Gabora, L., & Aerts, D. (2002). Contextualizing concepts using a
mathematical generalization of the quantum formalism. *Journal
of Experimental and Theoretical Artificial Intelligence, 14*(4), 327-358.

**Contextualizing Concepts
Using a Mathematical Generalization of the Quantum Formalism**

**Liane
M. Gabora and Diederik Aerts**

**ABSTRACT**

We outline the rationale
and preliminary results of using the state context property (SCOP) formalism,
originally developed as a generalization of quantum mechanics, to describe the
contextual manner in which concepts are evoked, used and combined to generate
meaning. The quantum formalism was developed to cope with problems arising in
the description of (i) the measurement process, and
(ii) the generation of new states with new properties when particles become
entangled. Similar problems arising with concepts motivated the formal
treatment introduced here. Concepts are viewed not as fixed representations,
but entities existing in states of potentiality that require interaction with a
context—a stimulus or another concept—to 'collapse' to an
instantiated form (e.g. exemplar, prototype, or other possibly imaginary
instance). The stimulus situation plays the role of the measurement in physics,
acting as context that induces a change of the cognitive state from
superposition state to collapsed state. The collapsed state is more likely to
consist of a conjunction of concepts for associative than analytic thought
because more stimulus or concept properties take part in the collapse. We
provide two contextual measures of conceptual distance—one using collapse
probabilities and the other weighted properties—and show how they can be
applied to conjunctions using the pet fish problem.

*Keywords: *analytic
thought, associative hierarchy, associative thought, collapse, conceptual
distance, focus/defocus, guppy effect, state space, superposition

**1. INTRODUCTION**

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 Ôre-constructedÕ,
on the fly through the interaction between the cognitive state and the
situation or context.

Representational theories
have met with some success. They 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, increasingly, for both
theoretical and empirical reasons, they are coming under fire (e.g. Riegler *et al.*
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 Murphy and Medin 1985, Hampton 1987, Medin
and Shoben 1988, Gerrig and
Murphy 1992, Komatsu 1992). 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. A concept's meaning shifts depending on what other concepts it
arises in the context of (Reed 1972, Storms *et
al.* 1996, 1999, Wisniewski 1991, 1997).

This paper shows how formalisms designed to cope with
context and conjunction in the microworld may be a
source of inspiration for a description of concepts. In this *contextualized 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.

**2. Limitations of representational approaches**

We begin by briefly summarizing some of the most
influential representational theories of concepts, and efforts
to delineate what a concept is with the notion of conceptual distance.
We then discuss difficulties encountered with representational approaches in
predicting membership assessment for conjunctions of concepts. We then show
that representational theories have even more trouble coping with the
spontaneous emergence or loss of features that can occur when concepts combine.

**2.1 . Theories
of concepts and conceptual distance**

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 (or overviews see Smith and Medin 1981, Komatsu 1992).

A number of alternatives have been put forth.
According to the *prototype theory *(Rosch 1975,
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. The prototype consists of a set of features {*a** _{1,} a_{2,} a_{3
...}a_{M}*}

The
smaller the value of *d _{s}*

According to the *exemplar theory, (e.g. *Medin *et al. *1984, Nosofsky
1988, 1992, Heit and Barsalou
1996) a concept is represented by, not defining or characteristic features, but
a set of *instances *of it stored in memory. Thus each of the {*E _{1}, E_{2}, E_{3},ÉE_{N}*} exemplars has a set {

Once
again, the smaller the value of ds for a given item, the more representative it
is of the concept.

Note
that these theories have difficulty accounting for why items that are
dissimilar or even opposite might nevertheless belong together; for example,
why **white **might be more likely to be categorized with **black **than
with **flat, **or why **dwarf **might be more likely to be categorized
with **giant **than with say, **salesman. **The only way out is to give
the set of relevant 'measurements' or contexts the same status
as features, i.e. to lump together as features not only things like 'large' but
also things like 'has a size' or 'degree to which size is relevant'.

According to another approach to concepts, referred to
as the *theory theory, *concepts take the form
of 'mini-theories' (e.g. Murphy and Medin 1985) or
schemata (Rummelhart and Norman 1988), in which the
causal relationships amongst features or properties are identified. A
mini-theory contains knowledge concerning both which variables or measurements
are relevant, and the values obtained for them. This does seem to be a step
toward a richer understanding of concept representation, though many
limitations have been pointed out (see for example Komatsu 1992, Fodor 1994,
Rips 1995). Clearly, the calculation of conceptual distance is less
straightforward, though to us this reveals not so much a shortcoming of the
theory theory, but of the concept of conceptual
distance itself. In our view, concepts are not distant from one another at all,
but interwoven, and this interwoven structure cannot be observed directly, but
only indirectly, as context-specific instantiations. For example, the concept **egg
**will be close to **sun **in the context 'sunny side up' but far in the
context 'scrambled', and in the context of the Dr Suess book *Green Eggs and Ham *it acquires the
feature 'green'.

Yet another theory of concepts, which captures their
mutable, context-dependent nature, but at the cost of increased vagueness, is *psychological
essentialism. *The basic idea is that instances of a concept share a hidden essence which defines its true nature (e.g. Medin and Ortony 1989). In this
paper we attempt to get at this notion in a more rigorous and explicit way than
has been done.

*2.2.
Membership assessments for conjunctive
categories*

The
limitations of representational theories became increasingly evident through
experiments involving conjunctions of concepts. One such anomalous phenomenon
is the so-called *guppy effect, *where a guppy is *not *rated as a
good example of the concept **pet, **nor of the concept **fish, **but it *is
*rated as a good example of **pet fish **(Osherson
and Smith 1981).^{2} Representational theories
cannot account for this. Using the prototype approach, since a guppy is neither
a typical pet nor a typical fish, *d _{s} *for the guppy stimulus is large for both

The problem is not solved using techniques from fuzzy
set mathematics such as the *minimum rule model, *where the typicality of
a conjunction (conjunction typicality) equals the minimum of the typicalities of the two constituent concepts (Zadeh 1965, 1982). (For example, the typicality rating for **pet
fish **certainly does not equal the minimum of that for **pet **or **fish.)
**Storms *et al. *(2000) showed that a weighted and calibrated version of
the minimum rule model can account for a substantial
proportion of the variance in typicality ratings for conjunctions exhibiting
the guppy effect. They suggested the effect could be due to the existence of *contrast
categories, *the idea being that a concept such as **fruit **contains not
only information about fruit, but information about categories that are related
to, yet different from, fruit. Thus, a particular item might be a better
exemplar of the concept **fruit **if it not only has many features in common
with exemplars of **fruit **but also few features in common with exemplars
of **vegetables **(Rosch and Mervis,
1975). However, another study provided negative evidence for contrast
categories (Verbeemen *et al. *in press).

Nor does the theory theory
or essence approach get us closer to solving the conjunction problem. As
Hampton (1997) points out, it is not clear how a set of syntactic rules for
combining or interpreting combinations of mini-theories could be formulated.

