2. How is it possible to develop a general theory of logics, to
unify logics so various as quantum logic, erotetic logic or fuzzy
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
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,
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
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