4. Universal logic takes the notion of structure as a starting
point; but what is a structure, should not the notion of structure
be analyzed also from a logical viewpoint ? If so, aren’t we in the
sin of circularity ?
Here we are faced with some of the most favourite problems of logicians,
those who are fond with gossiping about Buridan’s donkey that bites
his own tails, the barber who shaves his own wife or the fool who claims not
to be a fool, and so on. I have to say that I’m hardly interested with such
problems, and here I agree with Wittgenstein when the latter suspected the
Paradox of the Liar to have absolutely no logical philosophical relevance. I
don’t intend to go any further into some Lacanian analysis, but it seems to
me that such problems are somehow infantile. Many paradoxes are nothing
but toys and those who play with them often have a mental age of six or
seven.
As it was rightly stressed by the very witty Baron of Chambourcy:
“Si les math´ematiques ne sont qu’un jeu, je pr´efere jouer a la poup´ee” (“If
mathematics is just a game, then I prefer to play with dolls”). The notion
of structure is much more than a mere toy, but that doesn’t prevent it
from being a funny thing. First and foremost, let us stress that the notion
of structure doesn’t reduce to the notion of mathematical structure and
therefore, any logicist who would reduce mathematics to logic couldn’t spell
out the concept of structure. The notion of structure largely goes beyond
the mathematical area, and Bourbaki said himself that he was influenced
by such linguists as Benveniste. During the sixties, “structuralism” was
meant as a large movement that mainly occurred in human sciences. But
structuralism as we understand it is something still larger that includes
linguistics, mathematics, psychology, and so on.
In his book entitled
Pens´ee formelle et sciences de l’homme, Granger
makes some rather interesting comments about the source of structuralism
in the wide sense. Now what concerns us are not so much historical
and sociological considerations about the development of structuralism,
but rather the issue of the ultimate view of structuralism as underlying
mathematical structuralism and universal logic.
The view is that there is no object in itself, that any object is defined
by the relations it bears with any other objects within a structure; that
is typically the analysis Saussure offers for language: nasty only makes
sense with respect to angry, nice, and so on. Moreover, any object x in
a structure can be identified with an object y in another structure if one
considers that both behave in a similar way within some similar structures.
This makes translations possible. If Quine had read Saussure, he would
have relativised his thesis about indeterminacy of language.
Contrarily to what one could expect, there is presently no general
mathematical theory of structures. Some elements can be found in Bourbaki,
universal algebra, category theory, or model theory, but nothing conclusive.
Universal logic can contribute itself to the development of a general
theory of structures in stating and solving such crucial issues as for example
identity between logical structures. When and how two mathematical
structures are identical is a problem of crucial import in the theory of structures.
The notion of isomorphism is too weak to be satisfactory. The point
is to be in position to identify structures of different sorts. In the history
of mathematics, a canonical example is identification between an idempotent
ring and a complemented distributive lattice by Marshall Stone, both
being two equivalent formulations of what is called a Boolean algebra. The
concept that helps to account for the identification as revealed by Stone
appears nowadays as a concept from model theory, namely: the notion of
expansion by definition.
Now it happens that when we try to apply such a concept to the
identity between logical structures, we are then faced with various problems
that betray its very deficiency. Thus we are led to put such a series of
questions as the following: do a structure and one of its expansion really
have one and the same domain ?
To sum up, universal logic conclusively helps to make think us about
the nature of a structure, and this is much more significant than to solve
paradoxes about donkeys or monkeys.
5. How and when does universal logic begin? Who is really
the pioneer of universal logic?
