The Logic of Transdisciplinarity (2)

8/9/2019 The Logic of Transdisciplinarity (2)

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THE LOGIC OF THE TRANSDISCIPLINARITY

Luiz de Carvalho

Recife PE 2007

MINISTRIO DA EDUCAOUNIVERSIDADE FEDERAL RURAL DE PERNAMBUCO


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THE LOGIC OF THE TRANSDISCIPLINARITY

Luiz de Carvalho

Recife PE 2007

MINISTRIO DA EDUCAOUNIVERSIDADE FEDERAL RURAL DE PERNAMBUCO

Proposta de projeto de tese para

doutoramento a ser apresentadaao Dr. Basarab Nicolescu com co-orientao dos professores Dr.Romildo Nogueira e Dr. GeorgeC. Jimenez.


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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 toposes2.6 The nonlocal and the inconsistency

3 Transdisciplinary logics fibrillation and algebraization

3.1 Possible translations semantics

3.2 Morphism between two signature systems in Hilbert style H 1 H 2

Conclusion


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Presentation: a brief historic-scientific introduction

The aim of this work is to offer a logic-formal and axiomaticcharacterization to the Transdisciplinarity Theory. However, an introduction totransdisciplinarity requires a brief reading of the historic-scientific events that

justifies the need for a new epistemic approach, convenient to the changesoccurred during almost two centuries. These changes proceeded in order tobreak up some or almost every classical logic principles, which have motivatedor motivate the scientific concepts up to now. Those principles are: the principleof identity, which supports that anything is itself, or formally, A is A; thenoncontradiction law, which supports that, between two contradictoryprepositions, one is false, or formally: P( P ); and the law of excludedmiddle, or formally, P P.

The history of the classical logic-deductive method starts with Aristotleand, since then it is improving, up to the beginning of modernity. Nevertheless,from nineteenth century, cracks in the classical Aristotelian-Euclidean thinkingbegin to appear, regarding the called non-Euclidean geometries. Euclid (330-277), with his Elements, gives a systematic form to Greek knowledge gatheringproposition 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 theemergence of the completeness and movement problems, linked to the

postulate five (that, if a straight line falling on two straight lines makes theinterior angles on the same side less than two right angles, the two straightlines, if produced indefinitely, meet on that side on which are the angles lessthan the two right angles), and to the congruence axiom, that is:

a) (Completeness): The Elementsaxiomatization is not enough to deduce thetheorems.

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 thisattempt would be impossible. Archimedes has stood up for the first tendencyand others, for different versions of the postulate five (given any straight lineand a point not on it, there "exists one and only one straight line which passes"through that point). Nonetheless, Lobachevski, Gaus and Bolyai have arguedthat geometry does not represent the physical reality, and that its postulatesand theorems are not necessarily true in the physical world.

The postulate five has been attacked, culminating in the Lobachevskispostulate (Lp): there exist two lines parallel to a given line through a given pointnot on the line. A completely new geometry emerged from this novel axiom,

presenting no contradictions regarding the Euclidean geometry. From these


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progresses, we could not infer that the axioms of mathematics are intuitiveconstructions based on some kind of non-mathematical evidence.

This way, the evolutive process of the deductive axiomatic system leavesthe 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 theconsistency of the axioms in a confident way? It is necessary to build upmathematics from itself. On the other hand, the congruence axioms in theElements book V, from where Eudoxus method of exhaustion and Cauchy-Riemann integral emerged (408 B.C. 355 B.C.), were initially inspiring to theconception of real numbers as a geometric abstration (Axiom XI every rightangle are similar); it was necessary to give consistency to the real, free from theinfinite approach of the infinitesimals, which Berkeley used to call absentmagnitude ghost; this was performed by Weierstrass, with his formal theory oflimits, and by Dedekind/Cantor, with his construction of real numbers based onnatural numbers, named arithmetization of analysis. Such constructions involve

some use of mathematical infinity. In this context, David Hilberts program havearisen: he wanted mathematics to be formulated on a solid and complete logicalfoundation.

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 classicaldynamics, in his PRINCIPIA, by three axioms or laws of motion. Newtonsaxiomatic counted on the infinitesimal calculus method. However, the presumedself-evidence of the three axiomatic principles, which validated Newtonianprinciples, particularly the fundamental assertion regarding the principle ofinertia and its parameter, absolute time and space, ended up as a naivepretension. Despite its basis problems, Newtonian model prevailed during the1700s with a scientific revolution that was born to fight Aristotle authority, butthat turned to be a closed and powerful model, whose principles evoke aprivileged observer, Laplaces 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 aninstrument for a new research program, science, Cantors theory of sets, inwhich elementary arithmetic is based, sustained the aristotelian classical logicas na underlying logic (DOTTAVIANO,1994).

