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Tag: Tabula Generalis

May 2, 2016

Eco: The Arbor Scientarium, 2

Ramon Llull, Arbor Scientiae, Rome, 1295

Ramon Llull, Arbor Scientiae, Rome, 1295. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

“Between the first and last versions of his art, Lull’s thought underwent a long process of evolution (described by Carreras y Artau 1939: I, 394), in order to render his art able to deal not only with theology and metaphysics, but also with cosmology, law, medicine, astronomy, geometry and psychology.

Increasingly, the art became a means of treating the entire range of knowledge, drawing suggestions from the numerous medieval encyclopedias, and anticipating the encyclopedic dreams of the Renaissance and the baroque.

All this knowledge, however, needed to be ordered hierarchically. Because they were determinations of the first cause, the dignities could be defined circularly, in reference to themselves; beyond the dignities, however, began the ladder of being. The art was designed to permit a process of reasoning at every step.

The roots of the Tree of Science were the nine dignities and the nine relations. From here, the tree then spread out into sixteen branches, each of which had its own, separate tree. Each one of the sixteen trees, to which there was dedicated a particular representation, was divided into seven parts–roots, trunk, major branches, lesser branches, leaves, fruits and flowers.

Eight of the trees clearly corresponded to eight of the subjects of the tabula generalis: these are the Arbor elementalis, which represents the elementata, that is, objects of the sublunary world, stones, trees and animals composed of the four elements; the Arbor vegetalis;  the Arbor sensualis; the Arbor imaginalis, which represents images that replicate in the mind whatever is represented on the other trees; the Arbor humanalis et moralis (memory, intellect and will, but also the various sciences and arts); the Arbor coelestialis (astronomy and astrology); the Arbor angelicalis; and the Arbor divinalis, which includes the divine dignities.

To this list are added another eight: the Arbor mortalis (virtues and vices); the Arbor eviternalis (life after death); the Arbor maternalis (Mariology); the Arbor Christianalis (Christology); the Arbor imperialis (government); the Arbor apostolicalis (church); the Arbor exemplificalis (the contents of knowledge); and the Arbor quaestionalis, which contains four thousand questions on the various arts.

To understand the structure of these trees, it is enough to look at only one–the Arbor elementalis. Its roots are the nine dignities and nine relations. Its trunk represents the conjoining of these principles, out of which emerges the confused body of primordial chaos which occupies space.

In this are the species of things and their dispositions. The principle branches represent the four elements (earth, air, fire and water) which stretch out into the four masses which are made from them (the seas and the lands).

The leaves are the accidents. The flowers are the instruments, such as hands, feet and eyes. The fruits represent individual things, such as stone, gold, apple, bird.

Calling this a “forest” of trees would be an improper metaphor: the trees overlay one another to rise hierarchically like the peaked roof of a pagoda. The trees at the lower levels participate in those higher up.

The vegetable tree, for example, participates in the tree of elements; the sensual tree participates in the first two; the tree of imagination is built up out of the first three, and it forms the base from which the next tree, the human one, will arise (Llinares 1963: 211-2).

The system of trees reflects the organization of reality itself; it represents the great chain of being the way that it is, and must metaphysically be. This is why the hierarchy constitutes a system of “true” knowledge.

The priority of metaphysical truth over logical validity in Lull’s system also explains why he laid out his art the way he did: he wished his system to produce, for any possible argument, a middle term that would render that argument amenable to syllogistic treatment; having structured the system for this end, however, he proceeded to discard a number of well-formed syllogisms which, though logically valid, did not support the arguments he regarded as metaphysically true.

For Lull, the significance of the middle term of the syllogism was thus not that of scholastic logic. Its middle term served to bind the elements of the chain of being: it was a substantial, not a formal, link.

If the art is a perfect language, it is so only to the extent to which it can speak of a metaphysical reality, of a structure of being which exists independently of it. The art was not a mechanism designed to chart unknown universes.

In the Catalan version of his Logica Algazelis, Lull writes, “De la logic parlam tot breau–car a parlor avem Deu.” (“About logic we will be brief, for it is to talk about God”).

