Ars Demonstrativa: A (mis)user’s manual
Zsuzsa Peter Fourth Year 2016 Francesca Hughes - A Prehistory of the Universal Discrete Machine Architectural Association
Considered to be the world’s first analog computer, Ramon Llull’s Ars Demonstrativa, developed between 1272 and 1308, proposed the mechanization of thought by the means of structuring the logic of argumentation. Departing from a finite set of elements that are clearly defined - the principles, and a precise and strict way of constructing arguments through syllogisms using his diagrams and combinatory logic, the Ars Demonstrativa presents itself as an efficient, compressed and hermetic system that nevertheless claims to be capable of investigating and addressing the entirety of human knowledge. These qualities – the compression of knowledge structured within a finite set of elements that point to the understanding of the world while procuring easy precision in interpretation and argumentation - are all similar traits to the aims set out by seventeenth century English logicians and linguists in the quest for a Universal Language. Driven by the desire to find a Universal Language that does not allow for misinterpretation and rhetorical corruption of meaning while offering the possibility of communication between different cultures, Francis Bacon proposed a language that would use Real Characters instead of nominal ones (words that only represent sounds) thus establishing a direct and indisputable correlation between the signifier and the signified. Eliminating terms that he deemed unclear, confusing and focusing only on the “radical words” or “essential notions”, his universal language promised the possibility to rediscover the world anew in its concrete reality, without the mediation by different levels of error present in natural languages. Taking Bacon’s ideas further, John Wilkins in his Essay Towards a Real Character (1668) proposed a language that not only corrects these errors imposed by the ‘gradual corruption’ of natural languages, but also within its structure, reveals the presence of correspondences and relationships between concepts and things through the use of a system of notation that is translated as the very definition of these terms. Using letters as conventional signs, Wilkins constructed an exhaustive vocabulary that was formed by radical terms such as God, The World, Stone, Flower, Dwelling, Military, etc. each assigned a group of two letters (Ba, Be, Bi). Further marked by subsequent consonants that represent types of differences in relation to the radical term, secondary terms (Bab, Bad, etc.) are constructed and so on, forming the definition of each term both in their meaning and notation as a branching system - thus the language becomes not only representational but also an explorative and synthetic tool for accessing the nature of the world. Taking language perhaps to its logical end, Cave Beck proposed a numerical dictionary using numbers from 0 to 9 that in their sequential arrangement from 1 to 10.000 could relate to all the necessary, primitive terms used in a language – completely eliminating words and their promiscuity from language, foreshadowing the discrete logical systems to come.
Ba
Bab Bad Bag Bap Bat Bac Baz Bas Ban
The idea of bringing closer the nature and structure of language to that of mathematical notations and their operability in calculating the truth was an idea Gottfried Leibniz also embraced, famously declaring in his letter to Mersenne: “because once we had realized this language, calculating and reasoning will be the same thing”1. This idea is at the heart of Llull’s Ars Demonstrativa (this similarity not being accidental as we know that Leibniz has been studying Llull’s writings). While being on one hand a machine of thought that allowed the simulation and production of human knowledge using artificial means, the Ars Demonstrativa also served a very specific purpose: the conversion of non-believers, specifically Muslims and Jewish, to adhere to the Christian faith. Thus it was not only a machine that would produce artificial intelligence but in the same time, and perhaps more importantly, would calculate incontestable truths – namely the dogmas of Christian religion: the Trinity and the Reincarnation.
1. Couturat, Opuscules et fragments inédits de Leibniz, 27-8.
However, this brings forwards the question: could there have been a machine constructed that disproves these Christian dogmas and becomes a tool for converting people to other religions, or for dismantling all faith? To what extent is the machine a closed system that presents the means by which all its users will arrive at the same conclusion or might it allow the liberty to truly acts as a tool for the user’s personal investigation and interpretation of the world? These questions are certainly applicable to the Ars Demonstrativa, where the notion of the Trinity is already embedded within the structure of the mechanism, despite it being a tool to prove its existence - reflected through the triadic structuring of the principles from figure T (difference-concordance-contrariety, beginning-end-middle) as well as the possibility of any statement having a true, false or intermediary state as an outcome (in opposition with later binary systems to come). In the same time, considering this question in a wider and more contemporary context we have to recognize that, even if not addressing such existential questions, our daily lives are certainly dictated by albeit somewhat faster data processing machines, and not dissimilar algorithms, whether we are searching for something on Google or trying to find the quickest route between two locations.