Classical physics successess, on the other hand, kept confirming thethree 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 amember of itself, it would qualify as a member of itself by the same definition,


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culminated with Hilbert program that aimed to prove that such problems couldbe surpassed through a suitable formalization that would allow the meta-theoretical demonstration of the arithmetical consistency and, as aconsequence, of mathematics ( HILBERT & BERNAYS, 1934).

During the first half of 20th

century, Kurt Gdel publishes hisincompleteness theorems, which would limit Hilbert program:

A: (First theorem) To every -consistent recursive class of formulas therecorrespond recursive class signs r, such that neither vGenr nor Neg(vGenr)belongs to Flg() (where vis the free variableof r).

B: (Second theorem) For any formal recursively enumerable (i.e. effectivelygenerated) theory T including basic arithmetical truths and also certain truthsabout formal provability, T includes a statement of its own consistency if andonly if T is inconsistent.

One of the consequences of these theorems is that any formal systemthat envolves elementary arithmetic, accepts true and nondemonstrablesentences. Newton C. A. da Costa and Francisco A. Dria (da COSTA andDORIA 1991) have extended this result to classical physics. Nevertheless, whatdoes Gdel theorems mean to science and, particularly, to Physics? Thescience philosopher Maria Luiza Dalla Chiara understands that, from the pointof view of the development of Logic theories in its non-independent relation toany research content, Gdel theorems have produced a plurality situation in thetreatment of some physical theory (DALLA CHIARA, 1980).

However, if Gdel theorems regards only to the formal aspect, in whatsense the theorems could be extended to the physical theories? Before wecontinue, lets take a look to another result of the Gdel theorems, theundecidable, which will make our question clearer:

C: For any consistent formal theory that proves basic arithmetical truths, it ispossible to construct an arithmetical statement that is true but not provable inthe theory. That is, any consistent theory of a certain expressive strength isincomplete. The meaning of "it is possible to construct" is that there is somemechanical procedure that produces another statement.

Albert Einstein, concerning the debate about Quantum Physicsincompleteness in the famous EPR argument (Einstein-Podolski-Rosen), whichwe are going to consider soon, give us the necessary variables to understandGdels results regarding physical theories, that Lokenath Debnarh synthesizedin these words:

The totality of concepts and propositions constitute a physical theoryprovided it satisfied two essential equirements: external justification, whichmeans the agreement between theory and experiment, and internal perfection,which implies that the theory can be inferred from the most general

principles without addition concepts and assumptions. (DEBNATH, 1982).


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What Einstein comprehends as a theorys inner perfection is itsconsistency as a mathematical structure, what would lead us to Gdels Atheorem. However, Ds external justificative would demand an axiomaticcharacterization of physical theories, which we can obtain from Patrick Suppesstructure (P SUPPES - 1967), which links mathematic theory and physical

phenomena:

A physical theory T is a triple T = M,D,R , where M is the theorysmathematical structure; D is the application domain and R is the set of rules thatlinks M and D, according to da Costa and Santanna (da COSTA andSANTANNA, 2006):

On the other hand, mathematical concepts in principle cannot beconsidered as a faithful picture of the physical world; but can, throughaxiomatization, be susceptible to the use of logic-mathematical tools asindecidable propositions like C one hold physical meaning. In fact, according toLycurgo (LYGURGO 2004), there is a connection between these results andwhat we can observe in the physical world; but what is relevant in completenesstheory, according to the definition that a physical theory is complete if everyelement 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 toGdels results. But theses results do not provide us hope to get a completeaxiomatization, as incompleteness is structural. According to Maria Luiza DallaChiara, the notion of an independent Logic is, historically connected to somekind of single logic (Dalla Chiara, 1980).

Nevertheless, in front of current demands, logic is no more characterizedregarding the valid formula concept and is perceived as consequence relationsaccording to Tarski, allowing a plural and interrelation treatment between logics.

At the same time, another revolution was happening in scientific researchworld, which would definitely place us in another perspective: QuantumMechanics. 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, Heisenbergsmatrix mechanics (1925) and Schrdingers undulatory mechanics (1926) andPaulis and Diracs theories. All these formulations incorporate, somehow, the

controversial phenomena that turn quantum mechanics in a singularity inscience history: wave-particle duality, its probabilistic aspect, and theinseparability concept.

