General System theory, Like-Quantum Semantics and Fuzzy Sets
Microsoft Word - Like-QuantumSemantics.doc
General System theory, Like-Quantum Semantics and Fuzzy Sets 
Ignazio Licata 
Isem, Institute for Scientific Methodology, Pa, Italy 
Ignazio.licata@ejtp.info 
Abstract: It is  outlined the possibility to extend the quantum formalism in relation to the 
requirements of  the  general systems theory. It can be done by using a  quantum semantics 
arising from the deep logical structure of quantum theory. It is so possible taking into account the 
logical openness relationship between observer and system. We are going to  show how 
considering the truth-values of quantum propositions within the context of the fuzzy sets is here 
more useful for systemics . In conclusion we propose an  example of   formal quantum coherence. 
Key-words: Quantum Theory; Fuzzy Sets; System Theory; Syntax and Semantics of Scientific 
Theories; Logical Openness. 
Published in Systemics of Emergence. Research and Development, Minati G., Pessa E., Abram M., 
Springer, 2006, pages 723-734. 
1.The role of syntactics and semantics in general system theory  
The omologic element breaks specializations up, forces taking into account different things at the same time, stirs up the 
interdependent game of the separated sub-totalities, hints at a broader totality whose laws are not the ones of its 
components. In other words, the omologic method is an anti-separatist and reconstructive one, which thing makes it 
unpleasant to specialists. 
F. Rossi-Landi 1985 
The systemic-cybernetic approach ( Wiener, 1961; von Bertalannfy,1968; Klir, 1991) requires a 
careful evaluation of epistemology as the critical praxis internal to the building up of the scientific 
discourse. That is why the usual referring to a “connective tissue” shared in common by different 
subjects could be misleading. As a matter of fact every scientific theory is the outcome of  a 
complex conceptual construction aimed to the problem peculiar features, so what  we are 
interested in is not a framework shaping an abstract super-scheme made by the “filtering” of the 
particular sciences, but a research focusing on the global and foundational characteristics of 
scientific activity in a trans-disciplinary perspective.   According to such view, we can understand 
the General System Theory (GST) by the analogy to metalogic. It deals with the possibilities and 
boundaries of various formal systems to a more higher degree than any specific structure.  
A scientific theory presupposes  a certain set of relations between observer and system, so 
GST has the purpose to investigate the possibility of describing the multeity of system-observer 
relationships. The GST main goal is delineating a formal epistemology to study the scientific 
knowledge formation, a science able to speak about science. Succeeding to outline such 
panorama will make possible analysing those inter-disciplinary processes which are more and 
more important in studying complex systems and they will be guaranteed the “transportability” 
conditions of a modellistic set from a field to another one. For instance, during a theory developing, 
syntax gets more and more structured by putting univocal constraints on semantics according to 
the operative requirements of the problem. Sometimes it can be useful generalising a syntactic tool 
in a new semantic domain so as to formulate new problems. Such work, a typically trans-
disciplinary one, can only be done by the tools of a GST able to discuss new relations between 
syntactics  (formal model) and semantics ( model usage). It is here useful to consider again the 
omologic  perspective, which not only  identifies analogies and isomorphisms in pre-defined 
structures, but aims to find out a structural and dynamical relation among theories to an higher 
level of analysis, so providing new use possibilities (Rossi-Landi, 1985). Which thing is particularly 
useful in studying complex systems, where the very essence of the problem itself makes a 
dynamic use of models necessary to describe the emergent features of the system (Minati & 
Brahms, 2002; Collen, 2002).  
We want here to briefly discuss such GST acceptation, and then showing the possibility of 
modifying the semantics of  Quantum Mechanics (QM) so to get a conceptual tool fit for the 
systemic requirements.  
2. Observer as emergence surveyor and semantic ambiguity solver  
What we look at is not Nature in itself, but Nature unveiling to our questioning methods. 
W. Heisenberg, 1958 
A very important and interesting question in system theory can be stated as follows: given a set 
of measurement systems M and of theories T related to a system S, is it always possible to order 
them, such that Ti-1 �Ti, where the partial order symbol �  is used to denote the relationship 
“physically weaker than” ?  We shall point out that, in this case, the ith theory of the chain contains 
more information than the preceding ones. This consequently leads to a second key question: can 
an unique final theory Tf  describe exhaustively each and every aspect of system S ? From the 
informational and metrical side, this is equivalent to state that all of the information contained in a 
system S can be extracted, by means of adequate measurement processes.  
