Abstract
We present the fundamentals of the quantum theoretical approach we have developed in the last decade to model cognitive phenomena that resisted modeling by means of classical logical and probabilistic structures, like Boolean, Kolmogorovian and, more generally, set theoretical structures. We firstly sketch the operational-realistic foundations of conceptual entities, i.e. concepts, conceptual combinations, propositions, decision-making entities, etc. Then, we briefly illustrate the application of the quantum formalism in Hilbert space to represent combinations of natural concepts, discussing its success in modeling a wide range of empirical data on concepts and their conjunction, disjunction and negation. Next, we naturally extend the quantum theoretical approach to model some long-standing ‘fallacies of human reasoning’, namely, the ‘conjunction fallacy’ and the ‘disjunction effect’. Finally, we put forward an explanatory hypothesis according to which human reasoning is a defined superposition of ‘emergent reasoning’ and ‘logical reasoning’, where the former generally prevails over the latter. The quantum theoretical approach explains human fallacies as the consequence of genuine quantum structures in human reasoning, i.e. ‘contextuality’, ‘emergence’, ‘entanglement’, ‘interference’ and ‘superposition’. As such, it is alternative to the Kahneman–Tversky research programme, which instead aims to explain human fallacies in terms of ‘individual heuristics and biases’.
Similar content being viewed by others
Notes
A normalized function \(p: E \in \mathscr {A} \longrightarrow [0,1]\) is said ‘Kolmogorovian’ if it satisfies the following axioms: (i) \(p(\Omega )=1\) and (ii) \(p(\cup _{i}E_i)=\sum _{i}p(E_i)\), for every sequence \(\{E_i\}_i\) of pairwise disjoint events \(E_i\) (Kolmogorov 1933).
The monotonicty law of Kolmogorovian probability is globally expressed by the inequalities \(p(E_A \cap E_B)\le p(E_A),p(E_B)\le p(E_A\cup E_B)\).
We remind that an orthogonal projection operator is a liner operator which satisfies hermiticity, i.e. \(M^{\dag }=M\), and idempotency, i.e. \(M^2=M\cdot M=M\).
Indeed, \(|A\rangle\) and \(|B\rangle\) are orthogonal vectors, and also \(M|A\rangle\) and \((1\!\!1-M)|A\rangle\) and \(M|B\rangle\) and \((1\!\!1-M)|B\rangle\) are, representing three data points \(\mu (A)\), \(\mu (B)\) and \(\mu (A \ \mathrm{and} \ B)\) requires a Hilbert space of at least dimension 3.
The situation in which \(\mu (A)=0\) or \(\mu (B)=0\) requires some further technicalities and a more complex Hilbert space structure, the ‘Fock space’, which will be introduced later. We do not dwell on this aspect here, for the sake of brevity.
In Kolmogorovian probability, one proves the law of total probability, namely, \(p(E_A)=p(E_B)p(E_A|E_B)+p(E'_{B})p(E_A|E'_B)\), where \(E'_B=\Omega \setminus E_B\) denotes the ‘complement event’ with respect to \(E_B\).
References
Aerts, D. (2002). Being and change: Foundations of a realistic operational formalism. In D. Aerts, M. Czachor, & T. Durt (Eds.), Probing the structure of quantum mechanics: Nonlinearity, nonlocality, probability and axiomatics (pp. 71–110). Singapore: World Scientific.
Aerts, D., & Sozzo, S. (2011). Quantum structure in cognition. Why and how concepts are entangled. Quantum Interaction. Lecture Notes in Computer Science 7052, 116–127. Berlin: Springer.
Aerts, D., & Sozzo, S. (2016). Quantum structure in cognition: Origins, developments, successes and expectations. In Haven, E., & Khrennikov, A. (Eds.) The Palgrave handbook of quantum models in social science: Applications and grand challenges (pp. 157–193). London: Palgrave & Macmillan.
Aerts, D., Gabora, L., & Sozzo, S. (2013). Concepts and their dynamics: A quantum-theoretic modeling of human thought. Topics in Cognitive Science, 5, 737–772.
Aerts, D., Geriente, S., Moreira, C., & Sozzo, S. (2018). Testing ambiguity and Machina preferences within a quantum-theoretic framework for decision-making. Journal of Mathematical Economics. https://doi.org/10.1016/j.jmateco.2017.12.002.
Aerts, D., Sassoli de Bianchi, M., & Sozzo, S. (2016). On the foundations of the Brussels operational-realistic approach to cognition. Frontiers in Physics. https://doi.org/10.3389/fphy.2016.00017.
Aerts, D., Sassoli de Bianchi, M., Sozzo, S., & Veloz, T. (2018). Modeling meaning associated with documental entities: Introducing the Brussels quantum approach.