Quantum theory and its fundamental experiment, Youngs double-slitexperiment(http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html#TwoSlitsElectrons), which distinguishes quantum mechanics fromclassical mechanics through atomic phenomena explanation. This experimentprovide us with some odd results: if we call the first experiment open slit A andthe closed one, B, we have the following results, represented as logical

circuits: BABABA )( , that is, particles remain particles withpeculiar intensity distribution, and the same happens when B is open and A is


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closed, ABABAB )( . But what we note when we openboth A and B slits at the same time? Based upon previous experiments and onthe fact that the third one would keep distribution consistency, we would get thenatural sum of both beams projected in the screen, as each electron moving inits and well-drawn and fixed trajectory, which pass through the slit without

influencing other electrons that pass through the other slit. Therefore, we couldhave that, if this is valid for n experiment, so it is valid for n+1, as follows:

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

!)

2)

3)

4) 1 2

5) 1 3

6) 4 5

A B hiptese

A B A separao

A B B separao

A ModusPonens e

B Modusponens e

A B Introduo e

However, this is not what happens exactly. Electronic diffractionphenomenon shows a picture on the screen that occurs due to interference anddoes not correspond to the sum of the pictures produced by each of the slitsindividually, as the above hypothesis presents. Now lets imagine that thisphenomenon is consistent with the two first experiences, i.e., the thirdexperience maintains the properties of electrons pertaining to the previousexperiments. 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 deterioratetheir nuclei;

c) their speed should be higher than the speed of light C.

As these hypotheses are senseless, quantum mechanics is contradictoryin 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 thesame time. The first two experiments obviously suggest the continuity concept,according to Nicolescu: it is not possible to pass from one extreme to anotherwithout passing through the mid-space. However, the third experience breakscontinuity and places non-classical probability as a suitable mathematical tool todeal with diffraction interferences.

Diffraction phenomenon demonstrates that the continuity perceived in theprevious experiments with individual particles is not valid; because diffraction

phenomenon presents bindings where particles would not be splitted, accordingto the hypotheses shown above; this phenomenon is known as nonseparability


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or nonlocality. This condition made many researchers react, resulting in EPRParadox, or Einstein-Podolsky-Rosen Paradox. Essentially, EPR is a thoughtexperiment that demonstrates that the result of a measurement performed onone part A of a quantum system has a non-local effect on the physical reality ofanother distant part B. As far as one can judge, this situation is not possible

because special relativity asserts that no information can be trasmitted fasterthan light. Nevertheless, Jonh Bell, with his theorem, has proved an absolutedistinction between quantum and classical mechanics. Assuming separabilityand sustaining a, b, and c hypotheses, we would need hidden variables thatwould continuously determine experiments of diffraction, regarding a giveninterpretation of the quantum phenomena.

Bell proved that the hypothesis of local separability (a particle holdsdefined values that do not depend on a contextualization; and physical effectsspeed of propagation is finite) is not in accordance with quantum physics, thatis, the correlation coefficient C()N for some . It was born a new kind of causality:non-local causality. In our macrophysical world, if two objects interact in acertain moment in time and then stand back, they interact less. In quantumworld, entities keep interacting independently of the distant (NICOLESCU,2001).

Non-local causality idea questioned another classical science principle:the existence of only one level of reality, by which Laplaces demon, aprivileged observer, could be an invariant system from a number of generallaws (NICOLESCU,2001). Under this context, in one hundred years of scientificdevelopment, which meets a perspective pluralization, Transdisciplinarity andComplexity have emerged, as well as several perspectives with the sameobjective: adjust themselves to complexity and to the current disciplinary boom.

TRANSDISCIPLINARITY: COMPLEXITY, LOGIC OF THE INCLUDED THIRDAND LEVELS OF REALITY

What is transdisciplinarity/complexity as defined by Basarab Nicolescu? Thetransdisciplinarity is founded upon three pillars: levels of reality, logic of theincluded third and complexity. The revolution of quantum physics with itsproblems, especially the known measurement problem, plays central role in theconcept of transdisciplinarity and complexity. Avoiding go into metaphysicalsubtleties, which is as rigorous as calculation, for us is sufficient the idea oflevels of reality as an ensemble of invariant systems under the action of anumber of local general laws, or as something that offers resistance to ourexperiences. But we can use the principle that "the non-local is inconsistent withthe locality (NL)"and from this say that a level of reality is a structurally well-defined class, and is local, which plurality constitutes non-local classes that

made the complexity. The NL principle means that two levels of reality are


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different, if, passing from one to the other, there was a breaking in the laws anda break in fundamental concepts, (NICOLESCU, 2001).