**2.3.** *'Emergence' and loss of
properties during conjunction*

An
even more perplexing problem facing theories of concepts is that, as many
studies (e.g. Hastie *et al. *1990, Kunda *et
al. *1990, Hampton 1997) have shown, a conjunction often possesses features
which are said to be *emergent: *not true of its constituents. For
example, the properties 'lives in cage' and 'talks' are considered true of **pet
birds, **but not true of **pets **or **birds.**

Representational theories are not only incapable of *predicting
*what sorts of features will emerge (or disappear) in the conjunctive
concept, but they do not even provide a place in the formalism for the gain (or
loss) of features. This problem stems back to a limitation of the mathematics
underlying not only representational theories of concepts (as well as
compositional theories of language) but also classical physical theories. The
mathematics of classical physics 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 *X*_{1}^{ }is
the state space of the first subentity, and *X*_{2}^{
}the state space of the second, the state space of the joint entity is the
Cartesian product space . For this reason,
classical physical theories cannot describe the situation wherein two entities
generate a new entity with properties not strictly inherited from its
constituents.

One could try to solve the problem *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; for instance, every time
a conjunction like **pet bird **comes into existence. However, this happens
every time one generates a sentence that has not been used before, or even uses
the same sentence in a slightly different context. 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. This problem is hinted at by Boden
(1990), who uses the term *impossibilist**
creativity *to refer to creative acts that no 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.

**2.4.** *The** 'obligatory peeking' principle*

In
response to difficulties concerning the transformation of concepts, and how
mini- theories combine to form conjunctions, Osherson
and Smith (1981) suggested that, in addition to a modifiable mini-theory,
concepts have a stable definitional *core. *It is the core, they claim,
that takes part in the combining process. However, the notion of a core does
not straightforwardly solve the conjunction problem. Hampton (1997) suggested that
the source of the difficulty is that in situations where new properties emerge
during concept conjunction, one is making use of world knowledge, or
'extensional feedback'. He states: 'We cannot expect any model of conceptual
combination to account directly for such effects, as they clearly relate to
information that is obtained from another source—namely familiarity with
the class of objects in the world' (Hampton 1997: 148). Rips (1995) refers to this as the *No Peeking Principle. *Rips' own
version of a dual theory distinguishes between representations- of and
representations-about, both of which are said to play a role in conjunction.
However, he does not claim to have solved the problem of how to describe
concepts and their conjunctions, noting 'It seems likely that part of the
semantic story will have to include external causal connections that run
through the referents and their representations' (Rips 1995: 84).

Goldstone
and Rogosky's (in press) ABSURDIST algorithm is a
move in this direction. Concept meaning depends on a web of relations to other
concepts in the same domain, and the algorithm uses within-domain similarity
relations to translate across domains. In our contextualized approach, we take
this even further by incorporating not just pre-identified relations amongst
concepts, but new relations made apparent in the context of a particular
stimulus situation, i.e. the external world. We agree that it may be beyond our
reach to predict exactly how world knowledge will come into play in every particular
case. However, it is at least possible to put forth a theory of concepts that
not only *allows *'peeking', but in a natural (as
opposed to *ad hoc) *way provides a place for it. In fact, in our model,
peeking (from either another concept, or an external stimulus) is obligatory;
concepts require a peek, a context, to actualize them in some form (even if it
is just the most prototypical form). The core or essence of a concept is viewed
as a source of potentiality that requires some context to be dynamically
actualized, and that thus cannot be described in a context-independent manner
(except as a superposition of every possible context-driven instantiation of
it). In this view, each of the two concepts in a conjunction constitutes a
context for the other that 'slices through' it at a particular angle, thereby
mutually actualizing one another's potentiality in a specific way. As a
metaphorical explanatory aid, if concepts were apples, and the stimulus a
knife, then the qualities of the knife would determine not just which apple to
slice, but which direction to slice through it. Changing the knife (the
context) would expose a different face of the apple (elicit a different version
of the concept). And if the knife were to slash through several apples (concepts)
at once, we might end up with a new kind of apple (a conjunction).

**3.** **Two cognitive modes: analytic
and associative**

We
have seen that, despite considerable success when limited to simple concepts
like **bird, **representational theories run into trouble when it comes to
conjunctions like **pet bird **or even **green bird. **In this section we
address the question: why would they be so good for modeling many aspects of
cognition, yet so poor for others?

**3.1. Creativity and flat associative
hierarchies**

It
is widely suggested that there exist two forms of thought (e.g. James 1890,
Piaget 1926, Neisser 1963, Johnson-Laird 1983,
Dennett 1987, Dartnell 1993, Sloman
1996, Rips 2001a). One is a focused, evaluative *analytic mode, *conducive
to analysing relationships of *cause and effect. *The
other is an intuitive creative *associative mode *that provides access to
remote or subtle connections between features that may be *correlated *but
not necessarily causally related. We suggest that while representational
theories are fairly adequate for predicting and describing the results of
cognitive processes that occur in the analytical mode, their shortcomings are
revealed when it comes to predicting and describing the results of cognitive processes
that occur in the associative mode, due to the more contextual nature of
cognitive processes in this mode.

Since the associative model is thought to be more
evident in creative individuals, it is useful at this point to look briefly at
some of the psychological attributes associated with creativity. Martindale
(1999) has identified a cluster of such attributes, including defocused
attention (Dewing and Battye 1971, Dykes and McGhie 1976, Mendelsohn 1976), and high sensitivity
(Martindale and Armstrong 1974, Martindale 1977), including sensitivity to
subliminal impressions, that is, stimuli that are perceived but of which we are
not *conscious *of having perceived (Smith and Van de Meer 1994).

Another characteristic of creative individuals is that
they have *flat associative hierarchies *(Mednick
1962). The steepness of an individual's associative hierarchy is measured
experimentally by comparing the number of words that individual generates in
response to stimulus words on a word association test. Those who generate only
a few words in response to the stimulus have a *steep *associative
hierarchy, whereas those who generate many have a *flat *associative hierarchy. Thus, once such an individual has run
out of the more usual associations (e.g. **chair **in response to **table),
**unusual ones (e.g. elbow in response to **table) **come to mind.

It seems reasonable that in a state of defocused
attention and heightened sensitivity, more features of the stimulus situation
or concept under consideration get processed. (In other words, the greater the
value of *M *in equations (1) and (2) for prototype and exemplar
theories.) It also seems reasonable that flat associative hierarchies result
from memories and concepts being more richly etched into memory; thus there is
a greater likelihood of an associative link between any two concepts. The
experimental evidence that flat associative hierarchies are associated with
defocused attention and heightened sensitivity suggests that the more features
processed, the greater the potential for associations amongst stored memories
and concepts. We can refer to the detail with which items are stored in memory
as *associative richness.*

**3.2. Activation of conceptual space:
spiky versus flat**

We
now ask: how might different individuals, or a single individual under
different circumstances, vary with respect to degree of detail with which the
stimulus or object of thought gets etched into memory, and resultant degree of
associative richness?^{3} Each memory location
is sensitive to a broad range of features, or values of an individual feature
(e.g., Churchland and Sejnowski
1992). Thus although a particular location responds maximally to lines of a
certain orientation, it may respond somewhat to lines of a close orientation.
This is referred to as *coarse coding*.
It has been suggested that the coarseness of the coding—that is, the size
of the higher cortical receptive field—changes in response to attention (Kruschke 1993). Kruschke's
neural network model of categorization, ALCOVE, incorporates a selective attention
mechanism, which enables it to vary the number of dimensions the network takes
into account at a time, and thereby mimics some previously puzzling aspects of
human categorization. In neural networks, receptive field activation can be
graded using a radial basis function (RBF). Each input activates a hypersphere of hidden nodes, with activation tapering off
in all directions according to a (usually) Gaussian distribution of width ** σ**
(Willshaw and Dayan, 1990, Hancock

