Béziau 13 Questions about Universal Logic- Cont. preguntas 9 a 13 Bibliografía

9. What is the connection between universal logic and history

of logic?

Roughly speaking, there are two ways of doing history of logic or

history of science in general. The first can be called the philosophical

one: priority is given to texts and source materials, all the time is spent

describing who said what, who inspired who. The second, that can be

called the problematic one, consists in trying to understand what someone

understood from the perspective of a given problem.

The philological, bookworm’s approach, is fruitless and merely adds

some additional volumes that will serve as further food for worms. On the

other hand, the problematic approach is fruitful and brings theories back

to life, it constitutes some witty dialogue over the centuries. Such was

the move followed in logic by people like Jan Lukasiewicz and Abraham

Robinson. Lukasiewicz developed many-valued logics in order to solve the

problem of future contingents and determinism; whereas Robinson developed

non-standard analysis in order to explain infinitesimals. Here are two

great theories that brought some considerable advance to human mind,

whereas philologists have discussed during several centuries and are still

endlessly discussing about whether or not Aristotle did admit the principle

of bivalence, or whether it was Newton or Leibniz who developed infinitesimal

calculus.

History of science, the problematic one, is crucial for any science, since

each science is a historical process that expands throughout the ages but

not always in a linear way. One direction formerly discarded may well be

taken again later, as was the case with infinitesimals. Thus we have to keep

track to the past since it may always prompt inspiration.

Some people like van Heijenoort promoted the view that modern logic

entirely went as ready-made out of Frege-the-Genius’ head and represented

some fundamental break with all previous habits. Wittgenstein boasted

that he had never read Aristotle. It is true that to create something new

requires not to have the mind full with a host of outmoded theories, and no

Aristotelian professional philologist could have ever written the Tractacus

Logico-Philosophicus.

However, turning back to Lukasiewicz, we see that he developed equally

innovative views as compared with Frege and Wittgenstein while reading

Aristotle in Greek, but he read it critically and problematically. Lukasiewicz’s

book On Aristotle’s Principle of Contradiction, published in 1910, served

as a starting point for the Polish logical school Tarski originated from, a

school that dominated logic throughout the twentieth century and, as was

said earlier, Tarski can be properly seen as the major forerunner of universal

logic. Another emblematic character in the prehistory of universal logic,

namely Paul Hertz, considered that the cut rule from his abstract system

of logic was nothing but another formulation for the Barbara syllogism.

The problematic history of logic is part and parcel of universal logic.

From the standpoint of universal logic for example the square of oppositions

may be entirely reconsidered. Such a square displays a theory of oppositions

by distinguishing several types of opposition. Some much subtler theory can

be developed in the light of modern logic, first by turning the square into

a hexagon, following Robert Blanch, and then into a polyhedron. These

transformations are not mere geometrical ravings, given that a general

theory is thereby elaborated that connects various types of negations and

modalities. Such a problematic approach to the square of oppositions is

completely opposed to the philological one, in which one just quibbles about

small variations in the square of oppositions.

10. What is the connection between universal logic and natural

or informal logic? Is universal logic a theory of reasoning,

or argumentation?

Evidently classical logic is not a good account of our way of reasoning

in everyday life, so, many other logics were constructed, the so-called

non-classical ones that would give a better account for natural reasoning.

However, such logics as relevant or paraconsistent ones, are nothing else

than variants of classical logic, constructed from some similar ontological

ground and relying upon a formalist view of logic, among other things.

Some wanted to go further and out of the formal framework, namely those

working in informal logic or the theory of argumentation. The trouble is

that one runs the risk of being tied up again in natural language, while it

has nothing sacred as such.

Such a rejection of the formal, which brings very often back to the

cosy little nest of natural language, turns on some confusion in assimilating

the formalist doctrine with mathematics, a confusion generated by

formalists themselves. Now it is clear that mathematics don’t need to be

connected with the formalist doctrine, and a mathematical theory can be

well developed irrespective to this confused formalist jumble in which such

a sentence as

Santa Claus lives in Lapland is nothing but a sequence of

signs called “formula”.

