ABSTRACT: To cope with problems arising in the description of (1) contextual interactions, and (2) the generation of new states with new properties when quantum entities become entangled, the mathematics of quantum mechanics was developed. Similar problems arise with concepts. We use a generalization of standard quantum mechanics, the mathematical lattice theoretic formalism, to develop a formal description of the contextual manner in which concepts are evoked, used, and combined to generate meaning.
 

1. The Problem of Conjunctions
According to the classical theory of concepts, there exists for each concept a set of defining features that are singly necessary and jointly sufficient (e.g. Sutcliffe 1993). Extensive evidence has been provided against this theory (see Komatsu 1992 and Smith & Medin 1981 for overviews). Two major alternatives have been put forth. According to the prototype theory (Rosch, 1975a, 1978, 1983; Rosch and Mervis 1975), concepts are represented by a set of, not defining, but characteristic features, which are weighted in the definition of the prototype. A new item is categorized as an instance of the concept if it is sufficiently similar to this prototype. According to the exemplar theory, (e.g., Heit and Barsalou 1996; Medin et al. 1984; Nosofsky 1988, 1992) a concept is represented by, not defining or characteristic features, but a set of instances of it stored in memory. A new item is categorized as an instance of a concept if it is sufficiently similar to one or more of these previous instances. We use the term representational theories to refer to both prototype and exemplar theories since concepts take the form of fixed representations (as opposed to changing according to context).

Representational theories are adequate for predicting experimental results for many dependent variables including typicality ratings, latency of category decision, exemplar generation frequencies, and category naming frequencies. However, they run into problems trying to account for the creative generation of, and membership assessment for, conjunctions of concepts. They cannot account for phenomena such as the so-called guppy effect, where guppy is not rated as a good example of pet, nor of fish, but it is rated as a good example of pet fish (Osherson & Smith 1981). This is problematic because if (1) activation of pet does not cause activation of guppy, and (2) activation of fish does not cause activation of guppy, how is it that (3) pet fish, which activates both pet AND fish, causes activation of guppy? (In fact, it has been demonstrated experimentally that other conjunctions are better examples of the ‘guppy effect’ than pet fish (Storms et al. 1998), but since the guppy example is well-known we will continue to use it here as an example.)

Zadeh (1965, 1982) tried, unsuccessfully, to solve the conjunction problem using a minimum rule model, where the typicality of an item as a conjunction of two concepts (conjunction typicality) equals the minimum of the typicalities of the two constituents. Storms et al. (2000) showed that a weighted and calibrated version of this model can account for a substantial proportion of the variance in typicality ratings for conjunctions exhibiting the guppy effect, suggesting the effect could be due to the existence of contrast categories. However, another study provided negative evidence for contrast categories (Verbeemen et al., in press).

Conjunction cannot be described with the mathematics of classical physical theories because it only allows one to describe a composite or joint entity by means of the product state space of the state spaces of the two subentities. Thus if X₁ is the state space of the first subentity, and X₂ the state space of the second subentity, the state space of the joint entity is the Cartesian product space X₁ * X₂. So if the first subentity is ‘door’ and the second is ‘bell’, one can give a description of the two at once, but they are still two. The classical approach cannot even describe the situation wherein two entities generate a new entity that has all the properties of its subentities, let alone a new entity with certain properties of one subentity and certain of the properties of the other. The problem can be solved ad hoc by starting all over again with a new state space each time there appears a state that was not possible given the previous state space. However, in so doing we fail to include exactly those changes of state that involve the generation of novelty. Another possibility would be to make the state space infinitely large to begin with. However, since we hold only a small number of items in mind at any one time, this is not a viable solution to the problem of describing what happens in cognition.

2. The Problem of Contextuality
The problems that arise with conjunctions reflects a more general problem with representational theories (see Riegler, Peschl and von Stein 1999 for overview). As Rosch (1999) puts it, they do not account for the fact that concepts "have a participatory, not an identifying function in situations"; that is, they cannot explain the contextual way in which concepts are evoked and used (see also Gerrig and Murphy 1992; Hampton, 1987; Komatsu, 1992; Medin and Shoben, 1988; Murphy and Medin, 1985). Not only does a concept give meaning to a stimulus or situation, but the situation evokes meaning in the concept, and when more than one is active they evoke meaning in each other. It is this contextuality that makes them difficult to model when two or more arise together, or follow one another, as in a creative construction such as a conjunction, invention, or sentence.

