Llull, Leibniz, and the Logic of Discovery This version of the paper corrects typographical errors in "Llull and Leibniz: The Logic of Discovery," Catalan Review IV, Nos. 1–2 (July–December 1990), 75–83. John R. Welch Department of Philosophy Saint Louis University – Madrid Campus 28003 Madrid Spain jwelch7@slu.edu Abstract: Llull and Leibniz both subscribed to conceptual atomism: the belief that the majority of concepts are compounds constructed from a relatively small number of primitive concepts. Llull worked out techniques for finding the logically possible combinations of his primitives, but Leibniz criticized Llull's execution of these techniques. This paper argues that Leibniz was right about things being more complicated than Llull thought but that he was wrong about the details. The paper attempts to correct these details. Keywords: Llull, Ars magna, conceptual atomism, logic of discovery, Leibniz A logic conceived as a formal calculus is one thing; the same logic applied is another. When logic is applied, why is it applied? To what use is it put? We are perhaps most accustomed to answer in terms of justification; given a truth, logic may show us why it is true. But there is another answer, prominent from the early modern period down to the present day, which is that logic can also be used to discover. One of the pioneers of this point of view was Leibniz. His efforts to forge a logic of discovery were informed by a kind of conceptual atomism, a belief that the majority of concepts are compounds constructed from a relatively small number of primitives. He claimed that "a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of the alphabet and an analysis of the words made from them."1 Such an alphabet, as Leibniz says, could be used to judge and discover. Not only would it serve to demonstrate propositions already held to be true-a logic of justification-it could also be used to invent or discover new truths-a logic of discovery. What would a logic of discovery look like? Leibniz's answer, his ars inveniendo, was to have two parts: one combinatorial, to generate questions, and one analytic, to answer them.2 How closely Leibniz's thinking here parallels the Ars magna (hereafter the Art) of Ramón Llull is not sufficiently known. The parallel is not accidental, though the historical links between Llull, Lullism, and Leibniz are by no means simple.3 As a young man of twenty, Leibniz was both fascinated and repelled by the Art. The fascination came from Llull's having anticipated some of his leading ideas. The repulsion came from Llull's mathematical naiveté, a consequence of having lived some four centuries before the developments in combinatorial mathematics upon which Leibniz hoped to base his own logic of discovery. (Leibniz, in fact, was the first to use the term 'combinatorial' in its modern sense.) Both of Leibniz's reactions linger in the objectives of this note: to outline what it was about the Art that fascinated Leibniz, first of all; and, after correcting a mistake in Leibniz's critique of Llull, to extend it in a new direction. Llull anticipated Leibniz in the belief that human reason was a matter of combining a few primitive notions. To specify these notions, Llull devised a conceptual alphabet which, he believed, limned the basic structure of the universe. In the later, ternary phase of the Art (ca. 1290–1308), the alphabet takes the following form.4 2 Fig. A Fig. T Questions Subjects Virtues Vices and Rules B goodness difference whether? God justice avarice C greatness concordance what? angel prudence gluttony D eternity* contrariety of what? heaven fortitude lust E power beginning why? man temperance pride F wisdom middle how much? imaginative faith accidie G will end of what kind? sensitive hope envy H virtue majority when? vegetative charity ire I truth equality where? elementative patience lying K glory minority how? and with what? instrumentative pity inconstancy * or duration Each of the alphabet's six columns is meant to depict one of the universe's fundamental structural features. The first column (under 'Fig. A') lists the Llullian dignities, externalizations of the divine personality from which the world's goodness, greatness, duration, and so forth emanate in Neoplatonic fashion. The second column is composed of what Llull takes to be the primary logical categories. The third column details the kinds of questions that can be asked; the fourth, the medieval ontological hierarchy; and the fifth and sixth, the essential moral categories. Llull also anticipated Leibniz in recognizing that such an alphabet was the key to a logic of discovery. Moreover, the combinatory and analytic parts of Leibniz's ars inveniendi are clearly prefigured in the alphabet's function. Combining the "letters" of the alphabet, which were in fact words, produced "words", which were (roughly) sentences. Once the harvest of all logically possible combinations of the alphabet's letters was in (corresponding to the combinatory part of Leibniz's ars inveniendi), the Art was to be used to winnow the false combinations from the true (the analytic part). The