The idea of universal logic is to deal with any types of reasoning,

whether men’s, women’s or even dog’s ones, not by returning to the natural

language but by developing a mathematical theory free from the formalist

jumble.

What we must pay attention to, when developing a theory of reasoning,

is the connection between the problem at hand and this theory. It

turns out very often that the link between both is too smooth. It is typically

the case with relevant logic. The basic point in such a logic is to say

that some meaning connection should occur between premises and conclusion

of an argument ; now instead of rejecting the paradigm of structural

logics in which the substitution theorem holds, relevantist partisans go on

working within the traditional atomist formalistic framework and require

for premises and conclusion to have at least one atomic sentence in common.

That is a very narrowed and unsatisfactory way to account for the

meaning connection between premises and conclusion.

From the perspective of universal logic, there are much more elegant

and significant ways to proceed.

11. What are the applications of universal logic?

Universal logic considers the world of all possible logics and ways

to construct them, so that it gives a way out of many requirements and

problems.

Let us imagine a given Mr Ixman; he comes to see you, says he needs

a logic accounting for some given situation, say medicine, and gives you

an exposition of its typical problems. Universal logic gives rise to a quick

diagnosis. You see what is specific to the situation and what is universal,

common to some other sorts of reasoning, so that you are able to build a

logic that fits the bill. Mr Ixman points out to you the issue of contradictory

diagnosis, for instance, that one and the same symptom could be analysed

in a different ways by a physician, or even by different kinds of medicine,

and you see that therefore some paraconsistent logic should be used. He also

insists that we are only given incomplete sets of information in medicine and

any further information may lead to challenge the first diagnosis. Hence a

paraconsistent, paracomplete and non-monotonic logic will be needed. And

so on, so that after having listed all what Mr Ixman has to say you’ll be in

position to supply him with the proper tool for an analysis of reasoning in

medicine. For this purpose, you’ll have use general techniques that help to

construct various logics and to combine them.

Hence universal logic allows understanding some particular reasoning

in supplying one with a tool box that serves to construct a logic accounting

for that sort of reasoning; moreover, it allows locating such a new born in

connecting it with the set of conceivable reasonings. Such a technique as

combination of logics is very important. The art of combining logics is

somehow like that of setting mayonnaise: you have different ingredients

such as temporal, deontic or erotetic operators, for instance, and you want

to bring them together into one consistent whole that will account for some

particular reasoning.

Universal logic plays a crucial role with respect to AI, expert systems

and automated reasoning, since it helps to develop systems adapted to the

most various data: that is called ”logic engineering”. It is clear that some

given technique, some specific logic cannot solve every problem ; there

is no miraculous universal logic, a logic, gift of god that would apply to

any situation. However we can have a science, universal logic, that allows

proceeding in connection with reality because it happens to be itself in a

continuous interaction with reality. Universal logic is not a fixed theory,

it’s a progressive science in which the study of particular cases is always

significant for the development of abstract reasoning that, in turn, will be

fruitfully applied.

Universal logic is not cut off from reality, as is the case of Aristotelian

syllogistic or first-order logic. It is a useful theory.

12. Could you give an overview of the main problems and

prospects in universal logic?

First there is a series of questions about the nature of logical structures.

Several types of structures can be considered and, depending upon

the choice to be made, different results are obtained. For instance, classical

propositional logic is decidable as a structure with a unary predicate that

corresponds to the set of tautologies, but this is not so if it is considered

as a structure with a consequence operator or relation, with no restriction

on cardinality.

Another question may be then put, that is, the equivalence between

various logical structures. Can both structures be said to correspond to one

and the same logic while differing with respect to one fundamental property,

that is decidability? Another crucial question related to equivalence

between logical structures is the question of connections between different

logics: when can a logic be considered as weaker or stronger than another

one, as an extension of another one, as merging or being translatable into

another one?

