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In the histories of logic and of computer science which mention
precursors and pioneers, we often find mention of Leibniz, the great
l7th-century mathematician and philosopher, who was one of the first
to try to build a mechanical calculator, and who tried to formulate a
Mathesis universalis, a sort of scientific language which would permit
any two disputants to settle their differences merely by taking pencil
and paper and saying, "Let us calculate" But then we find that
Leibniz got certain important ideas from a character called Ramon
Llull who lived in the l3th century, who carne from a place called
Mallorca, and who spent his life trying to convert Muslims and
Jews. As if this weren't peculiar enough, when he has appeared in
modern treatises, it has usually been as the typical romantic genius or
in his case, medieval mystic who wasn't quite right in the head.
Even a man as sensible as Martin Gardner (1982) calls him
quixotic and paranoid!. So what Llull was up to is a question that
indeed needs a bit of clarification.
Perhaps the best way to begin is by trying to situate him in his time.
He was born here on Majorca around 1232, only two or three years
after the King of Aragon and Catalonia had recovered the island from
the Muslims. This meant that Llull grew up in an island that was
still strongly multicultural: Muslims continued to represent perhaps a
third of the population, and Jews, although a much smaller minority,
were an important economic and cultural force on the island. So when
at the age of thirty he was converted from a profligate youth and he
decided to devote his life to the service of the Church, it seemed
only logical to do so by trying to convert these "infidels", as they
were then called. And he decided to do this in three ways: (1) to
develop a system that his adversaries would find difficult to refute
(which is what we'll see in moment), and to try to persuade them of
the truth of Christianity instead of just trying to refute their own
doctrines, as his predecessors had done [1]; (2) to be willing
to risk his life in proselytizing among Muslims and Jews (he in fact
made three trips to North Africa); and (3) to try to persuade
Kings and Popes of the need for setting up language schools for
missionaries, for which purpose he traveled many times throughout
France and Italy. He lived to 83 or 84, an incredible age when
the average life-span was around 40, dying in 1316 [2].
Now this situation has presented historians with two serious
paradoxes. The first is that, if he was principally interested in
converting Muslims and Jews, what could this possibly have to do with
his being a pioneer of computer science? It would seem doubtful that
l3th-century unbelievers would have wanted to listen to arguments that
looked forward to Bill Gates, or that modern computer scientists
would deem their profession useful for the persuasion of Muslims and
Jews of the truths of Christianity. The second paradox is that the
system Llull thought up doesn 't look like anything his contemporaries
were using, nor can it be considered really acceptable to modern
logic. This second paradox has caused enormous problems for historians
of logic. Those equipped with a knowledge of medieval logic who try to
tackle Llull are disagreeably surprised to find him discussing either
not at all or passing over very superficially the topics they feel he
should be discussing, and which they know his contemporaries were
discussing. Those who try to tackle it from the point of view of
modern formal logic are understandably put off by his basing his system
on an extreme Platonic realism, and thus making it depend primarily on
meaning rather than form. Llull himself was aware of these problems,
and carefully tried to explain that his system was neither logic nor
metaphysics. But that only helps us to understand what it isn't;
what it is is something I will try in very broad outline to explain
now. But before doing so, I would ask you to suspend, for the
moment at least, your highly trained and normally indispensable sense
of disbelief, and only start applying it again when we've seen a bit
of the inside of the edifice Llull constructed, because if not,
we'll never get past the front door.
The first thing we have to face is the problem of his trying to
persuade unbelievers. From the outset Llull realized that previous
attempts had failed because people had based their arguments on sacred
texts. Christians argued positively trying to explain the truths of
the Bible, or negatively trying to point out the errors in the
Qu'ran or in the Talmud. Such discussions, however, invariably
became bogged down in arguments as to which texts were acceptable to
whom, and how to interpret them. Since it was clearly impossible for
opposing sides to agree on these points, such discussions never got
anywhere. Participants invariably left them with a feeling of having
tried unsuccessfully to walk uphill in sand.
