Figure 10.4
Once the combinatory system has been set in motion, we proceed to what Llull calls the “evacuation of the chambers.” For example, taking the BC chamber, and referring to the Tabula Generalis, we first read chamber BC according to the Absolute Principles and we obtain Bonitas est magna, then we read it according to the Relative Principles and we obtain Differentia est concordans (Ars magna II, 3). In this way we obtain twelve propositions: Bonitas est magna, Diffferentia est magna, Bonitas est differens, Differentia est bona, Bonitas est concordans, Differentia est concordans, Magnitudo est bona, Concordantia est bona, Magnitudo est differens, Concordantia est differens, Magnitudo est concordans, Concordantia est magna. Returning to the Tabula Generalis and assigning to B and C the corresponding questions (utrum or “whether” and quid or “what”) with their respective answers, we can derive, from the twelve propositions, twenty-four questions (of the type Utrum Bonitas sit magna? [Whether Goodness is great?] and Quid est Bonitas magna? [What is great Goodness?]) (see Ars magna VI, 1).
QUARTA FIGURA. In this case the mechanism is mobile, in the sense that we have three concentric circles decreasing in circumference, placed one on top of the other, and usually held together at the center with a knotted string. Revolving the smaller inner circles, we obtain triplets (see Figure 10.5).
These are produced from the combination of nine elements into groups of three, without the same element being repeated twice in the same triplet or chamber. Llull, however, adds to each triplet the letter t—an operator by which it is established that the letters that precede are to be read with reference to the first column of the Tabula Generalis, as Principles or Dignities, whereas those that follow are to be read as Relative Principles. Since the t changes the meaning of the letters, as Platzeck (1954: 140–143) explains, it is as if Llull were composing his triplets by combining, not three, but six elements (not merely BCD, for instance, but BCDbcd). The combinations of six elements into groups of three give (according to the rules of the combinatory system) twenty chambers.
Figure 10.5
Consider now the reproduction in Figure 10.6 of the first of the tables elaborated by Llull to exploit to the full the possibilities of his fourth figure (each table being composed of columns of twenty chambers each). In the first column we have BCDbcd, in the second BCEbce, in the third BCFbcf, and so on and so forth, until we have obtained eighty-four columns and hence 1,680 chambers.
If we take, for instance, the first column of the Tabula Generalis, the chamber bctc (or BCc) is to be read as b = bonitas, c = magnitudo, c = concordantia. Referring to the Tabula Generalis, the chambers that begin with b correspond to the first question (utrum), those that begin with c to the second question (quid), and so on. As a result, the same chamber bctc (or BCc) is to be read as Utrum bonitas in tantum sit magna quod contineat in se res concordantes et sibi coessentiales (“Whether goodness is great insofar as it contains within it things in accord with it and coessential to it”).
Figure 10.6
Quite apart from a certain arbitrariness in “evacuating the chambers,” in other words, in articulating the reading of the letters of the various chambers into a discourse, not all the possible combinations (and this observation is valid for all the figures) are admissible. After describing his four figures in fact, Llull prescribes a series of Definitions of the various terms in play (of the type Bonitas est ens, ratione cujus bonum agit bonum [“Goodness is something as a result of which a being that is good does what is good”]) and Necessary Rules (which consist of ten questions to which, it should be borne in mind, the answers are provided), so that such chambers generated by the combinatory system as contradict these rules must not be taken into consideration.
This is where the first limitation of the Ars surfaces: it is capable of generating combinations that right reason must reject. In his Ars magna sciendi, Athanasius Kircher will say that one proceeds with the Ars as one does when working out combinations that are anagrams of a word: once one has obtained the list, one excludes all those permutations that do not make up an existing word (in other words, twenty-four permutations can be made of the letters of the Italian word ROMA, but, while AMOR, MORA, ARMO, and RAMO make sense in Italian and can be retained, meaningless permutations like AROM, AOMR, OAMR, or MRAO can, so to speak, be cast aside). In fact Kircher, working with the fourth figure, produces nine syllogisms for each letter, even though the combinatory system would allow him more, because he excludes all the combinations with an undistributed middle, which precludes the formation of a correct syllogism.10
This is the same criterion followed by Llull, when he points out, for example, in Ars magna, Secunda pars principalis, apropos of the various ways in which the first figure can be used, that the subject can certainly be changed into the predicate and vice versa (for instance, Bonitas est magna and Magnitudo est bona), but it is not permitted to interchange Goodness and Angel. We interpret this to mean that all angels are good, but that an argument that asserts the “since all angels are good and Socrates is good, then Socrates is an angel” is unacceptable. In fact we would have a syllogism with an unquantified middle.
But the combinatory system is not only limited by the laws of the syllogism. The fact is that even formally correct conversions are only acceptable if they predicate according to the truth criteria established by the rules—which rules, it will be recalled, are not logical in nature but philosophical and theological (cf. Johnston 1987: 229). Bäumker (1923: 417–418) realized that the aim of the ars inveniendi (or art of invention) was to set up the greatest possible number of combinations among concepts already provided, and to draw from them as a consequence all possible questions, but only if the resulting questions could stand up to “an ontological and logical examination,” permitting us to discriminate between correct combinations and false propositions. The artist, says Llull, must know what is convertible and what is not.
Furthermore, among the quadruplets tabulated by Llull there are—by virtue of the combinatory laws—a number of repetitions. See, for example, in the columns reproduced in Figure 10.6, the chamber btch, which recurs in the second place in each of the first seven columns, and which in the Ars magna (V, 1) is translated as utrum sit aliqua bonitas in tantum magna quod sit differens (“whether a certain goodness is great insofar as it is different”) and in XI, 1, by the rule of obversion, as utrum bonitas possit esse magna sine distinctione (“whether goodness can be great without being different”)—permitting a positive answer in the first case and for a negative one in the second. The fact that the same demonstrative schema should appear several times does not seem to worry Llull, and the reason is simple.
He assumes that the same question can be resolved both by each of the quadruplets in the single column that generates it and by all the other columns. This characteristic, which Llull sees as one of the virtues of the Ars, signals instead its second limitation: the 1,680 quadruplets do not generate original questions and do not provide proofs that are not the reformulation of previously tried and tested arguments. Indeed, in principle the Ars allows us to answer in 1,680 different ways a question to which we already know the answer—and it is not therefore a logical tool but a dialectical tool, a way of identifying and remembering all the useful ways to argue in favor of a preestablished thesis. To such a point that there is no chamber that, duly interpreted, cannot resolve the question to which it is adapted.
All of the above-mentioned limitations become evident if we consider the dramatic question utrum mundus sit aeternus, whether the world is eternal. This is a question to which Llull already knows the answer, which is negative, otherwise we would fall into the same error as Averroes. Seeing that the term eternity is, so to speak, “explicated” in the question, this allows us to place it under the letter D in the first column of the Tabula Generalis (see Figure 10.1). However, the D, as we saw in the second figure, refers to the contrariety between sensitive and sensitive, intellectual and sensitive, and intellectual and intellectual. If we observe the second figure, we see that the D is joined by the same triangle to B and C. Moreover, the question begins with utrum, and, on the basis of the Tabula Generalis, we know that the interrogative utrum refers to B. We have therefore found the column in which to look for the arguments: it is the one in which B, C, and D all appear.
At this point all that is needed to interpret the letters is a good rhetorical ability, and, working on the BCDT chamber, Llull draws the conclusion that, if the world were eternal, since we already know that Goodness is eternal, it should produce an Eternal Goodness, and therefore evil would not exist. But, Llull observes, “evil does exist in the world, as we know from experience. Therefore we
