetic feelings as to how difficult life was for a medieval copyist: the pen blunted, the ink blotted, the writing blurred. Gayà [13, p. 23] tactfully calls the impression that AEP conveys as being de caligrafía más oscura.

AEP appears to have been written by a skilled hand, as a quick, cursory excerpt. The small cursive, with many abbreviations, is in parts hard to read. The number of lines per page changes repeatedly, the rather complicated figure on folio 11r is drawn free-hand. There are no references to the author of the text, nor to the copyist of the manuscript. However, on folios 13r-15v of the codex appears a third Llull text, Liber de accidente et substantia (Bo IV.75: 1313). This text, though by a different hand, says that it is a Raimundo compilatus.

Formerly, Cod. Vat. lat. 9332 was owned by Pier Leoni (d. 1492), court physician of Lorenzo de´ Medici. It is most likely that it was Leoni himself who copied the Ars notatoria and the Artifitium electionis personarum. This is the conclusion arrived at by Ruysschaert [38, p. 57], on the basis of a comparison of the hands in Cod. Vat. lat. 11585, 9414, 9425 and in Reginensis 1773, all from Leoni´s library. Note that Ruysschaert´s singular, le premier texte, comprises a plural, the Ars notatoria and the Artifitium electionis personarum. For more on Leoni and his relations to Llull´s work see Rotzoll [37].

For B24, Chapter 24 of the novel Blaquerna (Bo II.A.17: 1283), we rely on the Catalan manuscript, Codex Hispanicus 67 of the Bayerische Staatsbibliothek, Munich. The volume consists of 268 folios (paper), and dates from the end of the fourteenth or the beginning of the fifteenth century. Authors occasionally quote the obsolete call number Cod. Hisp. 610, from p. 91 in volume 7 of the 1858 (reprinted in 1971) Catalogus codicum manu scriptorum Bibliothecae Regiae Monacensis. The transcription in Section 4 below was kindly provided by Soler i Llopart [15]. We also consulted the transcription of Galmés/Ferrà [17], the English translations of Peers [14], and McLean/London [24, 25], and the French translation of Llinarès [18].

DAE (Bo III.38: 1299) is dated rather exactly by Llull himself: it was written on 1 July 1299, in Paris. The codex unicus preserving this text is the Codex Cusanus 83 in the Library of the Sankt Nikolaus-Hospital/Cusanusstift in Bernkastel-Kues. Honecker [11, p. 571, n. 23] considers it almost certain that it was written by Nicholas Cusanus. This view is reconfirmed by the more recent findings that Cusanus spent the spring of 1428 in Paris, copying parts of Llull´s works available in the Paris libraries. Roth [36, Sections A.I and A.V] reviews the current state of research on the relation of Cusanus to Llull, and also gives a description of Cod. Cus. 83. For the transcription and the translations we consulted Honecker [10, pp. 308-309], and McLean/London [24, 25].

In our transcriptions, numbers in small print are the line counts in the manuscripts; most of the paragraph breaks are ours. In the translations, we follow the transcriptions rather closely except for adding punctuation. A few additions and omissions that we considered helpful are indicated by brackets [ ] . In the main text, bracketed numbers refer to reference items at the end of the paper. Llull´s electoral systems rely on his Third Figure, the quaternary 16 candidate version being part of AEP and the ternary 9 candidate version heading DAE. These tables are built up from entities which we call cells, following today´s statistical terminology. While Llull´s notion of ars carries a rather broad meaning, see Bonner [1, vol. I, p. 62], we translate ars eleccionis simply by electoral system, for the purposes of the present paper.

3. Artifitium electionis personarum
(Biblioteca Apostolica Vaticana, Cod. Vat. lat. 9332, f. 11r-12v)

4. En qual manera Natana fo eleta a abadessa
(Bayerische Staatsbibliothek, Cod. Hisp. 67, f. 32v-34r)

5. De arte eleccionis
(Sankt Nikolaus-Hospital / Cusanusstift, Cod. Cus. 83, f. 47v-48r)

6. Llull´s electoral systems

6.1. The trail to truth

Llull´s electoral systems in AEP, B24 and DAE have in common that they are intended for the election of church officials in clerical communities. In AEP there are 16 persons or a few more who elect a prelate, without any further indication as to who they are or how they are selected. In B24 it is 20 nuns of a convent who elect a new abbess. DAE has 9 persons or a few more electing an abbott, prior, or bishop.