Ramon Llull Ars Demonstrativa -User’s manual-
In the attempt to find a universal way by which people of different religions might reason and ultimately arrive at the conclusion that the doctrines of the Christian religion are undoubtedly true (and that therefore there is no other true religion), Ramon Llull in 1305-81 developed his Ars Demonstrativa. Constructed upon common notions present in all major religions, both Christian, Muslim and Jewish, the Ars functions as a combinatory generative logic that on the one hand is capable of providing answers to any question through the use of generated combinations - that act as middle terms in the logical construction of an argument based on syllogisms2– while on the other hand is also an inquisitive tool into the nature of the world via progressive explorations of the mechanism of the Ars itself.
MECHANISM OF THE ARS 1.The Alphabet The Ars Demonstrativa uses 9 letters, from B to K and it is defined by polysemy – each of these letters can stand for different notions, depending on their use in certain figures or their position in combinations. The multiple meanings of the letters are shown in this alphabet:
Figure A
Figure T
Questions and Rules
Subjects
Virtues
Vices
B C D
goodness
difference
whether?
God
justice
avarice
greatness
concordance
what?
angel
prudence
gluttony
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
clarity
ire
I
truth
equality
where?
elementative
patience
lying
K
glory
minority
pity
inconstancy
how? and with istrumentative what?
*or duration
The principles (absolute – A, or relative – T) are the notions used in later figures.
1. After an illuminating experience on Mount Randa in Majorca, in about the year 1272, Ramon Llull came to see his life mission the propagation of Christian faith. Since then had the belief that arguments based on necessarily true principles, that of God, could be the only way to convince people of other faith to adhere to Christianity. Thus the beginning of the development of the Ars Demonstrativa is marked by this moment and all of Ramon Llulls life was dedicated to it, culminating in writing the Ars Magna (or otherwise known as the Ars Generalis Ultima) between 1308-1305 as the most complex document describing the Ars Demonstrativa. 2. Syllogism is a deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion (for example All men are mortal. John is a man. Therefore John is mortal.)
2.Figure A Figure A shows the circular placement of the principles defined in column A of the alphabet, all of them interconnected. This interconnection through graphic lines underlines the grammatical transposition of these principles – Goodness is Great, Greatness is Good and so on. As the principles in figure A show the attributes of God, their convertibility become God’s prime definition
K
B
glory
tru th
I
goo dn es s
virtue
H
C
ss atne gre
A w
G ill
D
etern ity r we po
E
wisdom
F
3.Figure T
Figure T focuses on the arrangement of the relative principles defined in the second column of the alphabet. Albeit positioned again in a circle, each attribute thus occupying equal status, a substructure of relations can be observed in the diagram showing interconnection between groups of three principles: difference-concordance-contrariety3, beginning-middle-end and majority-equality-minority. Furthermore, each of these principles is further divided on the concentric circles into three species in relation to the subject it might apply to – for example there is a difference in the notion of beginning in relation to cause, quantity or time.
cinjuncti on menurati on extreme s
l and sens. sensua
en d e
mid.
sensual and intel.
contr.
.
al qu
. nc
l. inte and al l. nte ctu i lle nd te la ens. in ua ds ns an se al su en
st. ub ds d. an cci e da nc id. an e acc c nd a nt
H
su bs ta su bs ta ac n cid e
I
substan c e and sub substan st. c e a nd acc acciden id. t an d a cci d. s
T
maj .
ctual and intel. intelle
f.
se cau ntity qua e tim
d if
E
min.
