Luiz de Carvalho
Recife – PE 2007
Proposta de projeto de tese para
doutoramento a ser apresentada
ao Dr. Basarab Nicolescu com coorientação dos professores Dr.
Romildo Nogueira e Dr. George
C. Jimenez.
Luiz de Carvalho
Recife – PE 2007
Category Theory Language
1 Category theory and local and nonlocal toposes
1.1 Local theory
1.2 Local language interpretation
1.3 Toposes and levels of reality
1.4 Local toposes sheaves
1.5 Logic fibrillation
2 Logic of transdicisplinarity/complexity
2.1 Intuitive characteristics, language, and formalization
2.2 Set theory paradoxes and type theory
2.3 Ascending hierarchy of types
2.4 Higher-order logic
2.4 Complex plurality and algorithmic complexity
2.6 Law of excluded middle: local and nonlocal semantic toposes
2.6 The nonlocal and the inconsistency
3 Transdisciplinary logic´s fibrillation and algebraization
3.1 Possible translations semantics
3.2 Morphism between two signature systems in Hilbert style H 1 ⊕ H 2
Presentation: a brief historic-scientific introduction
The aim of this work is to offer a logic-formal and axiomatic
characterization to the Transdisciplinarity Theory. However, an introduction to
transdisciplinarity requires a brief reading of the historic-scientific events that
justifies the need for a new epistemic approach, convenient to the changes
occurred during almost two centuries. These changes proceeded in order to
break up some or almost every classical logic principles, which have motivated
or motivate the scientific concepts up to now. Those principles are: the principle
of identity, which supports that anything is itself, or formally, A is A; the
noncontradiction law, which supports that, between two contradictory
prepositions, one is false, or formally: ¬( P ∧ ¬P ); and the law of excluded
middle, or formally, P ∨ ¬ P.
The history of the classical logic-deductive method starts with Aristotle
and, since then it is improving, up to the beginning of modernity. Nevertheless,
from nineteenth century, cracks in the classical Aristotelian-Euclidean thinking
begin to appear, regarding the called non-Euclidean geometries. Euclid (330277), with his Elements, gives a systematic form to Greek knowledge gathering
proposition and demonstration in a deductive way, according to Aristotle lesson,
which poses that, in a T theory, a proposition (P) and its denial, named non-P
( ¬ P), are not deductive, and that only P or non-P are demonstrable.
Logic is the pratical study of the valid arguments and, for centuries,
according to the Elements, the geometric model theory remained till the
emergence of the completeness and movement problems, linked to the
postulate five (that, if a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles, the two straight
lines, if produced indefinitely, meet on that side on which are the angles less
than the two right angles), and to the congruence axiom, that is:
a) (Completeness): The Elements axiomatization is not enough to deduce the
b) (Movement): the congruence axioms depend on the movement concept.
For many years, two tendencies have been trying to solve these
problems: adding more axioms to complete geometry or showing that this
attempt would be impossible. Archimedes has stood up for the first tendency
and others, for different versions of the postulate five (given any straight line
and a point not on it, there "exists one and only one straight line which passes"
through that point). Nonetheless, Lobachevski, Gaus and Bolyai have argued
that geometry does not represent the physical reality, and that its postulates
and theorems are not necessarily true in the physical world.
The postulate five has been attacked, culminating in the Lobachevski´s
postulate (Lp): there exist two lines parallel to a given line through a given point
not on the line. A completely new geometry emerged from this novel axiom,
presenting no contradictions regarding the Euclidean geometry. From these
progresses, we could not infer that the axioms of mathematics are intuitive
constructions based on some kind of non-mathematical evidence.
This way, the evolutive process of the deductive axiomatic system leaves
the aristotelic grammatical and discursive scope to feature itself as a reasoning
algebra with G. Boole (1815-1864). After that, how is it possible to set up the
consistency of the axioms in a confident way? It is necessary to build up
mathematics from itself. On the other hand, the congruence axioms in the
Element´s book V, from where Eudoxus´ method of exhaustion and CauchyRiemann integral emerged (408 B.C. – 355 B.C.), were initially inspiring to the
conception of real numbers as a geometric abstration (Axiom XI – every right
angle are similar); it was necessary to give consistency to the real, free from the
infinite approach of the infinitesimals, which Berkeley used to call absent
magnitude ghost; this was performed by Weierstrass, with his formal theory of
limits, and by Dedekind/Cantor, with his construction of real numbers based on
natural numbers, named arithmetization of analysis. Such constructions involve
some use of mathematical infinity. In this context, David Hilbert´s program have
arisen: he wanted mathematics to be formulated on a solid and complete logical
The axiomatic method has been encouraged with these developments,
proceeding with this perspective, which was also adopted by Isaac Newton
(1642-1727), who classified, in a axiomatic and deductive way, the classical
dynamics, in his PRINCIPIA, by three axioms or laws of motion. Newton´s
axiomatic counted on the infinitesimal calculus method. However, the presumed
self-evidence of the three axiomatic principles, which validated Newtonian
principles, particularly the fundamental assertion regarding the principle of
inertia and its parameter, absolute time and space, ended up as a naive
pretension. Despite its basis problems, Newtonian model prevailed during the
1700s with a scientific revolution that was born to fight Aristotle authority, but
that turned to be a closed and powerful model, whose principles evoke a
privileged observer, Laplace´s demon, that, according to Basarab Nicolescu,
validated the consistency of the newborn science postulates:
(I) The existence of universal laws, mathematically characterized
(II)The discovery of these laws by means of scientific experience
(III) The perfect reproducibility of the experimental data
According to what would assure the consistency of mathematics as an
instrument for a new research program, science, Cantor´s theory of sets, in
which elementary arithmetic is based, sustained the aristotelian classical logic
as na underlying logic (D’OTTAVIANO,1994).
