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Quantum Structure in Cognition

Diederik Aerts

arXiv:0805.3850v2math-phquant-ph

TL;DR

The paper addresses how concepts combine, carry meaning, and influence human thought. It develops quantum-based models of concept combinations and uses them to model experimental membership data, proposing a two-layer structure of human thought.

  • Problem

    Understanding how concepts combine to form meaningful sentences and texts, and how they communicate meaning between minds, remains a major challenge in studying human thought.

  • Method

    The paper uses SCOP and Hilbert-space quantum formalisms, including Fock-space modeling, to represent contextual influence and concept combinations in experimental data.

  • Results

    The quantum modeling accounts for Hampton’s membership-weight data involving overextension and underextension, including nonclassical disjunction weights such as µ(A or B) = 0.425 versus a relative classical weight of 0.25.

  • Takeaways & Limitations

    The proposed model supports a superposition of a classical logical layer and a quantum conceptual layer in human thought.

  • Takeaways & Limitations

    The modeling remains constrained by the need for detailed testing across alternative choices to determine whether Fock space is necessary for the disjunction effect.

Abstract

from arXiv · show

The broader scope of our investigations is the search for the way in which concepts and their combinations carry and influence meaning and what this implies for human thought. More specifically, we examine the use of the mathematical formalism of quantum mechanics as a modeling instrument and propose a general mathematical modeling scheme for the combinations of concepts. We point out that quantum mechanical principles, such as superposition and interference, are at the origin of specific effects in cognition related to concept combinations, such as the guppy effect and the overextension and underextension of membership weights of items. We work out a concrete quantum mechanical model for a large set of experimental data of membership weights with overextension and underextension of items with respect to the conjunction and disjunction of pairs of concepts, and show that no classical model is possible for these data. We put forward an explanation by linking the presence of quantum aspects that model concept combinations to the basic process of concept formation. We investigate the implications of our quantum modeling scheme for the structure of human thought, and show the presence of a two-layer structure consisting of a classical logical layer and a quantum conceptual layer. We consider connections between our findings and phenomena such as the disjunction effect and the conjunction fallacy in decision theory, violations of the sure thing principle, and the Allais and Elsberg paradoxes in economics.

Introduction

The paper develops a quantum-based theory of concept combinations to explain how concepts carry and communicate meaning. It links quantum effects in combinations to a two-layer structure of human thought and to related decision-theoretic and economic phenomena.

  • Motivation: The paper uses quantum mechanics as a mathematical instrument for modeling how concepts combine and communicate meaning.The approach treats concepts as context-sensitive entities whose states change under contextual influence.
  • Motivation: Hampton’s membership-weight data are investigated because deviations from classical set-theoretic rules may indicate quantum structure in cognition.The study targets both the modeling power of the scheme and the possibility of demonstrating genuine quantum structure.
  • Quantum modeling scheme: The model is aimed at meaning aspects of concepts and combinations rather than specific pure linguistic structures.Its general modeling power rests especially on an intrinsically contextual theory and changing concept states under context.
  • Related implications: The framework is connected to the disjunction effect, conjunction fallacy, violations of the sure thing principle, and Allais and Elsberg paradoxes.These connections extend the discussion to decision theory and economics, where quantum aspects have also been used for modeling.
  • Quantum modeling scheme: The proposed scheme combines contextual influence, superposition, interference, and quantum field-theoretic aspects to model concept-combination effects.Earlier work modeled contextual influence, while later work addressed emergence, interference, and field-theoretic structure.
  • Limitations: A stated limitation is that explicit Hilbert-space modeling shifted attention toward exemplar-linked typicality rather than feature-linked quantities.The earlier SCOP model treated both classes of quantities, but the concrete Hilbert-space model largely focused on exemplars.
  • Structure of human thought: The authors propose that human thought comprises a classical logical layer superposed with a quantum conceptual layer.Fock-space modeling of Hampton’s data is directly related to this layered hypothesis, in which concept combinations can emerge as individual entities.

1 A General Scheme for Quantum Modeling

This section introduces the general quantum-modeling scheme by first presenting Hampton’s experiments and selected data as the main experimental material.

  • 1 A General Scheme for Quantum Modeling: The general quantum-modeling scheme is introduced through an explanation of Hampton’s experiments and selected experimental data.These data provide the main experimental material for the discussion.