. be g
D
on
.
pr iva tio ter m in at pe i rfe ct i on
B
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C
n
ubst. ce and s stan sub and accid. e c n sta sub t and accid. iden c a c
co
G
K
F
3. Difference - a point or way in which people or things are dissimilar; contrariety - contrary opposition/ opposition or inconsistency between two things
4. Defintions Llull provides the definition of all 18 of principles used in figures A and T. They are the following4: 1. Goodness is that thing by reason of which good does good. 2. Greatness is that by reason of which goodness, duration, etc., are great. 3. Eternity or duration is that by reason of which goodness, etc., endure. 4. Power is that by reason of which goodness, etc., can exist and act. 5. Wisdom is that by reason of which the wise man understands. 6. Will is that by reason of which goodness, greatness, etc., are lovable or desirable. 7. Virtue is the origin of the union of goodness, greatness, and the other principles. 8. Truth is that which is true concerning goodness, greatness, etc. 9. Glory is that bliss in which goodness, greatness, etc., come to rest. 10. Difference is that by reason of which goodness, etc., are clearly distinguishable from one another. 11. Concordance is that by reason of which goodness, etc., accord in one or in several things. 12. Contrariety is the mutual opposition of certain things as a result of different goals. 13. Beginning is that which is found in everything where there is any question of priority. 14. Middle is the subject through which end influences beginning, and beginning re-influences end, and thus it participates in the nature of both. 15. End is that in which beginning comes to rest. 16. Majority is the image of the immensity of goodness, greatness, etc. 17. Equality is the subject in which the end of concordance, goodness, etc., comes to rest. 18. Minority is the thing close to nothingness In the Ars Magna (the expanded version of the Ars) Llull further develops the definitions by mixing the definitions of the principles with each other. Example: Truth combined with the sequence of principles5 truth + goodness
Truth is good by reason of goodness, while goodness is true by reason of truth; and if they convert, then the good and the true are always identical, but when they sometimes do not convert in some subject, one may falsely expect to obtain the truth in a way that is neither true nor virtuous. Here, we see that truth and goodness in man are not convertible.
truth + greatness Truth is great in greatness, which would not be so unless it had the verifier, verifiable and verifying. truth + duration
Truth endures through duration because the verifier, verifiable and verifying endure in it naturally, and sometimes also morally, when truth acts within itself by means of things external to it, as does radical moisture with nutritional moisture in sentient beings.
These definitions crucially stand to exclude any possible combination of letters that might contradict 4. Anthony Bonner, The Art and Logic of Ramon Llull (Leiden: Koninklijke Brill, 2007), 134-135. 5. Ramon Llull, Ars Generalis Ultima (Pisa, 1305-1308), trans. Yanis Dambergs, https://www.scribd.com/ doc/116720201/Raymond-Lull-Ars-Magna-Ars-Generalis-Ultima, 80.
5. Rules As seen in the Alphabet, each letter also signifies a question, considered to be a rule of inquiry, which, in their turn, has subspecies. For example Whether (letter B) has three subspecies – dubitative, affirmative and negative. Developed further in the Ars Magna, these rules combined with the principles, answered by Llull, provide certain statements that can also become part of the argumentation. Such an example would be Greatness combined with the rule C: “With the first species of rule C we ask what greatness is. We answer that it is the principle whose proper function is to magnify things and with which great things cause magnification. This is also corroborated by rule B. With the second species of rule C we ask what greatness has in itself. We answer that in corporeal things, it has its own co-essential magnifier, magnifiable and magnifying with which it causes substantial extension in subjects from which quantities arise with which greatness measures itself and the other principles in the corporeal subjects in which it exists. In a great spiritual subject, it causes great acts, namely great understanding, remembering and loving habituated with spiritual greatness, which is a spiritual principle; and the same applies to great bonifying, enduring, judging etc. as proved by rule B. With the third species we ask what greatness is in other things. We answer that greatness is the form informing everything that has great form and great matter, as the first species of rule D indicates. Further, with the fourth species of rule C we ask what greatness has in other things. We answer that it has great action and great passion with which it is naturally present in the great subjects in which it exists. The first species of rule D also indicates this.”6 6. Third Figure The third figure shows the binary combinations of principles from both figure A and T. As the order of letters here doesn’t matter, BC=CB and does not permit the double use of the same letters7, the figure is essentially a half matrix without the diagonal.
