General systems, classical systems, quantum systems and fuzzy systems: an introductory survey

Introduction

Scientific work has two components. In the theoretical part, scientists develop and formulate concepts, hypotheses and theories. Sometimes they can confirm their theoretical ideas and sometimes they have to reject them. In the other part, scientists design, plan, perform, control, and replicate experiments. To produce scientific knowledge, we need both parts: theoretical concepts and experimental systems.

Scientists observe real systems or phenomena and they measure data; they state laws and they introduce empirical theories that say that the laws hold for the data. That is to say, to study systems or phenomena in reality, we connect them with a theoretical structure. To this end, we give them a structure of their own. How to do that is not clear! This is one of the central problems in the philosophy of science: ‘The problem is that we create a connection between real systems and theoretical structures. We assume that this can be done. Without this assumption, it is senseless to talk about empirical science’ (Balzer 1982, p. 289). In the so-called structuralist view of scientific theories, these real systems connected with a theoretical structure were called ‘intended systems’ of the theory (Balzer et al. 1987). In their scientific research based on an intended system, scientists get a data structure and build a model that represents the structure of the system. Frequently we say that with a theory we have a ‘picture of reality’, but this is a very simple expression of what is done in scientific work.

A more sophisticated view is given in the proposal for ‘historical epistemology’ by Hans-Jörg Rheinberger, which focuses on ‘experimental systems’. ‘This notion is firmly entrenched in the everyday practice and vernacular of the twentieth-century life scientists, especially of biochemists and molecular biologists. Scientists use the term to characterize the scope, as well as the limits and the constraints, of their research activities. Ask a laboratory scientist what he is doing, and he will speak to you about his “system”. Experimental systems constitute integral, locally manageable, functional units of scientific research’ (Rheinberger 1997a, p. 246).

In this ‘historical epistemology’, we again find two parts of scientific research, called epistemic and technical, respectively. But Rheinberger emphasizes that there is no sharp boundary between them; moreover, this boundary is vague or fuzzy. Here is a brief sketch of his approach: ‘if there are concepts endowed with organising power in a research field, they are embedded in experimental operations. The practices in which the sciences are grounded engender epistemic objects, epistemic things as I call them, as targets of research’. Rheinberger (1992a, 1992b, 1997b) discussed these ‘fluctuating objects’ and ‘imprecise concepts’ – as he also called them – in detail in his historical work.

Without distaining the value of precision in science, Rheinberger (2000, p. 236) points out that ‘precision itself has historically changing boundaries’ and accentuates the value of imprecision, vagueness or fuzziness in science: ‘Assessing what it means to be fuzzy, instead of eliminating vagueness altogether and implementing precision, has become a major concern in fields such as AI-research. Lofti1 Zadeh claims that “there is a rapidly growing interest in inexact reasoning and processing of knowledge that is imprecise, incomplete or totally reliable. And it is in this connection that it will become more and more widely recognized that classical logical systems are inadequate for dealing with uncertainty and that something like fuzzy logic is needed for that purpose”’. Rheinberger (2000, p. 236) quoted this excerpt from Zadeh (1987) and then went back to the question of ‘Whether we need, in order to understand conceptual tinkering in research, more rigid metaconcepts than those first-order concepts that we, as epistemologists, analyse. I am inclined to deny this. Why should historians and epistemologists be less imprecise, less operational, and less opportunistic after all, than scientists?’