The fundamental proposition for reductionism is, in fact, the idea that such a theory chain will 
be sufficient to give a coherent and complete description for a system S. Reductionism, in the light 
of our definitions, coincides therefore with the highest degree of semantic space “compression”; 
each object  D ∈ Ti in S has a definition in a theory Ti belonging to the theory chain, and the latter 
is - on its turn - related to the fundamental explanatory level of the “final” theory Tf. This implies that 
each aspect in a system S is unambiguously determined by the syntax described in Tf. Each 
system S can be described at a fundamental level, but also with many phenomenological 
descriptions, each of these descriptions can be considered an approximation of the “final” theory.  
Anyway, most of the “interesting” systems we deal with cannot be included in this chained-
theory syntax compatibility program: we have to consider this important aspect for a correct 
epistemic definition of systems “complexity”. Let us illustrate this point with a simple reasoning, 
based upon the concepts of logical openness and intrinsic emergence (Minati, Pessa, Penna, 
1998; Licata, 2003b). 
Each measurement operation can be theoretically coded on a Turing machine. If a coherent 
and complete fundamental description Tf exists, then there will also exist a finite set - or, at most, 
countably infinite - of measurement operations M which can extract each and every single 
information that describes the system S. We shall call such a measurement set Turing-observer. 
We can easily imagine Turing-observer as a robot that executes a series of measurements on a 
system. The robot is guided by a program built upon rules belonging to the theory T. It can be 
proved, though, that this is only possible for logically closed systems, or at most for systems with a 
very low degree of logical openness. When dealing with highly logically open systems, no recursive 
formal criterion exists that can be as selective as requested (i.e., automatically choose which 
information is relevant to describe and characterize the system, and which one is not), simply 
because it is not possible to isolate the system from the environment. This implies that the Turing-
observer hypothesis does not hold for fundamental reasons, strongly related to Zermelo-Fraenkel's 
choice axiom and to classical Godel's decision problems. In other words, our robot executes the 
measurements always following the same syntactics, whereas the scenario showing intrinsic 
emergence is semantically modified. So it is impossible thinking to codify any possible 
measurement in a logically open system! 
The observer therefore plays a key rule, unavoidable as a semantic ambiguity solver: only the 
observer can and will single out intrinsic-observational emergence properties ( Bass & 
Emmeche,1997; Cariani, 1991), and subsequently plan adequate measurement processes to 
describe what  – as a matter of fact- have turned in  new systems. System complexity is 
structurally bound to logical openness and is, at the same time, both an expression of highly 
organized system behaviours (long-range correlations, hierarchical structure, and so on) and an 
observer's request for new explanatory models.  
So, a GST  has to allow - in the very same theoretical context – to deal with the observer as an 
emergence surveyor in a logical open system. In particular, it is clear that the observer itself is a 
logical open system. 
Moreover, it has to be pointed out that the co-existence of many description levels – compatible but 
not each other deductible – leads to intrinsic uncertainty situations, linked to the different 
frameworks  by which a system property can be defined. 
3. Like-quantum semantics  
I’m not happy with all the analyses that go with just the classical theory, because nature isn’t classical, damm it, 
and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful 
problem, because it doesn’t look so easy. Thank you. 
R. P. Feyman, 1981 
When we modify and/or amplify a theory so as to being able to speak about different systems 
from the ones they were fitted for, it could be better to look at the theory deep structural features so 
as to get an abstract perspective able to fulfil the omologic approach requirements, aiming to point 
out a non-banal conceptual convergence. 
As everybody knows, the logic of classical physics is a dichotomic language (tertium non 
datur), relatively orthocomplemented and able to fulfil the weak distributivity relations by the logical 
connectives AND/OR. Such features are the core of the Boolean  commutative elements of this 
logic because disjunctions and conjunctions are symmetrical and associative operations. We shall 
here dwell on the systemic consequences of these properties. A system S can get or not to get a 
given property P. Once we fix the P truth-value it is possible to keep on our research over a new 
preposition P subordinated to the previous one’s truth-value. Going ahead, we add a new piece of 
information to our knowledge about the system. So the relative orthocomplementation axiom 
grants that we keep on following a successions of steps, each one making our uncertainty about 
the system to diminish or, in case of a finite amount of steps, to let us defining the state of the 
system by determining all its properties. Each system’s property  can be described by a countable 
infinity of atomic propositions. So, such axiom plays the role of a describable axiom for classical 
systems.  