So Llull decided to try something completely abstracted from the
specific beliefs of any one religion, based only on whatever beliefs or
areas of knowledge they had in common. All three religions, for
instance, were monotheistic, and none of them could deny that this one
God of theirs had a series of positive attributes: goodness,
greatness, eternity, etc. They also shared a common heritage of
Greek science which taught them about the earth at the center of a
universe with seven planets rotating around it, and that this earth of
ours was composed of four elements, fire, earth, air and water. And
the framework in which all three philosophized about the world was that
of Aristotle. Finally, all could agree more or less about what
constituted virtues and vices.
What Llull then set out to do was to show how one could combine these
theological, scientific and moral components to produce arguments that
at least couldn't be rejected outright by his opponents. It was
furthermore clear that if he was going to set up an Ars combinatoria,
as later generations called it, its components would have to be finite
in number and clearly defined. Since they were like the premises of
his arguments, everybody had to be quite clear as to what they were and
how they functioned. Saying that people retained visual images better
than words, he decided to present his system graphically. This he did
in two stages: the first version of his system had twelve or more
figures, and he finally had to jettison it in the face of contemporary
complaints about its being too complicated [3]. The second version
in which the figures were reduced to four is the one for which he was
chiefly known in the l6th and l7th centuries, and which we will
present here. The final version of this second system found expression
in two works: the "Ars generalis ultima" [4], along with a much
shortened introductory version of same, the Ars brevis [5], which
follows the longer one chapter by chapter, but in outline form.
These works begin with an "Alphabet" giving the meaning of nine
letters, in which he says, "B signifies goodness, difference,
whether?, God, justice, and avarice. C signifies...", and so
on, all of which can best be set out in a table [6].
Notice first of all, as always with Llull, the letters don't
represent variables, but constants. Here they're connected by lines
to show that in the Divinity these attributes are mutually
convertible. That is to say that God's goodness is great, God's
greatness is good, etc. This, in turn was one of Llull's
definitions of God, because in the created world, as we all know too
well, people's goodness is not always great, nor their greatness
particularly good, etc. Now such a system of vertices connected by
lines is what, as mathematicians, you will of course recognize as a
graph. This might seem to be of purely anecdotal interest, but as we
shall see in a moment, the relational nature of Llull's system is
fundamental to his idea of an Ars combinatoria.
The components of the second column are set out in a Second Figure,
or Figure T [7]
Here we have a series of relational principles related among themselves
in three groups of three, hence the triangular graphs. The first
triangle has difference, concordance, and contrariety; the second
beginning, middle, and end; and the third majority, equality, and
minority. The concentric circles between the triangles and the outer
letters show the areas in which these relations can be applied. For
example, with the concept of difference, notice how it can be applied
to sensual and sensual, sensual and intellectual, etc. "Sensual"
here means perceivable by the senses, and Llull explains in the Ars
brevis, that: "There is a difference between sensual and sensual,
as for instance between a stone and a tree. There is also a difference
between the sensual and the intellectual, as for instance between body
and soul. And there is furthermore a difference between intellectual
and intellectual, as between soul and God".
Here Llull explains that B C, for instance, implies four
concepts: goodness and greatness (from Figure A), and difference
and concordance (from Figure T), permitting us to analyze a phrase
such as "Goodness has great difference and concordance" in terms of
its applicability in the areas of sensual/sensual,
sensual/intellectual, and intellectual/intellectual. It
furthermore, as he points out, permits us to do this systematically
throughout the entire alphabet. This is important, because one of the
ways in which Llull conceived his Art as "general" was precisely in
its capacity to explore all the possible combinations of its
components.
Now as mathematicians, you will recognize this figure as a half
matrix, and you will also see that, in relation to the graph of the
First Figure, it is an adjacency matrix. Because such a matrix is
symmetrical (in Llull's case this means he makes no distinction
between B-C and C-B), he saw no reason to reproduce the other
half; and because his graph admits no loops (that is, omits relations
such as B-B), he could also omit the principal diagonal.
If the Third Figure explores all possible binary combinations, the
Fourth Figure does the same for ternary combinations.