To deal with levels of reality from a logic point of view, that is our goal, isnecessary the locality theory, studied in the ambit of categories theory, the local

topos, which Lawvere gave an axiomatic characterization similar to the categoryof the sets, the called toposes, in reference to the toposes introduced byGrothrndieck for algebraic geometry. (LAWVERE, 1975). Then, levels of realityand levels of perception make a local topos. A topos form a kind of categorythat has the following properties similar to the category of sets:

- Should have a terminal object corresponding to the unitary element, theset ;

- Should be able to form the Cartesian product AB of two objects and theexponential object CB of functions of C on B;

- Should be able to form the Cartesian product AB of two objects and theexponential object CB of functions of B on C ;

- Must have an object of values of truth with a distinguished elementand acorresponding element sub-object one-by-one B of A and its characteristicfunction B: A such that, for any element aof A, abelongs to B if andonly if B(a)=, and must have an object of naturals N equipped with anelement 0N and the functions S:NN defined by recursion such that: for each

a A and each function h:AA, We can build a single function :NA suchthat (n)=hn(a) for any natural number n(LAMBEK, 2004).

Also according to Lawvere, these properties can be expressed in the languageof categories theory, not referring to elements, in a way that, is seen as anapplication 1 and 0 as 1 N. Thus, each topos is associated with atype theory () as its internal language. Inversely, for every type theory is associated a topos (), the topos generated by , known as Tarski-Lindenbaums category of . (LAMBEK, 2004). If we take the levels of realityand its perception as local topos, what relationship exists between complexityand transdisciplinarity? We can say that the possibility of dealing with multiple

levels of knowledge reality, which constitutes as a complex plurality in itself, isgiven by the logic of the included third without the need of included fourth, fifth,n.

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 theaxiom of regularity of ZF (Zermelo-Frankel), or, that any not empty set Xcontains an element Y that is disjunctive of X, formally:X (Y Y XYX I ); that axiom forbids circularities as the set of allsets that do not belongs to himself as a member, formally: M= AA/{ A} and

therefore belongs to itself, that is, the paradox of Russell. With the results ofthe theorem of Pierce of: toute polyade suprieure une triade peut tre


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analyse en terme de triades, mais une triade ne peut pas tre gnralementanalyse en termes de dyades" como faz Nicolescu (Nicolescu, 1998) andthose of N. Resch for polyvalent logic (MANY-VALUED LOGIC 1969),according to that, any system of logics Ln valents will contain tautologicalformulas 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 localtopos complete.

However, the transdisciplinary local toposes are generally incomplete, notworthing the generalized axiom of choice and could be formalized as intuitivelogic (J), classic while local levels of reality. What constitutes transdisciplinarylogic complexity, as we understand it, is the non-local interaction of localtoposes in accordance with NL principle. That is the sense of quantumdiscontinuity 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 Gdelian structure, thecorrelation of levels of reality, for example: What is the nature of the theorywhich can describe the passage from one level of reality to another? Is therecoherence or even a unity in the ensemble of levels of reality? What should bethe role of the subject-observer in the existence of an eventual unity of all thelevels of reality? Is there a privileged level of reality? Would be the unity ofknowledge, if any, of subjective or objective nature? What is the role of reasonin the eventual unity of knowledge? What would be, in the field of reflection andaction, the predictive power of the new model of reality? Would be possible tounderstand this present world? In the same text, Nicolescu refers to levels ofreality ensemble as an evolutive process that would possess a self-consistencywhich, by the second theorem of Gdel, this self-reference would imply in acontradiction and consequent separation in levels of reality undoing the sameself-consistency in local collapses in which worth the classical logics. As wesaid above, what is subjacent to this process of self-consistency is aninterpretation of measurement problem in quantum physics. Then in whatconsists 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 closedsystem is described by a state that evolves over time in a determinist way,

according Schrdinger equation, (r

, t). As we mentioned, this state providesonly the probability of obtaining different results from a measurement. By now,after measurement, the system is in another state, which depends on theoutcome, thus, in the course of measuring the system evolves in anindeterminist way, described in Von Neumann projection postulate or wavepacket collapse or state reduction. Therefore, the problem of measurementarises from this opposition between determinist and indeterminists evolutionassociated to projection postulate. And if we add two more hypotheses:

) the metric device belongs to the quantum state;