1.1 The Guppy Effect for Membership

The section explains graded membership and the guppy effect as deviations from classical conjunction and disjunction rules. It introduces the membership-weight measures used to characterize these effects.

  • Membership weights: Concept membership is represented as a graded weight between 0 and 1 rather than as a yes-or-no property.The endpoints represent membership and non-membership, while intermediate values represent graded membership.
  • Conjunction: Hampton’s conjunction experiments measured membership weights for items across individual concepts and their conjunctions.One example considers Cuckoo with respect to Bird, Pet, and Bird and Pet.
  • Conjunction: Overextension is the deviation in which conjunction membership exceeds what classical interpretation would predict.Hampton used this term for departures from the expected classical behavior of concept conjunctions.
  • Disjunction: Hampton’s disjunction example found Ashtray’s membership weight for Home Furnishings or Furniture to be 0.25, below its weights of 0.7 and 0.3 for the individual concepts.This is unexpected under the intuitive logical meaning of disjunction.
  • Experimental procedure: A typical guppy-effect experiment asks subjects to select values from −3 to +3, with larger positive values indicating greater typicality.The response scale operationalizes graded judgments of item membership.
  • Contextual influence: The theory interprets instability in graded concept structures as changes in a concept’s state under contextual influence.This reframes the instability discussed by Barsalou as behavior captured by the state-of-a-concept notion.
  • Conjunction: The guppy effect describes cases where an item is more typical of a conjunction than of either constituent concept alone.The Pet-Fish example motivates studying conjunctions that conflict with the fuzzy-set minimum rule.
  • Deviation measures: The conjunction minimum-rule deviation is ∆c = µ(A and B) − min(µ(A), µ(B)), while the disjunction maximum-rule deviation is ∆d = max(µ(A), µ(B)) − µ(A or B).These quantities characterize departures from the minimum and maximum rules, respectively.

1.2 Classical and Non Classical Data

The paper defines classical conjunction and disjunction data through measure-theoretic or Kolmogorovian probability structures. It argues that some Hampton data remain non-classical while other deviations can still be classically modeled.

  • Classical data: Classical conjunction and disjunction data are defined as data representable within a measure-theoretical or Kolmogorovian probability structure.The framework uses classical probability to determine which observed data can receive a classical set-theoretic model.
  • Classical data: A σ-algebra is a non-empty collection of subsets closed under complementation and countable unions.It forms a Boolean algebra completed to include countably infinite operations.
  • Classicality conditions: Conjunction data satisfying the fuzzy-set minimum rule are classical, and disjunction data satisfying the maximum rule are classical.These rules provide sufficient conditions for classicality within the broader measure structure.
  • Non-classical data: Hampton’s overextension and underextension cases remain non-classical under the measure-theoretic definition used here.The paper distinguishes these problematic cases from data that violate fuzzy-set rules but remain classically representable.
  • Non-classical data: Some Hampton conjunction and disjunction data are classical even though they do not satisfy the fuzzy-set minimum or maximum rules.Thus, violating a fuzzy-set rule is not by itself sufficient to establish non-classicality.

1.3 Classical and Non Classical Conjunction Data

Classical conjunction data are defined by whether membership weights can be represented in a Kolmogorovian probability space. Overextension guarantees non-classicality, while full classicality also requires a nonnegative Kolmogorovian conjunction factor.

  • Classical conjunction data: Classical conjunction data require membership weights for A, B, and A and B to correspond to events in a Kolmogorovian probability space.The conjunction weight is represented as the probability of the intersection of events for A and B.
  • Classical conjunction data: Theorem 1 characterizes classical conjunction data through inequalities on the three membership weights.These inequalities are equivalent to the conjunction weight not exceeding the minimum of the individual weights, together with an additional constraint.
  • Non-classical conjunction data: Overextension, defined by ∆c > 0, makes conjunction data non-classical and impossible to model with a Kolmogorovian probability space.Overextension occurs when the conjunction membership weight exceeds the relevant classical minimum rule.
  • Non-classical conjunction data: Classical conjunction data require both no overextension and a nonnegative Kolmogorovian conjunction factor kc.Thus, absence of overextension alone is insufficient for a classical representation.
  • Experimental pattern: Most non-classical conjunction items in Hampton’s data are overextended, while only a few violate the separate factor constraint.The overextended items correspond to the previously identified guppy-effect cases.