Whereas the argument for the establishment of ‘fuzzy epistemology’ in the philosophy of science is discussed in Seising (2009), this introductory editorial to the special issue focusing on Fuzzy and Quantum Systems addresses the fluctuating object or imprecise concept ‘system’ (or object) of a physical theory and the theoretical term(s) representing its ‘state’. To this end, we will look at two physical theories, classical mechanics and quantum mechanics. Of course, the quantities of both theories, including the state of their systems, are describable in terms of conventional mathematics. However, there are different interpretations of what the state of a system is according to the theory: whereas in classical mechanics the state of a system is given by the pair of two values of time-dependent variables, position and momentum, in quantum mechanics, these two quantities cannot be measured simultaneously, and consequently these values cannot constitute the state of a quantum mechanical system. Moreover, in the theory of quantum mechanics, a system's state is given by a ‘wave function’ Ψ. This new theoretical term in physics is an element of the abstract Hilbert space and Ψ is per se not observable, e.g. in an experiment. Of course, we can represent the abstract Hilbert space in the ‘picture’ of the position operator or in that of the momentum operator or other operators and then we can get a certain value for the classical observable position or momentum or other variables. We can also predict these values as results of an experiment in order to measure a position value, or of an experiment in order to measure a momentum value. But we cannot combine predictions of the values of position and momentum of a quantum mechanical system to describe the quantum mechanical state.

Therefore, the state of a quantum mechanical system is much more difficult to determine than that of classical systems. This is the meaning of Heisenberg's uncertainty principle.

From general systems theory and system theory to fuzzy systems

In May 1962, when the electrical engineer and system theorist Zadeh (1962, p. 857) described successful developments in his research area in an anniversary edition of the Proceedings of the IRE to mark the 50th year of the Institute of Radio Engineers (IRE), he specifically mentioned ‘General Systems Theory’: ‘Among the scientists dealing with animate systems, it was a biologist – Ludwig von Bertalanffy – who long ago perceived the essential unity of systems concepts and techniques in various fields of science and who in writings and lectures sought to attain recognition for “general systems theory” as a distinct scientific discipline. It is pertinent to note, however, that the work of Bertalanffy and his school, being motivated primarily by problems arising in the study of biological systems, is much more empirical and qualitative in spirit than the work of those system theorists who received their training in the exact sciences. In fact, there is a fairly wide gap between what might be regarded as “animate” system theorists and “inanimate” system theorists at the present time, and it is not at all certain that this gap will be narrowed, much less closed, in the near future. There are some who feel that this gap reflects the fundamental inadequacy of the conventional mathematics – the mathematics of precisely-defined points, functions, sets, probability measures, etc. – for coping with the analysis of biological systems, and that to deal effectively with such systems, which are generally orders of magnitude more complex than man-made systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions. Indeed, the need for such mathematics is becoming increasingly apparent even in the realm of inanimate systems, for in most practical cases the a priori data as well as the criteria by which the performance of a man-made system is judged are far from being precisely specified or having accurately-known probability distributions’.

When Zadeh established the theory of fuzzy sets and systems three years later, he proposed a solution for this problem. He introduced new mathematical entities, ‘fuzzy sets’, as classes or sets that ‘are not classes or sets in the usual sense of these terms, since they do not dichotomise all objects into those that belong to the class and those that do not’. In fuzzy sets ‘there may be a continuous infinity of grades of membership, with the grade of membership of an object x in a fuzzy set A represented by a number f _A (x) in the interval [0,1]’ (Zadeh 1965).

In April 1965, a Symposium on System Theory took place in Brooklyn, and Zadeh presented the participants with ‘A New View on System Theory’, where he introduced the term ‘fuzzy system’: A system S is a fuzzy system if (input) u(t), output y(t) or state s(t) of S or any combination of them ranges over fuzzy sets (Zadeh 1965, p. 33). He explained that ‘these concepts relate to situations in which the source of imprecision is not a random variable or a stochastic process but rather a class or classes which do not possess sharply defined boundaries’ (Zadeh 1965, p. 29).

From classical systems to quantum systems

In classical physics, the state of a ‘system’ (or ‘object’) is represented by a set of classical observables. For example, in Newtonian mechanics, a system is a particle with mass m and its state is given by the pair of values of the object's position vector r and its momentum vector p. These two vectors implicate all other properties of the object which are relevant in the Newtonian theory of mechanics. We can formulate that the state of a physical system is the collection of all the system's properties P _i . In order to represent these properties P _i in terms of the physical theory called ‘Newtonian’ (or classical) mechanics, we must determine the formally possible functions F _i in this mechanical theory, and in order to know the object's properties at a given point in time t, we must measure the values of these functions F _i . Thus, the representation of the ‘state of a classical object’ is related to the measuring process of the observer.

