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
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
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,
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
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
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
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
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
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
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
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
To sum up, we can distinguish five groups of research which are mutually
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.
This work was supported by a grant of the Swiss
National Science Foundation. Thanks to Fabien Schang for helping transcribing
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