1.4 Classical and Non Classical Disjunction Data

Classical disjunction data are characterized by Kolmogorovian representability of the individual and disjunctive membership weights. Underextension guarantees non-classicality, but classicality additionally requires a nonnegative disjunction factor.

  • Classical disjunction data: Classical disjunction data require membership weights for A, B, and A or B to correspond to events in a Kolmogorovian probability space.The disjunction weight is represented as the probability of the union of events for A and B.
  • Classical disjunction data: Theorem 4 characterizes classical disjunction data through inequalities on the three membership weights.These inequalities encode the classical constraints for a disjunction.
  • Non-classical disjunction data: Underextension, defined by ∆d > 0, makes disjunction data non-classical and impossible to model with a Kolmogorovian probability space.Underextension occurs when the disjunction membership weight violates the classical maximum rule.
  • Non-classical disjunction data: Classical disjunction data require both no underextension and a nonnegative Kolmogorovian disjunction factor kd.Therefore, absence of underextension alone does not establish classical representability.
  • Experimental pattern: Underextension is the commonest disjunction non-classicality in Hampton’s data, and many additional items have negative kd values.The data therefore contain more k-type non-classical items for disjunction than for conjunction.

1.5 Presenting the Quantum Modeling Scheme

The quantum modeling scheme represents concepts as states in a Hilbert space and concept disjunction as a normalized superposition. Membership probabilities are obtained through projection measurements, with interference producing deviations from the no-interference average.

  • Concept states: A concept’s state is represented by a unit ket vector encoding what the concept stands for with respect to relevant features and contexts.The Hilbert-space formalism supplies notions of length, orthogonality, and weight.
  • Measurement: The membership decision for an item is modeled by an orthogonal projection representing the alternatives of being or not being a member.Applying the projection to A, B, and A or B yields µ(A), µ(B), and µ(A or B).
  • Superposition and interference: The disjunction A or B is represented by the normalized superposition 1/√2(|A⟩+|B⟩) of orthogonal concept states.This construction treats the combined concept as a quantum superposition of the two component concepts.
  • Superposition and interference: The interference term ℜ⟨A|M|B⟩ shifts µ(A or B) away from the no-interference average 1/2(µ(A)+µ(B)).Superposition supplies the combined state, while interference supplies the deviation from the average.
  • C3 realization: The superposition-and-interference model is realized in the three-dimensional complex Hilbert space C3 for non-boundary individual membership weights.The construction excludes cases where µ(A) or µ(B) equals 0 or 1, which require a specific approach.
  • Examples: The C3 construction reproduces the observed disjunctive membership weight for concrete Hampton items, including Pencil Eraser and Field Mouse.Both examples are classically nonrepresentable because their disjunctive weight is below one individual concept weight.

1.6 Quantum Field Theory and Two Modes of Human Thought

The paper distinguishes two ways of processing a concept disjunction: a one-item conceptual combination and two parallel item-specific membership decisions. Hampton’s results are interpreted as showing both modes in superposition, motivating a Fock-space model of two thought layers.

  • The modeling problem: The normalized superposition for A or B resembles the double-slit situation, but its no-interference average differs from fuzzy-set and Kolmogorovian disjunction rules.This mismatch creates a modeling problem that the paper turns to quantum field theory to address.
  • Two-layer structure: The paper links the two modes to a two-layer structure of human thought, represented by a classical logical layer and a quantum conceptual layer.The distinction is complicated because creating an identical item within thought is described as an ongoing process.
  • Two modes of thought: The one-particle way treats A or B as a wholly new concept applied to one item, whereas the two-particle way evaluates two identical items against A and B separately.The Almond example illustrates these alternative decision processes for a disjunction.
  • Two modes of thought: Hampton’s experiments are interpreted as showing that both processing ways occur in superposition.Fock space is presented as the mathematical space capable of modeling this superposition.
  • Examples: For Apple, µ(A) = 1, µ(B) = 0, and µ(A or B) = 1, indicating dominance of the two-particle way in that comparison.The paper uses these membership weights to connect the experimental case to the proposed thought modes.
  • Examples: For Almond, µ(A) = 0.2, µ(B) = 0.1, and µ(A or B) = 0.425, with kd = −0.125, indicating a k-type non-classical item and strong dominance of the one-particle way.The disjunction weight exceeds both individual weights in this example, producing the reported negative kd.