In each of the three writings Llull quotes some of his prior work. AEP and B24 both mention the Ars compendiosa inveniendi veritatem (Bo II.A.1: c. 1274), see [32]. In addition, AEP refers in a somewhat cursory fashion to the Liber principiorum theologiae (Bo II.A.6: 1274-8), the Liber principiorum philosophiae (Bo II.A.7: 1274-8), and the Liber principiorum juris (Bo II.A.8: 1274-8). For an edition of these anlong with the Liber principiorum medicinae (Bo II.A.9: 1274-8) see [21]. B24 quotes the famous Llibre del gentil e dels tres savis (Bo II.A.4: 1274-6), see [16].

In DAE, Llull says that the Third Figure is taken from the Ars generalis. Platzeck [30, vol. II, p. 34, n. 85] identifies this reference with the Ars compendiosa (Bo III.36: 1299). Bonner [1, vol. I, p. 22, n. 88] points out, however, that from 1294 on Llull used the term Ars generalis as a generic reference to his system, the final version of which appeared in the Ars brevis (Bo III.77: 1308) and the Ars generalis ultima (Bo III.80: 1305-8). A version of the Third Figure is reproduced on plate XVII in Bonner [1, vol. I, after p. 582].

Self-references of authors to their own works have been popular from the middle ages till today and, besides advertising the author´s oeuvre, serve to claim an advanced level of competence. It would seem to us that, in the present case, Llull may want to send a more pointed message to his readers, emphasizing the ultimate function that he attributes to an electoral system. In our time we view an election as a fundamental building block of mass democracies, a means to collect the opinions of individual voters and to aggregate them across the entire society. This integrating function of popular elections in democratic societies would have appeared totally alien to Llull.

As a devoted Christian in his time, Llull´s concern was not the aggregation of many individual truths, but the discovery of the one and only truth existing, the truth of God. We understand Llull´s electoral systems as manifestations of his Ars generalis, and as such they are means to set mankind on a trail leading to the unique, divine truth: «By this method,» said Natana, «is found the truth; by this truth we will be able to find the sister who is most suitable and best to be our abbess.» (B24, f. 32v, l. 17 - f. 33r, l. 3).

6.2. The bookkeeping of sincerity

The Third Figure of the Ars generalis is displayed right at the beginning of DAE. It is an arrangement of the 36 pairs that can be formed from 9 candidates named b, c, ⋯, k. A larger version of the figure, for the 120 pairs from 16 candidates b, c, ⋯, r, is included on the first folio of AEP. The use of letters to indicate candidates serves a double purpose, of exhibiting the universal applicability of the system, and of producing a figure small enough to fit the limited space available.

The question arises, however, whether the sequence of letters is meant to indicate a ranking of the candidates. If so, the candidate with the first letter would hold the highest rank, and the candidate who is assigned the last letter would be ranked lowest. The other candidates would be ranked in-between, as determined by the order of the letters. If the letters were to indicate a ranking, then it is conceivable that Llull might have wished to design a biased electoral system favoring candidates of higher merits at the expense of candidates of lower merits. We maintain that such hypothesis, that the assignment of letters presupposes a ranking by merits, is inconsistent with Llull´s way of thinking.

Llull felt himself called upon to devote his life to spreading the Christian truth to the Muslim world. Pindl [16, p. 264] recalls a tragic incident from Llull´s autobiography, the Vita coetanea, as an episode of lasting effect, see [19, p. 279] or [1, vol. I, p. 21]: Llull and his Saracen teacher of Arabic enter into a fierce argument. The teacher draws his sword against his master. Llull successfully defends himself, and imprisons the teacher. While Llull contemplates what to do next, the discouraged Muslim hangs himself. Thus powerful demonstration of authority has led to a double defeat: the infidel is lost to everlasting damnation, and Llull failed to convert the disbelieving soul to his Christian God. Henceforth Llull builds on the force of arguments, not on the power of authority. He wishes to convert his opponents to Christianity through dialogue, not to dictate.

To this end the combinatorial aids of the Ars generalis serve as a warranty of fairness, so that no contraposition of opinions is omitted. For the electoral systems, the Third Figure is used to implement a bookkeeping of sincerity, that is, to secure a complete comparison of all arguments. Furthermore, Llull uses combinatorial structures quite generally to organize his philosophical view. But nowhere does Llull imply that the assignment of letters presupposes a rank order of arguments, see Platzeck,[30] Pring-Mill,[33] Hillgarth,[9] or Bonner.[1] Such a clandestine outmaneuvering of opponents is simply incompatible with a protagonist of the cross-cultural dialogue, as Llull is hailed by Pindl. [16, p. 259] We believe that it was entirely outside Llull´s way of thinking that the assignment of the letters might bias his electoral systems in one way or the other.