BC
CD
DE
EF
FG
GH
HI
BD
CE
DF
EG
FH
GI
HK
BE
CF
DG
EH
FI
GK
BF
CG
DH
EI
FK
BG
CH
DI
EK
BH
CI
DK
BI
CK
IK
BK
6. Ramon Llull, Ars Generalis Ultima (Pisa, 1305-1308), trans. Yanis Dambergs, https://www.scribd.com/ doc/116720201/Raymond-Lull-Ars-Magna-Ars-Generalis-Ultima, 94-95. 7. Because using the same letters would not permit the introduction of a middle term.
One such combination can lead to 12 possible sentences. If letters from figure A are capitals and letters from figure T are lowercase, these sentences from the combination BC would be the following: B= goodness / difference, C= greatness/ concordance BC Bc Bb CB Cb Cc bB bC bc cB cb cC
Goodness is great. Goodness is concordant. Goodness is different. Greatness is good. Greatness is different. Greatness is concordant. Difference is good. Difference is great. Difference is concordant. Concordance is good. Concordance is different. Concordance is great.
Applying the questions of both letters B (Whether?) and C (What?), each chamber generates 36 prepositions.8 Whether goodness is great? What goodness is great? 7. Fourth Figure The fourth figure is a paper mechanism (a wheel that anticipates the wheels in the combinatorial machines to come) consisting of three concentric circles with the letters arranged in them. This device allows for the generation of tertiary combinations between the principles.
B
C
K
B C
I
I
I
H
F F F
G
H
E
E
E
H
D
D
D
K
K
B
C
G
G
8.Each sentence is multiplied with the question, leading to 12 sentences + 12x2 questions = 36 preposi-
tions.
8. The Table The Table is formed out of rows (7 in the Ars Brevis, 84, the complete set in Ars Magna9) of tertiary combinations with non-repetitive letters. These combinations are then expanded into columns of all possible, precisely 20, variations between the letters, making a differentiation of their provenance from figure A or T based on the separating element of T that acts as the wall within the chambers, creating intervals in the tertiary combination – every letter that is before T is a principle from figure A, while every letter after denotes a principle from figure T. The only rule that is applied in generating these variations is that neither before nor after the separator T in the sequences the same letter cannot appear twice. Therefore, using tertiary combinations, a total of 1680/140 chambers have been generated, depending on their description in the Ars Magna or Ars Brevis, respectively. BCD
CDE
DEF
EFG
FGH
GHI
HIK
BCTB BCTC BCTD BDTB BDTC BDTD BTBC BTBD BTCD CDTB CDTC CDTD CTBC CTBD CTCD DTBC CTBD DTCD TBC
CDTC CDTD CDTE CETC CETD CETE CTCD CTCE CTDE DETC DETD DETE DTCD DTCE DTDE ETCD ETCE ETDE TCDE
DETD DETE DETF DFTD DFTE DFTF DTDE DTDF DTEF EFTD EFTE EFTF ETDE ETDF ETEF FTDE DTDF FTEF TDEF
EFTE EFTF EFTG EGTE_ EGTF EGTG ETEF ETEG ETFG FGTE FGTF FGTG FTEF FTEG FTFG GTEF GTEG GTFG TEFG
FGTF FGTG FGTH FHTF FHTG FHTH FTFG FTFH FTGH GHTF GHTG GHTH GTFG GTFH GTGH HTFG HTFH HTGH TFGH
GHTG GHTH GHTI GITG GITH GITI GTGH GTGI GTHI HITG HITH HITI HTGH HTGI HTHI ITGH ITGI ITHI TGHI
HITH HITI HITK HKTH HKTI HKTK HTHI HTHK HTIK IKTH IKTI IKTK ITHI ITHK ITIK KHTI KHTK KTIK THIK
As the first letter of the combinations always denotes the question, the translation of some of the combinations is as follows: BCTB BCTC BCTD BDTB BDTC BDTD BTBC BTBD BTCD CDTB CDTC CDTD
Whether a certain goodness is so great as to be different. Whether a certain goodness is so great as to be concordant. Whether a certain goodness is so great as to be contrary. Whether a certain goodness is so eternal as to be different. Whether a certain goodness is so eternal as to be concordant. Whether a certain goodness is so eternal as to be contrary. Whether a certain goodness is so different as to be concordant. Whether a certain goodness is so different as to be contrary. Whether a certain goodness is so concordant as to be different. What greatness is so eternal as to be different? What greatness is so eternal as to be concordant? What greatness is so eternal as to be contrary?