Much has been written about the analogy between Lull’s art and the kabbala. What distinguishes kabbalistic thought from Lull’s is that, in the kabbala, the combination of the letters of the Torah had created the universe rather than merely reflected it.

The reality that the kabbalistic mystic sought behind these letters had not yet been revealed; it could be discovered only through whispering the syllables as the letters whirled.

Lull’s ars combinatoria, by contrast, was a rhetorical instrument; it was designed to demonstrate what was already known, and lock it for ever in the steely cage of the system of trees.

Despite all this, the art might still qualify as a perfect language if those elementary principles, common to all humanity, that it purported to expound really were universal and common to all peoples.

As it was, despite his effort to assimilate ideas from non-Christian and non-European religions, Lull’s desperate endeavor failed through its unconscious ethnocentrism. The content plane, the universe which his art expounded, was the product of the western Christian tradition.

It could not change even though Lull translated it into Arabic or Hebrew. The legend of Lull’s own agony and death is but the emblem of that failure.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 67-9.

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May 2, 2016

Eco: The Alphabet and the Four Figures, 3

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Jonathan Swift, Gulliver’s Travels, 1892 George Bell and Sons edition, Project Gutenberg. Also see Jonathan Swift, Gulliver’s Travels, A.J. Rivero, ed., New York: W.W.Norton, 2001, Part III, chapter 5. Cited in Bethany Nowviskie, “Ludic Algorithms,” in Kevin Kee, ed., Pastplay: Teaching and Learning History with Technology, Ann Arbor, MI: University of Michigan Press, 2014. 

“It follows that Lull’s art is not only limited by formal requirements (since it can generate a discovery only if one finds a middle term for the syllogism); it is even more severely limited because the inferences are regulated not by formal rules but rather by the ontological possibility that something can be truly predicated of something else.

The formal rules of the syllogism would allow such arguments as “Greed is different from goodness — God is greedy — Therefore God is different from goodness.” Yet Lull would discard both the premises and the conclusion as false.

The art equally allows the formulation of the premise “Every law is enduring,” but Lull rejects this as well because “when an injury strikes a subject, justice and law are corrupted” (Ars brevis, quae est de inventione mediorum iuris, 4.3a).

Given a proposition, Lull accepts or rejects its logical conversion, without regard to its formal correctness (cf. Johnston 1987: 229).

Nor is this all. The quadruples derived from the fourth figure appear in the columns more than once. In Ars magna the quadruple BCTB, for example, figures seven times in each of the first seven columns.

In V, 1, it is interpreted as “Whether there exists some goodness so great that it is different,” while in XI, 1, applying the rule of logical obversion, it is read as “Whether goodness can be great without being different”–obviously eliciting a positive response in the first case and a negative one in the second.

Yet these reappearances of the same argumentative scheme, to be endowed with different semantic contents, do not bother Lull. On the contrary, he assumes that the same question can be solved either by any of the quadruples from a particular column that generates it, or from any of the other columns!

Such a feature, which Lull takes as one of the virtues of his art, represents in fact its second severe limitation. The 1,680 quadruples do not generate fresh questions, nor do they furnish new proofs.

They generate instead standard answers to an already established set of questions. In principle, the art only furnishes 1,680 different ways of answering a single question whose answer is already known.

It cannot, in consequence, really be considered a logical instrument at all. It is, in reality, a sort of dialectical thesaurus, a mnemonic aid for finding out an array of standard arguments able to demonstrate an already known truth.

As a consequence, any of the 1,680 quadruples, if judiciously interpreted, can yield up the correct answer to the question for which it is adapted.

See, for instance, the question “Whether the world is eternal” (“Utrum mundus sit aeternus“). Lull already knew the answer: negative, because anyone who thought the world eternal would fall into the Averroist error.

Note, however, that the question cannot be generated directly by the art itself; for there is no letter corresponding to world. The question is thus external to the art.

In the art, however, there does appear a term for eternity, that is, D; this provides a starting point.

In the second figure, D is tied to the relative principle contrarietas or opposition, as manifested in the opposition of the sensible to the sensible, of the intellectual to the sensible, and of the intellectual to the intellectual.

The same second figure also shows that D forms a triangle with B and C. The question also began with utrum, which appears at B under the heading Questiones in the tabula generalis. This constitutes a hint that the solution needs to be sought in the column in which appear B, C and D.