In medieval manuscripts, the outside circle is normally drawn on the
page, and the two inner ones are separate pieces of parchment or paper
held in place on top of it by a little piece of string, permitting them
to rotate in relation to each other and to the larger circle. In a
moment we'll see how he uses these ternary relations, but before going
on let me quote a book on logic for computer applications (Nerode and
Shore, 1993). Its authors say that one of the things lacking in
classical Aristotelian logic was the notion of a relation with many
arguments. His predicate relations P(x) were unary, and what he
missed was the basic building-block character of binary relations
R(x1y) and ternary relations S(x,y,z). This shows that
imbedded in what Künzel and Cornelius (1991) have called the
"hardware" of Llull's system we already have a full panoply of
binary and ternary relations.
Binary relations are worked out more extensively in a section he calls
"The Evacuation of the Third Figure". For the "compartment",
as he calls it, of B C, he not only uses "goodness" and
"greatness" from the First Figure, and "difference" and
"concordance" from the Second Figure, but also the first two
questions of the third column of the alphabet, those also corresponding
to the letters B C, which are "whether?" and "what?" This
means that for the combination of "goodness" and "greatness" one has
three possibilities, a statement and two questions:
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- Goodness is great.
- Whether goodness is great.
- What is great goodness?
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and so on for "goodness" and "difference", "goodness" and
"concordance", for a total of 12 propositions and 24 questions.
Ternary relations are worked out in a Table based on the Fourth
Figure:
The one we show here is the shortened form from the Ars brevis;
instead of 7 columns, the full form of the Ars generalis ultima has
84! Here the letter T acts as a separator: the letters that
precede it in any one compartment are from Figure A whereas those that
follow it are from Figure T. In addition the first letter can act as
an indicator of what question from the third column of the alphabet
should be considered. So, for instance, the ninth entry of the first
column, B T B D, could be translated as "Whether goodness
contains in itself difference and contrariety".
So much for the bare mechanics of the Art. Beyond that Llull
wanders even farther from the path of modern logic by basing his Art
not on the form of his propositions, but on the meaning of their
premises. It is therefore much more intensional than extensional.
How this side of his Art functions can perhaps best be explained by
making a brief excursion into Lullian definitions, and into the
questions and rules.
Now these definitions of his were based on how he felt the world
functioned. He proposed, in fact, a vision of reality which was as
novel as the system he built. He said that nothing whatever (and of
course for him, much less God) was inactive. Nothing just sat there
being itself; it also did whatever its nature called upon it to do.
He often used the analogy of fire which wasn't only a thing in
itself, but also was active in the production of heat. So also was
goodness not only a thing in itself, as, for instance, an essential
attribute of God, but it also produced goodness, and this in two
ways: interiorly making His greatness, etc. good, and exteriorly
creating the world's goodness (or lack of it where evil was
concerned). Here again he frequently used the analogy of fire, which
in itself creates a flame and heat, and exteriorly, as he said,
causes the water in a pot to boil. Moreover, anything active has to
have a point of departure (in the case of the thing that produces
good, he called it "bonificative"), an object which it affects
(the "bonifiable"), and the act itself going from one to the other
(that is, which "bonifies"). And it wasn't only God's
attributes that were active in this way; every rung of the scale of
being was similarly articulated with the three correlatives (as he
called them) of action. At the bottom of the ladder, fire had its
"ignificative", "ignifiable", and "ignifies", and in the
middle, the human mind had "intellective", "intelligible" or
"understandible", and "understanding". The world was thus for him
a vast dynamic web of ternary relations working both individually or
interiorly, as I said before, and exteriorly one upon the other. It
was this web of relations that was implied by his definitions. For
example, "goodness" the first component of Figure A, he defined
interiorly as "that thing by reason of which good does good". But
notice how the exterior definition of the second component,
"greatness", as "that by reason of which goodness, duration, etc.
are great", implies that even goodness could also be defined similarly
in terms of the other components of Figure A. So these definitions,
which to some commentators have seemed simply tautological, in fact
imply a dynamic reality articulated in a large web of interrelations.
