Brussels, Belgium, November 30, 2013 WHITHER QUANTUM STRUCTURES? QUANTUM LOGIC IN THE 21TH CENTURY THE QUANTUM CHALLENGE IN COGNITIVE SCIENCE AND DECISION THEORY SCIENCE AND DECISION THEORY SANDRO SOZZO (WITH D. AERTS) CENTER LEO APOSTEL (VUB BRUSSELS) SCHOOL OF MANAGEMENT (LEICESTER) www.le.ac.uk
FOUNDATIONS OF QUANTUM THEORY Non-Boolean quantum logic. The structural differences between classical and quantum theory, e.g., classical and quantum probability, Non-Kolmogorovian quantum probability. have been understood. Non-commutative algebra of quantum observables. Non-commutative algebra of quantum observables. Detection of genuine quantum aspects (interference, superposition, emergence, entanglement, incompatibility) in macroscopic physical systems and, more generally, outside the microscopic world. The identification of quantum structures outside the microscopic domain of quantum physics and the employment of the mathematical formalisms of quantum theory to model experimental data in social science is now a well established research field. 2
QUANTUM COGNITION Concept theory. Results have been obtained Behavioral economics. in the modeling of cognitive Decision theory. and decision processes. Animal behavior. Computer science. Finance. 3
SOME HIGHLIGHTS Books. First ideas. Media. Quantum Interaction workshops. Stanford (2007), Oxford (2008), Saarbrücken (2009), Washington (2010), Aberdeen (2011), Paris (2012), Leicester (2013), Filzbach (2014). 4
The Brussels team (S. Sozzo, D. Aerts, J. Broekaert and T. Veloz) has recently received an Outstanding Scholarly Contribution Award by the Institute for Advanced Studies in Systems Research and Cybernetics for his research on “The Quantum Challenge in Concept Theory and Natural Language Processing”. *D. Aerts, J. Broekaert, S. Sozzo, T. Veloz, “The Quantum Challenge in Concept 5 Theory and Natural Language Processing”, Int. J. IIAS Sys. Res. Cyb. 13 (1), pp. 13-17.
THE COMBINATION PROBLEM Progress in many fields (psychology, linguistics, AI, cognitive science) To understand the structure and dynamics of depends crucially on it. human concepts, how concepts combine to form sentences, and how meaning is expressed by such combinations, is one of the age-old challenges of Major scientific issues (text analysis, scientists studying the human mind. scientists studying the human mind. IR, IR, human-computer human-computer interaction) interaction) rely on a deeper understanding of how concepts combine. Much effort has been devoted to these matters, but very few substantial results have been obtained. However, models of concepts making use of the mathematical formalisms of quantum theory have been substantially more successful than classical approaches at modeling data generated in studies on combinations of two concepts. 6
QUANTUM MODELING OF CONCEPTS We put forward a quantum-theoretic modeling of how concepts combine, and identify the specific quantum aspects that contribute to the successful modeling of the collection of experimental data for the conjunction and the disjunction of two concepts (Hampton 1988a,b). We explain how the quantum effects of superposition, interference, emergence and contextuality give rise to a modeling of the overextension and the underextension of membership weights of give rise to a modeling of the overextension and the underextension of membership weights of exemplars with respect to the conjunction and the disjunction of concepts. We show how a Fock space modeling reveals human thought as a superposition of ‘quantum logical thought’ and ‘quantum emergent thought’. Remark. We have also identified an experimental violation of Bell’s inequalities for a specific concept combination, and elaborated a quantum representation for it, thus proving the entanglement of such combinations. 7
DIFFICULTIES OF EXISTING CONCEPT THEORIES Traditional (fuzzy set) approaches. A Classical view. All instances of a concept share a common concept is a container of instantiations. set of necessary and sufficient defining properties. Wittgenstein (1953). The meaning of concepts depends on the contexts in which they are used. Rosch (1973). Concepts exhibit graded typicality. Following Rosch, a probabilistic or fuzzy set approach was tried. Osherson & Smith (1981). People rate Guppy neither as a typical Pet nor as a Hampton (1988a,b). The membership typical Fish , but they rate it as a highly weight of an exemplar of a conjunction typical Pet-Fish (guppy effect). This (disjunction) of concepts is higher (lower) effect defies the fuzzy set modeling of than the membership weights of this typicality with respect to conjunction. exemplar for one or both of the constituent concepts (overextension, underextension). 8
