State Property Systems and Closure Spaces: Extracting the Classical en Nonclassical Parts

Aerts, Diederik; Deses, Didier

doi:10.1142/9789812778024_0006

Cited by 7 publications

(11 citation statements)

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“…Applying analogous techniques to those in Aerts et al (1999, 2002), it can be proven that λ (M) is a closure space. Further structural results about SCOP can be obtained along the lines of Aerts et al (2001 2005), Aerts and Deses (2002, 2005). Having introduced this closure space the supremum can be given a topological meaning, namely for e , f ∈M we have λ ( e ∨ f )= λ ( e )∪ λ ( f ) where λ ( e )∪ λ ( f ) is the closure of λ ( e )∪λ( f ), which is obtained exactly, in the case of the linear Hilbert space introduced in Aerts and Gabora (2005), by adding the superposition states to the set λ ( e )∪λ( f ).…”

Section: The Basic Structure Of Scopmentioning

confidence: 99%

“…Most of the results obtained in quantum axiomatics can be applied readily to concepts being considered as entities with properties. Borrowing from the study of State Property Systems (Aerts, 1999a, b, Aerts et al , 1999, 2002, 2001; Aerts and Deses, 2002, 2005) we introduce the function Equation 26, Equation 27 and Equation 28 that has been called the “Cartan Map” in the study of State Property Systems. Clearly we have for a ∈L Equation 29 and for a , b ∈L Equation 30 The Cartan Map introduces a closure space, namely κ (L).…”

Section: The Basic Structure Of Scopmentioning

confidence: 99%

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A theory of concepts and their combinations I

Aerts

Gabora²

2005

Kybernetes

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120

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If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -To elaborate a theory for modeling concepts that incorporates how a context influences the typicality of a single exemplar and the applicability of a single property of a concept. To investigate the structure of the sets of contexts and properties. Design/methodology/approach -The effect of context on the typicality of an exemplar and the applicability of a property is accounted for by introducing the notion of "state of a concept", and making use of the state-context-property formalism (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. Findings -The paper proves that the set of context and the set of properties of a concept is a complete orthocomplemented lattice, i.e. a set with a partial order relation, such that for each subset there exists a greatest lower bound and a least upper bound, and such that for each element there exists an orthocomplement. This structure describes the "and", "or", and "not", respectively for contexts and properties. It shows that the context lattice as well as the property lattice are non-classical, i.e. quantum-like, lattices. Originality/value -Although the effect of context on concepts is widely acknowledged, formal mathematical structures of theories that incorporate this effect have not been successful. The study of this formal structure is a preparation for the elaboration of a theory of concepts that allows the description of the combination of concepts.

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Section: The Basic Structure Of Scopmentioning

confidence: 99%

Section: The Basic Structure Of Scopmentioning

confidence: 99%

A theory of concepts and their combinations I

Aerts

Gabora²

2005

Kybernetes

Self Cite

120

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“…More concretely, the axiom of state determination in a state property system [1] is equivalent with the T 0 separation axiom of the corresponding closure space [44,45], and the axiom of atomicity in a state property system [1] is equivalent with the T 1 separation axiom of the corresponding closure space [46,47]. More recently it has been shown that 'classical properties' [4,6,8,9] of the state property system correspond to clopen (open and closed) sets of the closure space [48,49,50], and, explic-itly making use of the categorical equivalence, a decomposition theorem for a state property system into its nonclassical components can be proved that corresponds to the decomposition of the corresponding closure space into its connected components [48,49,50]. …”

Section: Identifying the Categorical Structurementioning

confidence: 99%

“…This is the reason the we prefer to call the type of classicality that we introduce here d-classicality. In [50] the structure related to d-classical entities is analyzed in detail.…”

Section: Definition 14 (Deterministic Statementioning

confidence: 99%

Being and Change: Foundations of a Realistic Operational Formalism

Aerts¹

2002

Probing the Structure of Quantum Mechanics

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The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms that have been developed in axiomatic quantum mechanics, where a specific type of quantum contextuality is modelled. However, because in general states also influence contextwhich is not the case in quantum mechanics -we need a more general setting than the one used there. Our focus on context as a fundamental concept makes it possible to unify 'dynamical change' and 'change under influence of measurement', which makes our approach also more general and more powerful than the traditional quantum axiomatic approaches. For this reason an experiment (or measurement) is introduced as a specific kind of context. Mathematically we introduce a state context property system as the structure to describe an entity by means of its states, contexts and properties. We also strive from the start to a categorical setting and derive the morphisms between * Published as: D. Aerts, "Being and change: foundations of a realistic operational formalism", in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation and Axiomatics, eds. D. Aerts, M. Czachor and T. Durt, World Scientific, Singapore (2002). 1 state context property systems from a merological covariance principle. We introduce the category SCOP with as elements the state context property systems and as morphisms the ones that we derived from this merological covariance principle. We introduce property completeness and state completeness and study the operational foundation of the formalism.

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“…We also want to study this "inverse" structure for the closure space that is connected through a categorical equivalence to the state property system, an equivalence of categories that has shown to be very fruitful for many other fundamental aspects of quantum axiomatics (Aerts et al, 1999a,b, in press-a,b;Aerts and Deses, 2002;van der Voorde, , 2001van Steirteghem, 2000). We also introduce two "weakest" ortho axioms to make the lattice of properties of our ortho state property system to be equipped with an orthocomplementation, a necessary structure for quantum axiomatics (Aerts, 1981(Aerts, , 1982(Aerts, , 1983Piron, 1976Piron, , 1989Piron, , 1990.…”

Section: Introductionmentioning

confidence: 99%

State Property Systems and Orthogonality

Aerts¹,

Deses

2005

Int J Theor Phys

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The structure of a state property system was introduced to formalize in a complete way the operational content of the Geneva-Brussels approach to the foundations of quantum mechanics (Aerts, D. the category of state property systems was proven to be equivalent to the category of closure spaces (Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., International Journal of Theoretical Physics, 38, 359-385, 1999; Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., The construct of closure spaces as the amnestic modification of the physical theory of state property systems, Applied Categorical Structures, in press). The first axioms of standard quantum axiomatics (state determination and atomisticity) have been shown to be equivalent to the T 0 and T 1 axioms of closure spaces (van Steirteghem, B.van der Voorde, A., Separation Axioms in Extension Theory for Closure Spaces and Their Relevance to State Property Systems, Doctoral Thesis, Brussels Free University, 2001), and classical properties to correspond to clopen sets, leading to a decomposition theorem into classical and purely nonclassical components for a general state property system (Aerts, D., van der Voorde, A., and Deses, D., Journal of Electrical Engineering, 52, 18-21, 2001; Aerts, D., van der Voorde, A., and Deses, D. International Journal of Theoretical Physics; Aerts, D. and Deses, D., Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation, and Axiomatics, World Scientific, Singapore, 2002). The concept of orthogonality, very important for quantum axiomatics, had however not yet been introduced within the formal scheme of the state property system. In this paper we introduce orthogonality in an operational way, and define ortho state property systems. Birkhoff's well known biorthogonal construction gives rise to an orthoclosure and we study the relation between this orthoclosure and the operational orthogonality that we introduced. KEY WORDS: orthogonality; state property systems; ortho state property systems; ortho axioms.

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State Property Systems and Closure Spaces: Extracting the Classical en Nonclassical Parts

Cited by 7 publications

References 8 publications

A theory of concepts and their combinations I

A theory of concepts and their combinations I

Being and Change: Foundations of a Realistic Operational Formalism

State Property Systems and Orthogonality

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