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.

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