Classical physics successess, on the other hand, kept confirming the
three postulates, even if the first postulate had never been definitively proved.
The arithmetization of analysis moviment, that, according sets theory would led
to the named sets paradoxes, or Cantor-Russell paradox, if such a set is not a
member of itself, it would qualify as a member of itself by the same definition,
culminated with Hilbert program that aimed to prove that such problems could
be surpassed through a suitable formalization that would allow the metatheoretical demonstration of the arithmetical consistency and, as a
consequence, of mathematics ( HILBERT & BERNAYS, 1934).
During the first half of 20th century, Kurt Gödel publishes his
incompleteness theorems, which would limit Hilbert program:
A: (First theorem) To every ω-consistent recursive class κ of formulas there
correspond recursive class signs r, such that neither vGenr nor Neg(vGenr)
belongs to Flg(κ) (where v is the free variable of r).
B: (Second theorem) For any formal recursively enumerable (i.e. effectively
generated) theory T including basic arithmetical truths and also certain truths
about formal provability, T includes a statement of its own consistency if and
only if T is inconsistent.
One of the consequences of these theorems is that any formal system
that envolves elementary arithmetic, accepts true and nondemonstrable
sentences. Newton C. A. da Costa and Francisco A. Dória (da COSTA and
DORIA 1991) have extended this result to classical physics. Nevertheless, what
does Gödel theorems mean to science and, particularly, to Physics? The
science philosopher Maria Luiza Dalla Chiara understands that, from the point
of view of the development of Logic theories in its non-independent relation to
any research content, Gödel theorems have produced a plurality situation in the
treatment of some physical theory (DALLA CHIARA, 1980).
However, if Gödel theorems regards only to the formal aspect, in what
sense the theorems could be extended to the physical theories? Before we
continue, let’s take a look to another result of the Gödel theorems, the
undecidable, which will make our question clearer:
C: For any consistent formal theory that proves basic arithmetical truths, it is
possible to construct an arithmetical statement that is true but not provable in
the theory. That is, any consistent theory of a certain expressive strength is
incomplete. The meaning of "it is possible to construct" is that there is some
mechanical procedure that produces another statement.
Albert Einstein, concerning the debate about Quantum Physic’s
incompleteness in the famous EPR argument (Einstein-Podolski-Rosen), which
we are going to consider soon, give us the necessary variables to understand
Gödel’s results regarding physical theories, that Lokenath Debnarh synthesized
in these words:
The totality of concepts and propositions constitute a physical theory
provided it satisfied two essential equirements: external justification, which
means the agreement between theory and experiment, and internal perfection,
implies that the theory can be inferred from the most general
principles without addition concepts and assumptions. (DEBNATH, 1982).
What Einstein comprehends as a theory’s inner perfection is its
consistency as a mathematical structure, what would lead us to Gödel’s A
theorem. However, D´s external justificative would demand an axiomatic
characterization of physical theories, which we can obtain from Patrick Suppes
structure (P SUPPES - 1967), which links mathematic theory and physical
A physical theory T is a triple T = 〈 M,D,R 〉 , where M is the theory’s
mathematical structure; D is the application domain and R is the set of rules that
links M and D, according to da Costa and Sant’anna (da COSTA and
SANT’ANNA, 2006):
On the other hand, mathematical concepts in principle cannot be
considered as a faithful picture of the physical world; but can, through
axiomatization, be susceptible to the use of logic-mathematical tools as
indecidable propositions like C one hold physical meaning. In fact, according to
Lycurgo (LYGURGO 2004), there is a connection between these results and
what we can observe in the physical world; but what is relevant in completeness
theory, according to the definition that “a physical theory is complete if every
element of the physical reality has a counterpart in the physical theory”, that is,
there is more in the world than the specific theory can anticipate, according to
Gödel’s results. But theses results do not provide us hope to get a complete
axiomatization, as incompleteness is structural. According to Maria Luiza Dalla
Chiara, the notion of an independent Logic is, historically connected to some
kind of single logic (Dalla Chiara, 1980).
Nevertheless, in front of current demands, logic is no more characterized
regarding the valid formula concept and is perceived as consequence relations
according to Tarski, allowing a plural and interrelation treatment between logics.
At the same time, another revolution was happening in scientific research
world, which would definitely place us in another perspective: Quantum
Mechanics. This theory is the best example of semi-complete theories´ blend.
Historically, we first have the quantum theories of Planck 1900), Einstein
(1905), Bohr (1913) e De Broglie (1924); in a posterior moment, Heisenberg’s
matrix mechanics (1925) and Schrödinger’s undulatory mechanics (1926) and
Pauli´s and Dirac´s theories. All these formulations incorporate, somehow, the
controversial phenomena that turn quantum mechanics in a singularity in
science history: wave-particle duality, its probabilistic aspect, and the
inseparability concept.
