Concept Combinations in a Quantum-Theoretic Framework

which consists of two sectors: ‘sector 1’ is a Hilbert space $\mathcal{H}$ , while ‘sector 2’ is a tensor product Hilbert space $\mathcal{H}\otimes \mathcal{H}$ .

Let us now consider the membership weights of exemplars of concepts and their conjunctions/disjunctions measured by Hampton [1, 2]. He identified systematic deviations from classical set (fuzzy set) conjunctions/disjunctions, an effect known as ‘overextension’ or ‘underextension’.

Let us start from conjunctions. It can be shown that a large part of Hampton’s data cannot be modeled in a classical probability space satisfying the axioms of Kolmogorov [5]. Indeed, the membership weights $\mu _{x}(A),\mu _{x}(B)$ and μ _x(A and B) of an exemplar x for the concepts A, B and ‘A and B’ can be represented in a classical probability model if and only if the following two conditions are satisfied (see [5] for a proof)

$\displaystyle\begin{array}{rcl} \varDelta _{x}^{c}& =& \mu _{ x}(A\ \mathrm{and}\ B) -\min (\mu _{x}(A),\mu _{x}(B)) \leq 0{}\end{array}$

(1)

$\displaystyle\begin{array}{rcl} 0 \leq k_{x}^{c}& =& 1 -\mu _{ x}(A) -\mu _{x}(B) +\mu _{x}(A\ \mathrm{and}\ B){}\end{array}$

(2)

Let us consider a specific example. Hampton estimated the membership weight of Mint with respect to the concepts Food, Plant and their conjunction Food and Plant finding μ _Mint(Food) = 0. 87, μ _Mint(Plant) = 0. 81, μ _Mint(Food and Plant) = 0. 9. Thus, the exemplar Mint presents overextension with respect to the conjunction Food and Plant of the concepts Food and Plant. We have in this case $\varDelta _{x}^{c} = 0.09\nleq 0$ , hence no classical probability model exists for these data.

Let us now come to disjunctions. Also in this case, a large part of Hampton’s data [2] cannot be modeled in a classical Kolmogorovian probability space, due to the following theorem. The membership weights μ _x(A), μ _x(B) and μ _x(A or B) of an exemplar x for the concepts A, B and ‘A or B’ can be represented in a classical probability model if and only if the following two conditions are satisfied (see [5] for a proof)

$\displaystyle\begin{array}{rcl} \varDelta _{x}^{d}& =& \max (\mu _{ x}(A),\mu (_{x}B)) -\mu _{x}(A\ \mathrm{or}\ B) \leq 0{}\end{array}$

(3)

$\displaystyle\begin{array}{rcl} 0 \leq k_{x}^{d}& =& \mu _{ x}(A) +\mu _{x}(B) -\mu _{x}(A\ \mathrm{or}\ B){}\end{array}$

(4)

Let us again consider a specific example. Hampton estimated the membership weight of Donkey with respect to the concepts Pet, Farmyard Animal and their disjunction Pet or Farmyard Animal finding μ _Donkey(Pet) = 0. 5, μ _Donkey(Farmyard Animal) = 0. 9, μ _Donkey(Pet or Farmyard Animal) = 0. 7. Thus, the exemplar Donkey presents underextension with respect to the disjunction Pet or Farmyard Animal of the concepts Pet and Farmyard Animal. We have in this case $\varDelta _{{\it \text{x}}}^{{\it \text{d}}} = 0.2\nleq 0$ , hence no classical probability model exists for these data.

It can be proved that a quantum probability model in Fock space exists for Hampton’s data, as follows [5, 12, 13].

Let us start from the conjunction of two concepts. Let x be an exemplar and let μ _x(A), μ _x(B), μ _x(A and B) and μ _x(A or B) be the membership weights of x with respect to the concepts A, B, ‘A and B’ and ‘A or B’, respectively. Let $\mathcal{F} = \mathcal{H}\oplus (\mathcal{H}\otimes \mathcal{H})$ be the Fock space where we represent the conceptual entities. The concepts A, B and ‘A and B’ are represented by the unit vectors $\vert A_{c}(x)\rangle$ , $\vert B_{c}(x)\rangle$ and $\vert (A\ \mathrm{and}\ B)_{c}(x)\rangle$ , respectively, where

$\displaystyle\begin{array}{rcl} \vert (A\ \mathrm{and}\ B)_{c}(x)\rangle & = & m_{c}(x)e^{i\lambda _{c}(x)}\vert A_{ c}(x)\rangle \otimes \vert B_{c}(x)\rangle \\ & & + n_{c}(x)e^{i\nu _{c}(x)} \frac{1} {\sqrt{2}}(\vert A_{c}(x)\rangle + \vert B_{c}(x)\rangle ){}\end{array}$

(5)

The numbers m _c(x) and n _c(x) are such that $m_{c}(x),n_{c}(x) \geq 0$ and $m_{d}^{2}(x)\! +\! n_{c}^{2}(x) = 1$ . The decision measurement of a subject who estimates the membership of the exemplar x with respect to the concept ‘A and B’ is represented by the orthogonal projection operator $M_{c} \oplus (M_{c} \otimes M_{c})$ on $\mathcal{F}$ , where M _cis an orthogonal projection operator on $\mathcal{H}$ . Hence, the membership weight of x with respect to ‘A and B’ is given by

$\displaystyle\begin{array}{rcl} & & \qquad \mu _{x}(A\ \mathrm{and}\ B) =\langle (A\ \mathrm{and}\ B)_{c}(x)\vert M_{c} \oplus (M_{c} \otimes M_{c})\vert (A\ \mathrm{and}\ B)_{c}(x)\rangle \\ & & = m_{c}^{2}(x)\mu _{ x}(A)\mu _{x}(B) + n_{c}^{2}(x)\Big(\frac{\mu _{x}(A) +\mu _{x}(B)} {2} + \mathfrak{R}\langle A_{c}(x)\vert M_{c}\vert B_{c}(x)\rangle \Big){}\end{array}$