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

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

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

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

Quantum Structure in Cognition, Origins, Developments, Successes and Expectations

We provide an overview of the results we have attained in the last decade on the identification of quantum structures in cognition and, more specifically, in the formalization and representation of natural concepts. We firstly discuss the quantum foundational reasons that led us to investigate the mechanisms of formation and combination of concepts in human reasoning, starting from the empirically observed deviations from classical logical and probabilistic structures. We then develop our quantum-theoretic perspective in Fock space which allows successful modeling of various sets of cognitive experiments collected by different scientists, including ourselves. In addition, we formulate a unified explanatory hypothesis for the presence of quantum structures in cognitive processes, and discuss our recent discovery of further quantum aspects in concept combinations, namely, 'entanglement' and 'indistinguishability'. We finally illustrate perspectives for future research.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1412.870