Quantum theory and its fundamental experiment, Young´s double-slit
html#TwoSlitsElectrons), which distinguishes quantum mechanics from
classical mechanics through atomic phenomena explanation. This experiment
provide us with some odd results: if we call the first experiment open slit A and
the closed one, B, we have the following results, represented as logical
circuits: A ∧ ¬B ↔ ¬( A → B) ↔ ¬¬A ∧ ¬B , that is, particles remain particles with
peculiar intensity distribution, and the same happens when B is open and A is
closed, B ∧ ¬A ↔ ¬( B → A) ↔ ¬¬B ∧ ¬A . But what we note when we open
both A and B slits at the same time? Based upon previous experiments and on
the fact that the third one would keep distribution consistency, we would get the
natural sum of both beams projected in the screen, as each electron moving in
its and well-drawn and fixed trajectory, which pass through the slit without
influencing other electrons that pass through the other slit. Therefore, we could
have that, if this is valid for ∀ n experiment, so it is valid for n+1, as follows:
!) A ∧ B [ hipótese]
2) A ∧ B → A [ separação ]
3) A ∧ B → B [ separação]
4) A [ ModusPonens ]1e2
5) B [ Modusponens ]1e3
6) A ∨ B [ Introdução ] 4e5
However, this is not what happens exactly. Electronic diffraction
phenomenon shows a picture on the screen that occurs due to interference and
does not correspond to the sum of the pictures produced by each of the slits
individually, as the above hypothesis presents. Now let’s imagine that this
phenomenon is consistent with the two first experiences, i.e., the third
experience maintains the properties of electrons pertaining to the previous
experiments. This could mean that the undulatory aspect observed is motivated
by the particles individually, suggesting some hypothesis:
a) electrons should irradiate electromagnetic waves continuously;
b) they would need a inexhaustible energy source in order to do not deteriorate
their nuclei;
c) their speed should be higher than the speed of light C.
As these hypotheses are senseless, quantum mechanics is contradictory
in comparison with classical electrodynamics, which proposes that the
undulatory aspect is consequence of the electron trajectory.
We can conclude that light presents particle and wave features at the
same time. The first two experiments obviously suggest the continuity concept,
according to Nicolescu: it is not possible to pass from one extreme to another
without passing through the mid-space. However, the third experience breaks
continuity and places non-classical probability as a suitable mathematical tool to
deal with diffraction interferences.
Diffraction phenomenon demonstrates that the continuity perceived in the
previous experiments with individual particles is not valid; because diffraction
phenomenon presents bindings where particles would not be splitted, according
to the hypotheses shown above; this phenomenon is known as nonseparability
or nonlocality. This condition made many researchers react, resulting in EPR
Paradox, or Einstein-Podolsky-Rosen Paradox. Essentially, EPR is a thought
experiment that demonstrates that the result of a measurement performed on
one part A of a quantum system has a non-local effect on the physical reality of
another distant part B. As far as one can judge, this situation is not possible
because special relativity asserts that no information can be trasmitted faster
than light. Nevertheless, Jonh Bell, with his theorem, has proved an absolute
distinction between quantum and classical mechanics. Assuming separability
and sustaining a, b, and c hypotheses, we would need hidden variables that
would continuously determine experiments of diffraction, regarding a given
interpretation of the quantum phenomena.
Bell proved that the hypothesis of local separability (a particle holds
defined values that do not depend on a contextualization; and physical effects
speed of propagation is finite) is not in accordance with quantum physics, that
is, the correlation coefficient C(θ)<N for local hidden variable theories and to
quantum mechanics C(θ)>N for some θ. It was born a new kind of causality:
non-local causality. In our macrophysical world, if two objects interact in a
certain moment in time and then stand back, they interact less. In quantum
world, entities keep interacting independently of the distant (NICOLESCU,
Non-local causality idea questioned another classical science principle:
the existence of only one level of reality, by which Laplace´s demon, a
privileged observer, could be an invariant system from a number of general
laws (NICOLESCU,2001). Under this context, in one hundred years of scientific
development, which meets a perspective pluralization, Transdisciplinarity and
Complexity have emerged, as well as several perspectives with the same
objective: adjust themselves to complexity and to the current disciplinary boom.
What is transdisciplinarity/complexity as defined by Basarab Nicolescu? The
transdisciplinarity is founded upon three pillars: levels of reality, logic of the
included third and complexity. The revolution of quantum physics with its
problems, especially the known measurement problem, plays central role in the
concept of transdisciplinarity and complexity. Avoiding go into metaphysical
subtleties, which is as rigorous as calculation, for us is sufficient the idea of
levels of reality as an ensemble of invariant systems under the action of a
number of local general laws, or as something that offers resistance to our
experiences. But we can use the principle that "the non-local is inconsistent with
the locality (NL)" and from this say that a level of reality is a structurally welldefined class, and is local, which plurality constitutes non-local classes that
made the complexity. The NL principle means that two levels of reality are
different, if, passing from one to the other, there was a breaking in the laws and
a break in fundamental concepts, (NICOLESCU, 2001).
