l:        The Ars Magna of Ramon Lull

ear the city of Palma, on the island of Majorca, largest of the Balearic isles off the eastern coast of Spain, a huge saddle-shaped mountain called Mount Randa rises abruptly from a monotonously level ridge of low hills. It was this desolate mountain that Ramon Lull, Spanish theologian and visionary, climbed in 1274 in search of spiritual refreshment. After many days of fasting an contemplation, so tradition has it, he experienced a divine illumination in which God revealed to him the Great Art by which he might confound infidels and establish with certainty the dogmas of his faith. According to one of many early legends describ­ing this event, the leaves of a small lentiscus bush (a plant still flourishing in the area) became miraculously engraven with letters from the alphabets of many languages. They were the languages in which Lull's Great Art was destined to be taught.

 

After his illumination, Lull retired to a monastery where he com­pleted his famous Ars magna, the first of about forty treatises on the working and application of his eccentric method. It was the earliest attempt in the history of formal logic to employ geometrical dia­grams for the purpose of discovering nonmathematical truths, and the first attempt to use a mechanical device-a kind of primitive logic machine-to facilitate the operation of a logic system.

 

Throughout the remainder of Lull's colorful, quixotic life, and for centuries after his death, his Art was the center of stormy con­ttroversy. Franciscan leaders (Lull belonged to a lay order of the movement) looked kindly upon his method, but Dominicans tended to regard it as the work of a madman. Gargantua, in a letter to his son Pantagruel (Rabelais, Gargantua and Pantagruel, Book 11, Chapter 8), advises him to master astronomy "but dismiss astrology and the divinitory art of Lullius as but vanity and imposture." Fran­cis Bacon similarly ridiculed the Art in two passages of almost identical wording, one in The Advancement of Learning (Book 11), the other in De augmentis scientiarum, a revised and expanded ver­sion of the former book. The passage in De augmentis (Book VI, Chapter 2) reads as follows:

 

And yet I must not omit to mention, that some persons, more ostentatious than learned, have laboured about a kind of method not worthy to be called a legitimate method, being rather a method of imposture, which neverthe­less would no doubt be very acceptable to certain meddling wits. The object of it is to sprinkle little drops of science about, in such a manner that any sciolist may make some show and ostentation of learning. Such was the Art of Lullius: such the Typocosmy traced out by some; being nothing but a mass and heap of the terms of all arts, to the end that they who are ready with the terms may be thought to understand the arts themselves. Such collections are like a fripper's or broker's shop, that has ends of everything, but nothing of worth.

 

Swift is thought to have had Lull's Art in mind when he described a machine invented by a professor of Laputa (Gulliver's Travels, Part III, Chapter 5). This contrivance was a 20-foot square frame containing hundreds of small cubes linked together by wires. On each face of every cube was written a Laputan word. By turning a crank, the cubes were rotated to produce random combinations of faces. Whenever a few words happened to come together and make sense, they were copied down; then from these broken phrases erudite treatises were composed. In this manner, Swift explained, "the most ignorant person at a reasonable charge, and with a little bodily labour, may write books in philosophy, poetry, politics, law, mathematics, and theology, without the least assistance from genius or study."

 

On the other hand we find Giordano Bruno, the great Renais­sance martyr, speaking of Lull as "omniscient and almost divine," writing fantastically elaborate treatises on the Lullian Art, and teaching it to wealthy noblemen in Venice where it had become a fashionable craze. Later we find young Leibnitz fascinated by Lull's method. At the age of nineteen he wrote his Dissertio de arte combinatoria (Leipzig, 1666), in which he discovers in Lull's work the germ of a universal algebra by which all knowledge, including moral and metaphysical truths, can some day be brought within a single deductive system.' * "If controversies were to arise," Leibnitz declared in an oft-quoted passage, "there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit_-down to their slates, and to say to each other (with a friend to witness, if they liked) : Let us calculate"

 