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) The composed system, object/device may be closed on the environment, so,its evolution should be deterministic, but, state reductions will continue to occurduring 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 howduring measurement, a quantum superposition may become in states that donot superpose. Von Neumann answer is that projection postulate accompanyingany act of measurement, formally:

): ( ) ( )

Where omega is the unitary operator and the state vector

is expressed by superposition 11 + 22 where 1 and 2 are self-state of an observable which would reduce to ): 1 2 depending onobservation/measurement result, and that the probability of each outcome is )

2 21 2a a The unitary operator and their inverse adjunct is equal to

1

From that solution, two problems appear: one of CHARACTERIZATION, i is,what are the conditions for apply the projection postulate, and being it ainherent condition for measurement, what will characterize a measurementand/or observation? The other problem would be COMPLETENESS: Couldprojection postulate be derived from other quantum physics principles with aphysical model suitable for the measurement process? Without going intodetails of various currents that proposed solve this problem, we see that a newthermodynamic axiom, linking a large number of particles and incoherent states,would have been introduced to the problem of completeness.

SUCH ADJUNCTION, CALLED THERMODYNAMIC AMPLIFICATION, COULDCHARACTERIZE MORE SPECIFICALLY THE ATOMIC PHENOMENASOBSERVATION BASED ON RECORDS OBTAINED THROUGHAMPLIFICATION DEVICES WITH IRREVERSIBLE WORK. This programwould be later unsolved, so that, the problem of completeness is unsolvableand that the thermodynamic models do not provide exact solution to themeasurement problem. If the completeness problem was solvable, it wouldprovide an example of measurement with certain conditions:

- a) a quantum state can be attributed to the macroscopic measuringequipment, (device state in a Hilbert space of finite dimension)

- b) The composite quantum system (device/object) can be considered closedin relation to environment and the rest of the universe, (the composed systemevolution is unitary)

- c) The different measurement results correspond to distinct final states of thedevice; (pointer states)

- d) A more precise specification of this pointers supposition constitutes acondition of solvability;


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- e) Should be precisely defined what is meant by measuring, (a measurementis a subclass of interactions governing the evolution of a composed system,device/object, that should be able to distinguish between classes of objectstates for which the medium values of a self-adjunct operator are different. Inaddition to this definition, there are the measurement classes of repeatable and

predictable. A repeatable measurement is such that, shortly after being madeand repeated, the result certainly is the same, or, the final state of the objectafter the measurement is the self-state corresponding to self-value obtained asa result. The predictable ones are that for each possible outcome there is astate of the object that leads to a predictable result.) A measurement satisfyingthese conditions, would give a solution for the completeness problem.Meanwhile, even as (OSVALDO 1992, pp. 177-217), the evidence is that suchmeasurement cannot be defined, and then, the completeness problem being asunsolved.

Being:

Unitary condition

Measurement

Solvability

The a, b and c hypothesis

The evidence of insolvability in general can be represented as:

,,

Accepting that does not exist e then does not exist measurements thatmeets the condition of solvability, ,,.

The problem of measurement received several proposals of solution, and one ofthem is the theory of open systems, in which the compound system, thatincludes object/device, cannot be completely isolated from the environment. In

this conception, there is an approach in which the reduction process would bean objective physical phenomenon that would transcend the measuring act,transforming itself in the collapse or decoherence problem, or the problem ofpotentialities update, ontologies of powers that could demand asupplementation. In this perspective is settled the Gdelian self-consistentuniverse of Nicolescu. Thus, the question for us is: Is the inconsistencyassociated to reality levels ensemble evolution self-referentialty, the sameinconsistency of the correlation between locality and non-locality? If the answeris yes, we have from the second theorem of Gdel, that levels of realityensemble are inconsistent. If the answer is no, we have the followinghypotheses:


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- Or the correlation local/ non-local would of be of extension and therefore will

worth : x x , , or a conservative interpretation of the logicalstructure would be extended to other structures or toposes, violating the NLprinciple, and the ensemble of levels of reality would be inconsistent;

- Or, self-referentialty not infringes the axiom of regularity and the correlationlocal/non-local would be inconsistent.

