Theories of concepts have by and large been representational theories. By this we mean that concepts are seen to take the form of fixed mental representations, as opposed to being constructed, or ‘re-constructed’, on the fly through the interaction between the cognitive state and the situation or context.
For theoretical and empirical reasons [1], representational theories are not the most adequate setup to embed the desired notion of basic unit of thought in the definition of what is a concept. The drawbacks of current theories of concepts reside in the idea of imagine the reality of concepts as a fixed (Cartesian) one: where there is a unique measure of similarity for concepts and the combination of two or more different concepts is generated assuming they are independent.

Limitations of Previous Approaches

Context dependence when measuring concepts: There have been established many notions to relate a concept with the possible elements that belong to it: Typicality, Membership, Representativeness, etc. On the other hand, in order to study how we associate concepts in meaningful ways, different notions of conceptual similarity have been established. However, all these notions have been proved to be context dependent, i.e. the outcome of the quantities above mentioned notably vary when the concept(s) is (are) considered in different situations. For example, in figure 1, we see a table where a typicality value (ranged from 0 to 7) is assigned to the exemplar 'Cowboy hat' of the concept HAT, in 5 different contexts. The considered contexts are: The hat, Worn to be funny, Worn for protection, Worn in the south, Not worn by a person.   

Fig 1. Typicality of concept's exemplar is context dependent

Concept combination: PET-FISH problem: In 1981, Osherson and Smith considered the conceptual combination of Pet and Fish into the conjunction Pet-Fish, and measured the typicality of different exemplars with respect to the concepts Pet, Fish and their conjunction Pet-Fish [6]. They observed that exemplars such as Guppy and Goldfish give rise to a typicality with respect to the conjunction Pet-Fish which is much bigger than would be expected if the conjunction of Pet and Fish were treated from the perspective of classical fuzzy set theory. Indeed, from a classical perspective, i.e. when modeled by fuzzy set theory applying the maximum rule for conjunction, the typicality of an exemplar with respect to the conjunction of two concepts should not exceed the maximum of the typicalities of this exemplar with respect to both concepts apart. This is, however, what  happens in the case of the conjunction Pet-Fish, i.e. the typicality of Guppy with respect to Pet-Fish is estimated by subjects to be bigger than with respect to Pet as well as with respect to Fish. The effect present in the Pet-Fish problem is often referred to as the Guppy effect.


The Quantum Approach

The quantum approach for concepts assumes that context influence actively the meaning of a concept. Indeed, we assume that meaning emerges from the interaction between concept and context. When no contextual influence applies to the concept, we assume the concept is in a potentiality state formed by the superposition of multiple possible states, and the interaction of the concept with a particular context makes it to collapse to a certain actual state, where its properties and potential associations are determined. The transition from the potential to the actual state is a probabilistic process [1-5].

This is possible in a quantum theoretical setup as follows: We assume states of a concept are unitary vectors in a Hilbert space and contextual influence is modeled by a linear operator acting on this Hilbert space [3].

The contextual influence is modeled by the application of the corresponding linear operator to the concept state, inducing the collapse of the potential states into one of the context-dependent eigen-states. This approach is inherently context-dependent, and solves the inconsistencies on similarity functions [2-4].

Furthermore, the PET-FISH problem and other concept combination effects [5] can be solved by assuming the concept combination in a quantum theoretical setup. Concepts when joined, can exhibit interference and entanglement properties. This phenomena permit explain the unexpected situations observed in the experiments.


References

1. Gabora, L. & Aerts, D. Contextualizing concepts using a mathematical generalization of the quantum formalism. J. Theor. Artif. Intell., 14: 327–358, 2002.
2. Aerts, D. and Gabora, L. A state-context-property model of concepts and their combinations I: The structure of the sets of contexts and properties. Kybernetes, 34(1/2): 151 - 175, 2005.
3. Aerts, D. and Gabora, L. A state-context-property model of concepts and their combinations II: A Hilbert space representation. Kybernetes, 34(1/2): 176 - 204, 2005.
4. Gabora, L. & Aerts, D. A model of the emergence and evolution of integrated worldviews. Journal of Mathematical Psychology, 53: 434 - 451, 2009.
5. Aerts, D. Quantum structure in cognition. Journal of Mathematical Psychology, 53: 314- 348, 2009.
6. Osherson, D. N., & Smith, E. E. On the adequacy of prototype theory as a theory of concepts. Cognition, 9: 35 - 58, 1981.