THEORY-255: In Aerts's subspace model of Hampton's membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice

Source

Aerts (2009), LIT-934, Journal of Mathematical Psychology 53(5), read in full in NOTE-808: Theorems 1–6 (§§1.3–1.4, Appendices A–B), §2.2 (eqs. 47–56), Theorem 9 (§2.3, Appendix E), §§3.1–3.3 and Tables 3–4.

What was actually shown

Two kinds of non-classical item. For an item’s weights µ(A), µ(B) and µ(A and B) there is a Kolmogorovian space with µ(A and B) the probability of the intersection event iff ∆_c = µ(A and B) − min(µ(A), µ(B)) ≤ 0 and k_c = 1 − µ(A) − µ(B) + µ(A and B) ≥ 0 (Theorems 1–3), and dually for “or” (Theorems 4–6). Hampton’s data fail both ways. Overextension or underextension (∆-type) is the commoner, the guppy effect proper: Cuckoo is rated 0.575 a Pet but 0.842 a “Bird and Pet”. A negative k (k-type) occurs in many disjunction items: Almond is 0.2 a Fruit, 0.1 a Vegetable and 0.425 a “Fruit or Vegetable”.

The lattice can carry the k-type items. Represent A and B as 2-dimensional subspaces of ℝ⁴ meeting in one axis, tilted by an angle θ (eqs. 47–48). The condition k_c ≥ 0 is replaced by q_conj(A, B, θ) ≥ 0 (eq. 54), which reduces to k_c at θ = 90° and is satisfiable for other θ by items with k_c < 0. Horse Cart and Dishwasher for Machine and Vehicle are fitted at θ = 80.9026°. The subspaces are then non-commuting and their lattice is not Boolean.

The lattice cannot carry the ∆-type items. Theorem 9: if concepts are subspaces, “A and B” is A ∩ B and “A or B” is A + B, then for every unit vector x, ‖P_{A∩B}x‖² ≤ ‖P_A x‖² and ‖P_A x‖² ≤ ‖P_{A+B}x‖², so ∆_c ≤ 0 and ∆_d ≤ 0. This could have failed only if projection weights were not monotone in the subspace, and they are.

What carries them instead. The model adds a second ℝ⁴ and a fixed 2-dimensional “emergent concept” subspace C per pair, so that µ(A and B) = x²_AB + c₅² + c₆² and µ(A or B) = x²_AB + x²_AB′ + x²_A′B + c₅² + c₆² (eqs. 61–62). C is not A ∩ B or A + B. With it, every item in Hampton’s lists gets a vector (Tables 3–4).

Where negation enters. Only in §3.3. The conjunction model for A, B is the disjunction model for “hypothetical concepts” A′, B′ with µ(A′) = 1 − µ(A), µ(B′) = 1 − µ(B) and µ(A′ or B′) = 1 − µ(A and B) (eqs. 119–125). A′ is the orthogonal complement of A, and the emergent subspace of “A′ or B′” is the complement of C in the second sector. So the model is closed under De Morgan with the orthocomplement as negation, by construction.

What this does not say

• That concepts form an orthocomplemented lattice whose negation was tested. No negated concept is measured. The orthocomplement in §3.3 is a calculational device, and its weight 1 − µ is assumed.
• That the emergent-concept model is right. It is a fit: one free parameter per item, two angles per pair, fixed by a stipulated “modularity condition” (eq. 117; unrelated to the modular law) and a halfway rule for classical items. No held-out prediction is made. The two-layer reading of thought is an interpretation.
• That the impossibility is special to subspaces, or to the orthocomplement. This is the record’s inference, not the paper’s statement: the proof of Theorem 9 uses only that the weight is order-preserving and that “and” is a lower bound of both concepts. So the same follows for a Boolean algebra, an orthomodular lattice or a Heyting algebra with any monotone valuation. That generalisation is checked in the note of 2026-10-19 below.
• That the data are contextual. Non-classical here means no single Kolmogorovian space in which “A and B” is the intersection event. Drop that identification, as Contextuality-by-Default does (LIT-264), and the criterion changes.

Connections

No relations are declared.

• QUESTION-055. The question asks which negation a non-Boolean lattice of communicative categories carries. This account adds a constraint that the question does not state: if categories blend as Hampton’s concepts do, with a blend counted as a better member than one of its parts, the blend cannot be the lattice meet under any of the three negations. It does not choose among them, and so is not declared as an answer.
• CLAIM-119. Its concept lattice would need a combination operation other than the meet to hold such blends. CLAIM-038’s context-indexed object, and the Heyting logic of THEORY-037, face the same limit, because their meets are monotone too.
• THEORY-041. Aerts’s “modularity condition” is not the modular law; the finite-dimensional subspace lattice he uses is modular for the ordinary reason.
• CLAIM-041 and THEORY-013. Consistent with both: this is a claim about which representations fit, not a demonstration of contextuality.
• LIT-340, LIT-312 (NOTE-294, NOTE-299). Aerts and Gabora posit an orthocomplemented lattice of contexts and use a commuting, Boolean model. This paper is where non-commuting subspaces are actually put to work on concept data, and it confines them to the k-type items.

Note of 2026-10-19: the generalisation, checked

Written for the promotion audit’s batch 018 (record/promotion-audit/decisions/THEORY-255.yaml).

1. Let (L, ≤) be any lattice and µ: L → [0, 1] any order-preserving weight. No negation, complement or distributive law is assumed.
2. A ∧ B ≤ A and A ∧ B ≤ B, so µ(A ∧ B) ≤ min(µ(A), µ(B)): ∆_c ≤ 0 when “and” is the meet. No overextension.
3. A ≤ A ∨ B and B ≤ A ∨ B, so µ(A ∨ B) ≥ max(µ(A), µ(B)): ∆_d ≤ 0 when “or” is the join. No underextension.
4. Theorem 9 is the case of subspaces of ℝⁿ with µ(S) = ‖P_S x‖². That weight is order-preserving, because S ⊆ T gives P_S = P_S P_T and so ‖P_S x‖ ≤ ‖P_T x‖.
5. So a model that reproduces Cuckoo (µ(Pet) = 0.575, µ(Bird and Pet) = 0.842) must drop conjunction-as-meet or the monotone weight. Aerts drops the first, with an emergent subspace.
6. The k-type clause is possible with the meet kept. In ℝ², take lines A and B at angle θ < 90°, so A ∧ B = {0}, and x the unit vector on their bisector. Then µ(A) = µ(B) = cos²(θ/2) > ½ and µ(A ∧ B) = 0, so k_c = 1 − 2cos²(θ/2) < 0. Aerts’s fitted pair (Machine, Vehicle at θ = 80.9026° in ℝ⁴) is an instance. That every k-type item in Hampton’s data can be fitted this way is not shown: NOTE-808 grades it moderate, worked for one pair.

Note of 2026-10-20: the wider claim, filed Proposed

Under ADR-047 v4 the narrowing-loss sweep (batch 070) filed THEORY-807 as Proposed, extending this entry. It carries Aerts’s emergence reading, which this entry’s title carried before batch 018 narrowed it to what Theorem 9 forces: that the guppy-type deviations arise because a combined concept is partly a new, emergent concept, with thought running in a classical and a quantum layer. Theorem 9 alone forces only that conjunction is not the meet or that the weight is not monotone, which is why the reading is a separate, Proposed thesis. This entry is its evidence and is not changed by the filing.