The unconstrained use of such kind of axiom tends to hide the conceptual problems spreading 
up from the fact that every description implies a context, as we have seen in the case of  Turing-
observer analysis, and it seems to imply that systemic properties are independent of the observer, 
it surely is a non-valid statement when we deal with  open logical systems. In particular, the 
Boolean features point out that it is always possible carrying out exhaustively a synchronic 
description of the properties of a systems. In other words, every question about the system is not 
depending on the order we ask it and it is liable to a fixed answer  we will indicate as 0- false / 1-
true. It can be suddenly noticed that the emergent features otherwise get a diachronic nature and 
can easily make such characteristics not taken for granted. By using Venn diagrams it is possible 
providing a representation of the complete descriptiveness of a  system ruled by classical logics. If 
the system’s state is represented by a point and a property of its by a set of points, then it is always 
possible a complete “blanketing” of the universal set I, which means the always universally true 
proposition. (see fig. 1).  
The quantum logics shows deep differences which could be extremely useful for our goals 
(Birkhoff & von Neumann, 1936; Piron, 1964). At the beginning it was born to clarify some QM’s 
counter-intuitive sides , later it has developed as an autonomous field greatly independent from the 
matters which gave birth to it. We will abridge here the formal references to an essential survey, 
focusing on some points of general interest in systemics. 
The quantum language is a non-Boolean orthomodular structure, which is to say it is relatively 
orthocomplemented but non-commutative, for the crack down of the distributivity axiom. Such thing 
comes naturally from the Heisenberg Indetermination Principle  and binds the truth- value of an 
assertion to the context and the order by which it has been investigated (Griffiths, 1995). A well-
known example is the one of a particle’s spin measurement along a given direction. In this case we 
deal with semantically well defined possibilities and yet intrinsically uncertain. Let put xΨ  
the spin measurement along the direction x. For the indetermination principle the value yΨ  will be 
totally uncertain, yet the proposition  yΨ =0 ∨   yΨ =1  is necessarily true. In general, if P is a 
proposition , (-P ) its negation and Q the property which does not commute with P, then we will get 
a situation that can be represented by a “patchy” blanketing of the set I (see fig.2). 
Such configuration finds its essential meaning just in its relation with the observer. So we can 
state that when a situation can be described by a quantum logics, a system is never completely 
defined a priori. The measurement process by which the observer’s action takes place is a choice 
fixing some system’s characteristics and letting other ones undefined. It happens just for the nature 
itself of the observer-system  inter-relationship. Each observation act  gives birth to new descriptive 
possibilities. The proposition Q – in the above example – describes properties that cannot be 
defined by any implicational chain of propositions P. Since the intrinsic emergence cannot be 
regarded as a system property independent of the observer  action- as in naïve classical  
emergentism - , Q can be formally considered the expression of an emergent property. Now we are 
strongly tempted to define as emergent the undefined proposition of quantum-like anti-
commutative language. In particular, it can be showed that a non-Boolean and irreducible 
orthomodular language arises infinite propositions. It means that for each couple of propositions P1 
and P2   such that non of them imply the other , there exists infinite propositions Q which imply  P1  
∨  P2  without necessarily implying the two of them separately: tertium datur. In a sense, the 
disjunction of the two propositions gets more information than their mere set-sum, that is the 
entirely opposite of what happens in the Boolean case. It is  now easy to comprehend the deep 
relation binding the anti-commutativity, indetermination principles and system’s holistic global 
structure. A system describable by a Boolean structure can be completely “solved” by analysing 
the sub-systems defined by a fit decomposition process( Heylighen, 1990; Abram, 2002). On the 
contrary, in the anti-commutative case studying any sub-system modifies the entire system in an 
irreversible and structural way and produces uncertainty correlated to the gained information, 
which think makes absolutely natural extending the indetermination principles to a big deal of 
spheres of strong interest for systemics (Volkenshtein , 1988). 
A particularly key-matter is how to conceptually managing the infinite cardinality of emergent 
propositions in a lik-quantum semantics. As everybody knows traditional QM refers to the 
frequentistic probability worked out within the Copenhagen Interpretation (CIQM). It is essentially a 
sub specie probabilitatis Boolean logics extension. The values between [ ]1,0  - i.e. between the 
completely and always true proposition I and the always false one O – are meant as expectation 
values, or the probabilities associated to any measurable property. Without dwelling on the 
complex – and as for many questions still open – debate on QM interpretation, we can here ask if 
the probabilistic acception of truth-values is the fittest for system theory. As it usually happens 
when we deal with trans-disciplinary feels, it will bring us to add a new, and of remarkable interest 
for the “ordinary” QM too, step to our search. 
