125 research outputs found

### Conjunction and Negation of Natural Concepts: A Quantum-theoretic Modeling

We perform two experiments with the aim to investigate the effects of
negation on the combination of natural concepts. In the first experiment, we
test the membership weights of a list of exemplars with respect to two
concepts, e.g., {\it Fruits} and {\it Vegetables}, and their conjunction {\it
Fruits And Vegetables}. In the second experiment, we test the membership
weights of the same list of exemplars with respect to the same two concepts,
but negating the second, e.g., {\it Fruits} and {\it Not Vegetables}, and again
their conjunction {\it Fruits And Not Vegetables}. The collected data confirm
existing results on conceptual combination, namely, they show dramatic
deviations from the predictions of classical (fuzzy set) logic and probability
theory. More precisely, they exhibit conceptual vagueness, gradeness of
membership, overextension and double overextension of membership weights with
respect to the given conjunctions. Then, we show that the quantum probability
model in Fock space recently elaborated to model Hampton's data on concept
conjunction (Hampton, 1988a) and disjunction (Hampton, 1988b) faithfully
accords with the collected data. Our quantum-theoretic modeling enables to
describe these non-classical effects in terms of genuine quantum effects,
namely `contextuality', `superposition', `interference' and `emergence'. The
obtained results confirm and strenghten the analysis in Aerts (2009a) and Sozzo
(2014) on the identification of quantum aspects in experiments on conceptual
vagueness. Our results can be inserted within the general research on the
identification of quantum structures in cognitive and decision processes.Comment: 32 pages, standard latex, no figures, 16 tables. arXiv admin note:
text overlap with arXiv:1311.6050; and text overlap with arXiv:0805.3850 by
other author

### Quantum Structure in Economics: The Ellsberg Paradox

The 'expected utility hypothesis' and 'Savage's Sure-Thing Principle' are
violated in real life decisions, as shown by the 'Allais' and 'Ellsberg
paradoxes'. The popular explanation in terms of 'ambiguity aversion' is not
completely accepted. As a consequence, uncertainty is still problematical in
economics. To overcome these difficulties a distinction between 'risk' and
'ambiguity' has been introduced which depends on the existence of a
Kolmogorovian probabilistic structure modeling these uncertainties. On the
other hand, evidence of everyday life suggests that context plays a fundamental
role in human decisions under uncertainty. Moreover, it is well known from
physics that any probabilistic structure modeling contextual interactions
between entities structurally needs a non-Kolmogorovian framework admitting a
quantum-like representation. For this reason, we have recently introduced a
notion of 'contextual risk' to mathematically capture situations in which
ambiguity occurs. We prove in this paper that the contextual risk approach can
be applied to the Ellsberg paradox, and elaborate a sphere model within our
'hidden measurement formalism' which reveals that it is the overall conceptual
landscape that is responsible of the disagreement between actual human
decisions and the predictions of expected utility theory, which generates the
paradox. This result points to the presence of a quantum conceptual layer' in
human thought which is superposed to the usually assumed classical logical
layer', and conceptually supports the thesis of several authors suggesting the
presence of quantum structure in economics and decision theory.Comment: 8 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1105.1814, arXiv:1104.1459, arXiv:1105.181

### On the Notion of Proposition in Classical and Quantum Mechanics

The term proposition usually denotes in quantum mechanics (QM) an element of
(standard) quantum logic (QL). Within the orthodox interpretation of QM the
propositions of QL cannot be associated with sentences of a language stating
properties of individual samples of a physical system, since properties are
nonobjective in QM. This makes the interpretation of propositions
problematical. The difficulty can be removed by adopting the objective
interpretation of QM proposed by one of the authors (semantic realism, or SR,
interpretation). In this case, a unified perspective can be adopted for QM and
classical mechanics (CM), and a simple first order predicate calculus L(x) with
Tarskian semantics can be constructed such that one can associate a physical
proposition (i.e., a set of physical states) with every sentence of L(x). The
set $P^{f}$ of all physical propositions is partially ordered and contains a
subset $P^{f}_{T}$ of testable physical propositions whose order structure
depends on the criteria of testability established by the physical theory. In
particular, $P^{f}_{T}$ turns out to be a Boolean lattice in CM, while it can
be identified with QL in QM. Hence the propositions of QL can be associated
with sentences of L(x), or also with the sentences of a suitable quantum
language $L_{TQ}(x)$, and the structure of QL characterizes the notion of
testability in QM. One can then show that the notion of quantum truth does not
conflict with the classical notion of truth within this perspective.
Furthermore, the interpretation of QL propounded here proves to be equivalent
to a previous pragmatic interpretation worked out by one of the authors, and
can be embodied within a more general perspective which considers states as
first order predicates of a broader language with a Kripkean semantics.Comment: 22 pages. To appear in "The Foundations of Quantum Mechanics:
Historical Analysis and Open Questions-Cesena 2004", C. Garola, A. Rossi and
S. Sozzo Eds., World Scientific, Singapore, 200