The real starting point is in the 1920’s, when Hertz on the one hand
and Tarski on the other hand make a jump into abstraction and are interested
with general theories that give rise to the study and development
of various systems. Tarski’s stance is a characteristic one: whereas
Lukasiewicz develops many-valued logic for the philosophical purpose to
solve questions about determinism, the former takes this as a tool in order
to elaborate a general theory of logic. Lindenbaum goes towards such
a trend, too, while proving several crucial theorems. In Poland, Lo´s and
Suszko pursue this line after the World War II, namely with their joint
paper “Remarks on sentential logics” in which they introduce the notion of
structural logic. While Lo´s gave up to logic and turned to economy, Suszko
pursued his works and developed with Bloom et Brown what he called “abstract
logic”. After his death, these works were pursued by Czelakowski in
Poland and by Font and Jansana in Barcelona.
One word should be said within this Polish trend about the French
logician Jean Porte, whose book entitled Recherches sur la th´eorie g´en´erale
des systemes formels was published in 1965 and contained some results from
the Polish school. Porte’s book is very interesting, because he clearly and
overtly argues for the independence of logic from the issue of mathematical
foundations, so that he rejects logic as metamathematics. On the other
hand, Porte distinguished logic from algebra, and that is not always the case
with Polish people who regrettably tend to assimilate logic with universal
algebra. Porte was a PhD student of Ren´e de Possel, one of those who
founded Bourbaki. Porte’s book didn’t have much influence unfortunately,
and this may be for several reasons: he was a forerunner, the book is
written in French and hasn’t been translated, Porte went to Africa and
stayed there many years in isolation from the community of logicians.
It is in the 1980’s that the trend of universal logic actually became
prominent. Issues about mathematical foundations were already eclipsed
in logic at that time. Logic was revived by some “practical” questions
from AI, linguistics and computer science. Many non-classical logics were
considered: non-monotonic logics, substructural logics, together with all
the conceivable variants of modal logics. General techniques of systematisation
started to be developed. Either old techniques were studied again
and reworked such as logical matrices, consequence operator (as used by
Makinson for investigating AGM theory of belief revisions as well as nonmonotonic
logics), sequent calculus (substructural logics); or new techniques
were developed such as LDS (Dov Gabbay’s Labelled Deductive
Systems).
Apart from some very active and dynamic groups, like Gabbay’s in
London and van Benthem’s in Amsterdam, some works from isolated people
like Epstein, Cleave, or Koslov should also be mentioned.
6. How did you come to universal logic?
I explained this at length in a paper entitled “From paraconsistent
logic to universal logic”. So I’ll merely sum up. During the eighties, I
was studying logic in Paris and observed the rise of all these new logics.
On the one hand, I attended a logical course with Jean-Yves Girard who
presented us in a unified and comparative way classical, intuitionistic and
linear logics through the sequent calculus ; on the other hand, I attended a
course with Daniel Andler who presented us a complete list of the new logics
(default logics, and so on). I myself discovered by chance paraconsistent
logic from the Brazilian logician Newton da Costa, a very unknown logic
at that time, and I was particularly interested with it because I wanted to
know whether one could still consider as a logic one in which the principle of
contradiction does not hold. Then I was quickly convinced that one could,
and was increasingly concerned with general techniques as used to generate
this sort of logic, especially with the theory of valuation as developed by
da Costa and on which I worked with him during a first stay in Brazil,
in 1991. Then all followed in a quick and natural way: I found Porte’s
book that contained some similar ideas to mine, and this ensured me in my
own researches. Then I went to Poland, in order to get acquainted with
Polish works da Costa had told me about and Porte mentioned in his book.
During my stay in 1993 at the University of Wroc law, Poland, I decided
to employ the expression “universal logic” that would appear later in the
title of my PhD, written in 1994 and defended in 1995 in the department
of mathematics at the University of Paris 7 under the supervision of Daniel
Andler.
Then the story goes on all over the world. I travelled a lot, and
the view of universal logic made its way too. The 1st World Congress of
Universal Logic took place in Montreux in Spring 2005, gathering about
200 logicians from 40 different countries. The book Logica Universalis was
launched on that occasion by Birkh¨auser.
