Brussels, Belgium, November 30, 2013

WHITHER QUANTUM STRUCTURES?

QUANTUM LOGIC IN THE 21TH CENTURY

THE QUANTUM CHALLENGE IN COGNITIVE

SCIENCE AND DECISION THEORY

www.le.ac.uk

SANDRO SOZZO

(WITH D. AERTS)

CENTER LEO APOSTEL (VUB BRUSSELS)

SCHOOL OF MANAGEMENT (LEICESTER)

SCIENCE AND DECISION THEORY

The structural differences between

classical and quantum theory, e.g.,

classical and quantum probability,

have been understood.

FOUNDATIONS OF QUANTUM THEORY

Non-commutative algebra of quantum observables.

Non-Kolmogorovian quantum probability.

Non-Boolean quantum logic.

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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.

Non-commutative algebra of quantum observables.

Results have been obtained

in the modeling of cognitive

and decision processes. Decision theory.

Animal behavior.

Behavioral economics.

QUANTUM COGNITION

Concept theory.

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Computer science. Finance.

First ideas.

Books.

SOME HIGHLIGHTS

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Quantum Interaction workshops. Stanford (2007), Oxford (2008), Saarbrücken (2009), Washington

(2010), Aberdeen (2011), Paris (2012), Leicester (2013), Filzbach (2014).

Media.

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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

Theory and Natural Language Processing”, Int. J. IIAS Sys. Res. Cyb. 13 (1), pp. 13-17.

THE COMBINATION PROBLEM

To understand the structure and dynamics of

human concepts, how concepts combine to form

sentences, and how meaning is expressed by such

combinations, is one of the age-old challenges of

scientists studying the human mind.

Progress in many fields (psychology,

linguistics, AI, cognitive science)

depends crucially on it.

Major scientific issues (text analysis,

IR, human-computer interaction)scientists studying the human mind.

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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.

IR, human-computer interaction)

rely on a deeper understanding of

how concepts combine.

QUANTUM MODELING OF CONCEPTS

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

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).

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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.

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.

We show how a Fock space modeling reveals human thought as a

superposition of ‘quantum logical thought’ and ‘quantum emergent thought’.

Classical view. All instances of a concept share a common

set of necessary and sufficient defining properties.

Wittgenstein (1953). The meaning of concepts depends on the contexts in which they are used.

DIFFICULTIES OF EXISTING CONCEPT THEORIES

Traditional (fuzzy set) approaches. A

concept is a container of instantiations.

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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

typical Fish, but they rate it as a highly

typical Pet-Fish (guppy effect). This

effect defies the fuzzy set modeling of

typicality with respect to conjunction.

Hampton (1988a,b). The membership

weight of an exemplar of a conjunction

(disjunction) of concepts is higher (lower)

than the membership weights of this

exemplar for one or both of the constituent

concepts (overextension, underextension).

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.

NOVELTIES OF THE BRUSSELS APPROACH

A concept is considered as an entity in a

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A concept is considered as an entity in a

specific state, and not, as in the classical

view, as a container of instantiations.

A context is a factor that influences the concept,

and changes its state, and is formed by

conceptual landscapes surrounding the concept.

Typicality is an observable

quantity, with different values for

different states of the concept.

A SCoP formalism was elaborated which

models any kind of entity in terms of

states, contexts and properties.

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 pGuppy of Pet (Fish)

has a low typicality in absence of context, while it scores a

high typicality under the context eFish (ePet).

pppp eˆˆ 1 →→

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We built an explicit quantum

representation in a complex Hilbert space

of the experimental data on the concepts

Pet, Fish and their conjunction Pet-Fish.

highpeppep

lowpppp

pppp

pppp

FishPetGuppyPetFishGuppy

FishFishGuppyPetPetGuppy

Guppye

FishGuppyFish

Guppye

PetGuppyPet

PetFish

FishPet

)ˆ,,(),ˆ,,(

)ˆ,1,(),ˆ,1,(

ˆˆ

ˆˆ

1

1

µµ

µµ

→→

→→

*D. Aerts, L. Gabora (2005a,b), “A Theory of Concepts and Their Combinations

I & II”, Kybernetes 34, pp. 167-191; 192-221.

WHY A QUANTUM FORMALISM IS SO EFFICIENT?

When a subject is asked to estimate the membership (or

the typicality) of an exemplar with respect to one (or

more concepts), contextual influence (of a cognitive type)

and a transition from potential to actual occur in which an

outcome is actualized from a set of possible outcomes.

In a quantum measurement

process, the measurement context

actualizes one possible outcome

and provokes an indeterministic

change of state of the microscopic

quantum particle that is measured.

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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

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.

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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

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15*D. Aerts, L. Gabora, S. Sozzo (2013), “Concepts and Their Dynamics: A

Quantum Theoretic Modeling of Human Thought”, Top. Cogn. Sci. 5, 337-372.

A QUANTUM MODEL FOR THE DISJUNCTION

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17*D. Aerts, L. Gabora, S. Sozzo (2013), “Concepts and Their Dynamics: A

Quantum Theoretic Modeling of Human Thought”, Top. Cogn. Sci. 5, 337-372.

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.

UNDERLYING MECHANISMS IN HUMAN THOUGHT

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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”,

J. Math. Psychol. (in print).

AN EXPLANATORY HYPOTHESIS

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The effects identified in concept research have their counterparts in other domains of cognitive science.

