Can quantum probability provide a
new direction for cognitive modeling?
Emmanuel M. Pothos
Department of Psychology, City University London, London EC1V 0HB,
United Kingdom
emmanuel.pothos.1@city.ac. uk
http://www.staff.city.ac.uk/∼sbbh932/
Jerome R. Busemeyer
Department of Psychological and Brain Scienc es, Indiana University,
Bloomington, IN 47405
jbusemey@indiana.edu
http://mypage.iu.edu/∼jbus emey/home.html
Abstract: Classical (Bayesian) probability (CP) theory has led to an influential research tradition for modeling cognitive processes.
Cognitive scientists have been trained to work with CP principles for so long that it is hard even to imagine alternative ways to
formalize probabilities. However, in physics, quantum probability (QP) theory has been the dominant probabilistic approach for
nearly 100 years. Could QP theory provide us with any advantages in cognitive modeling as well? Note first that both CP and QP
theory share the fundamental assumption that it is possible to model cognition on the basis of formal, probabilistic principles. But
why consider a QP approach? The answers are that (1) there are many well-established empirical findings (e.g., from the influential
Tversky, Kahneman research tradition) that are hard to reconcile with CP principles; and (2) these same findings have natural and
straightforward explanations with quantum principles. In QP theory, probabilistic assessment is often strongly context- and order-
dependent, individual states can be superposition states (that are impossible to associate with specific values), and composite systems
can be entangled (they cannot be decomposed into their subsystems). All these characteristics appear perplexing from a classical
perspective. However, our thesis is that they provide a more accurate and powerful account of certain cognitive processes. We first
introduce QP theory and illustrate its application with psychological examples. We then review empirical findings that motivate the
use of quantum theory in cognitive theory, but also discuss ways in which QP and CP theories converge. Finally, we consider the
implications of a QP theory approach to cognition for human rationality.
Keywords: category membership; classical probability theory; conjunction effect; decision making; disjunction effect; interference
effects; judgment; quantum probability theory; rationality; similarity ratings
1. Preliminary issues
1.1. Why move toward quantum probability theory?
In this article we evaluate the potential of quantum prob-
ability (QP) theory for modeling cognitive processes.
What is the motivation for employing QP theory in cogni-
tive modeling? Does the use of QP theory offer the
promise of any unique insights or predictions regarding
cognition? Also, what do quantum models imply regarding
the nature of human rationality? In other words, is there
anything to be gained, by seeking to develop cognitive
models based on QP theory? Especially over the last
decade, there has been growing interest in such models,
encompassing publications in major journals, special
issues, dedicated workshops, and a comprehensive book
(Busemeyer & Bruza 2012). Our strategy in this article is
to briefly introduce QP theory, summarize progress with
selected, QP models, and motivate answers to the above-
mentioned questions. We note that this article is not
about the application of quantum physics to brain physi-
ology. This is a controversial issue (Hammeroff 2007; Litt
et al. 2006) about which we are agnostic. Rather, we are
interested in QP theory as a mathematical framework for
cognitive modeling. QP theory is potentially relevant in
any behavioral situation that involves uncertainty. For
example, Moore (2002) reported that the likelihood of a
“yes” response to the que stions “Is Gore honest?” and “Is
Clinton honest?” depends on the relative order of the ques-
tions. We will subsequently discuss how QP principles can
provide a simple and intuitive account for this and a range
of other findings.
QP theory is a formal fram ework for assigning probabil-
ities to even ts (Hughes 1989; Isham 1989). QP theory can
be distinguished from quantum mechanics, the latter being
a theory of physical phenomena. For the present purposes,
it is sufficient to consider QP theory as the abstract foun-
dation of quantum mechanics not specifically tied to
physics (for more refined characterizations see, e.g., Aerts
& Gabora 2005b; Atmanspacher et al. 2002; Khrennikov
2010; Redei & Summers 2007). The development of
quantum theory has been the result of intense effort
from some of the greatest scientists of all time, over a
period of >30 years. The idea of “quantum” was first pro-
posed by Planck in the early 1900s and advanced by Ein-
stein. Contributions from Bohr, Born, Heisenberg, and
Schrödinger all led to the eventual formalization of QP
BEHAVIORAL AND BRAIN SCIENCES (2013) 36, 255–327
doi:10.1017/S0140525X12001525
© Cambridge University Press 2013 0140-525X/13 $40.00 255
theory by von Neumann and Dirac in the 1930s. Part of the
appeal of using QP theory in cognition relates to confidence
in the robustness of its mathematics. Few other theoretical
frameworks in any science have been scrutinized so inten-
sely, led to such surprising predictions , and, also, changed
human existence as much as QP theory (when applied to
the physical world; quantum mechanics has enabled the
development of, e.g., the transistor, and, therefore, the
microchip and the laser).