result, the wheat, would be the sum total of the most general truths about the world-the definitive philosophy. But in De arte combinatoria, Leibniz faults Llull's execution of the combinatorial part of this task.5 Llull considered only binary and ternary combinations of the letters of the alphabet, but unary all the way up through nonary combinations are possible.6 Therefore, Leibniz argued, the 9 letters of each column can be combined in 29 – 1 = 511 possible ways. And, since there are 6 columns, Llull's simple alphabet yields the astounding number of 5116 = 17,804,320,388,674,561 possible combinations. Actually, Leibniz's figure is either too low or too high. The total number k of unary through n-ary combinations that can be obtained from n things without repetition is given by k = 2n – 1. But Leibniz proceeds, in effect, by applying this formula to only the first column of the alphabet, obtaining 511, and raising that result to the sixth power. To see that this skews the results, the reader might try following Leibniz's procedure to answer two questions about the model M: . f e d c b a i) How many unary through n-ary combinations without repetitions are there where n = 6 (the number of letters in M)? Applying (1) in Leibniz's fashion to the first column of M yields 7, and squaring it gives 49. But the correct procedure of applying (1) to the entire matrix gives 26 – 1 = 63 combinations, making 49 too low. ii) Where n = 6, how many unary through ternary 3 combinations without repetitions are there? The number of combinations without repetitions of n things taken r at a time is given by . r-nr n = C rn, )!(! ! Hence there are 6 unary + 15 binary + 20 ternary = 41 combinations, making 49 too high. The same thing happens with Llull's alphabet. If Leibniz wanted the total number of unary through n-ary combinations without repetitions where n = 55 (the number of letters in the alphabet), that number is 255 – 1, all of 39 orders of magnitude larger than the figure in De arte combinatoria. On the other hand, if he wanted the total number of unary through nonary combinations without repetitions for the same number of letters, that is a number on the order of 109, which is 7 orders of magnitude smaller than Leibniz's figure. Nevertheless, Leibniz was right about things being much more complicated than Llull thought. In the remainder of this note, I offer a very modest second to Leibniz's critique. Instead of the alphabet, however, I will focus on the table that appears for the first time in the Taula general (1293) and remains intact down to the Ars generalis ultima and the Ars brevis (both 1308), the final versions of the Art. The table was designed with two very different functions in mind: to automatically provide a middle term for a sound categorical syllogism on any subject whatsoever, and to exhaustively tabulate the ternary combinations of the first two columns of the alphabet. The remarks that follow concern only the second of these functions. Llull's table was generated from the Fourth Figure of his Art, which is reproduced below. As one can see, the Fourth Figure is composed of three concentric circles, each compartmentalized by the variables B through K from the extreme left of the alphabet. The outer circle is to be thought of as fixed and the two inner circles as movable so as to produce the various ternary combinations of variables. In the manuscripts and some of the earliest printed editions, the inner circles really did move; they were cut out and joined to the center of the outer circle by a thread knotted at both ends. The Fourth Figure was thus a primitive logical machine. Here is how it generates the table. Given the 9 variables B through K, there are 84 = 3)!-(93! 9! ternary combinations without repetitions of variables. They are as follows. 1 BCD 2 BCE 3 BCF 4 BCG 5 BCH 6 BCI 7 BCK 8 BDE 9 BDF 10 BDG 11 BDH 12 BDI 13 BDK 14 BEF 15 BEG 16 BEH 17 BEI 18 BEK 19 BFG 20 BFH 21 BFI 22 BFK 23 BGH 24 BGI 25 BGK 26 BHI 27 BHK 28 BIK 29 CDE 30 CDF 31 CDG 32 CDH 33 CDI 34 CDK 35 CEF 36 CEG 4 37 CEH 38 CEI 39 CEK 40 CFG 41 CFH 42 CFI 43 CFK 44 CGH 45 CGI 46 CGK 47 CHI 48 CHK 49 CIK 50 DEF 51 DEG 52 DEH 53 DEI 54 DEK 55 DFG 56 DFH 57 DFI 58 DFK 59 DGH 60 DGI 61 DGK 62 DHI 63 DHK 64 DIK 65 EFG 66 EFH 67 EFI 68 EFK 69 EGH 70 EGI 71 EGK 72 EHI 73 EHK 74 EIK 75 FGH 76 FGI 77 FGK 78 FHI 79 FHK 80 FIK 81 GHI 82 GHK 83 GIK 84 HIK Each of these combinations is incorporated in the table at the head of one of 84 columns. The remainder of each column is composed of 19 variations on the combination at its head and the letter T. The complete table has 84 × 20 = 1680 compartments, therefore. For our limited purposes, however, the abbreviated table from the Ars brevis will suffice.7 It lists only 7 of the 84 columns. BCD CDE DEF EFG FGH GHI HIK BCTB CDTC DETD EFTE FGTF GHTG HITH BCTC CDTD DETE EFTF FGTG GHTH HITI BCTD CDTE DETF EFTG FGTH GHTI HITK BDTB CETC DFTD EGTE FHTF GITG HKTH BDTC CETD DFTE EGTF FHTG GITH HKTI BDTD CETE DFTF EGTG FHTH GITI HKTK BTBC CTCD DTDE ETEF FTFG GTGH HTHI BTBD CTCE DTDF ETEG FTFH GTGI HTHK BTCD CTDE DTEF ETFG FTGH GTHI HTIK CDTB