Not surprisingly, the instructions of how to assign letters to persons turn out to be cursory and incidental. In addition, the relevant section in AEP (f. 11v, l. 11-14) is somewhat hard to decipher. We understand that somebody, a primus of whatever sort, takes the lead to randomly attribute letters to persons. A brief examination then makes sure that all electors remember their letters. B24 makes mention neither of the letters nor of the assignment process. We take this as an indication of how unimportant this step is, in Llull´s view. In DAE (f. 48r, l. 9-10) Llull praises his electoral system because in the election a comparison is made between all persons who exist in the chapter, that is, because all chapter members are treated equally.

In the sequel our analysis is based on the assumption that all pairwise comparisons are of equal value, so that the decision between the first and the second candidates, b and c, enters the final evaluation with exactly the same weight as the decision between the first and the last candidates, b and k (or b and r).

6.3. The evaluation of a single pairwise comparison

All three variants of Llull´s electoral system build on pairwise comparisons of two candidates at a time. They differ in whether an exhaustive, or a partial series of pairwise comparisons is stipulated. In the present subsection we discuss the evaluation of a single pairwise comparison.

The instruction in AEP (f. 12r, l. 9-10) is clearest: [Then] a dot is placed by the letter assigned to the person who has the most votes. Thus the victorious candidate scores one point, and what is recorded is this winning point and not the number of votes that led to the victory. We are reminded of an ordeal by judicial duel, in which the truth emerges on the side of the victor no matter how easy or how hard he had to fight. This way of arriving at a decision is concordant with Llull´s belief in the uniqueness of the divine truth: among two candidates, one must be more worthy than the other.

The electors are humans tempted by sin, however, and it is not guaranteed that they recognize the truth in any case and at any time. Therefore Llull includes a rule for the situation when the electors fail to recognize the truth and two candidates are tied with the same number of votes (AEP, f. 12r, l. 11-12): If now one [person] has as many votes as another, then a dot is placed by both letters of this cell. By awarding a winning point to each of the two candidates, they remain equal between each other and yet gain an advantage over the other candidates.

The corresponding passage in B24 (f. 33v, l. 16-17) is briefer: and in secrecy let it be written down who has the most votes. We think that the meaning of the phrase is identical to the ruling in AEP. For at the beginning of B24 (f. 32v, l. 14-15), Llull claims to propose a new electoral method based on a system and figures. In the middle (B24, f. 33v, l. 3), he states that the electors compare each person with every other . Morever (B24, f. 33v, l. 24-25), a total of 36 cells are generated. Clearly this points to the Third Figure, even more so since in the other two texts the figure is reproduced in its entirety. From this we conclude that `written down´ means `marked´: `and in secrecy let the letter of the person be marked who has the most votes´. With this interpretation, B24 defines the same rule as does AEP. We cannot imagine that the quoted passage would support an extended meaning, to write down the number of votes of the victorious candidate, let alone two numbers, one for each candidate.

B24 does not contain a rule on how to break a tie. Llull may have hoped that his insistence on an odd number of electors (B24, f. 33r, l. 15) makes a tie-breaking rule superfluous. This is true for the majority of comparisons, when two of the seven electors are compared by the remaining five, or when the two additional candidates are compared by all seven electors. However, the other cases are left dangling in the air. When one of the electors and one of the additional candidates are compared, then there are six electors and a tie with three votes for each candidate becomes possible. Such ties are not accounted for, in B24.

Nor does the last text of the electoral trilogy, DAE, feature a tie-handling rule even though here it is needed most. DAE promotes a partial system of pairwise comparisons, based on successive eliminations. Therefore each pairwise comparison must leave a well-defined victor, who then proceeds to the next round. Other than in B24, no mention is made that an odd number of electors would be advantageous. Nor would the AEP rule, that tied candidates are treated equally, be of any help since just one candidate can proceed to the next round. In our view, the absence of a tie-breaking rule in DAE constitutes a fatal omission. If DAE was practiced as is, then the divine truth must have shone rather brightly in order to keep the electoral business on the road of decidability.

6.4. The overall evaluation of the election

The global evaluation of all pairwise comparisons is settled most clearly again in AEP (f. 12r, l. 20-21): Once the examination of all cells has been completed, the dots of each letter must be counted. The overall winner of the election is the one who scores the greatest number of winning points, that is, who wins most of the pairwise comparisons. Two extra rules take care of the possibility that two or more candidates are tied with the same number of winning points. Firstly, another election is to be conducted among those tied. Secondly, if yet again a tie emerges, lots are thrown.