These speculations of Leibnitz's have led many historians to credit Lull with having foreshadowed the development of modern symbolic logic and the empiricist's dream of the "unit of science." Is such credit deserved? Or was Lull's method little more than t e fantastic work of a gifted crank, as valueless as the geometric de­signs of medieval witchcraft? Before explaining and attempting to evaluate Lull's bizarre, now forgotten Art, it will perhaps be of interest to sketch briefly the extraordinary, almost unbelievable career of its inventor .²

 

Ramon Lull was born at Palma, probably in 1232. In his early teens he became a page in the service of King James I of Aragon and soon rose to a position of influence in the court. Although he married young and had two children, his life as a courtier was no­toriously dissolute. "The beauty of women, O Lord," he recalled at the age of forty, "has been a plague and tribulation to my eyes, for because of the beauty of women have I been forgetful of Thy great goodness and the beauty of Thy works."

 

The story of Lull's conversion is the most dramatic of the many picturesque legends about him, and second only to Saint Augus­tine's as a celebrated example of a conversion following a life of indulgence. It begins with Lull's adulterous passion for a eauti u and pious married woman who failed to respond to his overtures. One day as he was riding a horse down the street he saw the lady enter church for High Mass. Lull galloped into the cathedral after her, only to be tossed out by irate worshippers. Distressed by this scene, the lady resolved to put an end to Lull's campaign. She invited him to her chamber, uncovered the bosom that he had been praising in poems written for her, and revealed a breast partially

 

·          Superscript numbers designate references, to be found at the ends of chapters.

 

 

 

 

 

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4          Logic Machines, Diagrams, and Boolean Algebra

consumed by cancer. "See, Ramon," she exclaimed, "the foulness of this body that has won thy affection! How much better hadst thou done to have set thy love on Jesus Christ, of Whom thou mayest have a prize that is eternal!"

Lull retired in great shame and agitation. Shortly after this inci­dent, while he was in his bedroom composing some amorous lyrics, he was startled by a vision of Christ hanging on the Cross. On four later occasions, so the story goes, he tried to complete the verses, and each time was interrupted by the same vision. After a night of re­morse and soul searching, he hurried to morning confession as a penitent, dedicated Christian.

Lull's conversion was followed by a burning desire to win nothing less than the entire Moslem world for Christianity. It was an obses­sion that dominated the remainder of his life and eventually brought about his violent death. As the first necessary step in this ambitious missionary project, Lull began an intensive study of the Arabic language and theology. He purchased a Moorish slave who lived in his home for nine years, giving him instruction in the language. It is said that one day Lull struck the slave in the face after hearing him blaspheme the name of Christ. Soon thereafter the Moor re­taliated by attacking Lull with a knife. Lull succeeded in disarming him and the slave was jailed while Lull pondered the type of punish­ment he should receive. Expecting to be put to death, the Moor hanged himself with the rope that bound him.

Before this unfortunate incident, Lull had managed to finish writing, probably in Arabic, his first book, the Book of Contempla­tion. It is a massive, dull work of several thousand pages that seeks to prove by "necessary reasons" all the major truths of Christianity. Thomas Aquinas had previously drawn a careful distinction be­tween truths of natural theology that he believed could be estab­lished by reason, and truths of revelation that could be known only by faith. Lull found this distinction unnecessary. He believed that all the leading dogmas of Christianity, including the trinity and in­carnation, could be demonstrated by irrefutable arguments, although there is evidence that he regarded "faith" as a valuable aid in under­standing such proofs.

Lull had not yet discovered his Great Art, but the Book of Con­templation reveals his early preoccupation with a number symbol­ism that was characteristic of many scholars of his time. The work

The Ars Magna of Ramon Lull    5

is divided into five books in honor of the five wounds of Christ. Forty subdivisions signify the forty days Christ spent in the wilder­ness. The 366 chapters are designed to be read one a day, the last chapter to be consulted only in leap years. Each chapter has ten paragraphs (the ten commandments); each paragraph has three parts (the trinity), making a total of thirty parts per chapter (the thirty pieces of silver). Angles, triangles, and circles are occasionally introduced as metaphors. Of special interest to modern logicians is Lull's practice of using letters to stand for certain words and phrases so that arguments can be condensed to almost algebraic form. For example, in Chapter 335 he employs a notation of 22 symbols and one encounters passages such as this:

If in Thy three properties there were no difference . . . the demonstra­tion would give the D to the H of the A with the F and the G as it does with the E, and yet the K would not give significance to the H of any defect in the F or the G; but since diversity is shown in the demonstration that the D makes of the E and the F and the G with the I and the K, therefore the H has certain scientific knowledge of Thy holy and glorious Trinity.3

There are unmistakable hints of paranoid self-esteem in the value Lull places on his own work in the book's final chapter. It will not only prove to infidels that Christianity is the one true faith, he asserts, but it will also give the reader who follows its teaching a stronger body and mind as well as all the moral virtues. Lull ex­presses the wish that his book be "disseminated throughout the world," and he assures the reader that he has "neither place nor time sufficient to recount all the ways wherein this book is good and great."

These immodest sentiments are characteristic of most eccentrics who become the founders of cults, and it is not surprising to hear similar sentiments echoed by disciples of the Lullian Art in later centuries. The Old Testament was regarded by many Lullists as the work of God the Father, the New Testament, of God the Son, and the writings of Lull, of God the Holy Spirit. An oft-repeated jingle proclaimed that there had been three wise men in the world­Adam, Solomon, and Ramon:

Tres sabios hubo en el mundo, Addn, Solomon y Raymundo.

6          Logic Machines, Diagrams, and Boolean Algebra

_Lull's subsequent writings are extraordinarily numerous although many of them are short and there is much repetition of material and rehashing of old arguments. Some early authorities estimated that he wrote several thousand books. Contemporary scholars consider this an exaggeration, but there is good reason to think that more than two hundred of the works attributed to him are his (the alchemical writings that bear his name are known to be spurious). Most of his books are polemical, seeking to establish Christian doctrines by means of "necessary reasons," or to combat Averroism, Judaism, and other infidel doctrines. Some are encyclopedic sur­veys of knowledge such as his 1,300-page Tree of Science in which he finds himself forced to speak "of things in an abbreviated fashion." Many of his books are in the form of Socratic dialogues. Others are collections of terse aphorisms, such as his Book of Proverbs, a collection of some 6,000 of them. Smaller treatises, most of which concern the application of his Great Art, are devoted to almost every subject matter with which his contemporaries were concerned-astronomy, chemistry, physics, medicine, law, psychol­ogy, mnemonics, military tactics, grammar, rhetoric, mathematics, zoology, chivalry, ethics, politics.

Very few of these polemical and pseudo-scientific works have been translated from the original Catalan or Latin versions, and even in Spain they are now almost forgotten. It is as a poet and writer of allegorical romances that Lull is chiefly admired today by his countrymen. His Catalan verse, especially a collection of poems on The Hundred Names of God, is reported to be of high quality, and his fictional works contain such startling and imaginative concep­tions that they have become an imperishable part of early Spanish literature. Chief of these allegorical books is Blanquerna, a kind of Catholic Pilgrim's Progress.' The protagonist, who closely resembles the author, rises through various levels of church organization until he becomes Pope, only to abandon the office, amid much weeping of cardinals, to become a contemplative hermit.

The Book of the Lover and the Beloved, Lull's best known work, is contained within Blanquerna as the supposed product of the hermit's pen.' More than any other of Lull's works, this book makes use of the phrases of human love as symbols for divine love-a prac­tice as common in the Moslem literature prior to Lull's time as it was later to become common in the writings of Saint Theresa and

The Ars Magna of Ramon Lull    7

other Spanish mystics. Amateur analysts who enjoy looking for erotic symbols will find The Book of the Lover and the Beloved a fertile field. All of Lull's passionate temperament finds an outlet here in his descriptions of the intimate relationship of the lover (himself) to his Beloved (Christ).

In Lull's other great work of fantasy, Felix, or the Book of Marvels, we find him describing profane love in scenes of such repulsive realism that they would shock even an admirer of Henry Miller's fiction. It is difficult not to believe that Lull's postconversion attitude toward sex had much to do with his vigorous defense of the doctrine of the immaculate conception at a time when it was opposed by the Thomists and of course long before it became church dogma.

