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”.