The last rule sounds surprising since the Christian Church condemned the throwing of lots ever since the soldiers threw lots at the feet of the crucified Jesus to divide his garments among them (St John 19,24). Maleczek [23, p. 129] reports that the Church banned lots from electoral decisions in 1223. Did Llull realize that the inclusion of lots was detrimental to promoting his system to a clerical clientele? In any case, in B24 and DAE he does not follow up on the idea. Remarkably, in the political world of today, every electoral law we know of includes a final clause to break eventual ties - by drawing lots.

In B24 (f. 33v, l. 26-27) the winner of the election is the candidate who has the most votes in the most cells. We think that this ambiguous instruction is meant to set the same rule as in AEP, that the candidate with the largest number of winning points across all pairwise comparisons is the winner of the election. For the case of a tie between two or more candidates, only the first tie-handling rule of AEP is repeated, to conduct a further election among those tied.

In contrast to the exhaustive schemes in AEP and B24, of considering all pairwise comparisons, DAE advocates a partial system of successive eliminations. A candidate who loses in a pairwise comparison is eliminated from further consideration. Only the victorious candidate enters into the next round, and the last round´s victor is the overall winner of the entire election. We are reminded of the devout Christian knight who, in Chapter 64 of Blaquerna, proves the Christian truth to the Saracen king by fighting duels with all the knights of the king´s court, one after the other, see [31, p. 82] and [1, vol. II, p. 754, n. 44].

Today´s understanding of electoral systems focuses on the freedom with which individuals cast their votes; then the exhaustive system of AEP and B24, and the DAE system of successive eliminations may lead to diverging results. However, Llull coped with devising a decision procedure that would unerringly converge to the one and only divine will. He may have hoped that his goal would not be hampered by changing some electoral details here and there. In fact, it is easy to propose some hypotheses under which he could rightly hope so.

6.5. A hypothetical convergence of systems

Formally, the exhaustive comparisons of AEP and B24 and the partial comparisons of DAE are different systems, possibly leading to different results. Under mild hypotheses, however, the differences vanish and the final conclusions coincide. For an illustration, let us agree on the following two hypotheses,

• (i)  that the truth set by God is unique, and
• (ii) that, in any decision instance, it discloses itself to at least half of the electorate.

Llull would have emphatically supported our assumptions, or so we believe. Under hypotheses (i) and (ii), the electoral systems of AEP, B24, and DAE lead to the same result. We forgo a stringent proof of this claim, instead illustrating it by example.

Suppose there are nine electors b, c, d, e, f, g, h, i, k, each of whom also stands as candidate. By hypothesis (i) we may assume that there is a unique true ranking (*) in which God sees the candidates. In this example, we suppose it is given as follows:

• (*)    e < g < k < b < d < i < h < c < f

That is, e is the weakest candidate, g is the second weakest candidate, and so on up to c, the second most worthy candidate, and finally f, the worthiest candidate.

We assume that the six electors b, c, d, e, f, g wisely see the candidates in the same order that is determined by the true ranking (*). The other three electors, h, i, k, deviate from the divine will (*) and base their decisions on the following individual rankings:

• Elector h:    d < i < h < k < e < f < g < b < c
• Elector i:     h < d < g < k < b < f < c < e < i
• Elector k:    h < e < k < d < i < f < c < g < b

When two of the six wise electors are up for comparison, they are not permitted to vote. Therefore the least favorable decision instance still has four wise electors as opposed to three individualists, so that our assumptions satisfy hypothesis (ii).

Now the election begins. Round 1 features the pairwise comparison between b and c. The electors d, e, f, g check the ranking (*) to find that b ranks below c, and hence vote for c. So do electors h and i. Only elector k, upon checking his or her individual preference scheme, ranks b ahead of c, whence this vote goes to b. In total, c beats b by 6 to 1 votes. Candidate c records a winning point, and proceeds to the next round.

Following the rules of DAE, round 2 compares c and d. Everybody ranking c above d, candidate c wins with 7 to 0 votes. Thus d is eliminated. Candidate c enters round 3, together with e. It transpires that c beats e by 6 to 1 votes, whence e is out.

In round 4 the margin of victory turns out to be as narrow as possible. For, according to the electoral rules, candidates c and f are excluded from voting. This leaves just four electors blessed with the insight into the true ranking (*); they vote for f. When electors h, i,k check their individual preferences, each finds c ahead of f. Hence f beats c by 4 to 3 votes, and the victor of round 4 is f.

Moreover, f also wins rounds 5 through 8, against g, h, i, k. Thus eight pairwise comparisons, out of a total of 36 possible pairings, establish the end result: the overall winner is f, in agreement with the divine ranking (*).

