**Contextualizing Concepts**

**Liane Gabora and Diederik Aerts**

Center Leo Apostel for Interdisciplinary
Studies (CLEA)

Free University of Brussels (VUB)

Krijgskundestraat 33, Brussels

B1160, Belgium, EUROPE

lgabora@vub.ac.be
, diraerts@vub.ac.be

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
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.

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).

Aerts, D. 1991. A mechanistic classical laboratory situation violating
the Bell inequalities with 2sqrt(2), exactly 'in the same way’ as its violations
by the EPR experiments. *Helvetica Phyica Acta* 64: 1-23.

Aerts, D. 1993a. Quantum Structures Due to Fluctuations of the Measurement
Situations. *International Journal of Theoretical Physics* 32:2207–2220.

Aerts, D. 1993b. Classical Theories and Nonclassical Theories as a Special
Case of a more General Theory. *Journal of Mathematical Physics* 24:
2441-2453.

Aerts, D., Aerts, S., Broekaert, J. & Gabora, L. 2000a. The Violation
of Bell Inequalities in the Macroworld. *Foundations of Physics* 30:1387—1414.

Aerts, D., Broekaert, J. & Gabora, L. 1999. Nonclassical Contextuality
in Cognition: Borrowing from Quantum Mechanical approaches to Indeterminism
and Observer Dependence. In (R. Campbell, Ed*.) Dialogues Proceedings
of Mind IV Conference*, Dublin, Ireland.

Aerts, D., Broekaert, J. & Gabora, L. 2000b. Intrinsic Contextuality
as the Crux of Consciousness. In (K. Yasue, Ed.) *Fundamental Approaches
to Consciousness*. John Benjamins Publishing Company.

Aerts, D. & Durt, T. 1994a. Quantum, Classical and Intermediate:
A Measurement Model. In *Proceedings of the International Symposium on
the Foundations of Modern Physics 1994*, Helsinki, Finland, eds. C.
Montonen *et al*., Editions Frontieres, Gives Sur Yvettes, France.

Aerts, D., Colebunders, E., Van der Voorde, A. & Van Steirteghem,
B. 1999. State Property Systems and Closure Spaces: a Study of Categorical
Equivalence. *International Journal of Theoretical Phys*ics 38:359-385.

Aerts, D. & Durt, T. 1994b. Quantum, Classical, and Intermediate,
an Illustrative Example. *Foundations of Physics* 24:1353-1368.

Aerts, D., D’Hondt, E. & Gabora, L. 2000c. Why the Disjunction in
Quantum Logic is Not Classical. *Foundations of Physics* 30:1473—1480.

Bell, J. 1964. On the Einstein Podolsky Rosen Paradox, *Physics*
1:195.

Boden, M. 1991. *The Creative Mind: Myths and Mechanisms*. Cambridge
UK: Cambridge University Press.

Foulis, D., Piron C., & Randall, C. 1983. Realism, Operationalism
and Quantum Mechanics. *Foundations of Physics* 13(813).

Foulis, D. & Randall, C. 1981. What are Quantum Logics and What
Ought They to Be?" In *Current Issues in Quantum Logic, *35, eds.
Beltrametti, E. & van Fraassen, B., New York NY: Plenum Press.

Gabora, L. 2001. *Cognitive mechanisms underlying the origin and evolution
of culture*. PhD. Diss., CLEA, Free University of Brussels.

Gerrig, R. & Murphy, G. 1992. Contextual Influences on the Comprehension
of Complex concepts. *Language and Cognitive Processes* 7:205-230.

Hampton, J. 1987. Inheritance of Attributes in Natural Concept Conjunctions.
*Memory
& Cognition* 15:55-71.

Heit, E. & Barsalou, L. 1996. The Instantiation Principle in Natural
Language Categories. *Memory* 4:413-451.

James, W. (1890/1950) *The Principles of Psychology*. New York:
Dover.

Jauch, J. 1968. *Foundations of Quantum Mechanics*, Reading Mass:
Addison-Wesley.

Johnson-Laird, P. N. 1983. *Mental Models*. Cambridge MA: Harvard
University Press.

Mackey, G. 1963. *Mathematical Foundations of Quantum Mechanics*.
Reading Mass: Benjamin.

Medin, D., Altom, M., & Murphy, T. 1984. Given versus Induced Category
Representations: Use of Prototype and Exemplar Information in Classification.
*Journal
of Experimental Psychology: Learning, Memory, and Cognition* 10:333-352.

Medin, D., & Shoben, E. 1988. Context and Structure in Conceptual
Combinations. *Cognitive Psychology* 20:158-190.

Murphy, G. & Medin, D. 1985. The Role of Theories in Conceptual
Coherence. *Psychological Review*, 92, 289-316.

Neisser, U. 1963. The Multiplicity of Thought. *British Journal of
Psychology* 54: 1-14.