To deal with levels of reality from a logic point of view, that is our goal, is
necessary the locality theory, studied in the ambit of categories theory, the local
topos, which Lawvere gave an axiomatic characterization similar to the category
of the sets, the called toposes, in reference to the toposes introduced by
Grothrndieck for algebraic geometry. (LAWVERE, 1975). Then, levels of reality
and levels of perception make a local topos. A topos form a kind of category
that has the following properties similar to the category of sets:
- Should have a terminal object corresponding to the unitary element, the
set 〈 ∗ 〉 ;
- Should be able to form the Cartesian product A × B of two objects and the
exponential object CB of functions of C on B;
- Should be able to form the Cartesian product A × B of two objects and the
exponential object CB of functions of B on C ;
- Must have an object of values of truth Ω with a distinguished element ┬ and a
corresponding element sub-object one-by-one B of A and its characteristic
function χ B: A → Ω such that, for any element a of A, a belongs to B if and
only if χ B(a)=┬, and must have an object of naturals N equipped with an
element 0 ∈ N and the functions S:N→N defined by recursion such that: for each
a ∈ A and each function h:A→A, We can build a single function φ:N→A such
that φ(n)=hn(a) for any natural number n (LAMBEK, 2004).
Also according to Lawvere, these properties can be expressed in the language
of categories theory, not referring to elements, in a way that, ┬ is seen as an
application 1 → Ω and 0 as 1 → N. Thus, each topos ∆ is associated with a
type theory λ (∆) → ρ as its internal language. Inversely, for every type theory ρ
is associated a topos π(ρ), the topos generated by ρ, known as TarskiLindenbaum’s category of ρ. (LAMBEK, 2004). If we take the levels of reality
and its perception as local topos, what relationship exists between complexity
and transdisciplinarity? We can say that the possibility of dealing with multiple
levels of knowledge reality, which constitutes as a complex plurality in itself, is
given by the logic of the included third without the need of included fourth, fifth,
What is supposed regarding levels of reality to meet the above requirements,
while local topos, is that he keeps doing some kind of coherence asserting the
axiom of regularity of ZF (Zermelo-Frankel), or, that any not empty set X
X ≠ ø→ ∃Y ( Y ∈ X ∩ Y I X ≈ ø); that axiom forbids circularities as the set of all
sets that do not belongs to himself as a member, formally: M= { A / A ∉A} and
therefore belongs to itself, that is, the paradox of Russell. With the results of
the theorem of Pierce of: toute polyade supérieure à une triade peut être
analysée en terme de triades, mais une triade ne peut pas être généralement
analysée en termes de dyades" como faz Nicolescu (Nicolescu, 1998) and
those of N. Resch for polyvalent logic (MANY-VALUED LOGIC 1969),
according to that, any system of logics Ln valents will contain tautological
formulas as theorems that are not necessarily in Ln+1 and by theorem of CL
deduction (classical logic) all tautology is a theorem of CL, we have the local
topos complete.
However, the transdisciplinary local toposes are generally incomplete, not
worthing the generalized axiom of choice and could be formalized as intuitive
logic (J), classic while local levels of reality. What constitutes transdisciplinary
logic complexity, as we understand it, is the non-local interaction of local
toposes in accordance with NL principle. That is the sense of quantum
discontinuity as Nicolescu means (NICOLESCU, 1998)
From this point, the issues rose by Nicolescu about the sense of opened unity
of the world, is essential to understand what he calls the Gödelian structure, the
correlation of levels of reality, for example: What is the nature of the theory
which can describe the passage from one level of reality to another? Is there
coherence or even a unity in the ensemble of levels of reality? What should be
the role of the subject-observer in the existence of an eventual unity of all the
levels of reality? Is there a privileged level of reality? Would be the unity of
knowledge, if any, of subjective or objective nature? What is the role of reason
in the eventual unity of knowledge? What would be, in the field of reflection and
action, the predictive power of the new model of reality? Would be possible to
understand this present world? In the same text, Nicolescu refers to levels of
reality ensemble as an evolutive process that would possess a self-consistency
which, by the second theorem of Gödel, this self-reference would imply in a
contradiction and consequent separation in levels of reality undoing the same
self-consistency in local collapses in which worth the classical logics. As we
said above, what is subjacent to this process of self-consistency is an
interpretation of measurement problem in quantum physics. Then in what
consists the measurement problem?
OSVALDO (1992, pp. 177-217) in its study of measurement problem,
characterized the quantum mechanics (QM) as a structure as follows: A closed
system is described by a state that evolves over time in a determinist way,
according Schrödinger equation, Ψ(r, t). As we mentioned, this state provides
only the probability of obtaining different results from a measurement. By now,
after measurement, the system is in another state, which depends on the
outcome, thus, in the course of measuring the system evolves in an
indeterminist way, described in Von Neumann projection postulate or wave
packet collapse or state reduction. Therefore, the problem of measurement
arises from this opposition between determinist and indeterminists evolution
associated to projection postulate. And if we add two more hypotheses:
α) the metric device belongs to the quantum state;
β) The composed system, object/device may be closed on the environment, so,
its evolution should be deterministic, but, state reductions will continue to occur
during measurements.