Following the rules of AEP and B24, another 28 rounds are called for until all pairwise comparisons are exhausted. In each case the two candidates standing for comparison are banned from voting, and the remaining electors check the preferences peculiar to them before they cast their votes. Then unus punctus is marked near the letter of the winning candidate. In the end, the electoral record looks as follows:

The final step is to count the number of winning points. Candidate f gains the largest number of winning points, 8, and hence is the winner of the election. Moreover, we may count the winning points for all other candidates, so as to arrange them in winning order: e is lowest with 0 winning points, g is second-lowest with 1 point, etc., thereby actually reproducing the true ranking (*). Thus Natana´s promise is fulfilled that, by this method, is found the truth.

From our modern viewpoint it may seem overly simplistic to postulate hypotheses under which Llull´s electoral systems cannot but find the truth. And evidently our two hypotheses are not exhaustive of Llull´s thinking, because then he would not have added additional rules to handle ties. On the other hand it would seem anachronistic to accuse Llull, who lived seven centuries ago, of not providing a stringent argument that we find pleasing by today´s standards. Llull may have well satisfied himself by checking a few sample situations, like the one we have discussed just now.

6.6. The reduction of complexity

For all of his life Llull endeavored to improve the structure of his Ars generalis, so as to make it more easily accessible to his audience. Llull´s contemporaries were intrigued by the complexity of his philosophy, and presumably also felt that his electoral systems were fairly complicated and complex. We think that the changes from AEP over B24 to DAE may well have been motivated by Llull´s desire to increase acceptability by decreasing complexity.

It is plainly evident that basing the electoral record on the Third Figure was overly demanding. None of its versions mentioned in the present paper is drawn without error. In AEP the cells bh and bi are labeled incorrectly, in DAE the cell ei is misnamed, and so is the cell EK in plate XVII of Bonner [1, vol. I, after p. 582]. Perhaps this is why Llull requires multiple copies of the control figures (AEP, f. 11v, l. 20-21), and perhaps this is why Natana hopes that an electoral error spares her the burden of becoming abbess (B24, f. 34r, l. 16-24).

The first electoral system, from AEP, needs a considerable amount of time to be carried through. For every pairwise comparison

• (1) two candidates leave the electoral hall,
• (2) the remaining 14 electors submit their votes,
• (3) the votes are totaled,
• (4) a winning point is marked in the electoral records, and
• (5) the candidates are called in again.

Assuming that the electors declare their votes with appropriate deliberation, a single round may well have taken from 5 to 10 minutes. With 120 pairwise comparisons, the election then lasts from 10 to 20 hours. An adjournment overnight, however, counteracts Llull´s concern to prevent fraud and simony, as expressed in AEP (f. 11r, l. 9) and DAE (f. 48r, l. 5-6).

B24 presents the accelerating idea of reducing the electoral community, by selecting a smaller electoral college. A college size of seven is taken to be optimum. In the least favorable case, two of the electors are up for comparison and hence cannot cast a vote themselves. This leaves five voters, and a winning majority must gather at least three votes. To claim legitimacy, a vote of three may have seemed sufficient to Llull, for where there are two or three gathered together in my name, there am I in the midst of them (Matt. 18,20), see also Ganzer [6, p. 76]. With an electoral college of seven, plus two additional candidates as in B24, the duration of the electoral business is shortened to last between 3 and 6 hours.

The third system, in DAE, does not decrease the number of electors. Instead it diminishes the number of pairwise comparisons. With a touch of ingenuity Llull turns away from an exhaustive scheme of pairwise comparisons, and replaces it by a partial system of successive eliminations. For n candidates, an exhaustive system needs (n-1)n/2 rounds, while a system of successive eliminations makes do with n-1 rounds. With 16 candidates, this is a spectacular reduction from 120 comparisons down to 15, with 9 candidates, from 36 down to 8. Since a system of successive eliminations does not require keeping a detailed electoral record, the election may need no more than one hour.

Curiously, though, Llull neglects the hazards of a successive eliminations system, which may suffer from a severe bias in favor of those who enter late at the expense of those who enter early. In summary, we conclude that the changes from AEP via B24 to DAE entail a remarkable acceleration of the electoral business.

6.7. The abandonment of saniority

In medieval electoral systems it was neither time nor number that in the end constituted the binding effect of the electoral outcome. The crucial requirement is saniority, that is, the claim of the majority that in the face of God they are more sound of mind and more pleasing than the minority. Maior et sanior pars is the motto, see Ganzer [6] and Schimmelpfennig [40], [41]. The majority is well advised not to just celebrate their numerical victory, but to also claim a level of superior quality. If the minority succeeds in occupying this position, they may insist on having won the election on the grounds of better insight, even though they fall short numerically. With the result of the election contested, the way out is to take recourse to an arbitrator. Thus the notion of saniority necessitates an authority outside the electorate.