Aerts, D., Sozzo, S., & Veloz, T. (2015). Quantum structure of negation and conjunction in human thought. Frontiers in Psychology. https://doi.org/10.3389/fpsyg.2015.01447.
Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics. Journal of Mathematical Physics, 27, 202–210.
Aerts, D. (1999). Foundations of quantum physics: A general realistic and operational approach. International Journal of Theoretical Physics, 38, 289–358.
Aerts, D. (2009). Quantum structure in cognition. Journal of Mathematical Psychology, 53, 314–348.
Aerts, D. (2009a). Quantum particles as conceptual entities: A possible explanatory framework for quantum theory. Foundations of Science, 14, 361–411.
Aerts, D., & Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes. Foundations of Science, 1, 85–97.
Aerts, D., Broekaert, J., Gabora, L., & Sozzo, S. (2013). Quantum structure and human thought. Behavioral and Brain Sciences, 36, 274–276.
Aerts, D., & Gabora, L. (2005). A theory of concepts and their combinations I: The structure of the sets of contexts and properties. Kybernetes, 34, 167–191.
Aerts, D., & Gabora, L. (2005). A theory of concepts and their combinations II: A Hilbert space representation. Kybernetes, 34, 192–221.
Aerts, D., Haven, E., & Sozzo, S. (2018). A proposal to extend expected utility in a quantum probabilistic framework. Economic Theory, 65, 1079–1109.
Aerts, D., & Sozzo, S. (2014). Quantum entanglement in conceptual combinations. International Journal of Theoretical Physics, 53, 3587–3603.
Aerts, D., & Sozzo, S. (2016). From ambiguity aversion to a generalized expected utility. Modeling preferences in a quantum probabilistic framework. Journal of Mathematical Psychology, 74, 117–127.
Aerts, D., Sozzo, S., & Veloz, T. (2015). New fundamental evidence of non-classical structure in the combination of natural concepts. Philosophical Transactions of the Royal Society A, 374, 20150095.
Aerts, D., Sozzo, S., & Veloz, T. (2015). Quantum structure in cognition and the foundations of human reasoning. International Journal of Theoretical Physics, 54, 4557–4569.
Aerts, D., Sozzo, S., & Veloz, T. (2015). Quantum nature of identity in human concepts: Bose-Einstein statistics for conceptual indistinguishability. International Journal of Theoretical Physics, 54, 4430–4443.
Alxatib, S., & Pelletier, J. (2011). On the psychology of truth gaps. In R. Nouwen, R. van Rooij, U. Sauerland, & H.-C. Schmitz (Eds.), Vagueness in communication (pp. 13–36). Berlin, Heidelberg: Springer.
Bruza, P. D., Wang, Z., & Busemeyer, J. R. (2015). Quantum cognition: A new theoretical approach to psychology. Trends in Cognitive Sciences, 19, 383–393.
Busemeyer, J. R., & Bruza, P. D. (2012). Quantum models of cognition and decision. Cambridge: Cambridge University Press.
Busemeyer, J. R., Pothos, E. M., Franco, R., & Trueblood, J. S. (2011). A quantum theoretical explanation for probability judgment errors. Psychological Review, 118, 193–218.
Costello, J., & Keane, M. T. (2000). Efficient creativity: Constraint-guided conceptual combination. Cognitive Science, 24, 299–349.
Dirac, P. A. M. (1958). Quantum mechanics (4th ed.). Oxford: Oxford University Press.
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economic, 75, 643–669.
Fisk, J. E. (2002). Judgments under uncertainty: Representativeness or potential surprise? British Journal of Psychology, 93, 431–449.
Fisk, J. E., & Pidgeon, N. (1996). Component probabilities and the conjunction fallacy: Resolving signed summation and the low component model in a contingent approach. Acta Psychologica, 94, 1–20.
Hampton, J. A. (1988a). Overextension of conjunctive concepts: Evidence for a unitary model for concept typicality and class inclusion. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 12–32.
Hampton, J. A. (1988b). Disjunction of natural concepts. Memory & Cognition, 16, 579–591.
Haven, E., & Khrennikov, A. Y. (2013). Quantum social science. Cambridge: Cambridge University Press.
Haven, E., & Khrennikov, A. (2016). Statistical and subjective interpretations of probability in quantum-like models of cognition and decision making. Journal of Mathematical Psychology, 74, 82–91.
Jauch, J. M. (1968). Foundations of quantum mechanics. Reading, MA: Addison Wesley.
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse Der Mathematik; translated as Foundations of Probability (p. 1950). New York: Chelsea Publishing Company.
Kühberger, A., Kamunska, D., & Perner, J. (2001). The disjunction effect: Does it exist for two-step gambles? Organization Behavior and Human Decision Processes, 85, 250–264.
Kvam, P. D., Pleskac, T. J., Yu, S., & Busemeyer, J. R. (2016). Interference effects of choice on confidence: Quantum characteristics of evidence accumulation. Proceedings of the National Academy of Sciences, 112, 10645–10650.
