Liane Gabora and Diederik Aerts
Center Leo Apostel for Interdisciplinary Studies (CLEA)
Free University of Brussels (VUB)
Krijgskundestraat 33, Brussels
B1160, Belgium, EUROPE
firstname.lastname@example.org , email@example.com
http://www.vub.ac.be/CLEA/liane/ , http://www.vub.ac.be/CLEA/aerts/
ABSTRACT: The mathematics of quantum mechanics was developed to cope with problems arising in the description of (1) contextual interactions, and (2) the generation of new states with new properties when particles become entangled. Similar problems arise with concepts. This paper summarizes the rationale for and preliminary results of using a generalization of standard quantum mechanics based on the lattice formalism to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. Concepts are viewed not as fixed representations but dynamically ‘re-constructed’ entities generated on the fly through interaction between cognitive state and situation or context..
Representational theories have met with some success. However increasingly, for both theoretical and empirical reasons, they are coming under fire (e.g. Riegler, Peschl and von Stein 1999; Rosch 1999). As Rosch 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 manner 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). This contextuality is the reason why representational theories cannot describe or predict what happens when two or more concepts arise together, or follow one another, as in the creative generation or interpretation of conjunctions of concepts.
This paper suggests how formalisms designed to cope with context and conjunction in the microworld may be adapted to the formal description of concepts. In this contextual theory, 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.
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 when it comes to conjunctions. 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 and 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).
The generation of conjunctions is even more problematic. 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, the state space of the joint entity is the Cartesian product space X1 x 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. These issues are hinted to by Boden (1990), who 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.
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
First we describe the pure quantum formalism, and then we briefly describe the generalization of it that we apply to the description of concepts.
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., H1ÄH2. 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.
The motivation behind these general formalisms was purely mathematical. They describe much more than is needed for quantum mechanics, and in fact, standard quantum mechanics and classical mechanics fall out as special cases (Aerts 1983b). It is slowly being realized that they have relevance to the macroscopic world (e.g. Aerts 1991; Aerts et al. 2000), and that they can 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.
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 or generalized quantum) formalism is necessary.
We now present three sources of theoretical evidence of the utility of the approach.
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, Broekaert, and Gabora 2000).
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