QP theory is, in principle, applicable not just in physics,
but in any science in which there is a need to formalize
uncertainty. For example, researchers have been pursuing
applications in areas as diverse as economics (Baaquie
2004) and information theory (e.g., Grover 1997; Nielsen
& Chuang 2000). The idea of using quantum theory in psy-
chology has existed for nearly 100 years: Bohr, one of the
founding fathers of quantum theory, was known to
believe that aspects of quantum theory could provide
insight about cogniti ve process (Wang et al., in press).
However, Bohr never made any attempt to provide a
formal cognitive model based on QP theory, and such
models have started appearing only fairly recently (Aerts
& Aerts 1995; Aerts & Gabora 2005b; Atmanspacher
et al. 2004; Blutner 2009; Bordley 1998; Bruza et al.
2009; Busemeyer et al. 2006b; Buseme yer et al. 2011;
Conte et al. 2009; Khrennikov 2010; Lambert-Mogiliansky
et al. 2009; Pothos & Busemeyer 2009; Yukalov & Sornette
2010). But what are the features of quantum theory that
make it a promising framework for understanding cogni-
tion? It seems essential to address this question before
expecting readers to invest the time for understanding
the (relatively) new mathematics of QP theory.
Superposition, entanglement, incompatibility, and inter-
ference are all related aspects of QP theory, which endow
it with a unique character. Consider a cognitive system,
which concerns the cognitive representation of some infor-
mation about the world (e.g., the story about the hypotheti-
cal Linda, used in Tversky and Kahneman’s [1983] famous
experiment; sect. 3.1 in this article). Questions posed to
such systems (“Is Linda feminist?”) can have different out-
comes (e.g., “Yes, Linda is feminist”). Superposition has to
do with the nature of uncertainty about question outcomes.
The classical notion of uncertainty concerns our lack of
knowledge about the state of the system that determines
question outcomes. In QP theory, there is a deeper
notion of uncertainty that arises when a cognitive system
is in a superposition among different possible outcomes.
Such a state is not consistent with any single possible
outcome (that this is the case is not obvious; this remarkable
property follows from the Kochen–Specker theorem).
Rather, there is a potentiality (Isham 1989, p. 153) for
different possible outcomes, and if the cognitive system
evolves in time, so does the potentiality for each possibility.
In quantum physics, superposition appears puzzling: what
does it mean for a particle to have a potentiality for different
positions, without it actually existing at any particular pos-
ition? By contrast, in psychology, superposition appears an
intuitive way to characterize the fuzziness (the conflict,
ambiguity, and ambivalence) of everyday thought.
Entanglement concerns the compositionality of complex
cognitive systems. QP theory allows the specification of
entangled systems for which it is not possible to specify a
joint probability distribution from the probability distri-
butions of the constituent parts. In other words, in entangled
composite systems, a change in one constituent part of the
system necessitates changes in another part. This can lead
to interdependencies among the constituent parts not poss-
ible in classical theory, and surprising predictions, especially
when the parts are spatially or temporally separated.
In quantum theory, there is a fundamental distinction
between compatible and incompatible questions for a cog-
nitive system. Note that the terms compatible and incompa-
tible have a specific, technical meaning in QP theory, which
should not be confused with their lay use in language. If
two questions, A and B, about a system are compatible, it
is always possible to define the conjunction between A
and B. In classical systems, it is assumed by default that
all questions are compatible. Therefore, for example, the
conjunctive question “ are A and B true” always has a yes
or no answer and the order between questions A and B
in the conjunction does not matter. By contrast, in QP
theory, if two questions A and B are incompatible, it is
impossible to define a single question regarding their con-
junction. This is because an answer to question A implies a
superposition state regarding question B (e.g., if A is true at
a time point, then B can be neither true nor false at the
EMMANUEL POTHOS studied physics at Imperial
College, during which time he obtained the Stanley
Raimes Memorial prize in mathematics, and continued
with a doctorate in experimental psychology at Oxford
University. He has worked with a range of compu-
tational frameworks for cognitive modeling, including
ones based on information theory, flexible represen-
tation spaces, Bayesian methods, and, more recently,
quantum theory. He has authored approximately sixty
journal articles on related topics, as well as on appli-
cations of cognitive methods to health and clinical psy-
chology. Pothos is currently a senior lecturer in
psychology at City University London.