Llull never makes any mention of saniority, yet he shows a great concern that the more worthy and more suitable candidate be elected. Above and beyond the vow binding the electors as nuns (B24) or brethren (DAE), he makes them take an oath to rank the candidates according to pertinent properties. AEP (f. 11v, l. 15-19) stipulates that three things should be considered: honesty and holiness of life, knowledge and wisdom, and a suitable disposition of the heart. B24 (f. 33v, l. 4-6) defines four conditions: love and knowledge of God, love and knowledge of the virtues, knowledge of and opposition to vices, and greatest suitability. These attributes are similar to those that are characteristic of saniority, see Maleczek [23, p. 122]. We mention in passing that AEP, with three attributes, is from the quaternary phase of Llull´s writings, while B24, with four attributes, is from the ternary phase.

In DAE (f. 47v, l. 23) the oath plainly demands to elect the better and more suitable person (personam meliorem et magis ydoneam). Thus Llull shifts the emphasis from worthiness of the electors (saniority), to worthiness of the candidates (idoneïty). We take it that in Llull´s writings the absence of the notion of saniority is a deliberate abandonment of the concept, not an accidental omission. Secret voting would have been another option to deal with the problem, and indeed was under consideration at that time. The 1283 Montpellier General Chapter of the Dominican Order introduced secrecy of the votes, see Maleczek [23, p. 129] and Gaudemet et al. [7, p. 332]. Secrecy does away with saniority, because a secret vote cannot be traced back to the voter and his or her state of soundness. Llull remains somewhat undecided on whether he favors secret voting or not. In DAE (f. 48r, l. 6-8) he finally advocates open voting, welcoming the social control that comes with it.

We believe that Llull, designing his system so as to lead to the more worthy candidate, wants to shield it from saniority and outside intervention. The trail to truth envisioned by Llull needs no recourse to a higher worldly authority. When asked why in his electoral systems he mentions the notion of saniority not even once, Llull might have coined the phrase, half a millennium before Laplace: Je n´ai pas besoin de cette hypothèse-là.

7. Discussion

The credit for the discovery of DAE in Cusanus´ library in Kues lies with Honecker. Regrettably, his comments on the electoral details are unreliable and have obscured the subject to date. We would like to correct his misjudgement of the overall evaluation of the election, his grossly misleading implication that all electoral systems lead to identical results, and his false statement on the transitivity of pairwise comparisons.

Honecker [11, p. 567] summarizes the global evaluation of a B24 election as follows: Wer nun dabei in allen Wahlgängen zusammen die meisten Stimmen erhalten, das heißt in den meisten Einzelwahlen gesiegt hat, gilt als gewählt. [The one who among all pairwise comparisons has totaled the most votes, that is, has won most of the individual comparisons, wins the election.] The conjunction das heißt [that is] is fatally misplaced, for the electoral system in the initial section of the sentence, building on vote totals, is different from the electoral system in the final section, counting victories. In our contemporary terminology the electoral system of adding votes is called the system of Borda, while the system of counting victories is named after Condorcet, see McLean/London [24, p. 106; 25, p. 34].

With his past as a courtier, Llull may have seen the electoral process as a voting tournament. His prescription that unus punctus is awarded to a duel´s victor is most clearly formulated in the tract AEP edited here for the first time and not accessible to Honecker. Yet the texts that Honecker did have available should have prevented him from going astray. The evaluation of a single pairwise comparison is summarized correctly [11, p. 567]: und es wird jedesmal geheim aufgeschrieben, wer bei jeder einzelnen Abstimmung die meisten Stimmen bekommt. [and each time it is recorded in secrecy who in each pairwise comparison gains the most votes.] If in every round the name of the winner is recorded rather than the score of votes, then it is impossible to finally total those nonrecorded votes.

Honecker´s misjudgment is presumably caused by the fact that in B24 the word veus stands for both the votes of the electors as well as the winning points of the candidates. Of course, it is irritating when one and the same word is used with distinct meanings. There are more examples, though, where Llull passes over a shift of meaning as he exposes his electoral systems in more and more detail: persona is a member of the convent as well as a member of the electoral college as well as a candidate, and nombre is the number of electors as well as the number of candidates as well as their numbering as well as the number of pairwise comparisons.