Due to the possible errors of measurement and systematic errors that occur in every experiment, we can attribute the classical probability of this as being the real value of all measured values of observables. Thus, the state of an object in Newtonian mechanics is given by the pair of values of position r and momentum p and their classical probability distributions.

The discovery of quantum mechanics in the first third of the twentieth century was a scientific revolution. A basic change took place in the relationship between the exact scientific theory of physics and the phenomena observed in basic experiments. Systems of quantum mechanics – ‘quantum systems’ – do not behave like systems of classical theories in physics – they are not particles and they are not waves; they are different. This change led to a new mathematical conceptual basis in physics.

With the contributions of Bohr, Born, de Broglie, Dirac, Heisenberg, Jordan, Pauli, Schrödinger, von Neumann and others, a new mathematical theory of atomic physics or rather quantum physics was created, the so-called quantum mechanics, that differs significantly from those of classical physics. The properties of quantum mechanical systems are completely new and are not comparable with those of classical theories such as Newton's mechanics or Maxwell's electrodynamics.

To enhance understanding of the non-classical peculiarities of quantum mechanics, Born (1926a,b) presented a new view of the quantum mechanical wave function Ψ and, therefore, of the ‘state’ of a quantum system by proposing that the quantum mechanical wave function Ψ is a ‘probability-amplitude’, i.e. the absolute square of its value equals the probability of it having a certain position or a certain momentum if we measure the position or momentum, respectively. Six years later, von Neumann (1932) published his Mathematical Foundations of Quantum Mechanics, in which he defined the quantum mechanical wave function Ψ – i.e. the probability amplitude – as a one-dimensional subspace of an abstract Hilbert space. This one-dimensional subspace of the Hilbert space is then defined as the ‘state function’ of a quantum mechanical system.

Accordingly, in quantum mechanics, we have to use a modified concept of the state: the state of a quantum mechanical system consists of the probability distributions of all the object's properties that are formally possible in this physical theory. Of course, we can represent the abstract Hilbert space in the ‘picture’ of the position operator or in that of the momentum operator or other operators and then the absolute square of this representation of the Ψ-function equals the probability density function that we could measure as a certain value of the classical observable position or momentum or other properties, depending on the experiment in question.

Unfortunately, there is no joint probability distribution for events in which both variables have a certain value simultaneously (even though they are classical observables), as there is no classical probability space that comprises all these events. Such pairs would describe classical probability distributions for classical states in physics. For that reason, quantum mechanics pertains to quantities that are not describable in terms of classical probability distributions.

To summarise: the theory of quantum mechanics is completely abstract; it is a theory of state functions that are abstract Hilbert space elements and that have no exact counterpart in reality. A state function of a quantum system embodies the classical probabilities of all properties of the object, but it does not deliver a classical joint probability distribution for all these properties. Therefore, we make the following claim: we need a radically different kind of mathematics, the mathematics of quantities which are not describable in terms of classical probability distributions. This is analogue to Zadeh's (1962, p. 857) requirement: ‘… we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions’, stated in his 1962 article. The theory of fuzzy sets and systems also appeared as a theory that pertains to quantities that are not describable in terms of classical probability distributions; therefore, it is quite natural to ask how this theory can be used to interpret the theory of quantum mechanics.

Quantum logic and quantum probability theory

After the establishment of quantum mechanics as a new physical theory, some approaches were put forward to achieve a new logic and later also a new – and of course non-classical – ‘probability theory’ to handle quantum mechanical propositions. We have already argued that these ‘propositions’ of quantum logic will be ‘predictions’ – of quantum mechanical ‘events’. Since predictions are targeted on future events, we cannot valuate them with the logical values of ‘true’ or ‘false’, but use values in-between that have been called (quantum) probabilities.