Whether or not human memory works like a RBF neural
network, the idea underlying them suggests a basis for the distinction between
associative and analytic modes of thought. We will use the terms spiky and flat
activation function to refer to the extent to which memory gets activated by
the stimuli or concepts present in a given cognitive state, bearing in mind
that this may work differently in human cognition than in a neural network.^{5}
The basic idea then is that when the activation function is spiky, only the
most typical, central features of a stimulus or concept are processed. This is
conducive to analytic thought where remote associations would be merely a
distraction; one does not want to get sidetracked by features that are
atypical, or modal (Rips 2001b), which appear only in imagined or
counterfactual instances. However, as the number of features or stimulus
dimensions increases, features that are less central to the concept that best
categorizes it start to get included, and these features may in fact make it
defy straightforward classification as strictly an instance of one concept or
another. When the activation function is relatively flat, more features are
attended and participate in the process of activating and evoking from memory; atypical as well as typical ones. Therefore, more memory
locations participate in the release of 'ingredients' for the next instant.
These locations will have previously been modified by (and can therefore be
said to 'store' in a distributed manner) not only concepts that obviously share
properties with the stimulus, but also concepts that are correlated with it in
unexpected ways. A flat activation function is conducive to creative,
associative thought because it provides a high probability of evoking one or
more concepts not usually associated with the stimulus.

Thus we propose that representational
theories—in which concepts are depicted as fixed sets of
attributes—are adequate for modeling analytical processes, which
establish relationships of cause and effect amongst concepts in their most
prototypical forms. However, they are not adequate for modeling
associative processes, which involve the identification of correlations amongst
more richly detailed, context-specific forms of concepts. In the associative
mode, aspects of a situation the relevance of which may not be readily
apparent, or relations to other concepts which have
gone unnoticed—perhaps of an analogical or metaphorical nature—can
'peek through'. A cognitive state in which a new relationship amongst concepts
is identified is a state of *potentiality, *in the sense that the newly
identified relationship could be resolved different ways depending on the
contexts one encounters, both immediately, and down the road. For example,
consider the cognitive state of the person who thought up the idea of building
a snowman. It seems reasonable that this involved thinking of snow not just in
terms of its most typical features such as 'cold' and 'white', but also the
less typical feature 'moldable'. At the instant of inventing **snowman **there
were many ways of resolving how to give it a nose. However, perhaps because the
inventor happened to have a carrot handy, the concept **snowman **has come
to acquire the feature 'carrot nose'.

**4.
A formalism that incorporates context**

We
have seen that models of cognition have difficulty describing contextual,
associative, or correlation-based processes. This story has a precedent.
Classical physics does exceedingly well at describing and predicting relationships
of *causation, *but it is much less powerful in dealing with results of
experiments that entail sophisticated relationships of *correlation. *It
cannot describe certain types of correlations that appear when quantum entities
interact and combine to form joint entities. According to the dynamical
evolution described by the Schrodinger equation, whenever there is interaction
between quantum entities, they spontaneously enter an entangled state that
contains new properties that the original entities did not have. The
description of this birth of new states and new properties required the quantum
mechanical formalism.

Another way in which the shortcomings of classical
mechanics were revealed had to do in a certain sense with the issue of
'peeking'. A quantum particle could not be observed without disturbing it; that
is, without changing its state. Classical mechanics could describe situations
where the effect of a measurement was negligible, but not situations where the
measurement intrinsically influenced the evolution of the entity. The best it
could do is avoid as much as possible any influence of
the measurement on the physical entity under study. As a consequence, it had to
limit its set of valuable experiments to those that have almost no effect on
the physical entity (called observations). It could not incorporate the context
generated by a measurement directly into the formal description of the physical
entity. This too required the quantum formalism.

In this section we first describe the pure quantum
formalism. Then we briefly describe the generalization of it that we apply to
the description of concepts.

**4.1. Pure
quantum formalism**

In
quantum mechanics, the state of a physical entity can change in two ways: (i) under the influence of a measurement context, and this
type of change is called *collapse, *and (ii) under the influence of the
environment as a whole, and this change is called *evolution. *A state is
represented by a unit vector of a complex Hilbert space **H**,^{ }which is a
vector space over the complex numbers equipped with an inproduct
(see Appendix I). A property of the quantum entity is described by a closed
subspace of the complex Hilbert space or by the orthogonal projection operator *P
*corresponding to this closed subspace, and a measurement context by a self-adjoint operator on the Hilbert space, or by the set of
orthogonal projection operators that constitute the spectral family of this
self-adjoint operator (see Appendix II). If a quantum
entity is in a state of, and a measurement context
is applied to it, the state changes to the state:

where *P *is the projector of the spectral family of
the self-adjoint operator corresponding to the
outcome of the measurement. This change of state is more specifically what is
meant by the term collapse. It is a probabilistic change and the probability
for state to change to state under the influence of the
measurement context is given by:

(4)

where is the
inproduct of the Hilbert space (see Appendix II).

The state prior to, and independent of, the measurement,
can be retrieved as a theoretical object—the unit vector of complex
Hilbert space that reacts to all possible measurement contexts in
correspondence with experimental results. One of the merits of quantum
mechanics is that it made it possible to describe the undisturbed and
unaffected state of an entity even if most of the experiments needed to measure
properties of this entity disturb this state profoundly (and often even destroy
it). In other words, the message of quantum mechanics is that it is possible to
describe a reality that only can be known through acts that alter this reality.

There is a distinction in quantum mechanics between
similarity in terms of which measurements or contexts are relevant, and
similarity in terms of values for these measurements (a distinction which we
saw in section two has not been present in theories of concepts). Properties
for which the same measurement—such as the measurement of spin—is
relevant are said to be *compatible *with respect to this measurement. One
of the axioms of quantum mechanics—called *weak modularity—*
is the requirement that orthogonal properties—such as 'spin up' and 'spin
down'— are compatible.

In quantum mechanics, the conjunction problem is
seriously addressed, and to some extent solved, as follows. When quantum
entities combine, they do not stay separate as
classical physical entities tend to do, but enter a state of *entanglement. *If
H_{1 }is the Hilbert space
describing a first subentity, and H_{2} the Hilbert space
describing a second subentity, then the joint entity
is described in the tensor product space** **H_{1} ** **H_{2}^{ }of the two Hilbert spaces H_{1}^{ }and H_{2}^{ }The tensor product always
allows for the emergence of new states—specifically the entangled
states—with new properties.

The presence of
entanglement—*i.e.* quantum
structure—can be tested for by determining whether correlation
experiments on the joint entity violate Bell inequalities (Bell 1964). Pitowsky (1989) proved that if Bell inequalities are
satisfied for a set of probabilities concerning the outcomes of the considered
experiments, there exists a classical Kolmogorovian
probability model that describes these probabilities. The probability can then
be explained as being due to a lack of knowledge about the precise state of the
system. If, however, Bell inequalities are violated, Pitowsky
proved that no such classical Kolmogorovian
probability model exists. Hence, the violation of Bell inequalities shows that
the probabilities involved are non-classical. The only type of
non-classical probabilities that are well known in nature are the quantum
probabilities.