After Lull's illumination on Mount Randa, his conviction grew steadily that in his Art he had found a powerful weapon for the conversion of the heathen. The failure of the Crusades had cast doubt on the efficacy of the sword. Lull was convinced that ra­tional argument, aided by his method, might well become God's new means of spreading the faith. The remainder of his life was spent in restless wandering and feverish activity of a missionary and evangelical character. He gave up the large estate he had inherited from his father, distributing his possessions to the poor. His wife and children were abandoned, though he set aside funds for their welfare. He made endless pilgrimages, seeking the aid of popes and princes in the founding of schools and monasteries where his Great Art could be taught along with instruction in heathen languages. The teaching of Oriental languages to missionaries was one of Lull's dominant projects and he is justly regarded as the founder of Oriental studies in European education.

The esoteric character of his Art seems to have exerted a strong magic appeal. Schools and disciples grew so rapidly that in Spain the Lullists became as numerous as the Thomists. Lull even taught on several occasions at the great University of Paris-a signal honor for a man holding no academic degree of any kind. There is an amusing story about his attendance, when at the Sorbonne, of a class taught by Duns Scotus then a young man fresh from triumphs at Oxford. It seems that Scotus became annoyed by the old man in his audience who persisted in making signs of disagreement with what was being said. As a rebuke, Scotus asked him the exceedingly

8          Logic Machines, Diagrams, and Boolean Algebra

elementary question, "What part of speech is `Lord'?" Lull im­mediately replied, "The Lord is no art, but the whole." then pro-cee ed to stand and deliver a loud an

fections o God. The story is believable because Lull always be­haved as a man possessed by inspired, irrefutable truth.

On three separate occasions Lull made voyages to Africa to clash verbal swords with Saracen theologians and to preach his views in the streets of Moslem cities. On the first two visits he barely escaped with his life. Then at the age of eighty-three, his long beard snow white and his eyes burning with desire for the crown of martyrdom, he set sail once more for the northern shore of Africa. In 1315, on the streets of Bugia, he began expounding in a loud voice the errors of Moslem faith. The Arabs were understandably vexed, having twice ousted this stubborn old man from their country. He was stoned by the angry mob and apparently died on board a Genoese merchant ship to which his bruised body had been carried.6 A legend relates that before he died he had a vision of the American conti­nent and prophesied that a descendant (i.e., Columbus) of one of the merchants would some day discover the new world.

"           . no Spaniard since," writes Havelock Ellis (in a chapter on Lull in his The Soul of Spain, 1908), "has ever summed up in his own person so brilliantly all the qualities that go to the making of Spain. A lover, a soldier, something of a heretic, much of a saint, such has ever been the typical Spaniard." Lull's relics now rest in the chapel of the church of San Francisco, at Palma, where they are venerated as those of a saint, in spite of the fact that Lull has never been canonized.

In turning now to an examination of the Great Art itself,' it is impossible, perhaps, to avoid a strong sense of anticlimax. One wishes it were otherwise. It would be pleasant indeed to discover that Lull's method had for centuries been unjustly maligned and that by going directly to the master's own expositions one might come upon something of value that deserves rescue from the oblivion into which it has settled. Medieval scholars themselves sometimes voice such hopes. "We have also excluded the work of Raymond Lull," writes Philotheus Boehner in the introduction to his Medieval Logic, 1952, "since we have to confess we are not sufficiently familiar with his peculiar logic to deal with it adequately, though we suspect that it is much better than the usual evaluation

The Ars Magna of Ramon Lull    9

by historians would lead us to believe." Is this suspicion justified? Or shall we conclude with Etienne Gilson (History of Christian Philosophy in the Middle Ages, 1955) that when we today try to use Lull's tables "we come up against the worst difficulties, and one cannot help wondering whether Lull himself was ever able to use them"?

