Contribution
The paper turns Hampton’s (1988a,b) data on membership weights for pairs of concepts and their conjunctions and disjunctions into a test of classical representability. It gives exact conditions for an item’s three weights to come from one Kolmogorovian probability space, and finds two distinct ways the data fail them. It then builds an explicit vector-space model in ℝ⁸ that fits every item, with fixed angles per pair of concepts. It shows, by a short proof, that the dominant failure (overextension of “and”, underextension of “or”) cannot come from representing concepts as subspaces combined by the lattice meet and join.
Key insight
There are two kinds of non-classicality in these data, and only one of them is a matter of the lattice. The k-type, where the weights are monotone but do not add up, can be had by tilting two subspaces so their lattice is not Boolean. The ∆-type, the guppy effect proper, violates monotonicity, and no lattice meet or join can violate monotonicity. Aerts therefore models “A and B” and “A or B” as a superposition of the logical combination with a new, emergent concept, which lives in a second sector of the space.
Assumptions
- A conjunction query measures the intersection event. “Classical conjunction data” are weights with a Kolmogorovian space and events E_A, E_B such that P(E_A) = µ(A), P(E_B) = µ(B) and P(E_A ∩ E_B) = µ(A and B) (eq. 3); for disjunction, P(E_A ∪ E_B) = µ(A or B) (eq. 8). The non-classicality results hold relative to this identification. One space is asked for per item, not per experiment.
- Membership weights are relative frequencies. Subjects rated items on −3 … +3; the weight is positive ratings over non-zero ratings (§1.1).
- Measurement is a projector. In §1.5 a concept is a unit vector |A⟩ and the membership test for item X is a projector M, so µ(A) = ⟨A|M|A⟩. In §§2–3 the roles swap: the item is a unit vector x and each concept a subspace, with µ the squared length of the projection (“Quantum Rule”, §2.1). The two representations are not reconciled in the paper.
- “A or B” as a one-particle state is (|A⟩ + |B⟩)/√2 with ⟨A|B⟩ = 0, by analogy with the double slit (§§1.5–1.6), and, in Fock space, as m e^{iθ}|A⟩⊗|B⟩ + n e^{iφ}(|A⟩ + |B⟩)/√2 with m² + n² = 1 (eq. 19).
- Weights 0 or 1 are excluded from the ℂ³ model and “call for a specific approach” (§1.5).
Key results
- Theorem 1 and 3 (conjunction). µ(A), µ(B), µ(A and B) are classical iff 0 ≤ µ(A and B) ≤ µ(A), µ(B) ≤ 1 and k_c = 1 − µ(A) − µ(B) + µ(A and B) ≥ 0 (eqs. 4–7). Holds when: one Kolmogorovian space per item, with the identification above. Proof by an explicit four-point space (Appendix A).
- Theorems 4 and 6 (disjunction). Classical iff µ(A), µ(B) ≤ µ(A or B) ≤ 1 and k_d = µ(A) + µ(B) − µ(A or B) ≥ 0 (eqs. 9–12). Appendix B.
- Theorems 2 and 5. Overextension, ∆_c = µ(A and B) − min(µ(A), µ(B)) > 0, and underextension, ∆_d = max(µ(A), µ(B)) − µ(A or B) > 0, are each sufficient for non-classicality. Examples: Cuckoo for Bird and Pet, 1, 0.575, 0.842 (§1.1); Ashtray for Home Furnishings or Furniture, 0.7, 0.3, 0.25. Almond for Fruits or Vegetables, 0.2, 0.1, 0.425, is k-type with k_d = −0.125 (§1.6).
- Interference model (eqs. 13–17). µ(A or B) = ½(µ(A) + µ(B)) + Re⟨A|M|B⟩, realised in ℂ³ with an explicit phase β. Holds when: the interference term can reach the data; it is bounded by the smaller of √(µ(A)µ(B)) and √((1−µ(A))(1−µ(B))), so items with one weight near 0 and the other near 1 (Cave, Apple, Gardening) have no such model (§1.7, Tables 1–2).
- Fock-space weights (eqs. 23, 26). µ(A and B) = m²µ(A)µ(B) + n²(½(µ(A)+µ(B)) + Re⟨A|M|B⟩); µ(A or B) = m²(µ(A)+µ(B)−µ(A)µ(B)) + n²(½(µ(A)+µ(B)) + Re⟨A|M|B⟩). With zero interference these are convex mixtures of the product rule and the average, which fit the items the ℂ³ model cannot.
- Hawaii problem (§1.8). Tversky and Shafir’s 0.54, 0.57 and 0.32 are reproduced in ℂ³ with β = 121.8967°: 0.555 + 0.4447 cos β = 0.32.
- Theorems 7 and 8. Every classical item is represented in ℝ⁴ with A and B coordinate subspaces meeting in one axis (eqs. 35–46).