**4.2.** *Generalized quantum formalism *

The standard quantum formalism
has been generalized, making it possible to describe changes of state of
entities with any degree of contextuality, whose structure
is not purely classical nor purely quantum, but something in between (Mackey
1963, Jauch 1968, Piron
1976, 1989, 1990, Randall and Foulis 1976, 1978, Foulis and Randall 1981, Foulis *et al.* 1983, Pitowsky
1989, Aerts 1993, 2002, Aerts
and Durt 1994 a, b). The generalizations of the
standard quantum formalism have been used as the core mathematical structure
replacing the Hilbert space of standard quantum mechanics the structure of *a lattice*, representing the set of
features or properties of the physical entity under consideration. Many
different types of lattices have been introduced, depending on the type of
generalized approach and on the particular problem under study. This has
resulted in mathematical structures that are more elaborate than the original
lattice structure, and it is one of them, namely the *state context property system*, or *SCOP*, that we take as a starting point here.

Let us
now outline the basic mathematical structure of a *SCOP*. It consists of three sets and two functions, denoted:

(Σ, M, L,,)
(5)

where: is
the set of possible states; M is the set of
relevant contexts; L is the lattice which describes the relational structure
of the set of relevant properties or features; is a
probability function that describes how a couple (*e, p*), where *p* is a
state, and *e* a context, transforms to
a couple (*f ,q*) where *q* is the new state (collapsed state for
context *e*), and* f* the new context; is the
weight or applicability of a certain property, given a specific state and
context. The structure L is that of a complete, orthocomplemented
lattice. This means that:

¥
A partial order
relation denoted on L representing that the implication of properties, i.e.
actualization of one property implies the actualization of another. For L we have:

(6)

¥
Completeness: infimum (representing the conjunction and denoted ) and supremum (representing the disjunction and
denoted ) exists for any subset of properties. 0, minimum
element, is the infimum of all elements of L and *I,* maximal element, is
the supremum of all elements of L.

¥
Orthocomplemented: an operation** ^{⊥}** exists, such that for we have:

(7)

Thus ** **is the ÔnegationÕ of.

¥
Elements of L are weighted. Thus for state,* p*, context *e* and
property *a *there exists weight , and for a *a*
L:

(10)

These
general formalisms describe much more than is needed for quantum mechanics, and
in fact, standard quantum mechanics and classical mechanics fall out as special
cases (Aerts 1983). For the *SCOP *description of a pure quantum entity, see Appendix III.

It is gradually being realized that the generalized
quantum formalisms have relevance to the macroscopic world (e.g. Aerts 1991, Aerts, Aerts et a/. 2000, Aerts, Broekaert *et al.*
2000). Their application beyond the domain that originally gave birth to them
is not as strange as it may seem. It can even be viewed as an unavoidable sort
of evolution, analogous to what has been observed for chaos and complexity
theory. Although chaos and complexity theory were developed for application in
inorganic physical systems, they quickly found applications in the social and
life sciences, and are now thought of as domain-general mathematical tools with
broad applicability. The same is potentially true of the mathematics underlying
the generalized quantum formalisms. Although originally developed to describe
the behavior of entities in the microworld, there is
no reason why their application should be limited to this realm. In fact, given
the presence of potentiality and contextuality in
cognition, it seems natural to look to these formalisms for guidance in the
development of a formal description of cognitive dynamics.

**5. Application of SCOP to concepts**

In
this section we apply the generalized quantum formalism—specifically the
SCOP—to cognition, and show what concepts reveal themselves
to be within this framework. To do this we must make a number of subtle but
essential points. Each of these points may appear strange and not completely motivated
in itself, but together they deliver a clear and
consistent picture of what concepts are.

We begin by outlining some previous work in this
direction. Next we present the mathematical framework. Then we examine more
closely the roles of potentiality, context, collapse and actualization. Finally
we will focus more specifically on how the formalism is used to give a measure
of conceptual distance. This is followed up in the next section, which shows
using a specific example how the formalism is applied to concept conjunction.

**5.1.
Previous work**

One
of the first applications of these generalized formalisms to cognition was
modeling the decision making process. Aerts and Aerts (1994) proved that in situations where one moves from
a state of indecision to a decided state (or vice versa), and the change of
state is context-dependent, the probability distribution necessary to describe
it is non-Kolmogorovian . 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 a cognitive state changes in a context-dependent
way to an increasingly specified conceptualization of a stimulus or situation.
Once again, context induces a non-deterministic change of the cognitive state
that introduces a non-Kolmogorivia n probability on
the state space. Thus, a non-classical (quantum or generalized quantum)
formalism is necessary.

Using an example involving the concept cat and
instances of cats, we proved that Bell inequalities are violated in the
relationship between a concept and specific instances of it (Aerts, Aerts *et al*. 2000). Thus we have evidence that this formalism reflects
the underlying structure of concepts. In (Aerts, DÕHondt *et al.*
2000) we show that this result is obtained because of the presence of EPR-type
correlations amongst the features or properties of concepts. The EPR nature of
these correlations arises because of how concepts exist in states of
potentiality, with the presence or absence of particular properties being
determined *in the process* of the
evoking or actualizing of the concept. In such situations, the mind handles
disjunction in a quantum manner. It is to be expected that such correlations
exist not only amongst different instances of a single concept, but amongst
different related concepts, which makes the notion of conceptual distance even
more suspect.

**5.2. Mathematical
framework**

In
the development of this approach, it became clear that to be able to describe
contextual interactions and conjunctions of concepts, it is useful to think not
just in terms of concepts *per se*, but
in terms of the cognitive states that instantiate them. Each concept is
potentially instantiated by many cognitive states; in other words, many
thoughts or experiences are interpreted in terms of any given concept. This is
why we first present the mathematical structure that describes an entire
conceptual system, or mind. We will then illustrate how concepts appear in this
structure. We use the mathematical structure of a state context property system
or SCOP:

(Σ, M, L, *μ*, )
(11)

where: is the set
of all possible cognitive states, sometimes referred to as conceptual space .
We use symbols *p, q, r,É
*to denote states; M is the set of relevant contexts that can influence
how a cognitive state is categorized or conceptualized. We use symbols *e, f, g,É* to denote contexts; L is the lattice which describes
the relational structure of the set of relevant properties or features. We use
symbols *a, b, c, É *to denote features
or properties; *μ* is a property
function that describes how a couple (*e,
p*)—where *p* is a state, and* e *a context—transforms to a
couple (*f, q*)*, *where *q* is the new
state (collapsed state for context *e*),
and *f *the new context; *v *is the weight or applicability of a
certain property, given a specific state and context.

By cognitive states we man states of the mind (the
mind being the entity that experiences them). Whereas the two sets and M, along with the function , constitute the set of possible cognitive states and
the contexts that evoke them, the set L and the function , describe properties of these states, and their
weights. In general, a cognitive state under
context *e* (the stimulus) changes to
state according
to probability function *μ*. Even if
the stimulus situation itself does not change, the change of state from *p* to *q*
changes the context (i.e. the stimulus is now experienced in the context of
having influenced the change of state from *p
*and *q*). Thus we have a new
context *f*. For a more detailed
exposition of SCOP applied to cognition, see appendix D.