Lull says that “the solution to such a question must be found in the first column of the table;” however, he immediately adds that, naturally, “it could be found in other columns as well, as they are all bound to each other.”

At this point, everything depends on definitions, rules, and a certain rhetorical legerdemain in interpreting the letters. Working from the chamber BCDT (and assuming as a premise that goodness is so great as to be eternal), Lull deduces that if the world were eternal, it would also be eternally good, and, consequently, there would be no evil.

“But,” he remarks, “evil does exist in the world as we know by experience. Consequently we must conclude that the world is not eternal.” This negative conclusion, however, is not derived from the logical form of the quadruple (which has, in effect, no real logical form at all), but is merely based on an observation drawn from experience.

The art may have been conceived as the instrument to use universal reason to show the Averroist Muslims the error of their ways; but it is clear that unless they already shared with Lull the “rational” conviction that the world cannot be eternal, they are not going to be persuaded by the art.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 62-4.

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May 2, 2016

Eco: The Alphabet and the Four Figures, 2

Raymond Llull, Combinations, Strasbourg ed 1598

Umberto Eco, The Search for the Perfect Language, 1995, pg. 60. Figure 4.2, a page of combinations from the Strasbourg edition of the Ars Magna of Raymond Llull, 1598. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

Taken in groups of 3, 9 elements generate 84 combinations–BCD, BCE, CDE, etc. If, in his Ars breu and elsewhere, Lull sometimes speaks of 252 (84*3) combinations, it is because to each triple can be assigned three questions, one for each of the letters of the triple (see also the Jesuit Athanasius Kircher, Ars magna sciendi, p. 14.

ArsMagnaSciendi1

Athanasius Kircher, Ars Magna Sciendi, Amsterdam, 1669. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

Each triple further generates a column of 20 combinations (giving a table of 20 rows by 84 columns) because Lull transforms the triples into quadruples by inserting the letter T. In this way, he obtains combinations like BCDT, BCTB, BTBC, etc. (See examples in figure 4.2, at the top of this page).

The letter T, however, plays no role in the art; it is rather a mnemonic artifice. It signifies that the letters that precede it are to be read as dignities from the first figure, while those that follow it are to be read as relative principles as defined in the second figure.

Thus, to give an example, the quadruple BCTC must be read: B (= goodness) + C (= greatness) and therefore (switching to the second figure) C (=  concordance).

Looking at the tabula generalis, we further notice that combinations with an initial B take the question utrum, those with an initial C take quid, etc. This produces from BCTC the following reading: “Whether goodness is great inasmuch as it contains in itself concordant things.”

This produces a series of quadruples which seem, at first sight, embarrassing: the series contains repetitions. Had repetitions been permissible, there would have been 729 triples instead of 84.

The best solution to the mystery of these repetitions is that of Platzek (1953-4: 141). He points out that, since, depending on whether it precedes or follows the T, a letter can signify either a dignity or a relation, each letter has, in effect, two values.

Thus–given the sequence BCTB–it should be read as BCb. The letters in upper case would be read as dignities, and the one in lower case as a relation. It follows that, in his 84 columns, Lull was not really listing the combinations for three letters but for six. Six different elements taken three at a time give 20 permutations, exactly as many appear in each column.

The 84 columns of 20 quadruples each yield 1,680 permutations. This is a figure obtained by excluding inversions of order.

At this point, however, a new question arises. Given that all these 1,680 quadruples can express a propositional content, do they all stand for 1,680 valid arguments as well?

ArsMagnaSciendi

Athanasius Kircher, Ars Magna Sciendi sive Combinatoria, Amsterdam, 1669. Frontispiece. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

Not at all, for not every sequence generated by the art is syllogistically valid. Kircher, in his Ars magna sciendi, suggests that one must deal with the resulting sequences as if they were anagrams: one starts by forming a complete list of all the possible arrangements of the letters of a particular word, then discards those that do not correspond to other existing words.

The letters of the Latin word ROMA, for example, can be combined in 24 different orders: certain sequences form acceptable Latin words, such as AMOR, MORA, RAMO; others, however, such as AOMR, OAMR, MRAO, are nonsense, and are, as it were, thrown away.