Now this definitional doctrine turns up under one of the questions of
the third column of the Alphabet of the Art. Not under the first
question of "whether?" which inquires into the possibility of a thing
existing, but under the second which asks "what" a thing is. This
question (or rule, as Llull also calls it) is divided into four
species. In the Ars brevis Llull uses the example of the intellect
instead of goodness to illustrate how it works, saying that "The
first [species] is definitional, as when one asks, What is the
intellect? To which one must reply that it is that power whose
function it is to understand". Notice how this is identical with that
of "goodness" as being "that thing by reason of which good does
good". The second species goes further and asks, "What does the
intellect have coessentially in itself? To which one must reply that
it has its correlatives, that is to say, intellective, intelligible,
and understanding, without which it could not exist, and would,
moreover, be idle and lack nature, purpose, and repose". This
refers, of course, to the ternary dynamic structure we already
mentioned. We're also by now familiar with the third species, which
is when one asks, "What is the intellect in something other than
itself? To which one must reply that it is good when understanding in
goodness, great when understanding in greatness, etc.". Here we
are with the equivalent of "greatness" as being "that by reason of
which goodness, duration, etc. are great" which we saw before. The
rest of the questions and rules continue in the same vein, carefully
distinguishing the different ways in which one can formulate questions
such as "of what?" which inquires about material qualities,
"why?" which asks about formal causes, "how much?" concerning
quantity, "of what kind?" concerning quality, and so oil.
So when Llull starts combining elements of the first two figures to
answer questions or make proofs, he carefully shores up his arguments
with the appropriate definitions and rules. 1 won't show you how this
works in practice, because it would involve delving into too many
minutiae of his explanations. I would just like to make a few general
remarks. The first to answer a doubt that has probably occurred to
you: how can Llull prove anything useful if, as I said before, he
limits himself to such divine attributes such as goodness, greatness,
etc., which seem hopelessly vague and general in nature? The answer
is that in the first place he occasionally lets one see how definitions
can be more widely applicable than they might seem. In the above
definition of the intellect; for instance, when he says "it is good
when understanding in goodness, great when understanding in greatness,
etc.", he adds "and [it is] grammatical in grammar, logical in
logic, rhetorical in rhetoric, etc.", so right away we are applying
these concepts to other fields. Secondly, notice how in the
Alphabet, the fourth column of "Subjects" is a ladder of being in
which "everything that exists is implied, and there is nothing that
exists outside it", as Llull says in the Ars generalis ultima
(IX, 1). The ninth chapter of that work offers a detailed study
of each rung in terms of the 18 principles of Figures A and T, and
in terms of the 9 rules. The last rung of Instrumentative includes
the moral instruments of the virtues and vices which appear in the last
two columns of the alphabet, and which, in his more popular works,
Llull uses as important tools of persuasion. In yet another adventure
into outside material, Llull presents a chapter on "Application"
which gives definitions of what he calls the "Hundred Forms" to
which the mechanisms of the Art can also be applied. Here he includes
every subject imaginable: physical, conceptual, geometrical,
cosmological, social, etc. Lest you think we're still operating in
a sort of misty area of vague generalities, let me offer the
counter-example of Form no. 96 on Navegation in the Ars generalis
ultima, which in fact consists of a little five-page manual with
worked examples of how to find your position at sea!
Your chief objection that this continual reference to the real world
(in the Platonic sense that Llull understood it) on which the Art
is firmly based, places it at an opposite pole from any kind of formal
logic is undeniable. As I said before, however, Llull was aware of
this point, and was at pains to make clear that his Art was neither
logic nor metaphysics. My feeling, however, is that the Platonic
basis of his system is not without historical or conceptual interest;
we must remember Leibniz's comment that if someone could reduce
Plato's thought to a system, he would render humanity a great service
[8]. Secondly, Llull's invention of an ars combinatoria as the
only possible way of dealing with interrelationships of Platonic
forms, was to have a considerable impact in the Renaissance, and
would, as my colleague Ton Sales will explain, have a decisive
influence on Leibniz.