4. A Fuzzy Interpretation of Quantum Languages 
A slight variation in the founding axioms of a theory can give way to huge changings  on  the frontier. 
S. Gudder, 1988 
The study of the structural and logical facets of quantum semanics does not provide any 
necessary indications about the most suitable algebraic space to implement its own ideas. One of 
the thing which made a big merit of such researches has been  to put under discussion the key role 
of Hilbert space. In our approach we have kept the QM “internal” problems and its extension to 
systemic questions well separated.  Anyway, the last ones suggest an interpretative possibility 
bounded to fuzzy logic, which thing can considerably affect the traditional QM too. The fuzzy set 
theory is , in its essence, a formal tool created to deal with information characterized with 
vagueness and indeterminacy. The by-now classical paper of  Lotfi Zadeh (Zadeh, 1965) brings to 
a conclusion an old tradition of logics, which counts Charles S. Peirce, Jan C. Smuts, Bertrand 
Russell, Max Black and Ian Lukasiewicz among its forerunners. At the core of the fuzzy theory  lies 
the idea that an element can belong to a set to a variable degree of membership; the same goes 
for a proposition and its variable relation to the  true and false logical constants. We  underline here 
two aspects of  particular interest for our aims. The fuzziness’ definition  concerns single elements 
and properties, but not a statistical ensemble, so it has to be considered a completely different 
concept from the probability one, it should –by now-  be widely clarified (Mamdani, 1977; Kosko, 
1990). A further essential – even maybe less evident – point is that fuzzy theory calls up a non-
algorithmic “oracle”, an observator  (i.e. a logical open system and a semantic ambiguity solver) to 
make a choice as for the membership degree. In fact, the most part of the theory in its structure is 
free-model; no equation and no numerical value create constraints to the quantitative evaluation, 
being the last one the model builder’s task. There consequently exists a deep bound between 
systemics and fuzziness  successfully expressed by the Zadeh’s incompatibility principle (Zadeh, 
1972) which satisfies our requirement for a generalized indeterminacy principle. It states that by 
increasing the system complexity (i.e. its logical openness degree), it will decrease our ability to 
make exact statements and proved predictions about its behaviour. There already exists many 
examples of crossing between fuzzy theory and QM (Dalla Chiara, Giuntini, 1995; Cattaneo, Dalla 
Chiara, Giuntini 1993). We want here to delineate the utility of fuzzy polyvalence for systemic 
interpretation of quantum semantics.  
Let us consider a complex system, such as a social group, a mind and a biological organism. 
Each of these cases show typical emergent features owed both to the interaction among its 
components and the inter-relations with the environment. An act of the observer will fix some 
properties and will let some others undetermined according to a non-Boolean logic. The recording 
of such properties will depend on the succession of the measurement acts and their very nature. 
The kind of complexity into play, on the other hand, prevents us by stating what the system state is 
so as to associate to the measurement of a property an expectation probabilistic value. In fact, just 
the above-mentioned examples are related to macroscopic systems for which the probabilistic 
interpretation of QM is patently not valid. Moreover, the traditional application of the probability 
concept implies the notion of  “possible cases”, and so it also implies a pre-defined knowledge of 
systems’ properties. However, the non-commutative logical structure here outlined does not 
provide any cogent indication on probability usage. 
Therefore, it would be proper to look at a fuzzy approach so to describe the measurement acts. 
We can state that given a generic system endowed with high logical openness and an indefinite set 
of properties able of describing it, each of them will belong  to the system in a variable degree. 
Such viewpoint expressing the famous theorem of fuzzy “subsetness” – also known as “the whole 
into the part” principle – could seem to be too strong , indeed it is nothing else than the most 
natural expression of the actual scientific praxis facing intrinsic emergent systems.  At the 
beginning,  we have at our disposal  indefinite information progressively structuring thanks to the 
feedback between models and measurements. It can be shown that any logically open model of 
degree n – where n  is an integer – will let a wide range of properties and propositions 
indeterminate  (the Qs in fig. 2).The above-mentioned model  is a “static” approximation of a 
process showing aspects of  variable closeness and openness. The latter ones varies in time, 
intensity, different levels and context. It is remarkable pointing out how such systems are “flexible” 
and context-sensitive, change the rules and make use of “contradictions” . This point  has to be 
stressed to understand the link between fuzzy logic and quantum languages. By increasing the 
logical openness and the unsharp properties of a system, it  will be less and less fit to be described 
by a Boolean logic. It brings as a consequence that for a complex system the intersection between 
a set (properties, propositions) and its complement is not equal to the empty set, but it includes 
they both in a fuzzy sense. So we get a polyvalent semantic situation which is well fitted for being 
described by  a quantum language. As for our systemic goal  it is the probabilistic interpretation to 
be useless, so we are going to build a fuzzy acception of the semantics of the formalism. In our 
case, given a system S and a property Q,, let Ψ be a  function which associates Q to S, the 
expression ( ) [ ]1,0∈Ψ QS  has not to be meant as a probability value, but as a degree of 
membership. Such union between the non-commutative sides of quantum languages and fuzzy 
polyvalence appears to be the most suitable and fecund for systemics. 