3. The Formalism
This story has a precedent. The same two problems–that of conjunctions of entities, and that of contextuality–arose in physics in the last century. Classical physics could not describe what happens when quantum entities interact. According to the dynamical evolution described by the Schrödinger equation, quantum entities spontaneously enter an entangled state that contains new properties the original entities did not have. To describe the birth of new states and new properties it was necessary to develop the formalism of quantum mechanics.

The shortcomings of classical mechanics were also revealed when it came to describing the measurement process. It could describe situations where the effect of the measurement was negligible, but not situations where the measurement intrinsically influenced the evolution of the entity; it could not incorporate the context generated by a measurement directly into the formal description of the quantum entity. This too required the quantum formalism.

First we describe the pure quantum formalism, and then we briefly describe the generalization of it that we apply to the description of concepts.

3.1 Pure Quantum Formalism
As in any mathematical model, we begin by cutting out a piece of reality and say this is the entity of interest, and these are its properties. The set of actual properties constitute the state of the entity. We also define a state space, which delineates, given how the properties can change, the possible states of the entity. A quantum entity is described using not just a state space but also a set of measurement contexts. The algebraic structure of the state space is given by the vector space structure of the complex Hilbert space: states are represented by unit vectors, and measurement contexts by self-adjoint operators.

One says a quantum entity is entangled if it is a composite of subentities that can only be individuated by a separating measurement. When such a measurement is performed on an entangled quantum entity, its state changes probabilistically, and this change of state is called quantum collapse.

In pure quantum mechanics, if H₁ is the Hilbert space representing the state space of the first subentity, and H₂ the Hilbert space representing the state space of the second subentity, the state space of the composite is not the Cartesian product, as in classical physics, but the tensor product, i.e., H₁ Ä H₂. The tensor product always generates new states with new properties, specifically the entangled states. Thus it is possible to describe the spontaneous generation of new states with new properties. However, in the pure quantum formalism, a state can only collapse to itself with a probability equal to one; thus it cannot describe situations of intermediate contextuality.

3.2 Generalized Quantum Formalism
The standard quantum formalism has been generalized, making it possible to describe entities with any degree of contextuality (Aerts 1993; Aerts and Durt 1994a, 1994b; Foulis and Randall 1981; Foulis et al. 1983; Jauch 1968; Mackey 1963; Piron 1976, 1989, 1990; Pitowsky 1989; Randall and Foulis 1976, 1978). The generalized formalisms use lattice theory to describe the states and properties of physical entities, and the result is referred to as a state property system. The approach is sufficiently general to be used to describe the different context-dependent states in which a concept can exist, and the features of the concept manifested in these various states.

4. Incorporating Contextuality into a Theory of Concepts
One of the first applications of these generalized formalisms to cognition was modeling the decision making process. Aerts and Aerts (1996) proved that in situations where one moves from a state of indecision to a decided state (or vice versa), the probability distribution necessary to describe this change of state is non-Kolmogorovian, and therefore a classical probability model cannot be used. Moreover, they proved that such situations can be accurately described using these generalized quantum mathematical formalisms. Their mathematical treatment also applies to the situation where the state of the mind changes from thinking about a concept to an instantiation of that concept, or vice versa. Once again, context induces a nondeterministic change of the state of the mind which introduces a non-Kolmogorivian probability on the state space. Thus, a nonclassical (quantum) formalism is necessary.