- k-type items (§2.2). Tilting A and B by an angle θ (eqs. 47–48) replaces k_c by a “quantum logic factor” q_conj(A, B, θ) = 1 − µ(A and B) − (a + b − 2√a√b cos θ)/sin²θ, with a = µ(A) − µ(A and B), b = µ(B) − µ(A and B) (eq. 54), and q_conj(A, B, π/2) = k_c. An item is representable iff q_conj ≥ 0, which holds for cos θ between the roots of eq. 56. Horse Cart and Dishwasher (Machine and Vehicle) are fitted together at θ = 80.9026°. The paper attributes this kind of non-classicality to “the non Boolean nature of the set of closed subspaces”.
- Theorem 9 (§2.3, Appendix E). If concepts are subspaces of ℝⁿ, “A and B” is A ∩ B and “A or B” is A + B, then ∆_c ≤ 0 and ∆_d ≤ 0 for every item. The proof is monotonicity of ‖P_S x‖² in S. So ∆-type items have no such model.
- The ℝ⁸ = ℝ⁴ ⊕ ℝ⁴ model (§§2.3–3.3). A and B get a component in a second ℝ⁴ (eqs. 57–58), and the combination adds a fixed 2-dimensional “emergent concept” subspace C there (eqs. 61–62), parametrised by θ and φ per pair. With 8 coordinates, 4 equations and C fixed per pair, each item has one free degree of freedom (§2.3). It is fixed by a “modularity condition”, µ(A)·µ_c(B) = µ(B)·µ_c(A) (eq. 117), and, for classical items, by setting the relative classical disjunction weight halfway between the measured and the maximal value (§3.4). Tables 3–4 give a vector for every item.
- Conjunction through complements (§3.3). For “A and B” the paper introduces “hypothetical concepts” A′, B′ with µ(A′) = 1 − µ(A), µ(B′) = 1 − µ(B), µ(A′ or B′) = 1 − µ(A and B), and uses the disjunction model for them (eqs. 119–125). A′ is the orthogonal complement of A in ℝ⁸, and the emergent subspace for “A′ or B′” is the orthogonal complement of C within the second ℝ⁴. So the conjunction model is the De Morgan dual of the disjunction model under the orthocomplement.
- Two layers of thought (§§1.6, 3.4). The first sector’s weights are read as “classical logical thought” and the second’s as “quantum conceptual thought”. For Sugar under Spices or Herbs (0, 0, 0.2), the relative classical disjunction weight is 0 and the relative quantum one is 1.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Weights for a pair and its conjunction (disjunction) have a Kolmogorovian model iff ∆ ≤ 0 and k ≥ 0 | strong | Theorems 1–6, proofs in Appendices A–B |
| C2 | Many of Hampton’s items fail it, mostly by over- or underextension, and several disjunction items by k_d < 0 | strong | Tables 1–4, computed from Hampton’s published weights |
| C3 | No subspace model with conjunction as meet and disjunction as join can fit ∆-type items | strong | Theorem 9, Appendix E |
| C4 | k-type items can be fitted by tilting subspaces, so their non-classicality is of “quantum logic nature” | moderate | §2.2, worked for Machine and Vehicle only; the general claim is asserted |
| C5 | An ℝ⁸ model with one emergent subspace per pair fits all of Hampton’s items | moderate | Tables 3–4; one free parameter per item and two angles per pair, fixed by stipulated conditions |
| C6 | Thought has two superposed layers, classical logical and quantum conceptual | weak | an interpretation of the two sectors, illustrated, not tested |
| C7 | The disjunction effect, conjunction fallacy and sure-thing violations are the same effects as under- and overextension | weak | one ℂ³ fit (Hawaii) and one transposed item (Diving Mask); analogy otherwise |
Method
Compute ∆ and k for each item from Hampton’s weights. Classify items as classical, ∆-type or k-type. Fit ℂ³ interference models where the interference term can reach the data, Fock-space convex weights where it cannot (Tables 1–2). Then build a single real model per pair in ℝ⁸: the first ℝ⁴ holds the classical part (Theorems 7–8, tilted by θ for k-type items), the second holds the emergent concept C, obtained by rotating the canonical basis by 90° in the (e₆, e₈) plane and by φ in the plane of e₅ and the bisector of e₆, e₇ (eqs. 79–86). Solve eqs. 108–111 by a graphical search followed by iteration to 10⁻¹⁵ (Figure 1).
Concepts
- Membership weight µ(A). The proportion of positive among non-zero ratings of an item for concept A.
- ∆-type non-classical item. Overextended for a conjunction or underextended for a disjunction: monotonicity fails.
- k-type non-classical item. Monotone, but k_c < 0 (conjunction) or k_d < 0 (disjunction): the weights cannot be partitioned into four non-negative cells.
