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Contextualizing concepts using a mathematical generalization of the quantum formalism
 Trends in Cognitive Science
, 2000
"... We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was deve ..."
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Cited by 73 (40 self)
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We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed representations, but entities existing in states of potentiality that require interaction with a context—a stimulus or another concept—to ‘collapse ’ to an instantiated form (e.g. exemplar, prototype, or other possibly imaginary instance). The stimulus situation plays the role of the measurement in physics, acting as context that induces a change of the cognitive state from superposition state to collapsed state. The collapsed state is more likely to consist of a conjunction of concepts for associative than analytic thought because more stimulus or concept properties take part in the collapse. We provide two contextual measures of conceptual distance—one using collapse probabilities and the other weighted properties—and show how they can be applied to conjunctions using the pet fish problem.
Foundations of Quantum Physics: a General Realistic and OPERATIONAL APPROACH
 INT. J. THEOR. PHYS
, 1999
"... We present a general formalism with the aim of describing the situation of an entity, how it is, how it reacts to experiments, how we can make statistics with it, and how it ‘changes ’ under the influence of the rest of the universe. Therefore we base our formalism on the following basic notions: (1 ..."
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Cited by 58 (39 self)
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We present a general formalism with the aim of describing the situation of an entity, how it is, how it reacts to experiments, how we can make statistics with it, and how it ‘changes ’ under the influence of the rest of the universe. Therefore we base our formalism on the following basic notions: (1) the states of the entity; they describe the modes of being of the entity, (2) the experiments that can be performed on the entity; they describe how we act upon and collect knowledge about the entity, (3) the probabilities; they describe our repeated experiments and the statistics of these repeated experiments, (4) the symmetries; they describe the interactions of the entity with the external world without being experimented upon. Starting from these basic notions we formulate the necessary derived notions: mixed states, mixed experiments and events, an eigen closure structure describing the properties of the entity, an ortho closure structure introducing an orthocomplementation, outcome determination, experiment determination, state determination and atomicity giving rise to some of the topological separation axioms for the closures. We define the notion of sub entity in a general way and identify the morphisms of our structure. We study specific examples in detail in the light of this formalism: a classical deterministic entity and a quantum entity described by the standard quantum mechanical formalism. We present a possible solution to the problem of the description of sub entities within the standard quantum mechanical procedure using the tensor product of the Hilbert spaces, by introducing a new completed quantum mechanics in Hilbert space, were new ‘pure’ states are introduced, not represented by rays of the Hilbert space.
A theory of concepts and their combinations I: the structure of the sets of contexts and properties
, 2005
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State property systems and closure spaces: Extracting the classical . . .
, 2002
"... In [1] an equivalence of the categories SP and Cls was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties [3, 4, 5, 6, 7, 8]. The category Cls consists of ..."
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Cited by 39 (31 self)
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In [1] an equivalence of the categories SP and Cls was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties [3, 4, 5, 6, 7, 8]. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics are equivalent with separation axioms on the corresponding closure space. More particularly it was proven that the axiom of atomicity is equivalent to the T1 separation axiom [9]. In the present article we analyze the intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space, elaborating results that appeared in [10, 11]. We introduce classical properties using the concept of super selection rule, i.e. two properties are separated by a superselection rule iff there do not exist ‘superposition states ’ related to these two properties. Then we show that the classical properties of a state property system correspond exactly to the clopen subsets of the corresponding closure space. Thus connected closure spaces correspond precisely to state property systems for which the
A Partial Order on Classical and Quantum States
, 2002
"... We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information ..."
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Cited by 27 (8 self)
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We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain theoretic structure. In the same
Being and change: foundations of a realistic operational formalism
 in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics
, 2002
"... 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 cultur ..."
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Cited by 21 (19 self)
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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 context – which 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
Concepts and their dynamics: A quantumtheoretic modeling of human thought
 Topics in Cognitive Science
, 2013
"... We analyze different aspects of our quantum modeling approach of human concepts, and more specifically focus on the quantum effects of contextuality, interference, entanglement and emergence, illustrating how each of them makes its appearance in specific situations of the dynamics of human concepts ..."
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Cited by 20 (17 self)
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We analyze different aspects of our quantum modeling approach of human concepts, and more specifically focus on the quantum effects of contextuality, interference, entanglement and emergence, illustrating how each of them makes its appearance in specific situations of the dynamics of human concepts and their combinations. We point out the relation of our approach, which is based on an ontology of a concept as an entity in a state changing under influence of a context, with the main traditional concept theories, i.e. prototype theory, exemplar theory and theory theory. We ponder about the question why quantum theory performs so well in its modeling of human concepts, and shed light on this question by analyzing the role of complex amplitudes, showing how they allow to describe interference in the statistics of measurement outcomes, while in the traditional theories statistics of outcomes originates in classical probability weights, without the possibility of interference. The relevance of complex numbers, the appearance of entanglement, and the role of Fock space in explaining contextual emergence, all as unique features of the quantum modeling, are explicitly revealed in this paper by analyzing human concepts and their dynamics.
Quantaloids describing causation and propagation for physical properties
 Foundations of Physics Letters
, 2001
"... We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems. ..."
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Cited by 20 (11 self)
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We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems.
Physical traces: Quantum vs. classical information processing
 In Proceedings of Category Theory and Computer Science 2002 (CTCS’02), volume 69 of Electronic Notes in Theoretical Computer Science. Elsevier Science
, 2003
"... a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets an ..."
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Cited by 19 (5 self)
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a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesian product as monoidal structure, both of them providing a socalled wavestyle GoI. The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing highlevel approaches to quantum information processing.