Then comes the question about the combination of two logics: how

can we form from two logics a third one that is their combination? Such

question is directly related to the former one, since combination is defined

very often as the smallest conservative extension of combined logics. Now

such a definition is unsatisfactory, because two logics may have no common

conservative extension while being combinable.

These three questions, that is, identity of logical structures, connections

between logical structures, and combination between logical structures,

are part of what may be called the heart of universal logic.

Further questions are somehow related to these, and other problems

will remain confuse as long as no satisfactory theory or clear insight will be

obtained for these questions. But to study such other less central problems

also gives rise to some evolvement, especially because any abstract theory

is not a pure abstraction but an abstraction of something else; to consider

what exemplifies abstraction is to make some advance in elaborating the

latter.

Therefore, it is also useful to work on the systematisation of some

classes of logics like modal, non-monotonic, paraconsistent logics, and so

on. This is indeed a dialectical movement between the general and the

particular, given that the basic concepts of universal logic are not only

designed from such specific classes but applied back in return.

Methods for generating various logics should be taken into account,

namely: logical matrices, tableaux, Kripke structures, proof systems, and

so on. Some attention will be paid also to the scope of validity and application

of important theorems like interpolation, definability, cut-elimination,

and so on. There is also the historical and philosophical dimension we

already mentioned.

To sum up, we can distinguish five groups of research which are mutually

interrelated:

1) Basic concepts (identity, extension, combination)

2) Systematic study of classes of logics

3) Tools and building methods for logics

4) Scope of validity of important theorems

5) Historical and philosophical aspects.

13. What is the future of universal logic?

Universal logic is about to expand naturally and will plausibly become

soon the mainstream in logic in a short time, supplanting “formal logic”,

“symbolic logic”, or “mathematical logic”. It helps logic and logicians to be

again meaningful. It helps logicians with very distinct concerns to keep in

touch together. At a certain time, logic splashed in every direction; at some

point it lost its way or specialized into unintelligible branches, except for

small circles of specialists or even only one guy. Thanks to universal logic,

logicians find themselves back in a common ground in which communication

is possible, because of the very nature of universal logic, namely: the study

of the most general and abstract properties of the various possible logics.

In concrete terms, a 2nd World School and Congress on Universal

Logic should take place in China in 2007 following the first event, 1stWorld

School and Congress on Universal Logic, that took place in Montreux in

spring 2005; the story should continue with biannual meetings. Concerning

publications, after the book Logica Universalis, published by Birk¨auser,

some other books should be published within the scope of a series Studies

in Universal Logic with the same editor. The launching of a new periodic

journal Logica Universalis is also projected with Birkh¨auser in 2007.

Acknowledgements

This work was supported by a grant of the Swiss

National Science Foundation. Thanks to Fabien Schang for helping transcribing

the text.

References

[1] Jean-Yves B´eziau,

Universal logic, Logica ’94, P. Kolar et V.

Svoboda (eds.), Acad´emie des Sciences, Prague (1994), pp. 73–93.

[2] Jean-Yves B´eziau,

Recherches sur la logique universelle, PhD Thesis,

Universit´e Denis Diderot (Paris 7), 1995.

[3] Jean-Yves B´eziau,

The philosophical import of Polish logic, [in:]

Methodology and Philosophy of Science at Warsaw University

M. Talasiewicz (ed), Varsovie (1999), pp. 109–124.

[4] Jean-Yves B´eziau,

From paraconsistent logic to universal logic,Sorites12 (2001), pp. 5–32.

[5] Jean-Yves B´eziau (ed),

Logica Universalis, Birkh¨auser, Basel 2001.

[6] Garret Birkhoff,

Universal algebra, [in:] Comptes Rendus duPremier Congres Canadien de Math´ematiques

, Presses de l’Universit´e de Toronto, Toronto, 1946, pp. 310–326.

[7] Jan Lukasiewicz et Alfred Tarski, Untersuchungen ¨uber den Aussagenkalk¨ul

, Comptes Rendus des s´eances de la Soci´et´e des Sciences

et des lettres de Varsovie XXIII, Classe III, 1930, pp. 30–50.