**5.3. How concepts appear in the formalism**

We denote
concepts by the symbols *A, B, C,É *and the set of all concepts A. A concept appears in the
formalism as a subentity of this entire cognitive system,
the mind.^{6} This means that if we
consider a cognitive state , for each concept A, there exists a
corresponding state *p _{A}*
of this concept. The concept A is described by its own
SCOP denoted ( Σ

5.3.1. *Instantiation of concept actualizes
potential. *For a set of concepts {*A _{l},
A_{2},É, A_{n,},..*.}, where

5.3.2. *Uninstantiated**
concepts retain potential. *Let us continue considering the specific
situation where state p instantiates concept *A _{m} *under context

5.3.3. *Concepts
as contexts and features.** *In addition to appearing as subentities instantiated by cognitive states, concepts appear
in the formalism in two other ways. First, they can constitute (part of) a
context M. Second, something that constitutes
a feature or property *L* in one
situation can constitute a concept in another; for instance, ÔblueÕ is a property
of the sky, but also one has a concept **blue. **Thus, the three sets M
and L, of a SCOP are all in some way
affected by concepts.

5.3.4. *Conjunctions of concepts.** *As mentioned
previously, the operation applied to pure quantum entities is the tensor
product. The algebraic operation we feel to be most promising for the
description of conjunction of concepts is the following. In a SCOP, there is a
straightforward connection between the state of the entity under consideration
at a certain moment, and the set of properties that are actual at that moment.^{7}
This makes it possible to, for a certain fixed property L, introduce
what is called the relative SCOP for *a*,
denoted (Σ, M, L, *μ*, )_{a}_{ }*. *Suppose that (Σ, M, L, *μ*, ) describes concept *A*, then (Σ,
M, L, *μ*, )_{a} * _{ }*describes
concept

**5.4.
Superposition, potentiality couples and change of
cognitive state**

We cannot
specify with complete accuracy (i) the content of
state *p*, nor (ii) the stimulus
situation it faces, context *e*, nor
(iii) how the two will interact. Therefore, any attempt to
mathematically model the transition from *p*
to *q* must incorporate the possibility
that the situation could be interpreted in many different ways, and thus many
different concepts (or conjunctions of them) being activated. Within the
formalism, it is the structure of the probability field that describes this. For a given state *p*, and another state and contexts
*e *and M*,* the
probability *μ* (*f, q, e ,p*)
that state *p* change *s* under the influence of context *e* to state *q* (and that *e* changes to *f* ) will often be different from zero.
In the quantum language, we can express this by saying that p is a
superposition state of all the states such that the probability *μ* (*f, q, e,
p*) is non-zero for some M*. *Note that
whether or not *p* is in a state of
potentiality depends on the context *e*.
It is possible that state *p* would be
a superposition state for *e* but not
for another context *f*. Therefore, we
use the term *potentiality couple (e, p).*

We stress
that the potentiality couple is different from the potentiality of a concept;
the potentiality couple refers to the cognitive state (in all its rich detail)
with respect to a certain context (also in all its rich detail), wherein a
particular instantiation of some concept (or conjunction of them) may be what
is being subjectively experienced. However, they are related in the sense that
the potentiality of *p* decreases if
concepts A evoked in it enter collapsed states.

5.4.1. *Collapse:
non-deterministic change of cognitive state. *Following the quantum terminology,
we refer to the cognitive state following the change of state under the
influence of a context as a collapsed state. Very often, though certainly not
always, a state *p* is a superposition
state with respect to context *e* and
it collapses to state *q*
which is an eigenstate^{8} with respect to *e*, but a superposition state with respect to the new context *f*. This is the case when couple (*e, p*) refers to conception of stimulus
prior to categorization, and couple (*f, q*)
refers to the new situation after categorization has taken place.

Recall that
a quantum particle cannot be observed or Ôpeeked atÕ without disturbing it;
that is, without inducing a change of state. Similarly, we view concepts as
existing in states of potentiality which require a
context—activation by a stimulus or other concept that constitutes
(perhaps partially) the present cognitive state—to be elicited and
thereby constitute the content (perhaps partially) of the next cognitive state.
However, just as in the quantum case, this ÔpeekingÕ causes the concept to
collapse from a state of potentiality to a particular context-driven
instantiation of it. Thus, the stimulus situation plays the role of the
measurement by determining which are the possible states that can be collapsed
upon; it ÔtestsÕ in some way the potentiality of the associative network,
forces it to actualize, in small part, what it is capable of. A stimulus is
categorized as an instance of a specific concept according to the extent to
which the conceptualization or categorization of it constitutes a context that
collapses the cognitive state to a thought or experience of the concept.

5.4.2. *Deterministic
change of cognitive state.** *A special case is when the
couple (*e, p*) is *not *a potentiality
couple. This means there exists a context *f*
and a state *q*, such that with
certainty couple (*e, p*) changes to
couple (*f, q*). In this case we call (*e, p*) a deterministic couple and *p* a deterministic state as a member of
the couple (*e, p*). An even more
special case is when the context *e*
does not provoke any change of the state *p*.
Then the couple (*e, p*) is referred to
as an eigencouple, and the state *p* an eigenstate as a member of the couple
(*e, p*).

5.4.3. *Retention of potentiality during collapse.** *For a given
stimulus* e*, the probability that the
cognitive state *p* will collapse to a
given concept *A* is related to the
algebraic structure of the total state context property system (Σ, M, L, *μ*, ) and most of all, to the probability field *μ*(*f,
q, e ,p*) that describes how the stimulus and the cognitive state
interact. It is clear that, much as the potentiality of a concept (to be
applicable in all sorts of contexts) is reduced to a single actualized
alternative when it collapses to a specific instantiation, the potentiality of
a stimulus (to be interpreted in all sorts of ways) is diminished when it is
interpreted in terms of a particular concept. Thus, in the collapse process,
the stimulus loses potentiality. Consider as an example the situation that one
sees a flower, but if one were to examine it more closely, one would see that
it is a plastic flower. One possibility for how a situation such as this gets
categorized or conceptualized is that extraneous, or modal, feature(s) are
discarded, and the cognitive state collapses completely to the concept that at
first glance appears to best describe it: in this case, **flower. **We can
denote this cognitive state Some of the
richness of the particular situation is discarded, but what is gained is a straightforward
way of framing it in terms of what has come before, thus immediately providing
a way to respond to it: as one has responded to similar situations in the past.
This is more likely if one is in an analytical mode and thus is small, such that one does not encode
subtle details (e.g. Ôthe flower is made of plastic').

However, a
stimulus may be encoded in richer detail such that, in addition to features
known to be associated with the concept that could perhaps best describe it,
atypical or modal features are encoded. This is more likely if one is in an
associative mode, and thus is large. Let us denote as the state of
perceiving something that is flower-like, but that appears to be Ômade of
plasticÕ. The additional feature(s) of *P _{2}*
may make it more resistant to immediate classification, thereby giving it
potentiality. In the context of wanting to make a room more cheerful it may
serve the purpose of a flower, and be treated as a flower, whereas in the
context of a botany class it will not. The state

5.4.4. *Loss
of potentiality through repeated collapse.** *It seems
reasonable that the presence of potentiality in a cognitive state for a certain
context is what induces the individual to continue thinking about, recategorizing, and reflecting on the stimulus situation.
Hence if the cognitive state is like *p _{2}*,
and some of the potentiality of the previous cognitive state was retained, this
retained potentiality can be collapsed in further rounds of reflecting. Thus a
stream of collapse ensues, and continues until the stimulus or situation can be
described in terms of, not just one concept (such as

The process
can continue, leading to a sequence of states *P _{3}, P_{4}, P_{5},É.
*With each iteration the cognitive state changes
slightly, such that over time it may become possible to fully interpret the
stimulus situation in terms of it. Thus, the situation eventually gets
interpreted as an instance of a new, more complex concept or category, formed spontaneously
through the conjunction of previous concepts or categories during the process
of reflection. The process is contextual in that it is open to influence by
those features that did not fit the initial categorization, and by new stimuli
that happen to come along.