•	Birkhoff and von Neumann (1936) proposed the introduction of a ‘quantum logic’, as the lattice of quantum mechanical propositions is not distributive and, therefore, not Boolean.
•	Mackey (1963) attempted to provide a set of axioms for the propositional system of predictions of the outcomes of experiments. He was able to show that this system is an orthocomplemented partially ordered set.

In these logico-algebraic approaches, the ‘probabilities’ of evaluating the predictions of the properties of a quantum mechanical system do not satisfy Kolmogorov's classical axioms. The well-known double-slit experiment shows that they are not additive and, due to their non-distributivity, it is indicated that the logico-algebraic structure of quantum mechanics is not a Boolean lattice and, therefore, it is more complicated than that of the classical probability space as it was defined by Kolmogorov.

•	Already in the 1960s, the philosopher and statistician Patrick Suppes (1961, 1966) discussed the ‘probabilistic argument for a non-classical logic of quantum mechanics’. He introduced the concept of a ‘quantum mechanical σ-field’ as an ‘orthomodular partial ordered set’ covering the classical σ-fields as substructures.
•	In the 1980s, a ‘quantum probability theory’ was proposed and developed by Gudder (1988) and Pitowski (1989).

The quantum mechanical lattice of predictions is Suppes's ‘quantum mechanical σ-field’; it is also called a ‘Boolean atlas’ because it can be restricted to a Boolean lattice corresponding to a given observable (and its compatible observables). For these reduced Boolean lattices, the values of the quantum probabilities became classical probabilities again, only applying to predictions of compatible observables.

Quantum logic and quantum probability theory represent important approaches to managing quantum mechanical uncertainties within the limits of conventional mathematics, and developments in the last decades have produced numerous and also very difficult results. Thus, quantum logic and quantum probability theory are new theories in classical mathematics that have become more and more complex. On the other hand, the new theory of fuzzy sets and systems became available at the same time and the question arose in the 1980s and 1990s of whether fuzzy sets could be useful in the interpretation of quantum mechanics. However, at that time this approach was not successful. The disappointing results may have stemmed from the fact that fuzzy set theory was not as well accepted as a mathematical tool as it is today and from a lack of interest in using the new theory on the part of theoretical physicists. But now the theory is broadly accepted, particularly in applied sciences and technology. In physics, the results of new experiments (Alain Aspect's test of Bell's inequality in 1982; Aspect et al. 1982a,b) and Anton Zeilinger's experiments on quantum teleportation since 1997 (Bouwmeester et al. 1997) have sparked a new debate on the interpretation of quantum mechanics and there is also growing interest in the theory of fuzzy sets in the field of the history of science (Seising 2007, 2008, 2009).

Linguistic variables and fuzzy states

In his ‘Outline of a New Approach to the Analysis of Complex Systems and Decision Processes’, Zadeh (1973) introduced ‘linguistic variables’, which are variables whose values may be terms in a specific natural or artificial language. To illustrate, the values of the linguistic variable ‘age’ might be expressible as young, very young, not very young, somewhat old and more or less young. These values are formed with the label old, the negation not and the hedges very, somewhat and more or less.

Linguistic variables became a proper tool for reasoning without exact values. Since in many cases it is either impossible or too time-consuming (and, therefore, too expensive) to measure or compute exact values, the concept of linguistic variables has been successful in many fuzzy application systems, e.g. in control and (medical) decision-making. Let us apply the concept of linguistic variables (1) in classical mechanics, where exact values of classical variables exist, and (2) in quantum mechanics, where exact values of classical observables do not exist.

The outcomes of a physicist's experiments have to be values of observables, i.e. an observing physicist assigns sharp values (and their classical probability distributions of the classical variables – e.g. its position r or its momentum p – due to the possible errors of measurement and the systematic errors that occur in every experiment)2 even to a quantum theoretical object. As we have argued above, this value is not sufficient to determine the quantum system's state, but is only one representation – among many others – and none of these representations of the state of the quantum system is complete!