Essentially, Lull's method was as follows. In every knowledge, he believed, there are a small number of simple basic principles or categories that must be assumed without question. By exhausting all possible combinations of these categories we are able to explore all the knowledge that can be understood by our finite minds. To construct tables of possible combinations we call upon the aid of both diagrams and rotating circles. For example, we can list

'Ar P 14

Li FA

E

Figure 1.           Figure 2.

two sets of categories in two vertical columns (Figure 1), then ex­haust all combinations simply by drawing connecting lines as shown. Or we can arrange a set of terms in a circle (Figure 2), draw con­necting lines as indicated, then by reading around the circle we quickly obtain a table of two-term permutations.

A third method, and the one in which Lull took the greatest pride, is to place two or more sets of terms on concentric circles as shown in Figure 3. By rotating the inner circle we easily obtain a table of combinations. If there are many sets of terms that we wish to combine, this mechanical method is much more efficient than the others. In Lull's time these circles were made of parchment or metal and painted vivid colors to distinguish different subdivisions of terms. There is no doubt that the use of such strange, m

 

 

 

 

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10                                Logic Machines, Diagrams, and Boolean Algebra

ings that greatly intrigued men of little learning, anxious to find a short-cut method of mastering the intricacies of scholasticism. We find a similar appeal today in the "structural differential" invented by Count Alfred Korzybski to illustrate principles of general seman­tics. Perhaps there is even a touch of the same awe in the reverence with which some philosophers view 'symbolic logic as a tool of philosophical analysis.

Before going into the more complicated aspects of Lull's method, let us give one or two concrete illustrations of how Lull used his circles. The first of his seven basic "figures" is called A. The letter "A," representing God, is placed in the center of a circle. Around the circum erence, inside sixteen compartments (or "cam­erae" as Lull called them), we now place the sixteen letters from B through R (omitting J which had no existence in the Latin of the time). _These letters stand for sixteen divine attributes-B for goodness (bonitas), C for greatness (magnitudo), D for eternity (eternitas), and so on. By drawing connecting lines (Figure 4) we obtain 240 two-term permutations of the letters, or 120 different combinations that can be arranged in a neat triangular table as shown below.

Figure 3.

BC BD CD

MN MO MP MQ MR NO NP NQ NR OP OQ OR PQ PR QR

BE

BF

BG

Bli

BI

BK

BL

BM

BN

BO

BP

BQ

BR

CE

CF

CG

CH

CI

CK

CL

CM

CN

CO

CP

CQ

CR

DE

DF

DG

DH

DI

DK

DL

DM

DN

DO

DP

DQ

DR

EF

EG

EH

El

EK

EL

EM

EN

EO

EP

EQ

ER

FG

FH

FI

FK

FL

FM

FN

FO

FP

FQ

FR

GH

GI

GK

GL

GM

GN

GO

GP

GQ

GR

HI

HK

HL

HM

HN

HO

HP

HQ

HR

IK

IL

IM

IN

IO

IP

IQ

IR

KL

KM

KN

KO

KP

KQ

KR

LM

LN

LO

LP

LQ

LR

Figures 4 to 9, left to right, top to bottom. (From the Enciclopedia universal ilustrada, Bar. celona, 1923.)

12                                Logic Machines, Diagrams, and Boolean Algebra

Each of the above combinations tells us an additional truth about God. Thus we learn that His goodness is great (BC) and also eternal (BD), or to take reverse forms of the same pairs of letters, His greatness is good (CB) and likewise His eternity (DB). Re­flecting on these combinations will lead us toward the solution of many theological difficulties. For example, we realize that predesti­nation and free will must be combined in some mysterious way be­yond our ken; for God is both infinitely wise and infinitely just; therefore He must know every detail of the future, yet at the same time be incapable of withholding from any sinner the privilege of choosing the way of salvation. Lull considered this a demonstration "per aequiparantium," or by means of equivalent relations. Instead of connecting ideas in a cause-and-effect chain, we trace them back to a common origin. Free will an predestination sprout from equally necessary attributes of God, like two twigs growing on branches attached to t e trun o a single tree.