5.5. *Contextual conceptual
distance*

We have claimed that for any
concept, given the right context, any feature could potentially become involved
in its collapse, and thus the notion of conceptual distance becomes less meaningful.
However, it is possible to obtain a measure of the distance between states of
concepts, potentiality states as well as collapsed states (which can be prototypes, exemplars, or
imaginary constructions), and this is what the formulas here measure.

5.5.1. *Probability
conceptual distance.** *First, we define what we
believe to be the most direct distance measure, based on the probability field *μ* (*f, q, e,
p*). This method is analogous to the procedure used for calculating distance
in quantum mechanics. We first introduce a reduced probability:

(12)

where:

and is the probability that state *p* changes to state *q* under the influence of context *e*.

The
calculation of probability conceptual distance is obtained using a
generalization of the distance in complex Hilbert space for the case of a pure
quantum situation, as follows:

We can also introduce the conceptual
angle between two states, again making use of the formula from pure quantum
mechanics:

We call *d*_{μ}*
*the probability conceptual distance, or the *μ* distance,
and * *the probability conceptual angle, or
the *μ* angle. For details, see appendix A
and equations (31) and (32), and remark that for unit vectors (31) reduces to
(15).

Let us
consider some special cases to see more clearly what is meant
by this distance and this angle. If *μ*(*q,
e, p*) = 0 we have *d** _{μ}*(

It is
important to remark that the distance *d** _{μ}*(

5.5.2. *Property
conceptual distance.** *In order to illustrate explicitly
the relationship between our approach and the distance measures provided by the
prototype and exemplar approaches described previously, we define a second
distance measure based on properties. This distance measure requires data on
the probability of collapse of a cognitive state under the influence of a
context to a cognitive state in which a particular feature is activated. In
order to define operationally what this data refers to, we describe how it
could be obtained experimentally. One group of subjects is asked to consider
one particular concept *A*, and this
evokes in them cognitive state *p*.
This state will be subtly different for each subject, depending on the specific
contexts which led them to form these concepts, but
there will be nevertheless commonalities. A second group of subjects is asked
to consider another concept *B*, which
evokes cognitive state *q*. Again, *q* will be in some ways similar and in
some ways different for each of these subjects. The subjects are then asked to
give an example of ÔoneÕ feature for each one of the considered concepts. Thus,
two contexts are at play: context *e*
that consists of asking the subject to give a feature of the concept focused on
in state *p*, and context *f* that consists of asking the subject to
give a feature of the concept focused on in state *q*. Thus we have two potentiality couples (*e, p*) and (*f, q*). Suppose
couple (*e, p*) gives rise to the list
of features {*b _{1}
,*

We call *d _{p}*

Remark that to compare
this distance *d _{p}* to the distance

We can also define a
property conceptual distance based on weights of properties. Given a set of
features {*a _{l },*

We call *d _{w}*

5.5.3. *Relationship
between the two distance measurements.** *It would be
interesting to know whether there is a relationship between the distance
measured using the probability field and the distance measured using weighted
properties. In pure quantum mechanics, these two distances are equal (see
Appendix III, equations (35) and (39)).

This could
be tested experimentally as follows. Subjects are asked to give a single
feature of a given concept. We call *e*
the context that consists of making this request. Since a concept *A* evokes slightly different cognitive
states *p* in different subjects, they
do not all respond with the same feature. Thus we obtain the set of features {*a _{l}, a_{2}, ..., a_{M}*}. We denote the cognitive state of a given
subject corresponding to the naming of feature

**6.
Application to the pet fish problem**

We now present theoretical evidence
of the utility of the contextual approach using the pet fish problem. Conjunctions
such as this are dealt with by incorporating context dependency, as follows: (i) activation of **pet **still rarely causes activation
of **guppy**, and likewise (ii)
activation of **fish** still rarely
causes activation of **guppy.** But now
(iii) **pet fish **causes activation of
the potentiality state **pet ***in the context of* **pet fish** AND **fish **in the
context of **pet fish**. Since for this
potentiality state, the probability of collapsing onto the state **guppy** is high, it is very likely to be
activated.

**6.1.
The probability distance**

Let us now
calculate the various distance measures introduced in
the previous section. We use equation (15) for the relevant
states and contexts involved:

where is the probability that state *p *changes to state *q *under the influence of context *e*. Two states and three contexts are at play if we calculate the
different distances *d _{m}*

The transition
probabilities are , the
probability that a subject answers ÔguppyÕ if asked to give an example of **pet,
*** , *the probability that the subject
answers ÔguppyÕ if asked to give an example of **fish, **and mthe probability
that the subject answers ÔguppyÕ if asked to give an example of **pet fish. **The
probability distances are then:

Since and are experimentally close to zero, while is close to 1, we have that * *and * *are close to Ã2 (the maximal
distance), and * *is close to zero .

**6.2.
The property distances**

We only calculate explicitly the
weight property distance *d _{w}*

Four states *p, q, r, s* and four contexts *e, f, g,
h *are at play. The states *p, q, r, s*
are the cognitive states consisting of **guppy, pet, fish **and **pet fish **respectively.
The contexts *e, f, g, h *are the experimental situations of being asked
to rate the typicality of **guppy **as an instance of these four concepts
respectively. For an arbitrary feature *a _{m}, *the weights to
consider are,,, and . The
distances are:

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, relationships 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 with respect to a one
context (for example*, *the distance
between **guppy **and the cognitive state of the subject prior to the
context of being asked to name a pet), and close to one another with respect to
another context (for example*, *the distance
between **guppy **and the cognitive state of the subject prior to the
context of being asked to name a pet fish).

7. **Summary and Conclusions**

Representational theories of concepts-such
as prototype, exemplar and schemata or theory-based theories-have been adequate
for describing cognitive processes occurring in a focused, evaluative,
analytical mode, where one analyses relationships of cause and effect. However,
they have proven to be severely limited when it comes to describing cognitive
processes that occur in a more intuitive, creative, associative mode, where one
is sensitive to and contextually responds to not just the most typical
properties of an item, but also less typical (and even hypothetical or
imagined) properties. This mode evokes relationships of not causation, but
correlation, such that new conjunctions of concepts emerge spontaneously. This
issue of conjunctions appears to have thrown a monkey wrench into concepts
research, but we see this as a mixed blessing. It brought to light two things
that have been lacking in this research: the notion of ÔstateÕ and a rigorous
means of coping with potentiality and context.