So, in the positive response view, the system broadly would be equal to thecorrelation local/non-local which collapses from itself-reproduction in consistentand incomplete local toposes. Answerig to questions above, we could say thattransdisciplinarity is consistent and inconsistent making worth the contradictorypairs that experience and scientific theory, after quantum physics, had sawappear: local causality and non-local; reversibility and irreversibility, separabilityand non-separability; wave and particle, continuity and discontinuity, symmetryand broken symmetry, and so on. But Nicolescu himself answered suchquestions, particularly the first, about the nature of the theory that coulddescribe the transition from one level to another, saying that no one managed tofind a mathematical formalism that allows the passage of a strict world toanother. But we must differentiate logical epistemic formalization from formalmathematical structuring of models theories. A logical epistemic formalizationhas no compromise with the stander algebraic model, such as the formalmathematical structure. Logical epistemic formalization works with theory ofcategories advanced features, creating logical-topological spaces of possibleworlds from the Tarskian logic conception as relations of consequence. In thisway, we believe that logic subjacent to transdisciplinarity, is a combination ofclassical and inconsistent logics, kind of not Hegelian temporal dialectic,because we are always in some reality, and that subject self-position, as Hegelbelieves, is derived. However, criticism to Hegelian succession of contradictionseems to lack sense, because, if there is a contradiction, at some lapse of timethis contradiction is simultaneous. In any way, transdisciplinarity self-referenceis not closed in the immanence of self- position of idealistic subject of self, thatputs the non-self and understand it. The complexity of transdisciplinary self-reference opens to the multiplicity of historical-existential experience as asubject pertain to the worlds that it is included. Two other properties are at thecore 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 relativeincommensurability of different languages and/or (levels of reality) and meaninghorizon.

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 StphaneLupasco international conference, where Nicolescu asks:


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- B.N: E, so, you were quite attracted by theorem of Gdel. Because theessential point is there: this organizational structure of included third logic isGdelian type. It never closes.

- E.M: Of course. But Gdel reaches to a principle of uncertainty. That is, he

says the system cannot give account of itself, with its own resources. It is notknown whether it is consistent and it is not known if it contains a contradiction.First point: the uncertainty. Second point: is possible, however, conceive aricher meta-system, able to understanding the system, and that, of course, itopens to infinite itself. But then, at this moment, I do not see the expression of aincluded third specific logic.

Nicolescu then retakes aspects of Gdels theorems, almost always overlookedin the analytical approaches, introducing the meta-theoric affirmation of T insideof T consistency, and therefore, their self-reference as a contradiction that,according to E. Morin, is a priori: Le thorme de Gdel nous dit qu'un systme

d'axiomes suffisamment riche conduit invitablement des rsultats soitindcidables, soit contradictoires. Cette dernire assertion est souvent oubliedans les ouvrages de vulgarisation de ce thorme.

So let's see: La is the arithmetic language, T an axiomatizable recursiveextension of PA, Peano arithmetic, PrT(x) a proof predicate for T and Com(T)the sentencePrT( 0=1).

We assume that T is consistent and TCom(T) and being a sentence in Lawhich meets T PrT( ). We say that is a model of T,and ispositive if and negative in other cases. If T is consistent, there is a model1 of T. If 1 is positive, then 1 Com(T+), then there is a 2 negativemodel of T which is a definable extension of 1. If not, 1 = 2. For thehypothesis that 2 Com(T), we have a model 3 of T which is a definableextension of 2. 2 if is negative, 2PrT(), so 3 . Meanwhile, 3 and 2 and 3 are an extension of 2, which is a contradictionwhere T theory proves its own consistency. (This evidence is due Jech, 1994)

Once self-consistency reintroduces the subject in the system by itselfformulated, Nicolescu complete its approach of included third logic with the

ontological conception of Lupasco: the subject of knowledge is implied in thelogic by itself formulated. The experience is the subject itself experience. Thecircular character of logic assertion, as its own logical experience, occurs fromcircular character of subject: to define the subject we must take into account allphenomena, elements, events, states and propositions that relate to our worldand, furthermore, the affectivity. An obviously impossible task: in the ontology ofLupasco, the subject never would be defined. This Gdelian structure alwaysopens, beyond the levels of reality, a NOT RESISTANCE ZONE, that we ratherto call, SEMIOTICALLY NOT FORMED FIELD, which is a field not finitelyalgebraizable 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 Kolmogorovcomplexity measurement apply to this semiotically not formed field? Is possible