NON-CLASSICAL EFFECTS IN DECISION

THEORY AND ECONOMICS

There is a whole set of findings in

decision theory that entail effects of a

In behavioral economics, similar effects have

been found that point to a deviation from

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The tendency was to consider these deviations from classicality as fallacies, or as effects.

What has been called a fallacy, an effect or a deviation, is a consequence of the dominant

dynamics and its nature is emergence, while what has been considered as a default to deviate

from, namely classical logical reasoning, is a consequence of a secondary form of dynamics.

decision theory that entail effects of a

very similar nature, e.g., the disjunction

effect and the conjunction fallacy.

been found that point to a deviation from

classical logical thinking when human decisions

are at stake (Allais, Ellsberg, Machina paradoxes).

EXPECTED UTILITY THEORY

The predominant model of decision making under uncertainty is expected utility theory

(EUT). Notwithstanding its mathematical tractability and predictive success, the structural

validity of EUT at the individual level is questionable. Indeed, examples exist in the literature

which show an inconsistency between real preferences and the predictions of EUT.

EUT was formally developed by von Neumann and Morgenstern. They presented a

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Knight had reserved the term risk for ventures that can be described by known (physical)

probabilities, and the term uncertainty to refer to situations in which agents did not know the

probabilities associated with each of the possible outcomes of an act. However, probabilities

in the von Neumann and Morgenstern modeling are objectively (physically) given.

EUT was formally developed by von Neumann and Morgenstern. They presented a

set of axioms that allow to represent decision-maker preferences over the set of

acts (functions from the set of states of the world into the set of consequences) by

the functional Ep u(.), for some real-valued utility function u on the set of

consequences and an objective probability measure p on the set of states of the

world. An important aspect of EUT concerns the treatment of uncertainty.

THE ELLSBERG PARADOX

Ellsberg's experiments showed that Knightian's distinction

is empirically meaningful. E.g., he presented the following

Later, Savage extended EUT allowing agents to construct their own subjective

probabilities when physical probabilities are not available. According to Savage's

model, this Knightian distinction seems then to be irrelevant.

The empirical result cannot be explained by EUT. In fact, a very frequent pattern of response

was that f1 is preferred to f2, and f4 is preferred to f3, thus violating the Sure-Thing Principle, an

important axiom of Savage’s model. In both cases the unambiguous bet is preferred to its

ambiguous counterpart, a phenomenon called by Ellsberg ambiguity aversion.

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is empirically meaningful. E.g., he presented the following

experiment. Consider one urn with 30 red balls and 60 balls

that are either yellow or black, in unknown proportion. One

ball will be drawn from the urn. Then, free of charge, a

person is asked to bet on one of the acts f1, f2, f3 and f4.

ALTERNATIVE APPROACHES

After Ellsberg, many extensions of EUT

have been developed to represent this

kind of preferences, all replacing the

Sure-Thing Principle by weaker axioms. Second order probabilities.

Variational preferences.

Max-min EU.

Choquet EU.

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The above models have been widely used in economics and finance,

but they are not absent of critics. And, more, none of them can

reproduce the results of an Ellsberg-type variant, the Machina paradox.

Ambiguity is defined as a situation without a probability model describing it as

opposed to risk, where such a model exists. It is however presupposed usually

that a classical probability model is considered, defined on a σ-algebra of events.

We have worked out a representation of the Ellsberg and Machina paradox

situations in the Hilbert space formalism of quantum theory.

QUANTUM MODEL FOR THE ELLSBERG SITUATION

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RESULTS IN THE CASE OF THE ELLSBERG PARADOX

i. Quantum representation of Ellsberg-type paradox situations.

ii. Modeling of a cognitive experiment with test subjects by the mathematical formalism

of quantum theory.

iii. Explanation of the paradox in terms of the overall conceptual landscape. Ambiguity

aversion is only one, albeit an important one, of the conceptual landscapes influencing

subjects’ decisions and determining deviations from the predictions of EUT.

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subjects’ decisions and determining deviations from the predictions of EUT.

iv. Detection of genuine quantum aspects in this Ellsberg situation: contextuality,

interference, incompatibility, superposition.

v. Unified quantum framework for the representation of the Ellsberg and Machina

paradox situations.

vi. Contextual risk can mathematically capture ambiguity-laden situations of the Ellsberg-

type.

*D. Aerts, S. Sozzo, J. Tapia (2013), “Identifying Quantum Structures in the

Ellsberg Paradox”, accepted in Int. J. Theor. Phys.

QUESTIONS AND FUTURE DEVELOPMENTS

(?) Why the quantum-mechanical formalisms are so efficient in these domains?

(??) Microscopic quantum processes in the human brain? Quantum consciousness?

(???) Really “quantum”? Alternative classical explanations?

Questions.

… Back to the foundations(i) Entanglement, no-signaling, indistinguishability.

(ii) Quantum computation and information with non-micro systems?

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Applications.

Simulation of mental processes,

AI, robotics.

Economics and finance. Black-Scholes

model, random walk hypothesis.

Quantum effects in biological systems.

SA, IR, WWW search.

… Back to the foundations

of quantum theory …

(i) Entanglement, no-signaling, indistinguishability.

(ii) Quantum computation and information with non-micro systems?

(iii) Quantum beyond Hilbert space?