First a few
words about the notion of ÔstateÕ. In representational approaches, a concept is represented by one or more of its states. A prototype, previously encountered exemplar, or theory description
merely one state of a concept. The competition between different
representational approaches seems to boil down to Ôwhich of the states of a
concept most fully captures the potentiality of the concept'? Since different
experimental designs elicit different context-specific instantiations of a
concept, it is not surprising that the states focused on in one theory have
greater predictive power in some experiments, while the states focused on in
another theory have greater predictive power in others. The true state of
affairs, however, is that none of the states can represent the whole of the
concept, just as none of the states of a billiard ball can represent the whole
of the billiard ball. The billiard ball itself is described by the structure of
the state space, which includes all possible states, given the variables of
interest and how they could change. If one variable is location and another
velocity, then each location-velocity pair constitutes a state in this state
space. To represent the whole of an entity-whether it be
a concept or a physical object-one needs to consider the set of all states, and
the structure this set has.

This is the motivation for describing the essence of a concept as
a potentiality state. The potentiality state can, under the influence of a
context, collapse to a prototype, experienced exemplar, or an imagined or
counterfactual instance. The set of all these states, denoted for a
concept *A**,
i*s the state space of concept *A*.
It is this state space , as a
totality, together with the set of possible contexts M, and these two sets structured
within the SCOP (, M**, **L, ,)* *that represents the concept. Hence a concept is represented by an
entire structure—including the possible states and their properties, and
the contexts that bring about change from one state to another—rather
than by one or a few specific state(s).

This brings
us to the notion of context. If a theory is deficient with respect to its
consideration of state and state space, it is not unlikely to be deficient with
respect to the consideration of context, since contexts require states upon
which to act. The contextualized approach introduced here makes use of a
mathematical generalization of standard quantum mechanics, the rationale being
that the problems of context and conjunction are very reminiscent to the
problems of measurement and entanglement that motivated the quantum formalism.
Below we summarize how these two problems manifest in the two domains of
physics and cognition, and how they are handled by quantum mechanics and its
mathematical generalizations.

á
**The measurement problem for quantum mechanics. **To know the
state of a micro-entity, one must observe or measure some property of it.
However, the context of the measurement process itself changes the state of the
micro-entity from superposition state to an eigenstate
with respect to that measurement. Classical physics does not incorporate a
means of modeling change of state under the influence of context. The best it
can do is to avoid as much as possible any influence of the measurement on the
physical entity under study. However, the change of state under the influence
of a measurement context—the quantum collapse—is explicitly taken
into account in the quantum mechanical formalism. The state prior to, and
independent of, the measurement, can be retrieved as a theoretical object-the
unit vector of complex Hilbert space-that reacts to all possible measurement
contexts in correspondence with experimental results. Quantum mechanics made it
possible to describe the real undisturbed and unaffected state of a physical
entity even if most of the experiments that are needed to measure properties of
this entity disturb this state profoundly (and often even destroy it).

á **The
measurement problem for concepts. **According to RipsÕ No Peeking
Principle, we cannot be expected to incorporate into a model of a concept how
the concept interacts with knowledge external to it. But *can *a concept
be observed, studied, or experienced in the absence of a context, something
external to it, whether that be a stimulus situation
or another concept? We think not. We adopt a Peeking Obligatory approach;
concepts require a peek-a measurement or context-to be elicited, actualized, or
consciously experienced. The generalization of quantum mechanics that we use
enables us to explicitly incorporate the context that elicits a reminding of a
concept, and the change of state this induces in the concept, into the formal
description of the concept itself. The concept in its undisturbed state can
then be ÔretrievedÕ as a superposition of its instantiations.

á **The
entanglement problem for quantum mechanics. **Classical physics could
successfully describe and predict relationships of causation. However, it could
not describe the correlations and the birth of new states and new properties
when micro-entities interact and form a joint entity. Quantum mechanics
describes this as a state of entanglement, and use of the tensor product gives
new states with new properties.

á **The
entanglement problem for concepts. **Representational theories could
successfully describe and predict the results of cognitive processes involving
relationships of causation. However, they could not describe what happens when
concepts interact to form a conjunction, which often has properties that were
not present in its constituents. We treat conjunctions as concepts in the context
of one another, and we investigate whether the relative SCOP might prove to be
the algebraic operation that corresponds to conjunction.

Note that the measurement/peeking
problem and the entanglement/conjunction problem both involve context. The measurement/peeking
problem concerns a context very rxternal to, and of a
different sort from, the entity under consideration: an observer or measuring
apparatus in the case of physics, and a stimulus in the case of cognition. In
the entanglement/conjunction problem, the context is the same sort of entity as
the entity under consideration: another particle in the case of physics, or
another concept in the case of cognition. The flip side of contextuality
is potentiality; they are two facets of the more general problem of describing
the kind of nondeterministic change of state that takes place when one has
incomplete knowledge of the universe in which the entity (or entities) of
interest, and the measurement apparatus, are operating.

The
formalisms of quantum mechanics inspired the development of mathematical
generalizations of these formalisms such as the State Context Property system,
or SCOP, with which one can describe situations of varying degrees of contextuality. In the SCOP formalism, pure classical structure
(no effect of context) and pure quantum structure (completely contextual) fall
out as special cases. Applying the SCOP formalism to concepts, pure analytic
(no effect of context) and pure associative (completely contextual) modes fall
out as special cases. In an analytic mode, cognitive states consist of
pre-established concepts. In an associative mode, cognitive states are likely
to be potentiality states (i.e. not collapsed) with respect to contexts. This
can engender a recursive process in which the content of the cognitive state is
repeatedly reflected back at the associative network until it has been
completely defined in terms of some conjunction of concepts, and thus
potentiality gets reduced or eliminated with respect to the context. Eventually
a new stimulus context comes along for which this new state is a superposition
state, and the collapse process begins again. It has been proposed that the
onset of the capacity for a more associative mode of thought is what lay behind
the origin of culture approximately two million years ago (Gabora 1998,
submitted), and that the capacity to shift back and forth at will from
analytical to associative thought is what is responsible for the unprecedented
burst of creativity in the middle/upper Paleolithic (Gabora, submitted).

We suggest
that the reason conjunctions of concepts can be treated as entangled states is
because of the presence of EPR-type correlations among the properties of
concepts, which arise because they exist in states of potentiality, with the
presence or absence of particular properties of a concept being determined *in
the process of *evoking or actualizing it. If, concepts are indeed
entangled, and thus for any concept, given the right context, any feature could
potentially become involved in its collapse, then the notion of conceptual
distance loses some meaning. What *can *be defined is not the distance
between concepts, but the distance between *states *of them.^{10} That said, the measure * *determines the distance between the
cognitive state prior to context (hence a potentiality state) to the state
after the influence of context (hence the collapsed state). The measure *d _{p}*

Preliminary
theoretical evidence was obtained for the utility of the approach, using the
pet fish problem. Conjunctions such as this are dealt with by incorporating
context-dependency, as follows: (i) activation of **pet
**still rarely causes activation of **guppy, **and likewise (ii) activation
of **fish **still rarely causes activation of **guppy. **But now (iii) **pet
fish **causes activation of the superposition state **pet ***in the
context of ***pet fish **AND **fish ***in the context of ***pet
fish. **Since for this superposition state 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 the sort of
widespread anomalies that arise with concepts, such as this situation wherein
an entity that is neither a good instance of *A* nor *B* is nevertheless a
good instance of the conjunction of *A*
and *B*.