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consider the transdisciplinary not-separability as a totality whose complement isnot 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 calledrandom if n K (n). Kolmogorov has proved that not random numbers set isrecursively enumerable, but is not recursive. Odifreddi (1989) considered this a

version of Gdels first theorem of incompleteness. Second Kikuchi (1997),complexity of Kolmogorov also leads to the second theorem of incompletenessof Gdel, extracted from the paradox of Berry, (What is the "first not namednumber with less than ten words?). Since Gdel has proved that the firsttheorem leads to the second. Then, to apply Kolmogorov complexity to levels ofreality ensemble and semiotically not formed field, we would have to considerlocal/non-local inconsistency null or levels of reality ensemble self-referential asa recursive extension, which would give to levels of reality ensemble a localcharacter as such is the Kolmogorov complexity, valid for the local topos andthe incompletenesses of Gdel. The question turns on know what type ofcontextuality 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 byvarious environmental aspects, but not by irreducible and specific nonpredictable from system and experimental disturbance interaction. In classicalcontextuality, Kolmogorov axioms are met, and a classic model of probabilitycan be used. However, when contextuality is intrinsic, the system and thedisturbance, both, have an internal relation of constitution, as such its interfacecreates a concrescence of emergency. The presence of intrinsic contextualitymeans that Kolmogorov axioms are not met. The Entanglement can be testedby determining whether correlation experiments in entities which being togetherinfringe the inequality of Bell (Bell 1964) Pitowsky,(1989) had proved that if theinequality of Bell are met for a range of probabilities concerning to consideredexperiment response, then there is a Kolmogorovan classical probability thatdescribes this probability. In this case, the probability can be described as alack of knowledgeabout the precise status of the system. In other hand, if theinequality of Bell is infringed, such classical Kolmogorovian probability is notvaluable; rather, Bells inequalities infringement proves that the probabilitiesinvolved are not classical. In transdisciplinary contextuality case, we believebeing this classical and non-classical at the same time, what seems to confirmLupascos 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 apotential, at the same time that some knowledge is updated. So, algorithmicalcomplexity pertains to transdisciplinary complexity as a local topos.

In the chapter "Transdisciplinarity and open unity of the world", which Nicolescucalls levels of reality ensemble, is in the Gdelian conditions of its structure, alocal topos in which is valuable Kolmogorovian classical probability. But, inother hand, in the "Included third. From quantum physics to ontology. Theproblem, if is not from translation, the relationship between a level of reality andanother, does not preserves the coherence, and then, the application ofKolmogorovian complexity to structurally incoherent levels of reality ensemble is

not valuable. So we can say that:


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Being Cn: the levels of reality ensemble;

and Cn Zr: the levels of reality ensemble union zone of non-resistance. Cnhas a coherent structure and worth the algorithmic complexity Cn Zr isstructurally incoherent and generally does not worth the algorithmic complexity.

The question that immediately arises is whether is possible, in general, acoherent level of reality ensemble Cn. In this case, the expression levels ofreality ensemble, in general, mean a local extension. Then, the self-consistencyrefers to Cn Zr as irreducibility of contradiction and relativity of consistency.The union operator ( ) here is being taken in Lupasco sense, ofdistinguishable non-separability A ( is actualization) P ( is notpotentiation), ie, Cn Zr = A P. This formulation presents us a problemabout the use of denial and implication in included third logical. For example: Isthe 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 intuitionistdenial 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 acontradiction makes use of classical denial by the principles of De Morgan

( ) ( ) A A A A , what suggests us that in A(X) there is inducedoperations PX, the set of subsets of X, of finites intersection and union whichare closed and form a Boole algebra. So there is not a necessary link betweenincompleteness, that we will call C and A(X), the opening of a topologicalspace, i is (C A(X)). Thus, we believe that the transdisciplinary localtoposes form incomplete closed spaces in which worth the classical logic withopen complement in which worth a weaker denial, possibly of intuitive typewhere (C A(X)). These spaces or logic fibrillation, ( CA(X)) (C A(X)) ( C A(X)) (C A(X)) = , Where is theintrinsic contextualization, forming a logic of contradictory. So, we have to:

1) ( C A(X)) ( C A(X))

2) ( C A(X)) (C A(X))

From 1) e 2) is made1

and so on.

OBJECTIVE

This work aims to investigate the possibility of a logical construction oftransdisciplinarity in the Hilbert style LT 1 . In particular, we believe that atransdisciplinary logical is formed from an intuitionist classical logicalcombination and non-classicals temporal inconsistent. Give an algebraicformalization of local/ temporal inconsistent interaction whose subjacent logic isintuitionist and non-local, making use of W. Carniellis possible translationssemantics and Tarskis concept of logic as consequence relations.