Honecker [11, p. 568] also erroneously equates the results from the electoral system in B24 with those from the system in DAE: Im Vergleich zur Wahl im Libre de Blancerna stellt der Wahlmodus des Traktats De arte electionis einen abgekürzten Prozeß dar. Die Zuverlässigkeit des Verfahrens wird davon aber nicht berührt. [Compared to the election in Blaquerna, the electoral system of the tract De arte electionis presents a shortened process. The reliability of the procedure, however, remains untouched.] Everything is lumped together, including the contributions of Cusanus. Peculiar to Cusanus is [11, p. 574] die `Punktwertung´, die das Verfahren, wenn man von der ars electionis ausgeht, nicht nur erheblich vereinfacht, sondern zugleich auch eine zuverlässige zahlenmäßige Auswertung der Abstimmung gestattet. [the `scoring scheme´ that not only simplifies the procedure considerably, as compared with the ars electionis, but at the same time permits a reliable numerical evaluation of the voting system.] We cannot understand these quotations other than that, to Honecker, these systems always lead to the same result and differ only in how to get there. This is grossly misleading. As a matter of fact, the electoral systems of Llull/Condorcet and of Cusanus/Borda occasionally lead to different results, though, of course, not always.

Finally, Honecker [11, S. 574, n. 34] makes a false or, at best, confusing statement on the transitivity of pairwise comparisons. Transitivity means that the following logical conclusion holds: if b beats c and c beats d, then b beats d. For a system of exhaustive pairwise comparisons, as in AEP and B24, this conclusion is false, as may be illustrated with an example of three electors X, Y, Z, and three candidates b, c, d, see McLean/London [25, p. 23]:

• Elector X:    d < c < b
• Elector Y:    b < d < c
• Elector Z:    c < b < d

Here b beats c by 2 to 1 votes and, with the same score, c wins against d. Yet the voting duel between b and d sees b not winning, but losing by 1 to 2 votes. Thus the Llull/Condorcet system suffers from occasional intransitivity. In contrast, the Cusanus/Borda system is free from this defect.

Blinded by the similarity of the various electoral systems, Honecker creates the impression - though he never explicitly says so - that the systems differ only in procedural details, but in the end lead to identical results. As we have pointed out, this is not true and, in particular, it inappropriately belittles the originality of Cusanus´ contributions. A more detailed discussion of Cusanus´ electoral system will be presented elsewhere [35].

McLean/London [25, p. 27], upon referring to the inauguration of an electoral college in B24, propose the following view: Llull seems to wish to compromise between democracy and giving a more decisive voice to better qualified electors. This may represent Llull´s attempt to compromise between the rival principles of maior pars and sanior pars. We disagree and believe, as detailed in Section 6.7, that idoneïty of the winning candidate is in Llull´s eyes superior to saniority of the supportive electorate. Moreover, a democracy-Llull who devises an electoral system in order to mediate between the equal rights of the many and the specific talents of the few is, in our opinion, light years remote from Llull´s Llull whom Bonner [1, vol. I, p. xii] tries to present.

One question remains open, namely whether Llull´s electoral systems are as free of bias as depicted in our Section 6.1. After all, Llull lived in a time when hierarchies were omnipresent in all parts of society, and honoring them might have gone without saying. The relevant paragraph in AEP (f. 11v, l. 11-14) remains somewhat enigmatic to us.

Nor does DAE (f. 47v, l. 16-17) provide a definite answer. The key passage says that the letter b signifies primam personam ⋯ que in ecclesia vacante prius fuit recepta. What does this mean? Honecker [11, p. 568], McLean/London [25, p. 29, n. 4], and Meuthen [27, p. 5] all consider, with varying degrees of confidence, that the phrase implies a ranking by seniority. They rely on the words in ecclesia vacante, a technical term from canon law, designating the church community who have lost their leader; note that the crucial word vacante is missing from n. 4 in McLean/London [25, p. 29]. Then prius fuit recepta might mean that the letters are assigned in the sequence in which the electors became members of that church community. Cusanus, in 1428 a canon lawyer recently graduated from university, may well have understood the phrase in this way. Llull, however, was neither a canon lawyer nor did he enjoy a university education.

What else could Llull have had in mind? Honecker [10, p. 308, n. 6] suspects that the DAE version copied by Cusanus stems from a Catalan original. His evidence consists of four substitutions of a long whip-s for the letter c, though it is not clear to us why this would point to the Catalan language. Nevertheless, the expunged essencia (DAE, f. 48r, l. 13-14) is more easily misread from a Catalan esglesia, than from a Latin ecclesia on its sixth occurrence. Did indeed Llull draft a Catalan original, 1299 in Paris, which somehow or other mutated into the 1428 Latin copy of Cusanus?