Fuzzy states of classical systems

First, we define the ‘fuzzy state variable’ of a physical system as a vector of linguistic variables instead of numerical variables. This definition yields the interpretation of the ‘fuzzy state’ of a classical physical object as a pair of the two linguistic variables – position and momentum.

In a manner similar to Zadeh's extension of systems to fuzzy systems, the definition of a linguistic variable operating on a fuzzy set and the assignment of membership degrees and elements of the term set of the linguistic variable, the ‘fuzzy state’ of a physical system is interpreted as a vector of linguistic variables instead of numerical variables.

A concrete system a has a certain number of properties P ⁱ , i ∈ {1,…,n}. In classical physics, we attach classical numerical variables (observables) to these properties that can be measured. To use the methods of fuzzy set theory, now we also attach a linguistic variable to represent the property P ⁱ . Linguistic variables operate on fuzzy sets and assign membership degrees and elements of a term set, for example, T() = {very small, small, big, very big, etc.}

We can imagine the n-tuple to be a ‘linguistic vector’ in an n-dimensional Cartesian space. The value LVP _n (a, t) for a system a at time point t is called the ‘linguistic state’ or ‘fuzzy state’ of this system at this time. During this time, the linguistic state of a system moves in the ‘linguistic state space’ or ‘fuzzy state space’ of the system.

In the case of a classical particle in Newtonian mechanics, the ‘fuzzy state’ is the pair (2-tuple) (LV _r, LV _p ) of the two linguistic variables, position LV _r and momentum LV _p , that operate on fuzzy sets and assign membership degrees and elements of a term set. For example, T(LV _r ) = {very small, small, null, big, very big, etc.} and T(LV _p ) = {very small, small, null, big, very big, etc.}.

Usually, the shape of the fuzzy set's membership functions is subjectively chosen or dependent on the problem. In a very special case, the membership function may have the shape of the classical Gaussian probability distribution, and thus the fuzzy state variable yields the probabilities of measuring the observables' position r and momentum p due to the calculation of errors. However, in general, membership functions of fuzzy sets do not represent probability distributions of measurement errors or randomness, but more general uncertainties that are deeply rooted in the absence of the theoretical concept's strict boundaries.

We know already that classical concepts such as position and momentum, which have strict boundaries in Newtonian mechanics, do not have such boundaries in the theory of quantum mechanics. Therefore, this pair of concepts does not match the quantum mechanical state variable.

Fuzzy states of quantum systems

Let us try to extend our approach of ‘fuzzy states’ of systems to quantum systems to include the assumption that the classical theoretical concepts are not the right concepts, but that we have no better concepts to interpret the outcomes of classical experiments.

In the case of a quantum system, the ‘fuzzy quantum state’ is an infinite-dimensional vector LVP _∞ (a, t) of linguistic variables in the abstract Hilbert space, with an infinite tuple of linguistic variables LV _i , but not all linguistic variables LV _i and LV _j are compatible, i.e. they have an uncertain relationship with each other, e.g. LV ₁ = LV _r (position observable) and LV _j  = LV _p (momentum observable).

In general, at one point in time, we can measure one of these, LV _i , and this measurement reduces the membership function to a numerical value of this observable that can be interpreted as a ‘singleton’ in the view of fuzzy sets. This is an effect that is known in conventional quantum mechanics as the ‘collapse’ of the quantum mechanical state function.

When the fuzzy state concept is used for quantum systems, there is no ‘collapse’ of the fuzzy quantum state. We still have its complete representation as the infinite tuple of the linguistic variables . However, after the measurement of observable , this component of the fuzzy quantum state is a special linguistic term, a singleton, that can be reduced to a crisp numeric term (and its classical probability distribution) that has been measured.

These preliminary ideas on a fuzzy approach to physics and particularly to quantum mechanics show that it is useful and meaningful to consider the uncertainty in quantum mechanics much deeper. Our rough idea using linguistic variables represents the minimum mathematical structure of uncertainty, whereas classical probability theory represents the extreme of having a maximum of logical structure as its base, i.e. the Boolean lattice. Between these two extremes, there is enough room for fuzzy logical considerations.