Lull’s own practice seems to suppose such a criterion. He says, for example, in his Ars magna, segunda pars principalis that in employing the first figure, it is always possible to reverse subject and predicate (“Goodness is great” / “Greatness is good”).

It would not, however, be possible to reverse goodness and angel, for while angel participates in goodness, goodness does not participate in angel, since there are beings other than angels which are good.

In other words, angel entails goodness but not vice versa. Lull also adds that the combination “Greed is good” is inherently unacceptable as well. Whoever wishes to cultivate the art, Lull says, must be able to know what is convertible and what is not.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 60-2.

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May 1, 2016

Eco: The Alphabet and the Four Figures

Illuminati_sacre_pagine_p.fessoris_amplissimi_magistri_Raymundi_Lull._Ars_magna,_generalis_et_vltima_-_quarucunq3_artium_(et)_scientiarum_ipsius_Lull._assecutrix_et_clauigera_-_(et)_ad_eas_aditum_(14591005828)

Raymond Llull (1232-1316), Ars magna, segunda figura, generalis et ultima, 1517, held in the Getty Research Institute and digitized by that institution in collaboration with the Internet Archive, generously posted on archive.org. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.

 

“The ars combinatoria of Lull employs an alphabet of nine letters–B to K, leaving out J–and four figures (see figure 4.1). In a tabula generalis that appears in several of his works, Lull set out a table of six groups of nine entities, one for each of the nine letters.

The first group are the nine absolute principles, or divine dignities, which communicate their natures to each other and spread throughout creation.

After this, there are nine relative principles, nine types of question, nine subjects, nine virtues and nine vices.

Lull specifies (and this is an obvious reference to Aristotle’s list of categories) that the nine dignities are subjects of predication, while the other five series are predicates. We shall see that subject and predicate are sometimes allowed to exchange their roles, while in other cases variations of order are not considered as pertinent.

First figure. This traces all the possible combinations between the dignities, thus allowing predications such as “Goodness [bonitas] is great,” “Greatness [magnitudo] is glorious,” etc.

Since the dignities are treated as nouns when they appear as a predicate, the lines connecting them can be read in both directions. The line connecting magnitudo and bonitas can, for example, be read as both “Greatness is good” and “Goodness is great.” This explains why 36 lines produce 72 combinations.

The first figure is designed to allow regular syllogisms to be inferred. To demonstrate, for example, that goodness can be great, it is necessary to argue that “all that is magnified by greatness is great–but goodness is what is magnified by greatness–therefore goodness is great.”

The first table excludes self-predications, like BB or CC, because, for Lull, there is no possibility of a middle term in an expression of the type “Goodness is good” (in Aristotelian logic, “all As are B–C is an A–therefore C is a B” is a valid syllogism because, following certain rules, the middle term A is so disposed to act as the, as it were, bond between B and C).

Second figure. This serves to connect the relative principles with triples of definitions. They are the relations connecting the divine dignities with the cosmos. Since it is intended merely as a visual mnemonic that helps to fix in the mind the various relations between different types of entity, there is no method of combination associated with the second figure.

For example, difference, concordance and opposition (contrarietas) can each be considered in reference to (1) two sensible entities, such as a plant and a stone, (2) a sensible and an intellectual entity, like body and soul, and (3) two intellectual entities, like the soul and an angel.

Third figure. Here Lull displayed all possible letter pairings. The figure contains 36 pairs inserted in what Lull calls the 36 chambers. The figure makes it seem that he intended to exclude inversions.

Yet, in reality, the figure does contemplate inversions in order, and thus the number of the chambers is virtually 72 since each letter is permitted to function as either subject or predicate (“Goodness is great” also gives “Greatness is good:” Ars magna, VI, 2).

Having established the combinations, Lull proceeds to what he calls the “evacuation of the chambers.” Taking, for example, chamber BC, we read it first according to the first figure, obtaining goodness and greatness (bonitas and magnitudo); then according to the second figure, obtaining difference and concordance, (differentia and concordantia: Ars magna, II, 3).