It is not necessary to say that this paradox is linked to the double slit experiment
and to its paradox wave/particle. To summarize, the question is to know how
during measurement, a quantum superposition may become in states that do
not superpose. Von Neumann answer is that projection postulate accompanying
any act of measurement, formally:
Ζ): φ (τ ) ≈ Ω φ (τ ) Where omega is the unitary operator and the state vector
is expressed by superposition φ ≈ α 1 φ1 +α 2 φ2 where φ1 and φ2
state of an observable which would reduce to
η): φ1 ∨ φ2
are self-
depending on
observation/measurement result, and that the probability of each outcome is θ)
a1 ∧ a2
The unitary operator and their inverse adjunct is equal to Ω∧ Ω 1
From that solution, two problems appear: one of CHARACTERIZATION, i is,
what are the conditions for apply the projection postulate, and being it a
inherent condition for measurement, what will characterize a measurement
and/or observation? The other problem would be COMPLETENESS: Could
projection postulate be derived from other quantum physics principles with a
physical model suitable for the measurement process? Without going into
details of various currents that proposed solve this problem, we see that a new
thermodynamic axiom, linking a large number of particles and incoherent states,
would have been introduced to the problem of completeness.
would be later unsolved, so that, the problem of completeness is unsolvable
and that the thermodynamic models do not provide exact solution to the
measurement problem. If the completeness problem was solvable, it would
provide an example of measurement with certain conditions:
- a) a quantum state can be attributed to the macroscopic measuring
equipment, (device state in a Hilbert space of finite dimension)
- b) The composite quantum system (device/object) can be considered closed
in relation to environment and the rest of the universe, (the composed system
evolution is unitary)
- c) The different measurement results correspond to distinct final states of the
device; (pointer states)
- d) A more precise specification of this pointers supposition constitutes a
condition of solvability;
- e) Should be precisely defined what is meant by measuring, (a measurement
is a subclass of interactions governing the evolution of a composed system,
device/object, that should be able to distinguish between classes of object
states for which the medium values of a self-adjunct operator are different. In
addition to this definition, there are the measurement classes of repeatable and
predictable. A repeatable measurement is such that, shortly after being made
and repeated, the result certainly is the same, or, the final state of the object
after the measurement is the self-state corresponding to self-value obtained as
a result. The predictable ones are that for each possible outcome there is a
state of the object that leads to a predictable result.) A measurement satisfying
these conditions, would give a solution for the completeness problem.
Meanwhile, even as (OSVALDO 1992, pp. 177-217), the evidence is that such
measurement cannot be defined, and then, the completeness problem being as
Σ – Unitary condition
Λ – Measurement
Ξ – Solvability
Φ – The a, b and c hypothesis
The evidence of insolvability in general can be represented as:
Φ,Σ,Ξ├ ┐Λ
Accepting that does not exist ΦeΣ then does not exist measurements that
meets the condition of solvability, Φ,Σ,Λ├ ┐Ξ.
The problem of measurement received several proposals of solution, and one of
them is the theory of open systems, in which the compound system, that
includes object/device, cannot be completely isolated from the environment. In
this conception, there is an approach in which the reduction process would be
an objective physical phenomenon that would transcend the measuring act,
transforming itself in the collapse or decoherence problem, or the problem of
potentialities update, ontologies of powers that could demand a
supplementation. In this perspective is settled the Gödelian self-consistent
universe of Nicolescu. Thus, the question for us is: Is the inconsistency
associated to reality levels ensemble evolution self-referentialty, the same
inconsistency of the correlation between locality and non-locality? If the answer
is yes, we have from the second theorem of Gödel, that levels of reality
ensemble are inconsistent. If the answer is no, we have the following
- Or the correlation local/ non-local would of be of extension and therefore will
worth ψ : x ∈ σ ↔ x ∈ τ ,ψ ├ σ ≈ τ , or a conservative interpretation of the logical
structure would be extended to other structures or toposes, violating the NL
principle, and the ensemble of levels of reality would be inconsistent;
- Or, self-referentialty not infringes the axiom of regularity and the correlation
local/non-local would be inconsistent.
So, in the positive response view, the system broadly would be equal to the
correlation local/non-local which collapses from itself-reproduction in consistent
and incomplete local toposes. Answerig to questions above, we could say that
transdisciplinarity is consistent and inconsistent making worth the contradictory
pairs that experience and scientific theory, after quantum physics, had saw
appear: local causality and non-local; reversibility and irreversibility, separability
and non-separability; wave and particle, continuity and discontinuity, symmetry
and broken symmetry, and so on. But Nicolescu himself answered such
questions, particularly the first, about the nature of the theory that could
describe the transition from one level to another, saying that no one managed to
find a mathematical formalism that allows the passage of a strict world to
another. But we must differentiate logical epistemic formalization from formal
mathematical structuring of models theories. A logical epistemic formalization
has no compromise with the stander algebraic model, such as the formal
mathematical structure. Logical epistemic formalization works with theory of
categories advanced features, creating logical-topological spaces of possible
worlds from the Tarskian logic conception as relations of consequence. In this
way, we believe that logic subjacent to transdisciplinarity, is a combination of
classical and inconsistent logics, kind of not Hegelian temporal dialectic,
because we are always in some reality, and that subject self-position, as Hegel
believes, is derived. However, criticism to Hegelian succession of contradiction
seems to lack sense, because, if there is a contradiction, at some lapse of time
this contradiction is simultaneous. In any way, transdisciplinarity self-reference
is not closed in the immanence of self- position of idealistic subject of self, that
puts the non-self and understand it. The complexity of transdisciplinary selfreference opens to the multiplicity of historical-existential experience as a
subject pertain to the worlds that it is included. Two other properties are at the
core of transdisciplinarity:
# The opening of reflexivity in infinite resource becomes totality indeterminacy:
the whole, the same, just like this while incomplete (inexhaustible);
# The opening of dialectic to difference, as an outcome of relative
incommensurability of different languages and/or (levels of reality) and meaning
But, is the sense of self-referential of Nicolescu the same as of Morin?