1.7 Fock Space and What About Conjunction

The section extends the quantum model of concept combinations to conjunctions using Fock space, combining one-particle and two-particle representations. This framework addresses membership patterns that interference alone cannot model and treats underextension and overextension as effects of emergent concepts.

  • Fock-space construction: The two-particle representation models concepts A and B as the tensor-product state |A⟩⊗|B⟩ and evaluates membership through decision measurements.For disjunction, a positive response includes the outcomes yes–yes, yes–no, and no–yes.
  • Emergent concepts: Underextension for disjunction and overextension for conjunction are attributed to the emergence of a new combined concept.The same effect is proposed for both patterns, including the concepts A or B and A and B.
  • Fock-space construction: Fock space combines a two-particle Hilbert space with a one-particle Hilbert space as F = (H ⊗ H) ⊕ H.The concept state is a normalized linear combination of the corresponding two-particle and one-particle states.
  • Fock-space construction: Fock-space modeling represents otherwise difficult membership values as a convex combination of the product and average of the two concept weights.The model uses convex weights m^2 and n^2 for conjunction and disjunction cases.
  • Limits of interference: Classical items with one membership weight near 0 and the other near 1 generally cannot be modeled using quantum interference alone but can usually be modeled in Fock space.Interference is small when weights are near 0 or 1, limiting the one-particle model's ability to reach the observed conjunction value.

1.8 Application to Decision Theory, Economics and Other Domains

The section applies the quantum conceptual framework to decision theory, economics, and related cognitive-science domains. It models the Hawaii disjunction effect and connects concept-combination effects with violations of established decision principles and semantic-analysis approaches.

  • Decision theory: The disjunction effect occurs when people prefer option x when they know either event A or event B will occur, but refuse x when they know neither event's outcome.The Hawaii problem provides the best-known example described in this section.
  • The Hawaii problem: 54% chose to buy the Hawaii vacation after passing and 57% after failing, but only 32% chose it without knowing the exam outcome.The section constructs a C3 quantum model for these experimental proportions.
  • The Hawaii problem: The quantum modeling scheme explains the Hawaii problem by treating the whole conceptual landscape as influencing the decision weights.The authors note that different contexts, such as a free relaxation weekend, could produce the opposite response pattern.
  • Model scope: Detailed testing across alternative experimental choices is needed to determine whether the Hawaii disjunction effect requires a Fock-space model beyond the single Hilbert-space model.The authors expect Fock space may be needed when modeling different data from the same decision situation.
  • Economics: For Diving Mask, Hampton measured µ(A) = 1, µ(B) = 1, and µ(A or B) = 0.95, illustrating a membership pattern that violates the sure thing principle.The example transposes the decision-theoretic principle to concept-membership judgments.
  • Cross-domain connections: The paper relates its effects to the disjunction effect, conjunction fallacy, and sure thing principle violations, and identifies analogous quantum structures in semantic-analysis models.These connections support the proposed link between conceptual structure and decision-making.

2 A Simple Quantum Model Illustrating the General Scheme

The model represents concepts and items geometrically in real vector spaces, using projection-based membership weights and distinct structures for classical, k-type, and Δ-type cases. It shows that fixed subspace representations capture classical and k-type data, whereas Δ-type non-classicality requires superposition and emergent concept formation.

  • Model construction: The real-vector-space model represents all tested items for a concept pair with fixed conjunction and disjunction representations.This differs from the standard interference model, where fixed rays do not necessarily determine one specific superposition ray.
  • 2.1 Quantum Modeling of the Classical Items: R4 represents classical items using concept subspaces, their conjunction as A ∩ B, and their disjunction as A + B.The conjunction is one-dimensional, while the disjunction is three-dimensional in the canonical construction.
  • Model construction: The model represents concepts as subspaces and items as unit vectors, with membership weights given by squared projection lengths.This replaces ordinary set representations with a real-vector-space quantum representation.
  • 2.2 Modeling k-Type Non Classical Items: Changing the angle between concept subspaces enables quantum representations of k-type non-classical items when the quantum logic factor is non-negative.For θ between θ− and θ+, qconj(A, B, θ) is non-negative and a representation exists.
  • 2.2 Modeling k-Type Non Classical Items: The new angled representation can represent classical items as well as k-type non-classical items, making it more general than the earlier classical construction.The paper gives θ = 80.9026° or θ = 99.0974° for this broader representation.
  • 2.3 Modeling Δ-type Non-Classical Items: The subspace-intersection and subspace-sum construction cannot represent Δ-type non-classical items because it enforces classical membership inequalities.The model therefore introduces a one-particle way in which a combined concept is a superposition of the logical combination and an emergent new concept.