From these two pairs we derive 12 propositions: “Goodness is great,” “Difference is great,” Goodness is different,” “Goodness is different,” “Difference is good,” “Goodness is concordant,” “Difference is concordant,” “Greatness is good,” “Concordance is good,” “Greatness is different,” “Concordance is different,” “Greatness is concordant,” and “Concordance is great.”

Going back to the tabula generalis in figure 4.1, we find that, under the next heading, Questiones, B and C  are utrum (whether) and quid (what). By combining these 2 questions with the 12 propositions we have just constructed, we obtain 24 questions, like “Whether goodness is great?,” or “What is a great goodness?” (see Ars magna, VI, 1).

In this way, the third figure generates 432 propositions and 864 questions–at least in theory. In reality, there are 10 additional rules to be considered (given in Ars magna, VI, iv).

For the chamber BC, for example, there are the rules B and C. These rules depend on the theological definition of the terms, and on certain argumentative constraints which have nothing to do with the rules of combination.

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Quarta figura, fourth figure.

Fourth figure. This is the most famous of the figures, and the one destined to have the greatest influence on subsequent tradition. In this figure, triples generated by the nine elements are considered.

In contrast to the preceding figures, which are simply static diagrams, the fourth figure is mobile. It is a mechanism formed by three concentric circles, of decreasing size, inserted into each other, and held together usually by a knotted cord.

If we recall that in the Sefer Yezirah the combination of the letters was visually represented by a wheel or a spinning disc, it seems probable that Lull, a native of Majorca, has been influenced here by the kabbalistic tradition that flourished in his time in the Iberian peninsula.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 56-60.

 

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May 1, 2016

Eco: The Ars Magna of Raymond Lull

Raymond Lull, Tabula Generalis, pg. 57, Eco, Search for a Perfect Language, 1995

Raymond Lull (1232-1316), Tabula Generalis, figure 4.1, Lull’s Alphabet, from Umberto Eco, The Search for a Perfect Language, Blackwell, Oxford, 1995, pg. 57. 

“A near contemporary of Dante, Ramòn Llull (Latinized as Lullus and Anglicized as Lull–and sometimes as Lully) was a Catalan, born in Majorca, who lived probably between 1232 (or 1235) and 1316.

Majorca during this period was a crossroads, an island where Christian, Jewish and Arab cultures all met; each was to play a role in Lull’s development. Most of his 280 known works were written initially in Arabic or Catalan (cf. Ottaviano 1930).

Lull led a carefree early life which ended when he suffered a mystic crisis. As a result, he entered the order of Tertian friars.

It was among the Franciscans that all of the earlier strands converged in his Ars magna, which Lull conceived as a system for a perfect language with which to convert the infidels. The language was to be a universal; it was to be articulated at the level of expression in a universal mathematics of combination; its level of content was to consist of a network of universal ideas, held by all peoples, which Lull himself would devise.

St. Francis had already sought to convert the sultan of Babylonia, and the dream of establishing universal concordance between differing races was becoming a recurrent theme in Franciscan thought. Another of Lull’s contemporaries, the Franciscan Roger Bacon, foresaw that contact with the infidels (not merely Arabs, but also Tartars) would require study of foreign languages.

The problem for him, however, was not that of inventing a new, perfect language, but of learning the languages that the infidels already spoke in order to convert them, or, failing that, at least to enrich Christian culture with a wisdom that the infidels had wrongfully appropriated (“tamquam ab iniustis possessoribus“).

The aims and methods of Lull and Bacon were different; yet both were inspired by ideals of universality and of a new universal crusade based on peaceful dialogue rather than on arms.

In this utopia the question of language played a crucial role (cf. Alessio 1957). According to legend, Lull was to die martyred at the hands of the Saracens, to whom he had appeared, armed with his art, believing it to be an infallible means of persuasion.

Lull was the first European philosopher to write doctrinal works in the vulgar tongue. Some are even in popular verses, so as to reach readers who knew neither Latin nor Arabic: “per tal che hom puscha mostrar / logicar e philosophar / a cels que win saben lati / ni arabichi” (Compendium, 6-9).

His art was universal not merely in that it was designed to serve all peoples, but also in that it used letters and figures in a way (allegedly) comprehensible even to illiterates of any language.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 53-4.

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