Lull's secon    gure concerns the soul and is designated by the letter S. Four differently colored squares are used to represent four different states of the soul. The blue square, with corners B, C, D, E, is a normal, healthy soul. The letters signify memory that remem­bers (B), intellect that knows (C), will that loves (D), and the union of these three faculties (E). The black square (FGHI) is the condition that results when the will hates in a normal fashion as, for example, when it hates sin. This faculty is symbolized by the letter H. F and G stand for the same faculties as B and C, and I for the union of F, G, and H. The red square (KLMN) denotes a con­dition of soul in which the memory forgets (K), the mind is ignorant (L), and the will hates in an abnormal fashion (M). These three degenerate faculties are united in N. The green square (OPQR) is the square of ambivalence or doubt. R is the union of a memory that retains and forgets (0), a mind that both knows and is ignorant (P), and a will that loves and hates (Q). Lull considered this last state the unhealthiest of the four. We now superimpose the four squares (Figure 5) in such a way that their colored corners form a circle of sixteen letters. This arrangement is more ingenious than one might at first suppose. For in addition to the four corner letters E, I, N, R, which are unions of the other three corners of their respective squares, we also find that the faculties O, P, and Q are unions of the three faculties that precede them as we move clock­

The Ars Magna of Ramon Lull                                                    13

wise around the figure. The circle of sixteen letters can now be ro­tated within a ring of compartments containing the same faculties to obtain 136 combinations of faculties.

It would be impossible and profitless to describe all of Lull's scores of other figures, but perhaps we can convey some notion of their complexity. His third figure, T, concerns relations between things. Five equilateral triangles of five different colors are super­imposed to form a circle of fifteen letters, one letter at each vertex of a triangle (Figure 6). As in the previous figure, the letters are in compartments that bear the same color as the polygon for which they mark the vertices. The meanings of the letters are: God, crea­ture, and o e ration (blue triangle);^- difference, similarity, con­trariety (green); be main , middle, end (red); m_aiority, equality, minority (yellow); affirmation, ne ag tion, and doubt (black). Ro­tating this circle within a ring bearing the same fifteen basic ideas (broken down into additional elements) gives us 120 combinations, excluding pairs of the same term (BB, CC, etc.) We are thus able to explore such topics as the beginning and end of God, differences and similarities of animals, and so on. Lull later found it necessary to add a second figure T, formed of five tinted triangles whose vertices stand for such concepts as before, after, superior, inferior, universal, particular, etc. This likewise rotated within a ring to pro­duce 120 combinations. Finally, Lull combined the two sets of concepts to make thirty in all. By placing them on two circles he obtained 465 different combinations.

Lull's fourth figure, which he called V, deals with the seven virtues and the seven deadly sins. The fourteen categories are ar­ranged alte a around a circle in red (sinful) and blue (virtu­ous) compartments (Figure 7). Drawing connecting lines, or ro­tating the circle within a similarly labeled ring, calls our attention to such questions as when it might be prudent to become angry, when lust is the result of slothfulness, and similar matters. Lull's figure X employs eight pairs of traditionally opposed terms, such as being (esse) and privation (privatio), arranged in alternate blue and green compartments (Figure 8). Figures Y and Z are undivided circles signifying, respectively, truth and falsehood. Lull used these letters occasionally in connection with other figures to denote the truth or falsehood of certain combinations of terms.

This by no means exhausts Lull's use of rotating wheels. Hardly

14                                   Logic Machines, Diagrams, and Boolean Algebra

a science or subject matter escapes his analysis by this method. He even produced a book on how preachers could use his Art to dis­cover new topics for sermons, supplying the reader with 100 sample sermons produced by his spinning wheels! In every case the tech­nique is the same: find the basic elements, then combine them me­chanically with themselves or with the elements of other figures. Dozens of his books deal with applications of the Art, introducing endless small variations of terminology and symbols. Some of these works are introductions to more comprehensive treatises. Some are brief, popular versions for less intellectual readers who find it hard to comprehend the more involved figures. For example, the cate­gories of certain basic figures are reduced from sixteen to nine (see Figure 9). These simpler ninefold circles are the ones encountered in the writings of Bruno, Kircher, and other Renaissance Lullists, in Hegel's description of the Art (Lectures on the History of Philos­ophy, Vol. 3), and in most modern histories of thought that find space for Lull's method. Two of Lull's treatises on his Art are written entirely in Catalan verse.