NOVELTIES OF THE BRUSSELS APPROACH The Brussels group followed the axiomatic and operational approaches to quantum theory, identifying situations in the macro world, i.e. not necessarily situations of quantum particles in the micro world, which revealed quantum structures. A concept is considered as an entity in a A concept is considered as an entity in a specific state, and not, as in the classical view, as a container of instantiations. A SCoP formalism was elaborated which A context is a factor that influences the concept, models any kind of entity in terms of and changes its state, and is formed by states, contexts and properties. conceptual landscapes surrounding the concept. Typicality is an observable quantity, with different values for different states of the concept. 9
THE GUPPY EFFECT IN SCOP The guppy effect is explained in the SCoP formalism by considering the conjunction Pet-Fish as Pet in the context Fish or Fish in the context Pet . A state p Guppy of Pet ( Fish ) has a low typicality in absence of context, while it scores a high typicality under the context e Fish (e Pet ). → → e e → → p p p p p p p p 1 1 ˆ ˆ ˆ ˆ Pet Fish Pet Guppy Pet Guppy → e → p p p p 1 ˆ ˆ Fish Pet Fish Guppy Fish Guppy µ µ p p p p low ˆ ˆ ( , 1 , ), ( , 1 , ) Guppy Pet Pet Guppy Fish Fish We built an explicit quantum representation in a complex Hilbert space µ µ p e p p e p high ˆ ˆ ( , , ), ( , , ) Guppy Fish Pet Guppy Pet Fish of the experimental data on the concepts Pet , Fish and their conjunction Pet-Fish . *D. Aerts, L. Gabora (2005a,b), “A Theory of Concepts and Their Combinations 10 I & II”, Kybernetes 34 , pp. 167-191; 192-221.
WHY A QUANTUM FORMALISM IS SO EFFICIENT? In a quantum measurement When a subject is asked to estimate the membership (or process, the measurement context the typicality) of an exemplar with respect to one (or actualizes one possible outcome more concepts), contextual influence (of a cognitive type) and provokes an indeterministic and a transition from potential to actual occur in which an change of state of the microscopic outcome is actualized from a set of possible outcomes. quantum particle that is measured. Both quantum and conceptual entities are realms of genuine potentialities, not of lack of knowledge of actualities. At variance with classical Kolmogorovian probability, quantum probability enables coping with this kind of contextuality and pure potentiality. *D. Aerts, J. Broekaert, L. Gabora, S. Sozzo (2013), “Quantum 11 Structure and Human Thought”, Behav. Br. Sci. 36, pp. 274-276.
THE CONJUNCTION OF TWO CONCEPTS Hampton’s data on concept conjunction (1988a) cannot be modeled within a classical probability theory. 12
THE DISJUNCTION OF TWO CONCEPTS Hampton’s data on concept disjunction (1988b) cannot be modeled within a classical probability theory. 13 *D. Aerts (2009), “Quantum Structure in Cognition”, J. Math. Psychol. 53, pp. 314-348.
A QUANTUM MODEL FOR THE CONJUNCTION 14
*D. Aerts, L. Gabora, S. Sozzo (2013), “Concepts and Their Dynamics: A 15 Quantum Theoretic Modeling of Human Thought”, Top. Cogn. Sci. 5, 337-372.
A QUANTUM MODEL FOR THE DISJUNCTION 16
*D. Aerts, L. Gabora, S. Sozzo (2013), “Concepts and Their Dynamics: A 17 Quantum Theoretic Modeling of Human Thought”, Top. Cogn. Sci. 5, 337-372.
UNDERLYING MECHANISMS IN HUMAN THOUGHT i. Our quantum-theoretic modeling in Fock space allows the faithful representation of Hampton's data (1988a,b), describing the deviations from classical logic and probability theory in terms of genuine quantum aspects. ii. Our approach successfully models the data collected by Alxatib and Pelletier (2011) on the so-called `borderline contradictions’ (a predicate like `tall man' admits borderline cases, that is, there are exemplars where it is unclear whether the predicate applies, and this ambiguity cannot be removed by specifying the exact height of the person. and this ambiguity cannot be removed by specifying the exact height of the person. iii. This theoretical framework can be further tested to model data coming from future cognitive experiments. One can then inquire into the existence of underlying mechanisms determining these deviations from classicality and, conversely, the effectiveness of a quantum-theoretic modeling. *D. Aerts, S. Sozzo (2011), “Quantum Structure in Cognition: Why and How Concepts Are Entangled”, Quantum Interaction 2011, LNCS 7052, pp. 116-127, Berlin: Springer. **S. Sozzo (2013), “A Quantum Probability Explanation for Borderline Contradictions”, 18 J. Math. Psychol. (in print).
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