Perhaps an original Catalan wording would be supportive of an alternative interpretation proposed by McLean/London [25, p. 29, n. 4]. Before starting the election, the electors celebrate of course a mass, see Viollet [43, p. 70]. Might the quote from DAE then be interpreted so that after the service the electors leave the church, and that the letters are assigned as they re-enter the empty building in random procession? The location where the election is to be held is never specified by Llull. Only in AEP (f. 12r, l. 3-4) is a domus mentioned although it remains open what exactly is meant by it. According to Schimmelpfennig [41, p. 191], the election of a bishop normally took place in the canon chancel of the cathedral, or in the chapter house. A linguistic analysis of Llull´s oeuvre may shed more light on the problem. For instance in Felix, Dame Reynard appeals to a church building [1, vol. II, p. 783]: An election was once being held in a cathedral church, and in the chapter there was disagreement over the election of a bishop. Later, though, the hermit refers to the church community [1, vol. II, p. 1067]: Two clerics came before a bishop asking for a church position which was vacant.

A more detailed semantic study may be also profitable in other ways. For example, apart from early years, Llull reserved the notion of dignitates for the divine virtues attributed to God, see Bonner [2]. The changing usage of the word is also noticeable in the electoral trilogy under discussion. When searching the transcriptions in www.uni-augsburg.de/llull/ for the occurrence of the string dign AEP scores seven hits, B24 two, and DAE none. Clearly, AEP was written before dignitas was restricted to its narrower meaning.

8. Résumé

Llull´s electoral systems have in common that they are exclusively based on pairwise comparisons between two candidates at a time, but they differ in detail. AEP and B24 describe systems of exhaustive pairwise comparisons, whereby AEP accepts everybody in the community as an elector, while B24 comprises a preliminary step of first selecting a smaller electoral college. In both systems the winner of the election is the candidate who is victorious in most of the pairwise comparisons. In contrast, DAE proposes a system of successive eliminations; the victor in the last pairwise contest is the winner of the whole election.

The newly edited manuscript AEP precedes B24 and DAE. It is longer than the two later texts, and contains more detailed instructions. All three writings reflect Llull´s fundamental view that every person will be rewarded with the divine truth provided he or she trusts in God and researches honestly. Llull developed his systems for clerical communities and, having fought vehemently throughout his life for founding new monasteries specializing in missionary work, he may have aimed at such institutions. However, whether the electoral systems have actually ever been used is not known.

Nor is it known what triggered Llull´s interest in the subject, and which sources he built on. In 1283 the Dominican Order held a General Chapter in Montpellier, and electoral matters were on the agenda, see Ganzer [6, p. 76]. In 1285 Guilelmus de Mandagoto (d. 1321), archdeacon in Nimes, published his influential Tractatus de electionibus novorum praelatorum, see Schimmelpfennig [40, p. 476]. Viollet [43, p. 88] points out that Mandagoto treats three systems, the elections per scrutinium, per compromissum, and per inspirationem, while the next generation canon Ioannis Andreae (1270-1348) recognizes only the first two methods. Llull fits in-between and anticipates Andreae, as DAE (f. 48r, l. 5) lists only the secret scrutinium and the election per compromissum.

• Günter Hägele and Friedrich Pukelsheim
• University Library and Institute of Mathematics
• University of Augsburg
• D-86135 Augsburg, Germany
•  
• Guenter.Haegele@Bibliothek.Uni-Augsburg.DE
• Friedrich.Pukelsheim@Math.Uni-Augsburg.DE

9. References

* Acknowledgments. This paper is the result of an interdisciplinary dialogue, and we are very grateful for the encouragement and assistance with which so many colleagues have helped us. We would like to extend our sincere thanks to A. Bonner (Palma de Mallorca), F. Domínguez (Freiburg im Breisgau), N.R. Draper (Madison, Wisconsin), A. Fidora (Frankfurt am Main), K. Flasch (Mainz), C. Lohr (Freiburg im Breisgau), E. Meuthen (Köln), U. Roth (Freiburg im Breisgau), H.G. Senger (Köln), A. Soler (Barcelona), B. Torsney (Glasgow), and to our Augsburg colleagues F. Abel (roman languages), H. Heinz (practical theology), H. Immenkötter (church history), W. Reif (computer science), T. Scheerer (spanish), B. Schimmelpfennig (medieval history), A. Unwin (statistics). Copies of the manuscripts were kindly provided by the Biblioteca Apostolica Vaticana (Città del Vaticano), the Raimundus-Lullus-Institute (Freiburg im Breisgau), the Bayerische Staatsbibliothek (Munich), and the Sankt Nikolaus-Hospital / Cusanusstift (Bernkastel-Kues).

Source: This is file www.math.uni-augsburg.de/htdocs/emeriti/pukelsheim/2001a.html,
adapted from Studia Lulliana 41 (2001) 3-38.