DETC EFTD FGTE GHTF HITG IKTH CDTC DETD EFTE FGTF GHTG HITH IKTI CDTD DETE EFTF FGTG GHTH HITI IKTK CTBC DTCD ETDE FTEF GTFG HTGH ITHI CTBD DTCE ETDF FTEG GTFH HTGI ITHK CTCD DTDE ETEF FTFG GTGH HTHI ITIK DTBC ETCD FTDE GTEF HTFG ITGH KTHI DTBD ETCE FTDF GTEG HTFH ITGI KTHK DTCD ETDE FTEF GTFG HTGH ITHI KTIK TBCD TCDE TDEF TEFG TFGH TGHI THIK 5 Llull uses T as an interpretive device: all variables appearing before it are to be interpreted by reading across the alphabet to its first column; all variables coming after it are interpreted by reading across to the second. Hence BCTB stands for 'goodness', 'greatness', and 'difference', while TBCD stands for 'difference', 'concordance', and 'contrariety'. What Llull has done, in effect, is to construct each column from the possible ternary combinations of the 6 "letters" that are the values of the variables at the head of the column. The column BCD, for example, is composed of the 20 combinations of the values of the variables B, C, and D. There are two critical points to be made about this table. The first is that Llull restricts himself unduly to combinations that, when interpreted, have no repetitions. All ternary combinations from the table are considered meaningful, with BCDT, for example, being interpreted as 'Goodness is as great as eternity'.8 But if that makes sense, so does BCCT, 'Goodness is as great as greatness'. Yet BCCT-and all the other combinations with repetitive values-are excluded from the table. If we include them, the total number of triples is not 84 but 93 = 729.9 The second point is that even if we assume only Llull's 84 combinations without repetitive values, the table still turns out to be more complicated than it appears. When Llull interprets BCDT, for example, as 'Goodness is as great as eternity', he ignores the fact that there are 5 other ways of ordering the variables and 5 other equally legitimate interpretations. BDCT Goodness is as eternal as greatness. CBDT Greatness is as good as eternity. CDBT Greatness is as eternal as goodness. DBCT Eternity is as good as greatness. DCBT Eternity is as great as goodness. He does not register the difference between the variables, where order does not matter (BCD = DCB), and the variables' interpretations, where order matters indeed ('Goodness is as great as eternity'  'Eternity is as great as goodness'). Thus one might expect some sort of mixup about combinations, which are not ordered, and permutations, which are. That is in fact what happens. Llull calculates the number of combinations of the 6 values taken 3 at a time: 20. = 3)!-(63! 6! What he should have done, however, is to calculate the number of permutations. The number of permutations without repetitions for n things taken r at a time is given by . rn n = P rn, )!-( ! Hence the number of permutations of the 6 values taken 3 at a time is: 120. = 3)!-(6 6! What would a corrected table, one with unique entries for all and only the 3-place permutations without repetitions of values, look like? It would be larger than Llull's original, of course. Since the problem is not the ternary combinations of 9 variables but the ternary permutations of their 18 values, the table would have 4896 = 3)!-(18 18! different compartments. The procedure I am recommending here is none other than Llull's in an analogous situation. To evacuate the binary combinations of variables from the Third Figure, he specifies the possible permutations of their values.10 6 The foregoing critique of both Llull and Leibniz admittedly concerns matters of detail. But it is submitted in the belief that one of the sources of philosophical progress is a clearer understanding of its past, and that the larger, still tantalizing project of a logic of discovery can be carried forward only through a mastery of such details, past as well as present. 7 NOTES 1 Leibniz, Mathematische Schriften, ed. C.I. Gerhardt (Berlin and Halle, 1849–63; rpt. Darmstadt: Hildesheim, 1962), vol. 7, p. 185. Emphasis in original. 2 Opuscles et fragments inédits de Leibniz, ed. Louis Couturat (Paris, 1903; rpt. Darmstadt: Hildesheim, 1961), p. 167. 3 Catherine Wilson, Leibniz's Metaphysics: A Historical and Comparative Study (Princeton and Manchester: Princeton and Manchester Univ. Presses, 1989), ch. 1. 4 The alphabet is taken from the Ars brevis. See Anthony Bonner's Selected Works of Ramon Llull (1232–1316) (Princeton: Princeton Univ. Press, 1985), vol. I, p. 581. On the various phases of the Art, see ibid., I, pp. 56–57. 5 Dissertatio de arte combinatoria in Philosophische Schriften, ed. Paul Ritter and Willy Kabitz (Berlin: Akademie, 1971), vol. I, p. 193. 6 The 9 unary combinations would have been useless to Llull, who was interested only in combination that, when interpreted, bear truth values. 7 Bonner, p. 597. 8 When interpreting the compartments at the head of the column, T is understood as the last letter. BCD, then, is interpreted as BCDT. 9 Prantl argued that this was the correct number in Geschichte der Logik im Abendlande (Leipzig, 1867; rpt. Graz: Akademische Druck-u. Verlagsanstalt, 1955), vol. III, p. 162 n 90. 10 Bonner, p. 598.