Despite our
critique of representational approaches, the approach introduced here was
obviously derived from and inspired by them. Like exemplar theory, it
emphasizes the capacity of concepts to be instantiated as different exemplars.
In agreement to some extent with prototype theory, experienced exemplars are
Ôwoven togetherÕ, though whereas a prototype is limited to some subset of all
conceivable features, a potentiality state is not. Our way of dealing with the
ÔinsidesÕ of a concept is more like that of the theory or schemata approach. An
instance is described as, not a set of weighted features, but a lattice that
represents its relational structure. The introduction of the notion of a
concept core, and the return of the notion of essence, have been useful for
understanding how what is most central to a concept could remain unscathed in
the face of modification to the concepts mini-theory. Our distinction between state
or instantiation and potentiality state is reminiscent of the distinction
between theory and core. However, the introduction of a core cannot completely
rescue the theory theory until serious consideration
has been given to state and context.

We end by asking:
does the contextualized approach introduced here bring us closer to an answer
to the basic question Ôwhat is a conceptÕ? We have sketched out a theory in
which concepts are not fixed representation s but entities existing in states
of potentiality that get dynamically actualized, often in conjunction with
other concepts, through a collapse event that results from
the interaction between cognitive state and stimulus situation or context. But
does this tell us what a concept really is? Just as was the case in physics a
century ago, the quantum formalism, while clearing out many troubling issues,
confronts us with the limitations of science. We cannot step outside of any
particular orientation and observe directly and objectively what a concept is.
The best we can do is reconstruct a concepts essence from the contextually
elicited ÔfootprintsÕ it casts in the cognitive states that make up a stream of
thought.

**Appendices**

**I.**
*Complex Hilbert space*

A
complex Hilbert space H is a set such that for two elements H** ^{ }**of
this set an operation 'sum' is defined, denoted x + y

In addition to the two operations of 'sum' and
'multiplication by a complex number', a Hilbert space has an operation that is
called the 'inproduct of vectors'. For two vectors H ^{ }the inproduct is denoted , and it is a complex number that has the following
properties. For H**, ^{ }**and
,

^{}

^{}

The
inproduct makes it possible to define an orthogonality relation on the set of vectors. Two vectors H** ^{ }**are
orthogonal, denoted , if and only if

which
consists of all the vectors orthogonal to all vectors in *A. *It is easy
to verify that * *is a subspace of H,** ^{ }**and we call it the orthogonal subspace to

There is one more property satisfied to make the
complex vectorspace with an inproduct
into a Hilbert space, and that is, for H* ^{ }*we have:

^{}

This
means that for any subset H, each vector H
can always be written as the superposition:

where * ^{ }*and

^{}

^{ }

^{ }

^{}

and for one vector H,** ^{ }**the measure of a length of this vector:

^{}

This
distance makes the Hilbert space a topological space (a metric space). It can
be shown that for H, we have that * ^{ }*is a topologically closed
subspace of H,

**II.*** Quantum mechanics in
Hilbert space*

In
quantum mechanics, the states of the physical entity under study are represented
by the unit vectors of a complex Hilbert space H**. ^{ }**Properties are represented
by closed subspaces of H

^{}

^{ }

and * *and . We call the vector *y *the projection of *x *on
*M, *and denote it (*x*)*, *and the vector *z *the projection of *x *on *M,
*and denote it *. *The weight
*v(**x, M) *of the property *M *for the
state *x *is then given by:

^{}

^{ }

The
vectors * *are also called the
collapsed vectors under measurement context . An arbitrary measurement
context *e *in quantum mechanics is represented by a set of closed
subspaces } (eventually
infinite), such that:

^{}

^{ }

^{}

The
effect of such a measurement context is that the state *x *that
the physical entity is in when the measurement context is applied collapses to
one of the states:

and the probability of this collapse is given by:

If
we compare (35) and (39) we see that for a quantum mechanical entity the weight
of a property *M *for a state *x *is equal to the probability that
the state *x *will collapse to the state *, *if the measurement context
is applied to this physical entity in
this state. That is the reason that it would be interesting to compare these
quantities in the case of concepts (see section 5.5.3).

**III.
***SCOP
systems of pure quantum mechanics. *

The
set of states ·_{Q} of a quantum entity is the set of unit vectors of
the complex Hilbert space H. The set of
contexts M_{Q}* ^{ }*of a
quantum entity is the set of measurement contexts, i.e. the set of sequences of closed subspaces of the Hilbert space H

^{}

^{ }

Such
a sequence is also called a *spectral family. *The word spectrum refers to
the set of possible outcomes of the measurement context under consideration. In
quantum mechanics, a state * *changes to another state * *under the influence of a context * ^{ }*in the following way. If is the
spectral family representing the context

and the
change of *x *to is called the quantum collapse. The probability of
this change is given by:

Remark that in quantum mechanics
the context *e *is never changed. This means that:

As a consequence, we have for the reduced probability
(see (12)):

A
property *a *of a quantum entity is represented
by a closed subspace *M *of the complex Hilbert space H. A property *a *represented by *M *always
has a unique orthogonal property * *represented by * *the orthogonal closed subspace of *M. *This
orthogonal property * *is the quantum-negation of the property *a. *The
weight *v(**p,a**)
*of a property *a *towards a state *p *is given by:

where *M
*represents *a *and *x *represents *p. *Remark that at first
sight, the weight does not appear to depend on a context, as it does for a
general state context property system. This is only partly true. In pure
quantum mechanics, the weights only depend on context in an indirect way,
namely because a property introduces a unique context, the context
corresponding to the measurement of this property. This
context is represented by the spectral family .

**IV. ***SCOP systems applied to cognition*

A state
context property system (Σ, M**, **L, ,) consists of three sets M and L* *and two functionsand *v.*

· is the set of cognitive states of the subjects under
investigation, while M ^{ }is the set of contexts that influence and change these
cognitive states. L represents properties or features of concepts. The
function is defined
from the set
M M ^{ }to the interval [0, 1] of real numbers, such that:

and** ** is the probability that the cognitive
state *p* changes to cognitive state *q* under influence of context *e* entailing a new context *f*.

We noted
that properties of concepts can also be treated as
concepts. Remark also that it often makes sense to treat concepts as features.
For example, if we say Ôa dog is an animalÕ, it is in fact the feature ÔdogÕ of
the object in front of us that we relate to the feature ÔanimalÕ of this same
physical object. This means that a relation like Ôdog is animalÕ can be
expressed within the structure L in our formalism.

This
relation is the first structural element of the set L, namely a
partial order relation, denoted A property L ÔimpliesÕ a
property L, and we denote , if only
if, whenever a is true then also *b* is
true. This partial order relation has the following properties. For L we have:

For a set of properties {*a _{i}*}
there exists a conjunction property denoted . This
conjunction property is true if
and only if all of the properties

The conjunction property defines
mathematically an infimum for the partial order
relation <. Hence we demand that each subset of L has an infimum in L, which makes L into a *complete lattice.*

Each
property *a*
also has the ÔnotÕ (negation) of this property, which we denote. This is mathematically
expressed by demanding that the lattice L be equipped with an orthocomplementation, which is a function from L to L such that
for L we have:

where 0 is the
minimal property (the infimum of all the elements of L), hence a
property that is never true. The makes L into a complete ortho-complemented
lattice.

The function
*v* is denned from the set M L to the interval [0, 1], and *v(**p, e, a)* is the weight of property *a* under context *e* for state *p*. For we have:

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