J
EROME BUSEMEYER received his PhD as a mathemat-
ical psychologist from University of South Carolina in
1980, and later he enjoyed a post-doctoral position at
University of Illinois. For 14 years he was a faculty
member at Purdue University. He moved on to
Indiana University, where he is provost professor, in
1997. Busemeyer’s research has been steadily funded
by the National Science Foundation, National Institute
of Mental Health, and National Institute on Drug
Abuse, and in return he served on national grant
review panels for these agencies. He has published
over 100 articles in various cognitive and decision
science journals, such as Psychological Review, as well
as serving on their editorial boards. He served as chief
editor of Journal of Mathematical Psychology from
2005 through 2010 and he is currently an associate
editor of Psychological Review. From 2005 through
2007, Busemeyer served as the manager of the Cogni-
tion and Decision Program at the Air Force Office of
Scientific Research. He became a fellow of the Society
of Experimental Psychologists in 2006. His research
includes mathematical models of learning and decision
making, and he formulated a dynamic theory of
human decision making called decision field theory.
Currently, he is working on a new theory applying
quantum probability to human judgment and decision
making, and he published a new book on this topic
with Cambridge University Press.
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
256
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
same time point). Instead, QP defines conjunction between
incompatible questions in a sequential way, such as “A and
then B.” Crucially, the outcome of question A can affect the
consideration of question B, so that interference and order
effects can arise. This is a novel way to think of probability,
and one that is key to some of the most puzzling predictions
of quantum physics. For example, knowledge of the pos-
ition of a particle imposes uncertainty on its momentum.
However, incompatibility may make more sense when con-
sidering cognitive systems and, in fact, it was first intro-
duced in psychology. The physicist Niels Bohr borrowed
the notion of incompatibility from the work of William
James. For example, answering one attitude question can
interfere with answers to subsequent questions (if they
are incompatible), so that their relative order becomes
important. Human judgment and preference often
display order and context effects, and we shall argue that
in such cases quantum theory provides a natural expla-
nation of cognitive process.
1.2. Why move away from existing formalisms?
By now, we hope we have convinced readers that QP
theory has certain unique properties, whose potential for
cognitive modeling appears, at the very least, intriguing.
For many researchers, the inspiration for applying
quantum theory in cognitive modeling has been the wide-
spread interest in cognitive models based on CP theory
(Anderson 1991; Griffiths et al. 2010; Oaksford & Chater
2007; Tenenbaum et al. 2011). Both CP and QP theories
are formal probabilistic frameworks. They are founded on
different axioms (the Kolmogorov and Dirac/von
Neumann axioms, respectively) and, therefore, often
produce divergent predictions regarding the assignment
of probabilities to events. However, they share profound
commonalities as well, such as the central objective of
quantifying uncertainty, and similar mechanisms for
manipulating probabilities. Regarding cognitive modeling,
quantum and classical theorists share the fundamental
assumption that human cognition is best understood
within a formal probabilistic framework.
As Griffiths et al. (2010, p. 357) note, “probabilistic
models of cognition pursue a top-down or ‘function-first’
strategy, beginning with abstract principles that allow
agents to solve problems posed by the world … and then
attempting to reduce these principles to psychological
and neural processes.” That is, the application of CP
theory to cognition requires a scientist to create hypotheses
regarding cognitive representations and inductive biases
and, therefore, elucidate the fundamental questions of
how and why a cognitive problem is successfully addressed.
In terms of Marr’s(1982) analysis, CP models are typically
aimed at the computational and algorithmic levels,
although perhaps it is more accurate to characterize them
as top down or function first (as Griffiths et al. 2010,
p. 357).
We can recognize the advantage of CP cogniti ve models
in at least two ways. First, in a CP cognitive model, the prin-
ciples that are invoked (the axioms of CP theory) work as a
logical “team” and always deductively constrain each other.