The contributions in this special issue

In the first contribution to this issue, Towards many-valued/fuzzy interpretation of quantum mechanics, Jarosław Pykacz interprets the ‘truth values’ of predictions about properties of quantum mechanical systems (objects) as degrees that are associated with the results of future experiments. In the second contribution, Thomas Vetterlein discusses the Boolean atlas structure as a ‘partial Boolean algebra’, a structure that was endowed to the set of closed subspaces of a Hilbert space in 1965 by Kochen and Specker. With this approach, lattice operations are permitted only in the case that the two subspaces commute. In his contribution, Partial quantum logics revisited, Vetterlein considers the set of effects, i.e. self-adjoint Hilbert space operators representing generalised yes-no experiments. The effect's eigenvalues may range throughout the interval [0, 1] and, therefore, this approach to quantum logic includes ‘fuzzy properties’ that are properties of a quantum system to some degree. Vetterlein treats the structured set of effects (a partial MV algebra) as a ‘fuzzy version’ of the set of closed subspaces of the Hilbert space and he also proves a representation theorem for partial Boolean algebras using the existence of automorphisms.

Anatolij Dvurečenskij and Flavia Ventriglia discuss another way to generalise the structure of Boolean algebras. In their article, Representations of pseudo Vitali spaces and Loomis–Sikorski Theorem, they introduce a non-commutative version of Vitali spaces, i.e. distributive lattices with partially defined compatible sums. They consider pseudo Vitali spaces with a strong unit and they show that every Dedekind-σ-complete Vitali space E is an epimorphic image of a Vitali space of bounded functions where the operations are defined by points. As E is an interval [0, u], it can be converted into a σ-complete MV-algebra that we can represent as an epimorphic image of a system of fuzzy sets.

A newer form of quantum logic has been created by the combination of quantum mechanics and information theory; it is called quantum computational logics. This view is also different from the old logico-algebraic approach that started with the sharp ‘quantum logic’ of Birkhoff and von Neumann, because instead of formalising the meanings of sentences through closed subspaces of a Hilbert space, in quantum computational logics meanings of sentences are formalised by quantum information units, i.e. qubits, quregisters, etc. Roberto Giuntini, Antonio Ledda, Giuseppe Sergioli and Francesco Paoli give an overview of this approach in Some generalizations of fuzzy structures in quantum computational logic, and they discuss future research perspectives of fuzzy extensions. ‘Quantum computation, which can be seen as an investigation of the notion of uncertainty in information theory, admits fuzzy-like structures as underlying algebras’ write the team of authors, Maria Luisa Dalla Chiara, Roberto Giuntini and Roberto Leporini, in their survey article on Holism, ambiguity and approximation in the logics of quantum computation. Because quantum information quantities are identified with the meanings of sentences, they treat quantum computational logics as a mathematical framework for an abstract theory of meanings that – as they hopefully expect – will find applications in social sciences, in medicine, and in the language of art.

In their paper, Quantum computation techniques for gauging the reliability of interval and fuzzy data, Luc Longpré, Christian Servin and Vladik Kreinovich present an application of quantum computation to spur on the computation of the reliability of given data from measurements or from expert estimates, both of which are never exact. Checking this reliability is an NP-hard problem, but quantum computations can speed up the required computation time.

Vladik Kreinovich, Ladislav Kohout and Eunjin Kim provide an interesting study in another article, Square root of ‘not’: a major difference between fuzzy and quantum logics. They show that quantum logic has a ‘square root of not’ – whereas operation and fuzzy logic has not. They argue that this result is most crucial for the drastic acceleration of quantum computations.

Notes

^1. Rheinberger wrote erroneously ‘Lofti’ instead of ‘Lotfi’ – what some authors have done as well before.

^2. In the next sections, we sometimes will omit this completion about the classical errors of measurement and the systematic errors.