Nosofsky, R. 1988. Exemplar-based Accounts of Relations between Classification,
Recognition, and Typicality. *Journal of Experimental Psychology: Learning,
Memory, and Cognition* 14:700-708.

Nosofsky, R. 1992. Exemplars, Prototypes, and Similarity Rules. In *From
Learning Theory to Connectionist Theory: Essays in Honor of William K.
Estes* 1:149-167. A. Healy, S. Kosslyn, & R. Shiffrin, eds. Hillsdale
NJ: Lawrence Erlbaum.

Osherson, D. & Smith, E. 1981. On the Adequacy of Prototype Theory
as a Theory of Concepts. *Cognition* 9:35-58.

Piaget, J. (1926) The Language and Thought of the Child. London: Routledge & Kegan Paul.

Piron, C. 1976. *Foundations of Quantum Physics*, Reading Mass.:
W. A. Benjamin.

Piron, C. 1989. Recent Developments in Quantum Mechanics. *Helvetica
Physica Acta*, 62(82).

Piron, C. 1990. *Mécanique Quantique: Bases et Applications*,
Press Polytechnique de Lausanne, Lausanne, Suisse.

Pitowsky, I. 1989. Quantum Probability - Quantum Logic, *Lecture Notes
in Physics 321*, Berlin: Springer.

Randall, C. & Foulis, D. 1976. A Mathematical Setting for Inductive
Reasoning, in *Foundations of Probability Theory, Statistical Inference,
and Statistical Theories of Science III*, 169, ed. Hooker, C. Dordrecht:
Reidel.

Randall, C. and Foulis, D. 1978. The Operational Approach to Quantum
Mechanics, in *Physical Theories as Logico-Operational Structures*,
C. Hooker, ed. Dordrecht, Holland Reidel, 167.

Reed, S. 1972. Pattern Recognition and Categorization. *Cognitive
Psychology* 3:382-407.

Riegler, A. Peschl, M. & von Stein, A. 1999. *Understanding Representation
in the Cognitive Sciences*. Dortrecht Holland: Kluwer.

Rosch, E. 1975. Cognitive Reference Points. *Cognitive Psychology*
7:532-547.

Rosch, E. 1978. Principles of Categorization. In *Cognition and Categorization*,
E. Rosch & B. Lloyd, Eds., 27-48. Hillsdale, NJ: Erlbaum.

Rosch, E. 1983. Prototype Classification and Logical Classification:
The two systems. In *New Trends in Conceptual Representation: Challenges
to Piaget’s Theory?* E. K. Scholnick, Ed., 73-86. Hillsdale NJ: Erlbaum.

Rosch, E., & Mervis, C. 1975. Family Resemblances: Studies in the
Internal Structure of Categories. *Cognitive Psychology* 7:573-605.

Rosch, E. 1999. Reclaiming Concepts. *Journal of Consciousness Studies*
6(11).

Sloman, S. 1996. The Empirical Case for Two Systems of Reasoning. *Psychological
Bulletin* 9(1):3-22.

Smith, E., & Medin, D. 1981. *Categories and Concepts*. Cambridge
MA: Harvard University Press.

Storms, G., De Boeck, P., Hampton, J., & Van Mechelen, I. 1999.
Predicting conjunction typicalities by component typicalities. *Psychonomic
Bulletin and Review* 4:677-684.

Storms, G., De Boeck, P., & Ruts, W. 2000. Prototype and Exemplar
Based Information in Natural Language Categories. *Journal of Memory
and Language* 42:51-73.

Storms, G., De Boeck, P., Van Mechelen, I. & Ruts, W. 1996. The
Dominance Effect in Concept Conjunctions: Generality and Interaction Aspects.
*Journal
of Experimental Psychology: Learning*, *Memory & Cognition*
22:1-15.

Storms, G., De Boeck, P., Van Mechelen, I. & Ruts, W. 1998. Not
Guppies, nor Goldfish, but Tumble dryers, Noriega, Jesse Jackson, Panties,
Car crashes, Bird books, and Stevie Wonder. *Memory and Cognition*
26:143-145.

Sutcliffe, J. 1993. Concepts, Class, and Category in the Tradition of
Aristotle. In I. Van Mechelen, J. Hampton, R. Michalski, & P. Theuns
(Eds.), *Categories and Concepts: Theoretical Views and Inductive Data
Analysis*, 35-65. London: Academic Press.

Verbeemen, T., Vanoverberghe, V., Storms, G., and Ruts, W. 2002. The Role of Contrast Categories in Natural Language Concepts. Forthcoming.

Wisniewski, E. 1997. When Concepts Combine. *Psychonomic Bulletin
& Review* 4:167-183.

Wisniewski, E. & Gentner, D. 1991. On the combinatorial semantics
of noun pairs: Minor and major adjustments. In G. B. Simpson (Ed.), *Understanding
Word and Sentence*, 241-284. Amsterdam: Elsevier.