3 Solving the Modeling of the Disjunction and Conjunction Data

The paper models Hampton’s conjunction and disjunction membership data by adding subspaces for the “new concept” aspects of concept combinations. The resulting classical and quantum components account for underextension and overextension patterns.

  • 3 Solving the Modeling of the Disjunction and Conjunction Data: The model introduces subspaces representing the “new concept” formed by conjunctions and disjunctions of concepts.The disjunction data are modeled first, then used to model the conjunction data.
  • 3 Solving the Modeling of the Disjunction and Conjunction Data: Orthogonal transformations construct the two-dimensional subspace C for the disjunction A or B using a basis rotation parameterized by angle φ.The construction combines a 90° rotation with rotations through 45° and φ.
  • 3.3 Working out the Conjunction Data: For Library, the relative classical weights satisfy the fuzzy-set minimum rule, while the relative quantum weight for A and B is 0.9503 and yields stronger overextension than the measured data.The classical component restricts the quantum overextension to the moderate overextension observed by Hampton.
  • 3.4 Classical Logical and Quantum Conceptual Thought: For k-type items such as MSG, the classical disjunction weight equals µ(A) + µ(B), whereas the measured disjunction weight is larger.For MSG, the classical weight is 0.25, compared with measured weights µ(A) = 0.15, µ(B) = 0.1 and µ(A or B) = 0.425.
  • 3.4 Classical Logical and Quantum Conceptual Thought: The model assigns substantial quantum contribution to several ∆-type items, indicating that the new-concept component strongly affects their disjunction judgments.Mustard, for example, has µq(total) = 0.6448.

4 Fundamentals of Concept Formation and Combination

The paper links overextension and underextension to concept formation, modeling both classical combinations and emergent conceptual wholes within a Fock-space framework. It extends this scheme toward large collections of concept combinations while noting that broader validity still requires quantitative testing.

  • 4.1 General Concept Formation: Underextension arises naturally when a disjunction forms a new concept whose items are less characteristic than they are of component concepts.The paper illustrates this with Barking, highly characteristic of Dog but less characteristic of Animal.
  • 4.1 General Concept Formation: Overextension arises naturally when a conjunction forms a new concept whose items are more characteristic of the conjunction than of individual component concepts.Humans Friend is presented as characteristic of Dog but not necessarily of individual characteristics such as Has Four Legs or Likes to Swim.
  • 4.1 General Concept Formation: Superposition adds an emergent state whenever concepts combine, treating new concept formation as comparable in importance to classical logical combination.The scheme distinguishes a two-particle logical way from a one-particle emergent way of combining concepts.
  • 4.2 The Modeling of Large Collections of Combinations of Concepts: Fock space represents combinations at multiple scales, from the whole combination as one concept to individual concepts and intermediate conceptual groupings.For a sentence, the model includes both a superposition describing one new concept and tensor-product states describing concepts individually or in grouped parts.
  • 4.2 The Modeling of Large Collections of Combinations of Concepts: The Fock-space construction represents each possibility of treating subsets of an n-concept combination as individual concepts.The example includes sentence fragments such as “The cat eats the food” and “While the child plays in the garden” as grouped conceptual units.
  • 4.2 The Modeling of Large Collections of Combinations of Concepts: Approximate modeling is needed because Fock space can become very extensive, with the guiding idea of retaining primarily relevant instantiations in a combination.The sentence example focuses on concepts such as Cat, Food, Child, and Garden as prominent elements of its scenery.
  • 4.2 The Modeling of Large Collections of Combinations of Concepts: The scheme is intended to extend from concept combinations to large collections including pieces of text, documents, and books.The paper presents this extension as a potential application of the Fock-space construction rather than as a completed validation.
  • 4.2 The Modeling of Large Collections of Combinations of Concepts: Broader validity remains to be tested through quantitative experiments measuring typicality and feature applicability, despite confidence based on Hampton’s data.Tables organize disjunction and conjunction cases using vectors in R8, relative classical and quantum weights, total quantum weight, and item-type labels.