### General Quantum Hilbert Space Modeling Scheme for Entanglement

We work out a classification scheme for quantum modeling in Hilbert space of
any kind of composite entity violating Bell's inequalities and exhibiting
entanglement. Our theoretical framework includes situations with entangled
states and product measurements ('customary quantum situation'), and also
situations with both entangled states and entangled measurements ('nonlocal box
situation', 'nonlocal non-marginal box situation'). We show that entanglement
is structurally a joint property of states and measurements. Furthermore,
entangled measurements enable quantum modeling of situations that are usually
believed to be 'beyond quantum'. Our results are also extended from pure states
to quantum mixtures.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1304.010

### Contextual Risk and Its Relevance in Economics

Uncertainty in economics still poses some fundamental problems illustrated,
e.g., by the Allais and Ellsberg paradoxes. To overcome these difficulties,
economists have introduced an interesting distinction between 'risk' and
'ambiguity' depending on the existence of a (classical Kolmogorovian)
probabilistic structure modeling these uncertainty situations. On the other
hand, evidence of everyday life suggests that 'context' plays a fundamental
role in human decisions under uncertainty. Moreover, it is well known from
physics that any probabilistic structure modeling contextual interactions
between entities structurally needs a non-Kolmogorovian quantum-like framework.
In this paper we introduce the notion of 'contextual risk' with the aim of
modeling a substantial part of the situations in which usually only 'ambiguity'
is present. More precisely, we firstly introduce the essentials of an
operational formalism called 'the hidden measurement approach' in which
probability is introduced as a consequence of fluctuations in the interaction
between entities and contexts. Within the hidden measurement approach we
propose a 'sphere model' as a mathematical tool for situations in which
contextual risk occurs. We show that a probabilistic model of this kind is
necessarily non-Kolmogorovian, hence it requires either the formalism of
quantum mechanics or a generalization of it. This insight is relevant, for it
explains the presence of quantum or, better, quantum-like, structures in
economics, as suggested by some authors, and can serve to solve the
aforementioned paradoxes.Comment: 6 pages, 2 figure

### A Contextual Risk Model for the Ellsberg Paradox

The Allais and Ellsberg paradoxes show that the expected utility hypothesis
and Savage's Sure-Thing Principle are violated in real life decisions. The
popular explanation in terms of 'ambiguity aversion' is not completely
accepted. On the other hand, we have recently introduced a notion of
'contextual risk' to mathematically capture what is known as 'ambiguity' in the
economics literature. Situations in which contextual risk occurs cannot be
modeled by Kolmogorovian classical probabilistic structures, but a
non-Kolmogorovian framework with a quantum-like structure is needed. We prove
in this paper that the contextual risk approach can be applied to the Ellsberg
paradox, and elaborate a 'sphere model' within our 'hidden measurement
formalism' which reveals that it is the overall conceptual landscape that is
responsible of the disagreement between actual human decisions and the
predictions of expected utility theory, which generates the paradox. This
result points to the presence of a 'quantum conceptual layer' in human thought
which is superposed to the usually assumed 'classical logical layer'.Comment: 6 pages, 1 figur

### Quantum Entanglement in Concept Combinations

Research in the application of quantum structures to cognitive science
confirms that these structures quite systematically appear in the dynamics of
concepts and their combinations and quantum-based models faithfully represent
experimental data of situations where classical approaches are problematical.
In this paper, we analyze the data we collected in an experiment on a specific
conceptual combination, showing that Bell's inequalities are violated in the
experiment. We present a new refined entanglement scheme to model these data
within standard quantum theory rules, where 'entangled measurements and
entangled evolutions' occur, in addition to the expected 'entangled states',
and present a full quantum representation in complex Hilbert space of the data.
This stronger form of entanglement in measurements and evolutions might have
relevant applications in the foundations of quantum theory, as well as in the
interpretation of nonlocality tests. It could indeed explain some
non-negligible 'anomalies' identified in EPR-Bell experiments.Comment: 16 pages, no figure

### Generalized Observables, Bell's Inequalities and Mixtures in the ESR Model for QM

The extended semantic realism (ESR) model proposes a new theoretical
perspective which embodies the mathematical formalism of standard (Hilbert
space) quantum mechanics (QM) into a noncontextual framework, reinterpreting
quantum probabilities as conditional instead of absolute. We provide in this
review an overall view on the present status of our research on this topic. We
attain in a new, shortened way a mathematical representation of the generalized
observables introduced by the ESR model and a generalization of the projection
postulate of elementary QM. Basing on these results we prove that the
Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality
and quantum predictions hold together in the ESR model because they refer to
different parts of the picture of the physical world supplied by the model.
Then we show that a new mathematical representation of mixtures must be
introduced in the ESR model which does not coincide with the standard
representation in QM and avoids some deep problems that arise from the
representation of mixtures provided by QM. Finally we get a nontrivial
generalization of the Luders postulate, which is justified in a special case by
introducing a reasonable physical assumption on the evolution of the compound
system made up of the measured system and the measuring apparatus.Comment: 24 pages, 1 figure, Found. Phy

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