In a revealing dialogue we get the difference of self-referential sense involved in
the "logic" of included third between B. Nicolescu and E. Morin, at Stéphane
Lupasco international conference, where Nicolescu asks:
- B.N: E, so, you were quite attracted by theorem of Gödel. Because the
essential point is there: this organizational structure of included third logic is
Gödelian type. It never closes.
- E.M: Of course. But Gödel reaches to a principle of uncertainty. That is, he
says the system cannot give account of itself, with its own resources. It is not
known whether it is consistent and it is not known if it contains a contradiction.
First point: the uncertainty. Second point: is possible, however, conceive a
richer meta-system, able to understanding the system, and that, of course, it
opens to infinite itself. But then, at this moment, I do not see the expression of a
included third specific logic.
Nicolescu then retakes aspects of Gödel’s theorems, almost always overlooked
in the analytical approaches, introducing the meta-theoric affirmation of T inside
of T consistency, and therefore, their self-reference as a contradiction that,
according to E. Morin, is a priori: Le théorème de Gödel nous dit qu'un système
d'axiomes suffisamment riche conduit inévitablement à des résultats soit
indécidables, soit contradictoires. Cette dernière assertion est souvent oubliée
dans les ouvrages de vulgarisation de ce théorème.
So let's see: La is the arithmetic language, T an axiomatizable recursive
extension of PA, Peano arithmetic, PrT(x) a proof predicate for T and Com(T)
the sentence
¬PrT( ┌0=1┐).
We assume that T is consistent and T├ Com(T) and being σ a sentence in La
which meets T ├ σ ↔ ¬PrT(┌ σ ┐). We say that Λ is a model of T,and is
positive if Λ ╞ σ and negative in other cases. If T is consistent, there is a model
Λ1 of T. If Λ1 is positive, then Λ1 ╞ Com(T+¬σ), then there is a Λ2 negative
model of T which is a definable extension of Λ1. If not, Λ1 = Λ2. For the
hypothesis that Λ2 ╞ Com(T), we have a model Λ3 of T which is a definable
extension of Λ2. Λ2 if is negative, Λ2╞ PrT(┌ σ ┐), so Λ3 ╞ σ. Meanwhile, Λ3
╞ ¬ σ and Λ2 ╞ ¬ σ and Λ3 are an extension of Λ2, which is a contradiction
where T theory proves its own consistency. (This evidence is due Jech, 1994)
Once self-consistency reintroduces the subject in the system by itself
formulated, Nicolescu complete its approach of included third logic with the
ontological conception of Lupasco: the subject of knowledge is implied in the
logic by itself formulated. The experience is the subject itself experience. The
circular character of logic assertion, as its own logical experience, occurs from
circular character of subject: to define the subject we must take into account all
phenomena, elements, events, states and propositions that relate to our world
and, furthermore, the affectivity. An obviously impossible task: in the ontology of
Lupasco, the subject never would be defined. This Gödelian structure always
opens, beyond the levels of reality, a NOT RESISTANCE ZONE, that we rather
to call, SEMIOTICALLY NOT FORMED FIELD, which is a field not finitely
algebraizable by AAL method (abstract algebraic logic) (W. Blok and D. Pigozzi,
1989) that we will describe soon. Here there is a problem on the complexity
measurement in the algorithmic complexity point of view. Will Kolmogorov
complexity measurement apply to this semiotically not formed field? Is possible
consider the transdisciplinary not-separability as a totality whose complement is
not computable? Intuitively speaking, Kolmogorov complexity of a number n,
denoted by K (n), is the size of a program that generates n ; and n is called
random if n ≤ K (n). Kolmogorov has proved that not random numbers set is
recursively enumerable, but is not recursive. Odifreddi (1989) considered this a
version of Gödel’s first theorem of incompleteness. Second Kikuchi (1997),
complexity of Kolmogorov also leads to the second theorem of incompleteness
of Gödel, extracted from the paradox of Berry, (“What is the "first not named
number with less than ten words?”). Since Gödel has proved that the first
theorem leads to the second. Then, to apply Kolmogorov complexity to levels of
reality ensemble and semiotically not formed field, we would have to consider
local/non-local inconsistency null or levels of reality ensemble self-referential as
a recursive extension, which would give to levels of reality ensemble a local
character as such is the Kolmogorov complexity, valid for the local topos and
the incompletenesses of Gödel. The question turns on know what type of
contextuality is involved the non-resistance area or the transdisciplinary
semiotically not formed field: quantum or classical?