Lambdin, C., & Burdsal, C. (2007). The disjunction effect reexamined: Relevant methodological issues and the fallacy of unspecified percentage comparisons. Organization Behavior and Human Decision Processes, 103, 268–276.
Lu, Y. (2015). The conjunction and disjunction fallacies: Explanations of the Linda problem by the equate-to-differentiate model. Integrative Psychological and Behavioral Science, 1–25.
Machina, M. J. (2009). Risk, ambiguity, and the dark-dependence axioms. American Economic Review, 99, 385–392.
Melucci, M. (2015). Introduction to information retrieval and quantum mechanics. Berlin: Springer.
Morier, D., & Borgida, E. (1984). The conjunction fallacy: A task specific phenomenon? Personality and Social Psychology Bulletin, 10, 243–252.
Moro, R. (2009). On the nature of the conjunction fallacy. Synthese, 171, 1–24.
Murphy, G. L., & Medin, D. L. (1985). The role of theories in conceptual coherence. Psychological Review, 92, 289–316.
Nosofsky, R. (1992). Exemplars, prototypes, and similarity rules. In Healy, A., Kosslyn, S., & Shiffrin, R. (Eds.), From learning theory to connectionist theory: Essays in honor of William K. Estes. Hillsdale, NJ: Erlbaum.
Nosofsky, R. (1988). Exemplar-based accounts of relations between classification, recognition, and typicality. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 700–708.
Osherson, D., & Smith, E. (1981). On the adequacy of prototype theory as a theory of concepts. Cognition, 9, 35–58.
Piron, C. (1976). Foundations of quantum physics. Reading, MA: Reading.
Pitowsky, I. (1989). Quantum probability, quantum logic. Lecture Notes in Physics (vol. 321). Berlin: Springer.
Pothos, E. M., & Busemeyer, J. R. (2013). Can quantum probability provide a new direction for cognitive modeling? Behavioral and Brain Sciences, 36, 255–274.
Pothos, E. M., Busemeyer, J. R., Shiffrin, R. M., & Yearsley, J. M. (2017). The rational status of quantum cognition. Journal of Experimental Psychology: General, 146, 968–987.
Rosch, E. (1973). Natural categories. Cognitive Psychology, 4, 328–350.
Rosch, E. (1978). Principles of categorization. In E. Rosch & B. Lloyd (Eds.), Cognition and categorization (pp. 133–179). Hillsdale, NJ: Lawrence Erlbaum.
Rosch, E. (1983). Prototype classification and logical classification: The two systems. In E. K. Scholnick (Ed.), New trends in conceptual representation: Challenges to Piaget theory? (pp. 133–159). New Jersey: Lawrence Erlbaum.
Rumelhart, D. E., & Norman, D. A. (1988). Representation in memory. In R. C. Atkinson, R. J. Hernsein, G. Lindzey, & R. L. Duncan (Eds.), Stevens handbook of experimental psychology. New Jersey: Wiley.
Savage, L. (1954). The foundations of statistics. New York: Wiley.
Shah, A. K., & Oppenheimer, D. M. (2008). Heuristics made easy: An effort-reduction framework. Psychological Bulletin, 134, 207–222.
Sozzo, S. (2014). A quantum probability explanation in Fock space for borderline contradictions. Journal of Mathematical Psychology, 58, 1–12.
Sozzo, S. (2015). Conjunction and negation of natural concepts: A quantum-theoretic modeling. Journal of Mathematical Psychology, 66, 83–102.
Thagard, P., & Stewart, T. C. (2011). The AHA! experience: Creativity through emergent binding in neural networks. Cognitive Science, 35, 1–33.
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124–1131.
Tversky, A., & Kahneman, D. (1983). Extension versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90, 293–315.
Tversky, A., & Shafir, E. (1992). The disjunction effect in choice under uncertainty. Psychological Science, 3, 305–309.
Van Dantzig, S., Raffone, A., & Hommel, B. (2011). Acquiring contextualized concepts: A connectionist approach. Cognitive Science, 35, 1162–1189.
Wang, Z., Solloway, T., Shiffrin, R. M., & Busemeyer, J. R. (2014). Context effects produced by question orders reveal quantum nature of human judgments. Proceedings of the National Academy of Sciences, 111, 9431–9436.
Wittgenstein, L. (1953/2001). Philosophical investigations. Blackwell Publishing.
Zadeh, L. (1982). A note on prototype theory and fuzzy sets. Cognition, 12, 291–297.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aerts, D., Sassoli de Bianchi, M., Sozzo, S. et al. Modeling Human Decision-Making: An Overview of the Brussels Quantum Approach. Found Sci 26, 27–54 (2021). https://doi.org/10.1007/s10699-018-9559-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10699-018-9559-x