By contrast, alternative cognitive modeling approaches
(e.g., based on heuristics) work “alone” and therefore are
more likely to fall foul of arbitrariness problems, whereby
it is possible to manipulate each principle in the model
independently of other principles. Second, neuroscience
methods and computational bottom-up approaches are
typically unable to provide much insig ht into the funda-
mental why and how questions of cognitive process (Grif-
fiths et al. 2010). Overall, there are compelling reasons
for seeking to under stand the mind with CP theory. The
intention of QP cognitive models is aligned with that of
CP models. Therefore, it makes sense to present QP
theory side by side with CP theory, so that readers can
appreciate their commonalities and differences.
A related key issue is this: if CP theory is so successful
and elegant (at least, in cognitive applications), why seek
an alternative? Moreover, part of the motivation for using
CP theory in cognitive modeling is the strong intuition sup-
porting many CP principles. For example, the probability
of A and B is the same as the probability of B and A
(Prob(A&B)=Prob(A&B)). How can it be possible that
the probability of a conjunction depends upon the order
of the constituents? Indeed, as Laplace (1816, cited in
Perfors et al. 2011) said, “probability theory is nothing
but common sense reduced to calculation.” By contrast,
QP theory is a paradigm notorious for its conceptual diffi-
culties (in the 1960s, Feynman famously said “I think I
can safely say that nobody understands quantum mech-
anics”). A classical theorist might argue that, when it
comes to modeling psychological intuition, we should
seek to apply a computational framework that is as intuitive
as possible (CP theory) and avoid the one that can lead to
puzzling and, superficially at least, counterintuitive predic-
tions (QP theory).
Human judgment, however, often goes directly against
CP principles. A large body of evidence has accumulated
to this effect, mostly associated with the influential research
program of Tversky and Kahneman (Kahneman et al.
1982;
Tversky & Kahneman 1973; 1974 ; Tversky & Shafir 1992).
Many of these findings relate to order/context effects, vio-
lations of the law of total probability (which is fundamental
to Bayesian modeling), and failures of compositionality.
Therefore, if we are to understand the intuition behind
human judgment in such situations, we have to look for
an alternative probabilistic framework. Quantum theory
was originally developed so as to model analogous effects
in the physical world and therefore, perhaps, it can offer
insight into those aspects of human judgment that seem
paradoxical from a classical perspective. This situation is
entirely analogous to that faced by physicists early in the
last century. On the one hand, there was the strong intui-
tion from classical models (e.g., Newtonian physics, classi-
cal electromagnetism). On the other hand, there were
compelling empirical findings that were resisting expla-
nation on the basis of classical formalisms. Therefore, phy-
sicists had to turn to quantum theory, and so paved the way
for some of the most impressive scientific achievements.
It is important to note that other cognitive theories
embody order/context effects or interference effects or
other quantum-like components. For example, a central
aspect of the gestalt theory of perception concerns how
the dynamic relationships among the parts of a distal
layout together determine the conscious experience corre-
sponding to the image. Query theory (Johnson et al. 2007)
is a proposal for how value is constructed through a series of
(internal) queries, and has been used to explain the endow-
ment effect in economic choice. In query theory, value is
constructed, rather than read off, and also different
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BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3 257
queries can interfere with each other, so that query order
matters. In configural weight models (e.g., Birnbaum
2008) we also encounter the idea that, in evaluating
gambles, the context of a particular probability-conse-
quence branch (e.g., its rank order) will affe ct its weight.
The theory also allows weight changes depending upon
the observer perspective (e.g., buyer vs. seller). Anderson’s
(1971) integration theory is a family of models for how a
person integrates information from several sources, and
also incorporates a dependence on order. Fuzzy trace
theory (Reyna 2008; Reyna & Brainerd 1995) is based on
a distinctio n between verbatim and gist information, the
latter corresponding to the general semantic qualities of
an event. Gist information can be strongly context and
observer dependent and this has led fuzzy trace theory to
some surprising predictions (e.g., Brainerd et al. 2008).
This brief overview shows that there is a diverse range of
cognitive models that include a role for context or order,
and a comprehensive comparison is not practical here.
However, when comparisons have been made, the results
favored quantum theory (e.g., averaging theory was shown
to be inferior to a matched quantum model, Trueblood &
Busemeyer 2011). In some other cases, we can view QP
theory as a way to formalize previously informal conceptual-
izations (e.g., for query theory and the fuzzy trace theory).
Overall, there is a fair degree of flexibility in the particu-
lar specifi cation of computational frameworks in cognitive
modeling. In the case of CP and QP models, this flexibility
is tempered by the requirement of adherence to the axioms
in each theory: all specifi c models have to be consistent
with these axioms. This is exactly what makes CP (and
QP) models appealing to many theorists and why, as
noted, in seeking to understand the unique features of
QP theory, it is most natural to compare it with CP theory.