In our approach, concepts are described using what to a first approximation can be viewed as an entangled states of exemplars, though this is not precisely accurate. For technical reasons (see Gabora 2001), the term potentiality state is used instead of entangled state. For a given stimulus, the probability that a potentiality state representing a certain concept will, in a given context, collapse to another state representing another concept is related to the algebraic structure of the total state space, and to how the context is represented in this space. The state space where concepts ‘live’ is not limited a priori to only those dimensions which appear to be most relevant; thus concepts retain in their representation the contexts in which they have, or even could potentially be, evoked or collapsed to. It is this that allows their contextual character to be expressed. The stimulus situation plays the role of the measurement context by determining which state is collapsed upon. Stimuli are categorized as instances of a concept not according to how well they match a static prototype or set of typical exemplars, but according to the extent to which they correspond to, and thereby actualize or collapse upon, one the potential interpretations of the concept. (As a metaphorical explanatory aid, if concepts were apples, and the stimulus a knife, then the qualities of the knife determine not just which apple to slice, but which direction to slice through it: changing the context in which a stimulus situation is embedded can cause a different version of the concept to be elicited.) This approach has something in common with both prototype and exemplar theories. Like exemplar theory, concepts consist of exemplars, but the exemplars are in a sense ‘woven together’ like a prototype.

5. Preliminary Theoretical Evidence of the Utility of the Approach
We present three sources of theoretical evidence of the utility of the approach.

5.1 A Proof that Bell Inequalities can be Violated by Concepts
The presence of entanglement can be tested for by determining whether correlation experiments on the joint entity violate Bell inequalities (Bell 1964). Using an example involving the concept cat and specific instances of cats we proved that Bell inequalities are violated in the relationship between a concept, and specific instances of this concept (Aerts et al. 2000a; Gabora 2001). Thus we have evidence that this approach indeed reflects the underlying structure of concepts.

5.2 Application to the Pet Fish Problem
We have applied the contextualized approach to the Pet Fish Problem (Aerts et al. 2000b; Gabora 2001). Conjunctions such as this are dealt with by incorporating context-dependency, as follows: (1) activation of pet still rarely causes activation of guppy, and likewise (2) activation of fish still rarely causes activation of guppy. But now (3) pet fish causes activation of the potentiality states petin the context of pet fish AND fish in the context of pet fish. Since for both, the probability of collapsing onto the state guppy is high, it is very likely to be activated. Thus we have a formalism for describing concepts that is not stumped by a situation wherein an entity that is neither a good instance of A nor B is nevertheless a good instance of A AND B.

Note that whereas in representational approaches relations between concepts arise through overlapping context-independent distributions, in the present approach, the closeness of one concept to another (expressed as the probability that its potentiality state will collapse to an actualized state of the other) is context-dependent. Thus it is possible for two states to be far apart from each other with respect to a one context (for example ‘fish’ and ‘guppy’ in the context of just being asked to name a fish), and close to one another with respect to another context (for example ‘fish’ and ‘guppy’ in the context of both ‘pet’ and being asked to name a ‘fish’). Examples such as this are evidence that the mind handles nondisjunction (as well as negation) in a nonclassical manner (Aerts et al. 2000b).

5.3 Describing Impossibilist Creativity
Boden (1990) uses the term impossibilist creativity to refer to creative acts that not only explore the existing state space but transform that state space. In other words, it involves the spontaneous generation of new states with new properties. In (Gabora 2001) the contextual lattice approach is used to generate a mathematical description of impossibilist creativity using as an example the invention of the torch. This example involves the spontaneous appearance of a new state (the state of mind that conceives of the torch) with a new property (the property of being able to move fire).

6. Research in Progress
We are comparing the performance of the contextualized theory of concepts with prototype and examplar theories using previous data sets for typicality ratings, latency of category decision, exemplar generation frequencies, category naming frequencies on everyday natural language concepts, such as ‘trees’, ‘furniture’, or ‘games’. The purpose of these initial investigations is to make sure that the proposed formalism is at least as successful as representational approaches for the simple case of single concepts. Assuming this to be the case, we will concentrate our efforts on conjunctions of concepts, since this is where the current approach is expected to supercede representational theories. We will re-analyze previously collected data for noun-noun conjunctions such as ‘pet fish’, and relative clause conjunctions such as ‘pets that are also fish’ (Storms et al. 1996). A new study is being prepared which will compare the proposed approach with representational approaches at predicting the results of studies using situations that are highly contextual. Typicality ratings for conjunctions will be compared with, not just their components, but with other conjunctions that share these components. (Thus, for example, does ‘brainchild’ share features with ‘childbirth’ or ‘brainstorm’? Does ‘brainstorm’ share features with ‘birdbrain’ or ‘sandstorm’?)

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