[8] Stephen L . Bloom, Donald J. Brown et Roman Suszko,

Some theorems on abstract logics, Algebra and Logic 9 (1970), pp. 165–168.

[9] Nicholas Bourbaki, L’architecture des math´ematiques, [in:] Lesgrands courants de la pens´ee math´ematique F. Le Lionnais (ed),1948, pp. 35–48.

[10] Donald J. Brown et Roman Suszko, Abstract logics, Dissertationes

Mathematicae 102 (1973), pp. 9–41.

[11] John P. Cleave, A study of logics, Clarendon, 1991 Oxford.

[12] Richard L. Epstein, The semantic foundation of logic, Kluwer,

1990 Dordrecht.

[13] Gehrard Gentzen, ¨ Uber die Existenz unabhangiger Axiomensysteme

zu unendlichen Satzsystemen, Matematische Annalen 107 (1932),

pp. 329–350.

[14] Gilles-Gaston Granger, Pens´ee formelle et science de l’homme,Aubier Montaigne, Paris.

[15] Paul Hertz, ¨ Uber Axiomensysteme f¨ur beliebige Satzsysteme,Matematische Annalen

101 (1929), pp. 457–514.

[16] Arnold Koslow, A structuralist theory of logic, CambridgeUniversity Press, 1992 New-York.

[17] Jerzy Lo´s et Roman Suszko,

Remarks on sentential logics, Indigationes

Mathematicae 20 (1958), pp. 177–183.

[18] Jean Porte, Recherches sur la th´eorie g´en´erale des systemes

formels et sur les systemes connectifs, Gauthier-Villars, Paris

et Nauwelaerts, Louvain 1965.

[19] Alfred Tarski, Remarques sur les notions fondamentales de la

m´ethodologie des math´ematiques

, Annales de la Soci´et´e Polonaise de Math´ematique

(1928), pp. 270–271.

[20] Alfred Tarski, ¨ Uber einige fundamenten Begriffe der Metamathematik,

Comptes Rendus des s´eances de la Soci´et´e des Sciences et

des lettres de Varsovie XXIII, Classe III (1930a), pp. 22–29.

[21] Alfred Tarski, Fundamentale Begriffe der Methodologie der deduktiven

Wissenschaften. I, Monatshefte f¨ur Mathematik und Physik

37 (1930b), pp. 361–404.

Béziau..13 Questions about Universal Logic - Cont- preguntas 4 a 9

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.

Béziau-13 Questions about Universal Logic- continuación. Preguntas 2 y 3

2. How is it possible to develop a general theory of logics, to

unify logics so various as quantum logic, erotetic logic or fuzzy

logic?

In order to solve such a question, we have to ask, how two different

systems can be considered both as logics, and this naturally leads to ask

what a logic is. That is the central point. The key of the problem.

Even a Girard, a Brady or a Hintikka would admit that, while anything

cannot be considered as a logic, there are different logics, their is not the

only one even if it appears to them as the only true one, as depicting the

reasoning most adequately. In fact their systems are like many other ones,

whether concerning their properties or the technicalities displayed in order

to elaborate them.

Hence, it seems natural to consider what is commonly shared by all

logical systems. Such is the approach of universal logic. Now what does

mean all logical systems: all systems called logical? Recognized as logical?

Or every possible and conceivable systems? What is the criterion according

to which we can say that such a thing is a logic and such another one has

nothing to do with a logic, is only a paralogic or something totally illogical?

Universal logic cannot be a descriptive theory: it cannot claim to describe

what is logical in a variety of systems considered as logics by the

people or the elite. No theory in human science is a purely descriptive one:

it seems impossible to account for an inconsistent variety of various viewpoints,

some of which appear to be completely arbitrary ones, unless some

very special logic is used for this purpose like Bychovsky’s paraconsistent

turbopolar logic.