In sum, a central aspect of this article is the debate about
whether psychologists should explore the utility of
quantum theory in cognitive theory; or whether the existing
formalisms are (mostly) adequate and a different paradigm
is not necessary. Note that we do not develop an argument
that CP theory is unsuitable for cognitive modeling; it
clearly is, in many cases. And, moreover, as will be dis-
cussed, CP and QP processes sometimes converge in
their predictions. Rather, what is at stake is whether
there are situations in which the distinctive features of
QP theory provide a more accurate and elegant explanation
for empirical data. In the next section we provide a brief
consideration of the basic mechanisms in QP theory.
Perhaps contrary to common expectation, the relevant
mathematics is simple and mostly based on geometry and
linear algebra. We next consider empirical results that
appear puzzling from the perspective of CP theory, but
can naturally be accommodated within QP models.
Finally, we discuss the implications of QP theory for under-
standing rationality.
2. Basic assumptions in QP theory and
psychological motivation
2.1. The outcome space
CP theory is a set-theoretic way to assign probabilities to
the possible outcomes of a question. First, a sample
space is defined, in which specific outcomes about a ques-
tion are subsets of this sample space. Then, a probability
measure is postulated, which assigns probabilities to dis-
joint outcomes in an additive manner (Kolmogorov 1933/
1950). The formulation is different in QP theory, which is
a geometric theor y of assigning probabilities to outcomes
(Isham 1989). A vector space (called a Hilbert space)is
defined, in which possible outcomes are represented as
subspaces of this vector space. Note that our use of the
terms questions and outcomes are meant to imply the tech-
nical QP terms observables and propositions.
A vector space represents all possible outcomes for ques-
tions we could ask about a system of interest. For example,
consider a hypothetical person and the general question of
that person’s emotional state. Then, one-dimensional sub-
spaces (called rays) in the vector space would correspond
to the most elementary emotions possible. The number
of unique elementary emotions and their relation to each
other determine the overall dimensionality of the vector
space. Also, more general emotions, such as happiness,
would be represented by subspaces of higher dimensional-
ity. In Figure 1a, we consider the question of whether a
Figure 1. An illustration of basic processes in QP theory. In Figure 1b, all vectors are co-planar, and the figure is a two-dimensional one.
In Figure 1c, the three vectors “Happy, employed,”“Happy, unemployed,” and “Unhappy, employed” are all orthogonal to each other, so
that the figure is a three-dimensional one. (The fourth dimension, “unhappy, unemployed” is not shown).
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
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BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
hypothetical person is happy or not. However, because it is
hard to picture high multidimensional subspaces, for prac-
tical reasons we assume that the outcomes of the happiness
question are one-dimensional subspaces. Therefore, one
ray corr esponds to the person definitely being happy and
another one to that person definitely being unhappy.
Our initial knowledge of the hypothetical person is indi-
cated by the state vector, a unit length vector, denoted as
|Ψ〉 (the bracket notation for a vector is called the Dirac
notation). In psychological applications, it often refers to
the state of mind, perhaps after reading some instructions
for a psychological task. More formally, the state vector
embodies all our current knowledge of the cognitive
system under consideration. Using the simple vector space
in Figure 1a, we can write |Ψ〉 = a|happy〉 + b|unhappy〉.
Any vector |Ψ〉 can be expressed as a linear combination of
the |happy〉 and |unhappy〉 vectors, so that these two
vectors form a basis for the two-dimensional space we
have employed. The a and b constants are called amplitudes
and they reflect the components of the state vector along the
different basis vectors.