9. Although the Ars Magna was intended to be the last writing describing the mechanism of the Ars Demonstrativa, due to complaints that it is too difficult to use because its complexity, Llull wrote the Ars Brevis, the shortened and simplified version of the Ars Magna, using less combinations.
9. Application to questions By posing a question, identifying the letter corresponding to it and the principles that can be associated with the alphabet if they are present, the prerequisites of choosing a camera or chamber are established – camera or chamber being terms set out by Llull for the spatial definition of the combinations, thus already embedding architecture within the logic machines, computers. All the terms implied by the camera act as middle terms with their according definitions and rules to construct an argument and find an answer. The application of the Ars for the question “Is the world eternal?” is exemplified by LLull (in relation to the camera BCD because the question whether suggests letter B, and the principle eternal letter D, while the third principle remains a random selection): “In answer to the question Whether the world is eternal? we say by B C D that it is not, because if it were eternal, its foundation (ratio) would be eternal and it would produce eternal good throughout eternity while greatness, by its definition, would magnify this good foundation from eternity and in eternity; and eternity would make this production last from eternity and in eternity, so that there could be no evil in the world, because good and evil are contrary. But there is evil in the world, as we know by experience. We therefore conclude that the world is not eternal. Moreover, rule B says that one must answer this question negatively, according to the definitions we have given and according to what we propose to say by Rules C and D, which is: if the world is eternal, its eternity is that which causes evil to endure as much as good, as shown by the first species of Rule C. By the first species of rule D, evil and good are equally primordial. No day is first or last. By the second species of Rule C and D the world is composed of good and evil in eternity and from eternity. By the thrid species of Rule C the world is infinite in eternity but finite in good and evil. By the fourth species of C the world has repose in things subjected to generation and decay, where generation is due to good and decay is due to evil. And by the second species of Rule D God’s eternity and goodness need evil in repose in causing the world’s eternity. And since all things are impossible, the answer to the question is clearly negative.”10
BCD D
Is the world eternal?
= eternity Eternity definition: Eternity or duration is that by reason of which goodness, etc., endure. its foundation (ratio) would be eternal and it would produce eternal good throughout eternity
C
= greatness Greatness definition: Greatness is that by reason of which goodness, duration, etc., are great. would magnify this good foundation from eternity and in eternity; and eternity would make this production last from eternity and in eternity so that there could be no evil in the world good
contrariety
evil
But there is evil in the world, as we know by experience We therefore conclude that the world is not eternal.
10. Anthony Bonner, The Art and Logic of Ramon Llull (Leiden: Koninklijke Brill, 2007), 151.
Ramon Llull Ars Demonstrativa -Misuser’s manual-
If one is inclined to misuse this device for conversion, there are two possible paths to take. One is defined by the in-adherence of words to logical operations, when they escape the system of syllogisms in the pursuit of producing truth - resulting in fallacies, while the other method focuses on the exploration of combinations that the generative logic does create, but the Ars Demonstrativa explicitly forbids.