One of Lull's ninefold circles is concerned with objects of knowl­edge-God, angel, heaven, man, the imagination, the sensitive, the negative, the elementary, and the instrumental. Another asks the nine questions-whether? what? whence? why? how great? of what kind? when? where? and how? Many of Lull's books devote con­siderable space to questions suggested by these and similar circles. The Book of the Ascent and Descent of the Intellect, using a twelve­fold and a fivefold circle in application to eight categories (stone, flame, plant, animal, man, heaven, angel, God) considers such scientific posers as: Where does the flame go when a candle is put out? Why does rue strengthen the eyes and onions weaken them? Where does the cold go when a stone is warmed?

In another interesting work Lull uses his Art to explain to a her­mit the meaning of some of the Sentences of Peter Lombard. The book takes up such typical medieval problems as: Could Adam and Eve have cohabited before they ate their first food? If a child is slain in the womb of a martyred mother, will it be saved by a bap­tism of blood? How do angels speak to each other? How do angels pass from one place to another in an instant of time? Can God make matter without form? Can He damn Peter and save Judas? Can a fallen angel repent? In one book, the Tree of Science, over

The Ars Magna of Ramon Lull                                                   15

four thousand such questions are raised! Sometimes Lull gives the combination of terms in which the answer may be found, together with a fully reasoned commentary. Sometimes he merely indicates the figures to be used, letting the reader find the right combinations for himself. At other times he leaves the question entirely un­answered.

The number of concentric circles to be used in the same figure varies from time to time-two or three being the most common. The method reaches its climax in a varicolored metal device called the figura universalis which has no less than fourteen concentric circles! The mind reels at the number and complexity of topics that can be explored by this fantastic instrument.

Before passing on to an evaluation o   ull's method, it should be mentioned that he also frequently employed the diagrammatic de­vice of_ the_ tree to indicate subdivisions of genera and species. For Lull it was both an illustrative and a mnemonic device. His Prin­ciples of Medicine, for example, pictures his subject matter as a tree with four roots (the four humors) and two trunks (ancient and modern medicine). The trunks branch off into various boughs on which flowers bloom, eacH flower                                             aving a symbo i           meaning r_, _exercise, food, sleep, etc. . Colored triangles, squares, and other Lullian figures also are attached to the branches.

None of Lull's scientific writings, least of all his medical works, added to the scientific knowledge of his time. In such respects he was neither ahead nor behind his contemporaries. Alchemy and geomancy he rejected as worthless. Necromancy, or the art of com­municating with the dead, he accepted in a sense common in his day and still surviving in the attitude of many orthodox churchmen; miraculous results are not denied, but they are regarded as demonic in origin. Lull even used the success of necromancers as a kind of proof of the existence of God. The fallen angels could not exist, he argued, if God had not created them.

There is no doubt about Lull's complete acceptance of astrology. His so-called astronomical writings actually are astrological, show­ing how his circles can be used to reveal various favorable and un­favorable combinations of planets within the signs of the zodiac. In one of his books he applies astrology to medicine. By means of the Art he obtains sixteen combinations of the four elements (earth, air, fire, water) and the four properties (hot, cold, moist, dry).

16                                Logic Machines, Diagrams, and Boolean Algebra

These are then combined in various ways with the signs of the zodiac to answer medical questions concerning diet, evacuation, prepara­tion of medicines, fevers, color of urine, and so on.