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METHODOLOGY

The methodology used in the logics combination to form a transdisciplinary logicis 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 L1 and L 2 , or can beanalytical (Splitting logic), for example: W. Carniellis possible translationssemantics:

L=L 1 ... L n in the case where L is known and its decomposition in moresimple logic.The question that arises before combining logics is: what kind of structure wewant to achieve with the concept of logical system: proof system as tabl,axiomatic system, natural deduction, or semantic methods as logical matrices,

valuations and Kripkes semantics? As usual in this kind of methodology, A.Tarskis concept of logic enables more resourcefulness in the combinedtreatment of logics, which is independent of the concept of valid formulas.What's common in any system of logic is the concept of logical consequencedenoted by defined from an ensemble of sentences or formulas of L.

Definition:

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 , as such U { } it is a formula set of L( C ).The elements , of R, the elements are called rules if = is called axiom.We can make a fibrillation of logic in Hilberts type to form transdisciplinaritylogic in such way that they are morphed in the systems of categories of HilbertHil. Therefore, the logic of the transdisciplinarity forms a composition of classiclogics and inconsistent in Hilberts style.

JUSTIFICATION

There is no doubt that the twentieth century was the center of greatchanges in all areas of knowledge, particularly in the scientific universe,resulting in remarkable victories in every technique and production fields. Inspite of that, it could be noted that knowledge splitting and compartmentation,inherited from a tradition shaped by the thoughts generated during the intersticebetween fifteenth and nineteenth centuries, was no longer enough to create theepistemological references required to solve the features of knowledge itself.

Many problems seemed to be out of the entangled theory systems, and akind of blindness hovered over the attempts to understand many naturesfundamental problems, as well as its more common problems. However,debates about topics like the structure of matter, the objectivity or the possible


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relationships between the observer and the levels of reality, brought to lightunsurpassable paradoxes. Therefore, new ideas emerged, as the Theory ofRelativity, 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 determiningconceptualization, as well as of the systems that configure the logicaloperations. This intends even to affect the designation structures of theintelligibility fundamental categories, like the mechanisms that operate theirapplication control.

It is accepted, therefore, that thinking about the complex systems impliesaccepting the need to overcome major challenges, starting from the implicationsof the multidimensional and hologramatic structures, or even theinterconnectivity and inseparability of the one and the multiple, which ispertinent to it.

Despite the expectation that open up new possibilities to understanding

and the sociocultural transformations that this could raise, it is known that itwould be impossible to embrace the study of the Complex Systems only startingfrom the established contemporary disciplinarities.

During the First World Congress of Transdisciplinarity, held in Conventode Arrbida, Portugal, from 2-7 November, 1994, a Letter of Intent wasprepared, outlining a set of fundamental principles, which exalted the need toconsolidate the transdisciplinar and transcultural thinking as the best way toapproach the different aspects of complexity in distinct systems (Nicolescu,2001).

Transdisciplinarity would then represent a conception of the researchbased 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, wheretransdisciplinarity emerges to build a new model to bring the realities of theobject of study closer (Hernndez, .....).

Turning to the Article 14 of the Transdisciplinarity Letter, whose termsrefer to argument rigidity as the best limit to the possible conceptual deviationsfrom the articulation of the problem situation data, we can justify the need forunfolding a logic-formal and axiomatic characterization of the TransdisciplinarityTheory, mainly when we consider the need for another epistemic perspectivesuitable 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 theprevailing operations, relevant and evident under its domain (exclusion-inclusion, disjunction-conjunction, implication-negation), it is intended, throughthis logical constructions, to foster the inclusion of other features to thetransdisciplinary way of thinking, like the Opening, comprehending theunknown, unexpected, and unpredictable acceptance; and Tolerance, as anexercise 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 thisproject, especially regarding the reiteration of almost every classical logicprinciples disruption, which sustain the ground theory in all the study fields

mentioned above.


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In Brazil, many research groups are involved in studing logic, standingout Prof. Newton da Costa group, which I am involved in, becoming specializedin non-classical logics. Nevertheless, regarding the indication of a wide workfield outlining, I intend, through this proposition, to look for deeperTransdisciplinarity subsidies from a more authentic source, hoping that it will

constitute, in a near future, an important reference to the development oftheoretical and technological studies on semiotics. Therefore, we expect thatthe outcomes from this project accomplishment allow a contextualization in thispromising scenario, and serve as a back up to exploration approaches of newhorizons, particularly those which find themselves immersed in the paradigmsinvisible zone, so that we can overcome the determinism of the currentexplanatory models, which are associated to the convictions and beliefssystems, in every scope. This must contribute to the collapse of the cognitiveand intellectual conformisms.

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