On the other hand, to develop a theory that would be a purely normative

one, imposing some viewpoint that has just a slight bearing to what

is ordinarily called logic or logics, wouldn’t appear to be satisfactory at all

unless it is some genial view that would give us a new insight, making us

realize that we were entirely mistaken. But if so, the theory would not be a

properly normative one, it will impose the force of a description we didn’t

already know. It cannot be said that the Einsteinian theory is more normative

than the Newtonian one. In any case we have to vacillate between

normative and descriptive. We have to be cautious concerning variety while

having some unitary view that doesn’t reduce to such a variety.

The basic view of universal logic is double, inspired both by Tarski

and Birkhoff. From the late twenties, Tarski suggested its theory of the

consequence operator that is a very general theory of the notion of logical

consequence, making abstraction of the logical operators. He thus made a

jump into abstraction. Laws of logic don’t appear any more as for example

laws concerning negation such as principles of contradiction or excluded

middle, but as laws ruling the notion of consequence: self-deducibility,

monotony, transitivity. However these very laws can be and have actually

been criticized, so that the view is to reject any law, any axiom, and even

those located at a more abstract level. This may appear as totally absurd,

prima facie

.

Then Birkhoff comes into play. He himself developed a general theory

of algebra from a primary notion of algebraic structure not obeying

any axiom, whereas its predecessors sought to unify algebra around such

very general laws as associativity or commutativity. But as he aptly said

himself, such a unification was no more possible to a certain stage, and

especially it was not possible to unify two large trends, algebras studied by

the Noether school, on the one hand, and, on the other hand, the Boolean

trend including the notion of lattice as developed in particular by Birkhoff

himself. Thus Birkhoff developed universal algebra without taking axioms

into account.

Such a surprising approach can be called a conceptual one, as opposed

to an axiomatic one. Category theory is itself more conceptual than

axiomatic. The point is not to produce a large axiomatic system like ZF

set-theory from which everything could be deduced; rather, it is to elaborate

some concepts that could serve to describe the whole of mathematical

phenomena in a unitary fashion.

The approach of universal logic is also a conceptual one, where the

point is to capture the whole logical phenomena, not to be looking for

some axiomatic Graal or genuine laws of thought or reality, from which

everything could be deduced.

3. What is meant exactly by a logic according to universal

logic ? You often refer to Bourbaki, although the latter is often

considered as a suspicious guy by logicians.

According to universal logic, a logic is a certain kind of structure. The

project of universal logic is in the spirit of modern mathematics. As it is

well known, from the 1930’s onward, Nicolas Bourbaki made the proposal

to reconstruct the entire mathematics through the notion of structure.

For Bourbaki, any mathematical object does only make sense from the

perspective of a structure or, better, of a set of structures. The number 4

does not exist in itself and per se, but as connected with other numbers

that form the entire structure of natural numbers. Now its existence is not

confined to the structure of natural numbers, it also extends to the structure

of integers, rational, real numbers, and so on. So such connections between

these various structures also characterize what the number 4 is.

Bourbaki’s insight consists in reconstructing every mathematical structure

from some “fundamental structures” or “mother structures” through

a crossing process, which gives rise to “cross-structures”. He distinguishes

between three sorts of basic structures, namely: algebraic structures, topological

structures and structures of order, and reconstructs the structure of

real numbers as a crossing between these three fundamental mother structures.

The idea of universal logic is that logical structures are fundamental

ones but departing from the Bourbakian trinity. Note that this is not

in opposition with the insight of the very famous General, given that he

admitted the plausible appearance of other core structures. What does

matter with such a perspective is that we argue against any reduction of

logic to algebra, since logical structures are differing from algebraic ones

and cannot be reduced to them. Universal logic is not universal algebra.

Some logicians are at a loss to understand this because two basic

trends are often contrasted in the history of modern logic, namely: Boolean

and Fregean trends, and one tends to assimilate any mathematization of

logic with the Boolean trend, the notion of Boolean algebra, or algebraic

logic. For some people, any structure is an algebraic structure. Historically,

algebraic structures certainly played a crucial role in promoting the notion

of structure, since someone like Glivenko used this word structure as a

synonym for lattice. But nowadays, such a confusion appears ridiculous

after Bourbaki and category theory.