Let us consider the traditional expression of quantum coherence (the property expressing the 
QM global and non-local characteristics, i.e. superposition principle, uncertainty, interference of 
probabilities), 2211 Ψ+Ψ=Ψ aa . In the fuzzy interpretation, it means that the properties 1Ψ e 2Ψ  
belong to Ψ with degrees of membership 1a  e 2a  respectively. In other words, for complex 
systems the Schrödinger’s cat can be simultaneously both alive and dead ! Indeed the recent 
experiments with SQUIDs and the other ones investigating the so-called macroscopic quantum 
states suggest a form of macro-realism quite close to our fuzzy acception (Leggett, 1980; Chiatti, 
Cini, Serva, 1995). It can provide in nuce an hint which could show up to be interesting  for the QM 
old-questioned interpretative problems. 
In general, let x be a position coordinate of a quantum object and Ψ  its wave function, 
( ) dVx 2Ψ is usually meant as the probability of finding the particle in a region dV of space.  On the 
contrary, in the fuzzy interpretation we will be compelled to look at  the Ψ  square modulus as the 
degree of membership of the particle to the region  dV of space. How unusual it may seem, such 
idea has not to be regarded thoughtlessly at. As a matter of fact, in Quantum Field Theory and in 
other more advanced quantum scenarios, a particle is not only a localized object in the space, but 
rather an event emerging from the non-local networks elementary quantum transition (Licata, 
2003a).  Thus, the measurement  is  a “defuzzification” process which, according to the stated, 
reduces the system ambiguity by limiting the semantic space and by defining  a fixed  information 
quantity. 
If we agree with such  interpretation we will easily and immediately realize that we will able to 
observate quantum coherence behaviours in non-quantum and quite far from the range of  Plank’s 
h constant  situations. We reconsider here a situation owed to Yuri Orlov (Orlov, 1997). 
Let us consider a Riemann’s sphere (Dirac, 1947) – see fig. 3 - and let assume that each point 
on the sphere represents a single interpretation of a given situation, i.e. the assigning of a  
coherent set  of truth-values to a given proposition. Alternatively, we can consider the choosing of 
a vector v  from the centre O to a point on the sphere as a logical definition of a world. If we 
choose a different direction, associated to a different vector w , we can now set the problem about 
the meaning of the amplitude between the logical descriptions of the two worlds. It is known that 
such amplitude is expressed by ( )ϑcos121 + , where ϑ  is the angle between the two 
interpretations. The amplitude corresponds to a superposition of worlds, so producing the typical  
interference patterns which in vectorial terms are related to v
w .  In this case, the traditional use of 
probability is not necessary because our knowledge of one of the two world  with probability equal 
to p =1 (certainity), say nothing us about the other one probability. An interpretation is not a 
quantum object  in the proper sense, and yet we are forced to formally introduce a wave-function 
and interference terms whose role is  very obscure a one. The fuzzy approach, instead, clarifies 
the quantum semantics of this situation by interpreting  interference as a measurement where the  
properties of the world wv wv Ψ+Ψ  are owed to the global and indissoluble (non-local)  
contribution of the  v  and w  overlapping. 
In conclusion, the generalized using of quantum semantics associated to new interpretative 
possibilities gives to systemics a very powerful tool to describe the observator-environment relation 
and to convey the several, partial attempts - till now undertaken - of applying  the quantum 
formalism to the study of complex systems into a comprehensive  conceptual root. 
ACKNOWLEDGEMENTS  
A special thank to Prof. G. Minati  for his kindness and his supporting during this paper drafting.  I 
owe a lot to the useful discussing on structural Quantum Mechanics and logics with my good 
friends Prof. Renato Nobili (who let me use the figs. 1 and 2 from his book “Dai Quark alla Mente”, 
to be published) and Prof. Eliano Pessa. Dedicated to M.V. 
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