In one sense, however, Llull's system was more abstract and more
amenable to analysis by modern mathematical methods. This was in his
attempt to systemize not only totally but even semi-mechanically its
all-embracing relational nature. This is, of course, what we mean
when we say that he developed an ars combinatoria. His use of graphs,
along with their alternate representation as matrices, to display the
relational structure of his system shows a certain understanding of the
general nature of the problem. But there is another aspect of his
system which also has curious modern parallels.
The first period of the Art, which we haven't touched on today,
very frequently developed its arguments or proofs by pairwise comparison
of concepts. Let me briefly show one such proof from a central work of
that period, the Ars demonstrativa. Here he always starts his proofs
with a series of concepts within what he calls a compartment (or
camera), as you can see in the figure.
The four words not presented by letter symbols come from a Figure X,
which disappeared from the later version of the Art, and as you can
see, they represent opposites, "privation" being a synonym of
"non-being". Notice also the words "contrariety" and
"concordance" written above the compartments, which you will
recognize as coming from Figure T.
Now A stands for God, and it is double because he is exploring two
hypotheses, a positive and negative. The positive one presents no
problem: if God exists, there exists a perfect being contrary to
privation (or non-being) and imperfection. If, however, God does
not exist, then all being has some imperfection, and the only thing
that's perfect is non-being or privation, which of course accords
with imperfection. Since the concordance of perfection and
imperfection is clearly contradictory, the existence of God has been
proved by reductio ad impossibile. I won't explain the second half of
the proof, except to say that it functions similarly.
As you can see the technique of beginning with a hypothesis and working
down a branching structure to a confirmation or refutation, bears a
certain resemblance to the tableaux methods of Gentzen, Beth and
Smullyan. Notice furthermore how it works by a series of pairwise
comparisons.
Which brings me to a further curious piece of evidence recently brought
to light by two English scholars. In the social sciences, the modern
deductive theory of voting was initiated in the 1950's by Arrow
(1951) and Black (1958), with techniques of paired
comparisons which in graph theory are called "tournaments". Now the
usual history of voting theory says that they were preceded by two
Frenchmen, Borda and Condorcet in the l8th century, whose
discoveries were forgotten and repeated from scratch by Lewis
Carroll, whose work was again utterly neglected. What Mclean &
London (1990 & 1992) have shown is that Condorcet and
Borda were preceded by half a millenium by Ramon Llull, "who made
one of the first systematic contributions to the deductive theory of
voting", and this with slightly varying systems presented in two
different works. One is aptly called the Ars electionis, but the
other one is, of all things, embedded in the novel Blaquerna, where
Llull uses it to explain how nuns should elect their abbess!
What's significant about this, it seems to me, is not so much Llull
as the neglected genius, but rather as a thinker with enough breadth of
vision to see in his discoveries a generality greater than the initial
uses for which they were intended. To a professional mathematician of
the late 20th century the connections between "tournaments", graph
theory and combinatorics is obvious, but that a l3th-century
Majorcan missionary should have seen the connection is, I think
interesting.
I would like to end on a more personal note, or what in the scientific
community could fall under the euphemistic heading of a call for
papers. On the negative side, we have shown that Lull's Art was
not a formal logic, but the positive side is unusual and still in many
ways in need of explanation. It was highly structured system, to the
point of being semi-mechanical. And the more one deals with it, the
more consistent and interesting it seems to become. Lastly, its
structure was relational and combinatorial, thus mirroring a world
which Llull saw as primarily relational. Might these factors not make
it possible to program at least part of the Art in a relational
language such as Prolog? And if so, might this not clarify to us,
that is, by putting it into modern terms the functioning of this
lath-century computer? The basic problem, as I see it, is that
here we have inherited an ancient computer made of parchment and ink,
but along the way the manual got lost. We have many of the materials
to make a new one, and if you ask, well, what use would it be, I
would answer what a professor from New York University answered some
years ago. He was an arachnologist, and when a reporter asked him
what good spiders were, he replied, "Spiders are damned
interesting, that's what good spiders are".
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