There is no indication that Ramon Lull, the Doctor Illuminatus as he was later called, ever seriously doubted that his Art was the product of divine illumination. But one remarkable poem, the Desconort ("Disconso ateness"), suggests that at times he may have been tormented by the thought that possibly his Art was worthless. The poem is ingeniously constructed of sixty-nine stanzas, each consisting of twelve lines that end in the same rhyme. It opens with Lull's bitter reflections on his failure for the past thirty years to achieve any of his missionary projects. Seeking consolation in the woods, he comes upon the inevitable hermit and pours out to him the nature of his sorrows. He is a lonely, neglected man. People laugh at him and call him a fool. His great Art is ridiculed and ig­nored. Instead of sympathizing, the hermit tries to prove to Ramon that he d_ eserves this s ridicule. If his books on the Art are read by men "as fast as a cat that runs through burning coals," perhaps this is because the dogmas of the church cannot be demonstrated by reason. If they could be, then what merit would there be in believ­ing them? In addition, the hermit argues, if Lull's method is so valu­able, how is it that the ancient philosophers never thought of it?

 And if it truly comes from God, what reason has he to fear it will ever be lost? ,

     Lull relies so eloquently to these objections that we soon find

the hermit begging forgiveness or all he has said, offering to join Ramon in his labors, and even weeping because he had not learned the Art earlier in life!

Perhaps the most striking illustration of how greatly Lull valued his method is the legend of how he happened to join the third order of Franciscans. He had made all necessary arrangements for his first missionary trip to North Africa, but at the last moment, tor­mented by doubts and fears of imprisonment and death, he allowed the boat to sail without him. This precipitated a mental breakdown that threw him into a state of profound depression. He was carried into a Dominican church and while praying there he saw a light like a star and heard a voice speak from above: "Within this order thou shalt be saved." Lull hesitated to join the order because he knew the Dominicans had little interest in his Art whereas the

The Ars Magna of Ramon Lull                                                   1]

Franciscans had found it of value. A second time the voice spoke from the light, this time threateningly: "And did I not tell thee that only in the order of the Preachers thou wouldst find salvation?" Lull finally decided it would be better to undergo personal damna­tion than risk the loss of his Art whereby others might be saved Ignoring the vision, he joined the Franciscans.

It is clear from Lull's writings that he thought of his method as possessin many values. The diagrams and circles aid the under­standing by making it easy to visualize the elements of a given argument. They have considerable mnemonic value, an aspect of his Art that appealed strongly to Lull's Renaissance admirers. They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combina­tions start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power.

Lull did not, however, regard his method as a substitute for the formal logic of Aristotle and the schoolmen. He was thoroughly familiar with traditional logic and his writings even include the popular medieval diagrams of immediate inference and the various syllogistic figures and moods. He certainly did not think that the mere juxtaposition of terms provided in themselves a proof by "necessary reasons." He did think, however, that by the mechanical combination of terms one could discover the necessary building blocks out of which valid arguments could then be constructed. Like his colleagues among the schoolmen, he was convinced that each branch of knowledge rested on a relatively few, self-evident princi­Ales w ich formed the structure o a now e ge in  e same way

that geometrical theorems were forme   out of basic axioms. It was natural for him to suppose that by ex austing the combinations of such principles one might thereby explore all possible structures of truth and so obtain universal knowledge.

There is a sense, of course, in which Lull's method of explora­tion does possess a formal deductive character. If we wish to exhaust the possible combinations of given sets of terms, then Lull's method

obviously will do this for us in an irrefutable way. Considered mathe­matically, the technique is sound, though even in its day it was es-

18                                Logic Machines, Diagrams, and Boolean Algebra

sentially trivial. Tabulating combinations of terms was certainly a familiar process to mathematicians as far back as the Greeks, and it would be surprising indeed if no one before Lull had thought of using movable circles as a device for obtaining such tables. Lull's mistake, in large part a product of the philosophic temper of his age, was to suppose that his combinatorial method had useful ap­plication to subject matters where today we see clearly that it does not apply. Not only is there a distressing lack of "analytic" structure in areas of knowledge outside of logic and mathematics, there is not even agreement on what to regard as the most primitive, "self-evi­dent" principles in any given subject matter. Lull naturally chose for his categories those that were implicit in the dogmas and opin­ions he wished to establish. The result, as Chesterton might have said, was that Lull's circles led him in most cases into roofs tat were circular. Other schoolmen were of course often guilty of question begging, but it was Lull