There is no good reason to say that any logic is an algebra, or algebraic.

For instance, to take such a connective as negation to be a function seems

to be quite arbitrary, given that negation can be equally seen as a relation.

Another pernicious assimilation is that of logical structures with ordering

structures: this leads one to think that the notion of logical consequence

has to be naturally transitive, but this is quite questionable.

In order to avoid any ambiguity, it should be said that the stance of

universal logic is a Neobourbakian and not a Bourbakian one, not only

because Bourbaki did not see logics as fundamental structures but he once

adopted some axiomatic-formalistic stance that is not ours and which is

quite independent of his informal conceptual stance, the stance we are

following was mainly expressed in his famous paper, “L’architecture des

math´ematiques”.

Peguntras sobre Lógica Universal a Jean-Yves Béziau

De a poco ire subiendo las preguntas que le hizo Linda Easthood a Béziau y sus respuestas.
La lógica universal no es principio lo que uno pensaría.... muy interesante!!!

1- . Although your proposal to develop a universal logic is

very appealing, isn’t it a utopian one? Isn’t it an absurd, or even

dangerous thing to believe that it would be possible to develop a

unique logic accounting for everything?

Let us immediately reject some misunderstanding; universal logic, as

I understand it, is not one universal logic. In fact, from the viewpoint of

universal logic the existence of one universal logic is not even possible, and

this is a result that can easily be shown. One might thus say somehow

ironically the following: according to universal logic there is no universal

logic.

Some people in some countries have always tried to elaborate a universal

system that would account for any sort of reasoning, or reasoning

as a whole. Aristotelian logic was depicted itself as a universal one. More

recently, first-order classical logic appeared to some as a universal system

accounting for mathematical reasoning as well as current one, that is, the

one used to buy your bread at the bakery.

But first-order classical logic was also criticized at length, whether

concerning its claim to describe mathematical reasoning or physical, computational,

current, philosophical ones, and the like. Many new logics were

further developed, namely: intuitionistic logic, combinatory logic, linear

logic, quantum logic, erotetic logic, modal logic, paraconsistent logic, polar

logic, relevant logic and so many others, all the more that each of these is

often to be divided into a disparate multiplicity, as in the case of modal

logics.

Among advocates of these logics, some forcefully believe that their

own logic is the best one, that it explains everything, solves everything,

so that their logic is universal, as was formerly the case with Stanis law

Le´sniewski or, more recently, with Jean-Yves Girard and its linear logic,

Jaakko Hintikka and its IF logic, and, even more explicitly, Ross Brady

with its relevant logic he squarely dubbed a “universal logic”.

Such a view is not shared by people working in quantum logic, for

example; indeed, these only want to account for one particular reasoning

related to one particular area, without ever claiming that such is the reasoning

we are using or should use whenever we go at the bakery. Now is

such a view consistent? Are we entitled to say the following: to each area,

to each situation, its own logic, or even to each group of persons, to each

individual, its own logic. So there would be a logic of chemistry, logic of

clouds, logic of sex, logic of women, logic of dogs, the logic of Bouvard and

the logic of P´ecuchet.

Actually, such a relativization of logic is equally absurd as the opposite

stance according to which only one logic could explain everything.

Obviously, there is also one intermediary situation according to which there

are neither only one nor thousand and one logics, but three or four: so is

the middle, not to say mediocre position of people who cut the cake into

three parts saying that there is the reasoning for formal sciences, on the one

hand, the reasoning for empirical sciences, on the other hand, and finally

the natural reasoning for daily life. Behind such a stance we see again the

old contradistinction between inductive logic and deductive logic.

The view of universal logic is that one plausibly can unify the large

kaleidoscopic variety of logics, while preserving their diversity. In the case of

universal logic, as opposed to those who support the view of one universal

logic, unity is entailed by diversity. Universal logic is not a logic but a

general theory of different logics. This general theory is no more a logic

itself than is meteorology a cloud.