THEORY-545: Aerts and Gabora's founding quantum model of concepts is a classical probability model written in a Hilbert space: every context it uses is a projector diagonal in one basis, so the contexts commute and generate a Boolean algebra, its entangled 'pet fish' state gives the same statistics as a classically correlated mixture for every context in the model, and its guppy weights are copied from single-concept ratings rather than predicted for the conjunction

Source

• Aerts & Gabora (2005), A theory of concepts and their combinations II: A Hilbert space representation, LIT-312, read in full in NOTE-299.
• Aerts & Gabora (2005), A theory of concepts and their combinations I: The structure of the sets of contexts and properties, LIT-340, read in full in NOTE-294. Both appeared on arXiv on 26 February 2004.

What was actually shown

Part I’s data and structure. Eighty-one convenience subjects rated 14 exemplars and 14 properties of “pet” under seven contexts. The shifts are large: dog goes from 0.12 under the unit context to 0.50 under “chewing a bone”; goldfish from 0.10 to 0.48 under “is a fish” (Tables 1–5). Part I then proposes that contexts and properties each form a complete orthocomplemented lattice. Completeness is supposed (“We suppose”, §3.2), and the orthocomplement is stated by axioms and illustrated by negated sentences. Its one mark of non-classicality is eq. 19: the eigenstates of a join can exceed the union of the joined contexts’ eigenstates.

Part II’s model. Each of 1,400 “basic contexts” for “pet” (408 for “fish”) is an orthonormal basis vector, the ground state is their uniform superposition, and each context is the projector onto a set of basis vectors (eqs. 25, 30, 36, 43, 60). The counts n(E_i) and n(X_ij) are chosen so that their ratios match the rounded frequencies (§3.1). Every Born probability is then a ratio of counts, for example µ(goldfish | e6) = 48/100. The authors say the count table “corresponds more or less to a set theoretical model of the experimental data, such that the Hilbert space model can considered to be a quantization … of this set theoretical model”.

The checks the readings made. NOTE-299 and NOTE-294 establish four facts about the model as published.

1. Commutative. Every projector used is diagonal in the basis B, so all contexts commute and generate a Boolean algebra. No incompatible context, order effect or interference term occurs.
2. Entanglement idle. The “pet fish” state Σ_u |u⟩⊗|u⟩/√100 (eq. 78) is used only with projectors diagonal in B⊗B. For all of them it gives the same statistics as the separable mixture Σ_u |u⟩⟨u|⊗|u⟩⟨u|/100.
3. Part I’s criterion is not a test. In any Hilbert space a family of commuting projectors generates a Boolean lattice, yet the eigenstates of P ∨ Q strictly exceed those of P and Q together. So eq. 19 cannot certify a non-Boolean logic of concepts.
4. Guppy imported. The guppy and goldfish weights in the compound, 0.46 and 0.48, are Part I’s ratings of “pet” under “The pet is a fish”. No typicality for the conjunction “pet fish” was measured, and the authors write of the single-concept version “Of course this is not the real guppy effect” (§5).

Two cells of the model are infeasible. n(X_ij) must not exceed n(E_i). Table 2 has 126 for parrot under e5 where n(E₁₇) = 98, and Table 5 has 39 for guppy where m(E₄₂) = 32. As printed neither model exists. NOTE-299 judges this a bookkeeping error, since the parrot row can be repaired.

What could have come out otherwise. The model could have used incompatible contexts, so that entangled and classically correlated states gave different predictions, or it could have predicted conjunction typicality out of sample. It does neither, so its fit could not have failed.

What this does not say

• That quantum models of concepts are classical. This is about the 2005 model. Aerts (2009) puts non-commuting subspaces to work on Hampton’s membership data, for the k-type items (THEORY-255).
• That the data show nothing. Context shifts the ratings of single exemplars and properties, and that is the experiment’s finding. Part I’s Table 5 has three p-values above 1 and four duplicated columns, and the full analysis is unpublished, so the significance claims cannot be checked from the paper.
• That the proposal is empty. Concepts as states changed by contexts, contexts as projectors, and the tensor product for combination are owned here, and Part II proposes tensor-product sentence meaning in 2004, before DisCoCat. They are proposals; the paper does not demonstrate them.
• That the Hilbert model reproduces property weights. Part II’s §5 says it does; no property projector is ever built.
• Anything about contextuality. “Contextual” in these papers means that the state changes with context, which shows up as shifting marginals. It is not Kochen–Specker contextuality, and the papers do not claim it.

Connections

• Contextuality-by-Default. Every effect in Part I’s data is a context-dependent marginal, which THEORY-013 separates from contextuality. The programme’s later CHSH claims (LIT-270, Rejected) rest on the same confusion.
• CLAIM-041 grounds its claim that a Hilbert-space representation alone does not establish non-classicality partly on LIT-312’s diagonal projectors; this account states that ground.
• The later model. THEORY-255 shows that in Aerts’s 2009 subspace model meet and join cannot produce the guppy-type deviations, and that the non-commuting structure is confined to the k-type items.
• Lattices. THEORY-041 records what Birkhoff and von Neumann’s modular law was; Part I never reaches modularity or orthomodularity.
• DisCoCat. The maximally correlated sentence state Σ|u⟩⊗|u⟩⊗|u⟩ is the image of the copy map of the basis, the Frobenius structure behind THEORY-017 and LIT-272. The connection is NOTE-299’s, not the paper’s.