A Appendix: Proof of Theorem 1:

The appendix constructs a four-outcome Kolmogorovian probability space for classical conjunction data and verifies that the resulting function is a valid probability measure.

  • Classical conjunction membership weights can be represented by events whose probabilities equal µ(A), µ(B), and µ(A and B).The construction uses the intersection event for the conjunction and derives the required probability inequalities.
  • The proof defines probabilities for arbitrary subsets of a four-element outcome space and verifies the probability-measure sum formula.The sum formula follows from the construction, while subset bounds are checked using the conjunction inequalities.
  • The constructed space reproduces µ(A and B), µ(A), and µ(B), establishing that these data are classical conjunction data.The appendix identifies P({1}) with µ(A and B), P({1,2}) with µ(A), and P({1,3}) with µ(B).

B Appendix: Proof of Theorem 4

The appendix applies the same Kolmogorovian construction to classical disjunction data, using union probabilities and the associated probability inequalities.

  • Classical disjunction membership weights can be represented by events whose probabilities equal µ(A), µ(B), and µ(A or B).The proof uses the union event for the disjunction and derives bounds on the event probabilities and their intersection.
  • The construction defines a probability measure on a four-element outcome space and verifies that every subset has probability within [0,1].The sum formula follows directly from the definition, while the subset bounds follow from the disjunction inequalities.

C Appendix: Proof of Theorem 7

The appendix verifies that the vector used in the model is well-defined, reproduces the conjunction membership weights, and has unit norm.

  • Theorem 1 ensures that the components xAB, xAB′, xA′B, and xA′B′ are well-defined.Their well-definedness follows from the previously established probability bounds.
  • The squared vector components reproduce the measured conjunction weights µ(A and B), µ(A), and µ(B).The appendix explicitly derives xAB^2 = µ(A and B), xAB′^2 = µ(A), and xA′B^2 = µ(B).
  • The remaining component makes x a unit vector in R4.The appendix sums the four squared components to obtain 1.

D Appendix: Proof of Theorem 8

The proof establishes that the constructed coordinates xAB, xAB′, xA′B, and xA′B′ are well-defined and satisfy the required membership-weight relations. It also shows that the resulting vector has unit norm.

  • Well-defined coordinates: Theorem 4 guarantees that xAB, xAB′, xA′B, and xA′B′ are well-defined.This follows from the previously established probability bounds and equations (133)–(136).
  • Membership relations: The constructed coordinates reproduce µ(A) and µ(B) through the relevant squared-coordinate relations.The proof verifies the relations corresponding to equations (35) and (36).
  • Disjunction relation: The construction also reproduces µ(A or B), establishing the relation corresponding to equation (38).The relevant squared-coordinate sum equals the disjunction membership weight.
  • Normalization: xA′B′ = µ(A and B) + 1 − µ(A and B) = 1, proving that x is a unit vector in R4.The final coordinate relation completes the normalization proof.

E Appendix: Proof of Theorem 9

The proof uses orthogonal projections to show that conjunction membership cannot exceed either constituent concept, while disjunction membership cannot be below either constituent. Therefore, the item is not Δ-type non-classical for either operation.

  • Projection representation: Representing item X by x ∈ Rn expresses each membership weight as the squared norm of an appropriate orthogonal projection.The projections correspond to A, B, A ∩ B, and A + B for the two concepts and their conjunction and disjunction.
  • Conjunction: µ(A and B) ≤ µ(A) and µ(A and B) ≤ µ(B), because A ∩ B is contained in both A and B.These inequalities imply Δc ≤ 0, so X is not a Δ-type non-classical item for conjunction.
  • Disjunction: µ(A) ≤ µ(A or B) and µ(B) ≤ µ(A or B), because both A and B are contained in A + B.These inequalities imply Δd ≤ 0, so X is not a Δ-type non-classical item for disjunction.
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