In classical contextuality, according to Aert,(2002), the result is affected by
various environmental aspects, but not by irreducible and specific non
predictable from system and experimental disturbance interaction. In classical
contextuality, Kolmogorov axioms are met, and a classic model of probability
can be used. However, when contextuality is intrinsic, the system and the
disturbance, both, have an internal relation of constitution, as such its interface
creates a concrescence of emergency. The presence of intrinsic contextuality
means that Kolmogorov axioms are not met. The Entanglement can be tested
by determining whether correlation experiments in entities which being together
infringe the inequality of Bell (Bell 1964) Pitowsky,(1989) had proved that if the
inequality of Bell are met for a range of probabilities concerning to considered
experiment response, then there is a Kolmogorovan classical probability that
describes this probability. In this case, the probability can be described as a
lack of knowledge about the precise status of the system. In other hand, if the
inequality of Bell is infringed, such classical Kolmogorovian probability is not
valuable; rather, Bell’s inequalities infringement proves that the probabilities
involved are not classical. In transdisciplinary contextuality case, we believe
being this classical and non-classical at the same time, what seems to confirm
Lupasco’s proposition cied by Nicolescu: quantic dialetic is a doubt expansion,
in this case is not a case of lack of knowledge, but context generation as a
potential, at the same time that some knowledge is updated. So, algorithmical
complexity pertains to transdisciplinary complexity as a local topos.
In the chapter "Transdisciplinarity and open unity of the world", which Nicolescu
calls levels of reality ensemble, is in the Gödelian conditions of its structure, a
local topos in which is valuable Kolmogorovian classical probability. But, in
other hand, in the "Included third. From quantum physics to ontology”. The
problem, if is not from translation, the relationship between a level of reality and
another, does not preserves the coherence, and then, the application of
Kolmogorovian complexity to structurally incoherent levels of reality ensemble is
not valuable. So we can say that:
Being Cn: the levels of reality ensemble;
and Cn ∪ Zr: the levels of reality ensemble union zone of non-resistance. Cn
has a coherent structure and worth the algorithmic complexity ∧ Cn ∪ Zr is
structurally incoherent and generally does not worth the algorithmic complexity.
The question that immediately arises is whether is possible, in general, a
coherent level of reality ensemble Cn. In this case, the expression levels of
reality ensemble, in general, mean a local extension. Then, the self-consistency
refers to Cn ∪ Zr as irreducibility of contradiction and relativity of consistency.
The union operator ( ∪ ) here is being taken in Lupasco sense, of
distinguishable non-separability ∧ A ( is actualization) ¬ ∧ P ( is not
potentiation), ie, Cn ∪ Zr = ∧ A ¬ ∧ P. This formulation presents us a problem
about the use of denial and implication in included third logical. For example: Is
the included third or not a negation of the principle of the excluded third (ETP)?
Apparently, the denial of the excluded third, is not equivalent to intuitionist
denial which is based on openings of a topological space X , A( X ) in which
implication definition ( → ) does not follows (∧, ∨, ¬) as in classical logic ;
¬( A ∧ ¬B) but not worth the double negation, ¬¬A ├ A.
In transdisciplinary logic, the denial of the excluded third when generates a
contradiction makes use of classical denial by the principles of De Morgan
¬( A ∨ ¬A) ≡ (¬A ∧ ¬¬A) , what suggests us that in A(X) there is induced
operations PX, the set of subsets of X, of finites intersection and union which
are closed and form a Boole algebra. So there is not a necessary link between
incompleteness, that we will call ¬C and A(X), the opening of a topological
space, i is ¬ (¬C → A(X)). Thus, we believe that the transdisciplinary local
toposes form incomplete closed spaces in which worth the classical logic with
open complement in which worth a weaker denial, possibly of intuitive type
where □(¬C→ A(X)). These spaces or logic fibrillation, ¬ ( ¬C→A(X)) ⊕ (
¬C→ A(X)) ⊕ ϖ ⊕ ( ¬C → A(X)) ⊕ ¬ (¬C → A(X)) = Ω , Where ϖ is the
intrinsic contextualization, forming a logic of contradictory. So, we have to:
1) Ω ├ ¬ ( ¬C → A(X)) ⊕ ( ¬C→ A(X)) ⊕ ϖ
2) Ω ├ ϖ ⊕
( ¬C → A(X)) ⊕ ¬ (¬C → A(X))
From 1) e 2) is made Ω1 and so on.
This work aims to investigate the possibility of a logical construction of
transdisciplinarity in the Hilbert style LT 1 . In particular, we believe that a
transdisciplinary logical is formed from an intuitionist classical logical
combination and non-classicals temporal inconsistent. Give an algebraic
formalization of local/ temporal inconsistent interaction whose subjacent logic is
intuitionist and non-local, making use of W. Carnielli’s possible translations
semantics and Tarski’s concept of logic as consequence relations.
The methodology used in the logics combination to form a transdisciplinary logic
is the categorical fibrillation. The process of logics composition can be synthetic
(Splicing logic), for example: fibrillation of D. Gabbay.
L= L 1 L 2 if L is our incognite, since it is known L 1 and L 2 , or can be
analytical (Splitting logic), for example: W. Carnielli’s possible translations
L=L 1 ... L n in the case where L is known and its decomposition in more
simple logic.