- Emergent concept. The 2-dimensional subspace C of the second ℝ⁴, representing “A or B” (or “A and B”) as a new concept, not as a logical combination.
- One-particle and two-particle ways. Judging membership in “A or B” as one new concept, or as two parallel judgements combined by truth table. They are the two sectors of Fock space.
- Modularity condition. Eq. 117: classical weights proportional to measured weights. It has nothing to do with the modular law of lattices.
Connections
The paper continues Gabora and Aerts (2002) and Aerts and Gabora (2005a,b, held here as LIT-340 and LIT-312, NOTE-294 and NOTE-299), and Aerts (2007a,b), which it summarises. Its data are Hampton (1988a,b), and its classical criterion is related to Pitowsky’s correlation polytopes through Aerts, Aerts and Gabora (2009). It cites Busemeyer, Matthew and Wang (2006), Franco (2007) and Khrennikov (2008) for quantum models of the disjunction effect, and Widdows and Peters (2003) and van Rijsbergen (2004, LIT-262) for orthogonal negation in retrieval. None of these lineages is declared on the LIT.
Bearing on the record
- QUESTION-055 (the negation of a category). The paper does not
answer it and does not ask it.
- Negation of a concept appears in only two places. One is the “no” outcome of a membership test, 1 − M. The other is the device of §3.3, where hypothetical complements A′, B′ with weights 1 − µ let the conjunction model be read off the disjunction model through the orthocomplement. No negated concept is measured.
- In §1.5 a concept is a state vector, which has no complement at all.
- The subspace lattice it does use (§2) is orthocomplemented and, being finite-dimensional, modular. But Theorem 9 shows that this lattice’s meet and join cannot represent the main data. What carries the guppy effect is the emergent subspace, which is not a lattice operation on A and B.
- The record’s inference: Theorem 9 needs only that membership is order-preserving and that “and” is a meet. So the same holds for a Boolean, orthomodular or Heyting lattice. If communicative categories show guppy-type effects, the choice among the three negations of QUESTION-055 cannot by itself account for them. This is filed as THEORY-255.
- LIT-934’s own body and status note say the paper models the data “with quantum, orthocomplemented structure”, one of the “candidate negations”. That holds only weakly. The orthocomplement appears in §3.3 as a calculational device, and the lattice whose negation it is fails for the ∆-type data.
- NOTE-294’s open question names “Aerts 2009” as where tests of a non-distributive concept lattice are “said to appear”. In part they do. §2.2 uses non-commuting subspaces (θ ≠ 90°), whose lattice is not distributive, and needs them for the k-type items. But no identity such as e ∧ (f ∨ g) = (e ∧ f) ∨ (e ∧ g) is tested directly, and the main effect is placed outside the lattice.
- THEORY-041. Aerts’s “modularity condition” (eq. 117) is a proportionality constraint on weights, not Birkhoff and von Neumann’s modular law. A citation that runs the two together would be wrong.
- CLAIM-041 and THEORY-013. Non-classicality here means “no single Kolmogorovian space in which ‘A and B’ is the intersection event”. That is not contextuality in the Contextuality-by-Default sense of LIT-264, and the paper does not claim it is. It is consistent with CLAIM-041: the paper’s evidence for quantum structure is that a quantum model fits where a classical one, so defined, cannot.
- CLAIM-119. If communicative categories behave like Hampton’s concepts, a concept lattice with meet as conjunction cannot model their overextended blends. That is a constraint on CLAIM-119’s lattice that the record had not stated.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Fit, not test. Each item has one free parameter after the per-pair angles are chosen, and the extra conditions that fix it are stipulated (eq. 117, and the halfway rule for classical items). The angles were found by search, and for two pairs only “a very narrow window” worked (§3.2). No held-out prediction is made.
- Classical items are recast as non-classical “in reality” (§3.4). That is a modelling choice, made because the data allowed a purely classical fit.
- The two-layer hypothesis and the decision-theory transfer are not tested; the paper says further experiments are needed (§1.8).
- Feature-linked quantities are not modelled, which the paper itself names as a limitation (Introduction).
- Slips in the text. After Theorem 4 the text writes “µ(A and B) ≤ min” where the disjunction condition is meant. Discus Throwing’s weights are attributed to Hampton (1988a) though they are disjunction data, and its measured µ(A or B) is given as 0.75 and then as “the original weight of 0.7” (§3.2). For MSG, µ(B) is written “µ(A) = µ_2c(A) = µ_rq(A) = 0.1” (§3.4).
Open questions
- Does any lattice-theoretic semantics, with conjunction a meet and a non-monotone membership function, reproduce overextension? Theorem 9 rules out monotone ones only.
- What would data on negated concepts (“not a Pet”) show? Whether µ(not A) = 1 − µ(A) holds, and whether “not A or not B” mirrors “A and B”, would test the orthocomplement that §3.3 assumes.