To determine the probability of the answer happy, we need
to project the state represented by |Ψ〉 onto the subspace for
“happy” spanned by the vector |happy〉. This is done using
what is called a projector, which takes the vector |Ψ〉 and
lays it down on the subspace spanned by |happy〉;thisprojec-
tor can be denoted as P
happy
. The projection to the |happy〉
subspace is denoted by P
happy
|Ψ〉=a |happy〉. (Here and
elsewhere we will slightly elaborate on some of the basic
definitions in the Appendix.) Then, the probability that
the person is happy is equal to the squared length of the
projection, ||P
happy
|Ψ〉||
2
. That is, the probability that the
person has a particular property depends upon the projec-
tion of |Ψ〉 onto the subspace corresponding to the prop-
erty. In our simple example, this probability reduces to
||P
happy
|Ψ〉||
2
=|a|
2
, which is the squared magnitude of
the amplitude of the state vector along the |happy〉 basis
vector. The idea that projection can be employed in psy-
chology to model the match between representations has
been explored before (Sloman 1993), and the QP cognitive
program can be seen as a way to generalize these early
ideas. Also, note that a remarkable mathematical result,
Gleason’s theorem, shows that the QP way for assigning
probabilities to subspaces is unique (e.g., Isham 1989,
p. 210). It is not possible to devise another scheme for
assigning numbers to subspaces that satisfy the basic
requirements for an additive probability measure (i.e.,
that the probabilities assigned to a set of mutu ally exclusive
and exhaustive outcomes are individually between 0 and 1,
and sum to 1).
An important feature of QP theory is the distinction
between superposition and basis states. In the abovemen-
tioned example, after the person has decided that she is
happy, then the state vector is |Ψ〉 = |happy〉; alternatively
if she decides that she is unhappy, then |Ψ〉 = |unhappy〉.
These are called basis states, with respect to the question
about happiness, because the answer is certain when the
state vector |Ψ〉 exactly coincides with one basis vector.
Note that this explains why the subspaces corresponding
to mutually exclusive outcomes (such as being happy and
being unhappy) are at right angles to each other. If a
person is definitely happy, i.e., |Ψ〉 = |happy〉, then we
want a zero probability that the person is unhap py, which
means a zero projection to the subspace for unhappy.
This will only be the case if the happy, unhappy subspaces
are orthogonal.
Before the decision, the state vector is a superposition of
the two possibilities of happiness or unhappiness, so that
|Ψ〉 = a|happy〉 + b|unhappy〉. The concept of superposition
differs from the CP concept of a mixed state. According
to the latter, the pers on is either exactly happy or exactly
unhappy, but we don’t know which, and so we assign
some probability to each possibility. However, in QP
theory, when a state vector is expressed as |Ψ〉 = a
|happy〉 + b|unhappy〉 the person is neither happy nor
unhappy. She is in an indefinite state regarding happiness,
simultaneously entertaining both possibilities, but being
uncommitted to either. In a superposition state, all we
can talk about is the potential or tendency that the
person will decide that she is happy or unhappy. Therefor e,
a decision, which causes a person to resolve the indefinite
state regarding a question into a definite (basis) state, is
not a simple read-out from a pre-existing definite state;
instead, it is constructed from the current context and
question (Aerts & Aerts 1995). Note that other researchers
have suggested that the way of exploring the available pre-
mises can affect the eventual judgment, as much as the pre-
mises themselves, so that judgment is a constructive
process (e.g., Johnson et al. 2007; Shafer & Tversky
1985). The interesting aspect of QP theory is that it
funda-
mentally requires a constructive role for the process of dis-
ambiguating a superposition state (this relates to the
Kochen–Specker theorem).
2.2. Compatibility
Suppose that we are interested in two questions, whether
the person is happy or not, and also whether the person
is employed or not. In this example, there are two out-
comes with respect to the question about happiness, and
two outcomes regarding employment. In CP theory, it is
always possible to specify a single joint probability distri-
bution over all four possible conjunctions of outcomes for
happiness and employment, in a particular situation. (Grif-
fiths [2003] calls this the unicity principle, and it is funda-
mental in CP theory). By contrast, in QP theory, there is
a key distinction between compatible and incompatible
questions. For compatible questions, one can specify a
joint probability function for all outcome combinations
and in such cases the predictions of CP and QP theories
converge (ignoring dynamics). For incompatible questions,
it is impossible to determine the outcomes of all questions
concurrently. Being certain about the outcome of one
question induces an indefinite state regarding the outcomes
of other, incompatible questions.
This absolutely crucial property of incompatibility is one
of the characteristics of QP theory that differentiates it
from CP theory. Psychologically, incompatibility between
questions means that a cognitive agent cannot formulate
a single thought for combinations of the corresponding out-
comes. This is perhaps because that agent is not used to
thinking about these outcomes together, for example, as in
the case of asking whether Linda (Tversky & Kahneman
1983) can be both a bank teller and a feminist. Incompatible
questions need to be assessed one after the other. A heuristic
guide of whether some questions should be considered
compatible is whether clarifying one is expected to interfere
with the evaluation of the other. Psychologically, the
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