The question that arises before combining logics is: what kind of structure we
want to achieve with the concept of logical system: proof system as tablê,
axiomatic system, natural deduction, or semantic methods as logical matrices,
valuations and Kripke’s semantics? As usual in this kind of methodology, A.
Tarski’s concept of logic enables more resourcefulness in the combined
treatment of logics, which is independent of the concept of valid formulas.
What's common in any system of logic is the concept of logical consequence
denoted by ├ defined from an ensemble of sentences or formulas of L.
A system of Hilbert is a pair H = ‹ C, R› as such C it is a signature and R one
set of pairs of the formula ‹ Γ,Ψ › as such Γ U { Ψ} it is a formula set of L( C ).
The elements ‹ Γ,Ψ› of R, the elements are called rules if Γ= is called axiom.
We can make a fibrillation of logic in Hilbert’s type to form transdisciplinarity
logic in such way that they are morphed in the systems of categories of Hilbert
Hil. Therefore, the logic of the transdisciplinarity forms a composition of classic
logics and inconsistent in Hilbert’s style.
There is no doubt that the twentieth century was the center of great
changes in all areas of knowledge, particularly in the scientific universe,
resulting in remarkable victories in every technique and production fields. In
spite of that, it could be noted that knowledge splitting and compartmentation,
inherited from a tradition shaped by the thoughts generated during the interstice
between fifteenth and nineteenth centuries, was no longer enough to create the
epistemological references required to solve the features of knowledge itself.
Many problems seemed to be out of the entangled theory systems, and a
kind of blindness hovered over the attempts to understand many nature’s
fundamental problems, as well as its more common problems. However,
debates about topics like the structure of matter, the objectivity or the possible
relationships between the observer and the levels of reality, brought to light
unsurpassable paradoxes. Therefore, new ideas emerged, as the Theory of
Relativity, the Quantum Mechanics, and the Complexity Theory, among others
(Nicolescu, 2003).
Anyway, the straighten up of the contents referring the Complexity
Theory suggests a revaluation of the systems for selecting and determining
conceptualization, as well as of the systems that configure the logical
operations. This intends even to affect the designation structures of the
intelligibility fundamental categories, like the mechanisms that operate their
application control.
It is accepted, therefore, that thinking about the complex systems implies
accepting the need to overcome major challenges, starting from the implications
of the multidimensional and hologramatic structures, or even the
interconnectivity and inseparability of the “one” and the “multiple,” which is
pertinent to it.
Despite the expectation that open up new possibilities to understanding
and the sociocultural transformations that this could raise, it is known that it
would be impossible to embrace the study of the Complex Systems only starting
from the established contemporary disciplinarities.
During the First World Congress of Transdisciplinarity, held in Convento
de Arrábida, Portugal, from 2-7 November, 1994, a Letter of Intent was
prepared, outlining a set of fundamental principles, which exalted the need to
consolidate the transdisciplinar and transcultural thinking as the best way to
approach the different aspects of complexity in distinct systems (Nicolescu,
Transdisciplinarity would then represent a conception of the research
based on a new comprehension milestone, shared among several disciplines,
and which is followed by mutual interpretation of the disciplinary epistemologies.
The cooperation, in this case, would be headed for problems solving, where
transdisciplinarity emerges to build a new model to bring the realities of the
object of study closer (Hernández, .....).
Turning to the Article 14 of the Transdisciplinarity Letter, whose terms
refer to argument rigidity as the best limit to the possible conceptual deviations
from the articulation of the problem situation data, we can justify the need for
unfolding a logic-formal and axiomatic characterization of the Transdisciplinarity
Theory, mainly when we consider the need for another epistemic perspective
suitable to the changes in the way of thinking, inherent of the last 200 years.
Further, considering that a logic system is always useful to choose the
prevailing operations, relevant and evident under its domain (exclusioninclusion, disjunction-conjunction, implication-negation), it is intended, through
this logical constructions, to foster the inclusion of other features to the
transdisciplinary way of thinking, like the “Opening”, comprehending the
unknown, unexpected, and unpredictable acceptance; and “Tolerance”, as an
exercise to recognize the right of the ideas and truths to be the opposite of ours.
Finally considering that logic keeps close connections with metaphysics,
mathmatics, philosophy and linguistics, we can appraise the impacts from this
project, especially regarding the reiteration of almost every classical logic
principles disruption, which sustain the ground theory in all the study fields
mentioned above.
In Brazil, many research groups are involved in studing logic, standing
out Prof. Newton da Costa group, which I am involved in, becoming specialized
in non-classical logics. Nevertheless, regarding the indication of a wide work
field outlining, I intend, through this proposition, to look for deeper
Transdisciplinarity subsidies from a more authentic source, hoping that it will
constitute, in a near future, an important reference to the development of
theoretical and technological studies on semiotics. Therefore, we expect that
the outcomes from this project accomplishment allow a contextualization in this
promising scenario, and serve as a back up to exploration approaches of new
horizons, particularly those which find themselves immersed in the paradigms
invisible zone, so that we can overcome the determinism of the current
explanatory models, which are associated to the convictions and beliefs
systems, in every scope. This must contribute to the collapse of the cognitive
and intellectual conformisms.
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