1. Fallacies Even though the Ars Demonstrativa uses a restricted number of terms, whether they can or cannot be converted1 in the context of particular sentences is left to the user’s discretion. Thus the system in itself does not exclude the formation of argumentations such as: All angels are good. Socrates is good. Therefore Socrates is an angel. Trying to prevent such ilogical constructions Llull develops a chapter in the Ars Magna entitled Detecting Fallacies where he offers methods of detection through the application of multiple combinations to the sentence examined. However, this way any argumentation would have to be reiterated and retested endlessly in order to confirm whether a certain preposition is true or not or whether a certain type of fallacy is present. Differentiating between fallacies relating to the word or the subject matter, Llull describes the following categories of erroneous arguments:2
Fallacies in the word 1. False Conclusions due to Equivocation3 or Ambiguity - relating to words with multiple meaning All dogs can bark. But the Dog Star is a dog, therefore a star can bark. 2. Amphyboly4 - relating to sentences with multiple meanings Whatever is Aristotle’s is owned by Aristotle. But this is Aristotle’s book, therefore it is owned by Aristotle. 3. The Fallacy of Composition - deception due to the multiple meanings of some sayings whose parts can be differently associated with one another Whatever has the possibility of being white can be white, but black has the possibility of being white, therefore black can be white. 4. The Fallacy of Division - deception due to the multiple meaning of a saying whose parts can be disassociated from each other in different ways All animals are either rational or irrational. But not all animals are rational, therefore all animal are irrational. 5. The Fallacy of Accent - deception arising from the fact that the same utterance pronounced with different emphasis can mean different things All the spectators at wrestling match who yell ‘Kill Bart!’ want Bart to lose. However, some of Bart’s most loyal fans are also yelling ‘Kill Bart!’. There fore some of Bart’s most loyal fans also want him to lose the match. 6. The Fallacy of Figures of Speech - due to similarity between utterances Everything you saw yesterday is what you see today. But yesterday you saw white, therefore you see white today. 1. Conversion in logic is the inference in which the subject and predicate are interchanged. 2. Ramon Llull, Ars Generalis Ultima (Pisa, 1305-1308), trans. Yanis Dambergs, https://www.scribd.com/ doc/116720201/Raymond-Lull-Ars-Magna-Ars-Generalis-Ultima,, 65-70. 3. Equivocation in linguistics is the ambiguity of words or phrases 4. Amphyboly in linguisticis an ambiguity which results from ambiguous grammar
Fallacies in the subject matter 1. The Fallacy of Accident - improper identification of two things as one on account of some accidental feature they have in common I know Socrates, but Socrates is arriving therefore I know who is arriving. 2. The Fallacy of Over Generalizing - statement made with reference to something taken in an absolute sense Negroes have white teeth, therefore Negroes are white. 3. The Fallacy of Ignoring the Elenchus5 - deception arising from ignoring the things needed to define an elenchus and especially the contradiction it entails (works in relation to a statement and syllogism or two syllogisms) Some animal is incorruptible. All things composed of contraries are corruptible. But all animals are composed out of contraries, therefore all animals are corruptible. 4. The Fallacy of Begging the Question - deception from proving something with the same expressed in different words Rational animals run, but men are rational animals, therefore men run. 5. The Fallacy of Affirming the Consequent - when the consequent is deemed identical to the antecedent Lions are animals, but you are an animal, therefore you are a lion. 6. Fallacy of the Specious Reason - when some preposition that merely appears to be a reason but has no valid bearing on the conclusion is inserted between the premises leading to the conclusion Soul and life are identical, but death and life are contraries, and generation and corrupt are contraries; now if death is corruption, then life is generation, and to live is to be generated. 7. The Complex Question Fallacy - giving a single answer to a question that inquires about several things Do you suppose that men and lions are rational animals? Men and lions are not rational animals, and therefore man is not a rational animal. 8. The Fallacy of Contradiction - deception that arises and derives from the above-mentioned fallacies that reach apparently contradictory conclusions No stone is sighted, but some stone is sighted, therefore some stone is both sighted and not sighted.