Finally, I want to precise my own contribution: it is difficult to say
who has created the expression “universal logic” or used it for the first time,
what I did is to use it to mean “a general theory of logics”. Furthermore,
there are several ways of framing a general theory of logics and, as we just
saw it, a large trend developed around this since twenty years. I do not
see universal logic as a general theory among others but as a concept, an
expression designed to depict such a whole trend.
7. Is universal logic a new way to view logic?
Surely. The view that dominated in the beginning of the 20th century
and still dominates in some way is a hybrid view in which some rather
different influences are mixed, namely: formalism, linguistics, and logical
atomism. This can be seen as a rather monstrous, inconsistent whole.
To give just one example: the traditional distinction between syntax and
semantics. What does it really mean? Does it have a foundation? And, if
so, which one? Syntax only means the construction of a formal language for
some, and for others it also includes what is called proof theory; for others,
like Chang and Keisler, it concerns all what is recursive, in particular they
call syntax the semantics of truth tables for sentential logic.
A more reasonable thing would be to make a distinction between
model theory and proof theory, but even such a distinction is questionable
because there are a lot of intermediary theories, e.g. Beth tableaux.
The path from proof theory to model theory could be said to be a continuous
one ; when one comes out from the land of proofs and enters into the
land of models, it is difficult to know, this is an issue we’ll leave for bald
persons who like to sit on heaps of rice.
What is crucial in universal logic is that logics are considered irrespective
of the way they are generated, so that one thus makes a jump into
abstraction. And this is not surprising at all, it’s the most natural thing
you could have. Classical propositional logic can be generated in a hundred
different ways, through Hilbert systems, Gentzen systems, tableaux, two-,
three- or infinite-valued semantics. What is this object that can be defined
in so much different ways? Everybody believes in it, and nobody would
venture to claim that classical propositional logic reduces to one particular
way of constructing it.
Universal logic consequently brings an answer to this question, saying
that classical propositional logic is a logical structure in just the same way
as intuitionistic or linear logic. Hence this helps to throw some light on the
connection between various ways to generate a given logic, as well as on
the relation between different logics.
8. What are philosophical consequences of universal logic?
They are tremendous, since universal logic gives a way to bring every
logical philosophical problem into some new light. Given that the traditional
view of logic is highly obscure, so is the philosophy connected to
it.
Let us take a canonical case, namely the famous distinction by Susan
Haack between logics as deviations from the classical one and logics as
conservative extensions from this same logic. Here is prima facie something
like a nice and easy thing to understand: for instance, the modal logic S5
is a conservative extension of classical logic since additional operators are
added that don’t alter the previous content, whereas intuitionistic logic is
a deviant logic with respect to classical logic because properties of negation
and implication are altered. However, such a nice distinction vanishes once
one sees that classical logic is definable within intuitionistic logic. Then
intuitionistic logic appears in some sense as a conservative extension of
classical logic.
The trouble with Haack’s distinction is that it doesn’t rest upon any
serious and systematic theory, but only on some ideas thrown in the air
and explained and justified with basic elementary examples. On this respect,
philosophers of logic are not so much different from other superficial
philosophers like Deleuze or Lacan. That logic is unclear itself is certainly
an excuse for them, but the task for a philosopher is to clear up confusion,
not to adorn it with nice concepts. Their behaviour is unproductive and
doesn’t bring any real understanding.
Philosophy and logic should not indeed be viewed separately. In order
to catch the difference between deviant logics and conservative extensions,
some news concepts and an entire theory are required, and universal logic
turns out to be a framework for this purpose. In order to construct such
a theory, one needs to be a philosopher, that is, to try to understand how
things are. Every good logician is a philosopher. Others are just applying
and reproducing some devices at their disposal. This equally holds for logic
and for science in general. On the other hand, any philosopher of science
who is not a scientist cannot be taken seriously; to borrow a favourite view
of Newton da Costa, it’s like a priest philosophizing about women. How to
take seriously a philosopher of logic who had never proved any theorem ?
He is a historian of logic, at the very best and, at worst, a charlatan who
talks about something he doesn’t understand.