2. Forbidden Chambers According to the mechanism of the Ars Demonstrativa, it is forbidden to generate tertiary combinations, otherwise called chambers, that utilize the same letter more than once from the Alphabet before or after the separator T, which denotes the boundary between principles from figure A and principles from figure T. To investigate the result of these forbidden chambers, the letter B has been chosen as a generic example, resulting in 4 prohibited combinations: BBTB, BTBB, TBBB, BBBT. If we translate B as a principle from figure A as a capital B (=goodness) and B as a principle from figure T as lowercase b (=difference), the BBb Bbb bbb BBB
5. a logical refutation
Whether a certain goodness is so good insofar as it is different. Whether a certain goodness is different from difference. Whether a certain difference is different from difference. Whether a certain goodness is so good as to be good.
Using the combined definitions of goodness and difference set out by Llull, these 4 sentences convert into each other, forming an infinite loop that works in both directions. BBB The definitions are the following:
BBb
Bbb
Goodness is clear without confusion because of difference, which is a good, distinct and clear reason for good to do good that is clearly distinct from itself.
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t Bu
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th i s re aso n
con clu d
on o
t
reas
e th a
Bbb
d by
ss
therefor e we ca
goo
odn e
e enc fer
ce is
t: ha
eren
yt sa
diff
is g o
dif
n ca
Bu t
that:
ct distin
of
we
Bb W B di het ffe he re r a nt c as ert to ain be g go ood od n . es
diff eren ce
ess is
so n rea
B
therefo re we c an say
dn goo
by
re efo er th
Bb
BBB Whether a certain goodness is so good as to be good.
But
at:
BBB
Difference is good by reason of goodness, while goodness is distinct by reason of difference; and thus, goodness and difference are reason for the subject in which they exist to produce distinct good by acting distinctly well.
While converting one sentence to another through the use of the definitions, in the case of the translation from BBb to bbb as well as from Bbb to BBB, being a two-step conversion (two principles need to be converted), an intermediary step emerges through the generated combinations of bBb, respectively BbB. bBb BbB
Whether a certain difference is good insofar as it is different. Whether a certain goodness is so different as to be good.
These combinations deliver the breaking of a further rule set out by the Ars Demonstrativa: that principles from figure T can only follow progressively principles from figure A - they cannot be intermixed as in the examples above.
Bibliography
Eco, Umberto: “On Llull, Pico, and Llullism”, From the Tree to the Labyrinth, Cambridge, Massachusetts: Harvard University Press, 2014. Bonner, Anthony: “What was Llull up to?”, Ramon Llull: From the Ars Magna to Artificial Intelligence, ed. Alexander Fidora and Carles Sierra, Barcelona: Artificial Intelligence Research Institute, 2011. Gardner, Martin: “The Ars Magna of Ramon Llull”, Logic Machines and Diagrams, New York: McGraw-Hill Book Company, Inc., 1958. Bonner, Anthony: The Art and Logic of Ramon Llull, Leiden: Koninklijke Brill, 2007. Llull, Ramon: Ars Generalis Ultima, Pisa: 1305-1308, trans. Yanis Dambergs, https://www.scribd.com/ doc/116720201/Raymond-Lull-Ars-Magna-Ars-Generalis-Ultima Llull, Ramon: Ars Brevis, Pisa: 1308, trans. Yanis Dambergs, https://www.scribd.com/document/127210673/ Ars-Brevis Yates, Frances A.: “Llullism as an Art of Memory”, The art of memory,London: The Bodley Head, 2014. Vega, Amador, ed.: The Thinking Machine. Ramon Llull and the Ars Combinatoria, Barcelona: Centre de Cultura Contemporania, 2016. Rossi, Paolo: “The Construction of a Universal Language”, Logic and the art of memory : the quest for a universal language, London : Continuum, 2006.