moldoveanu&langer_2002
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False Memories of the Future: A Critique of the Applications of Probabilistic Reasoning to the Study of Cognitive Processes
Mihnea MoldoveanuUniversity of Toronto
Ellen LangerHarvard University
The authors argue that the ways in which people—scientists and laymen—use probabilistic reasoning is
predicated on a set of often questionable assumptions that are implicit and frequently go untested. They
relate to the correspondence between the terms of a theory and the observations used to validate the
theory and to the implicit understandings of intention and prior knowledge that arise between the
conveyer and the receiver of information. The authors show several ways in which the use of probabilistic
reasoning rests on a priori commitments to a partitioning of an outcome space and demonstrate that there
are many more assumptions underlying the use of probabilistic reasoning than are usually acknowledged.
They unfold these assumptions to show how several different interpretations of the same results in
behavioral decision theory and cognitive psychology are equally well supported by “the facts.” They then
propose a more comprehensive approach to mapping cognitive processes than those currently used, onethat is based on the analysis of all of the relevant alternative interpretations presented in the article.
A man demonstrates his rationality not by a commitment to fixed
ideas, stereotyped procedures, or immutable concepts, but by the
manner in which, and the occasions on which, he changes those ideas,
procedures and concepts. ——Stephen Toulmin
Concepts, like individuals, have their histories, and are just as inca-
pable of withstanding the ravages of time as are individuals.
——Søren Kierkegaard
Sense—and this one ought to know—is always the nonsense that one
lets go. ——Odo Marquard
We see others not as they are, but as we are. ——The Talmud
When social scientists study the ways in which people make
decisions with yet-unknown consequences or judgments with in-
complete information, they commonly invoke a normative calculus
of belief in which statements can have subunitary truth values—
distributed between 0 and 1—that are required to obey the laws of
probability. The reasons for this requirement are only infrequently
scrutinized. Social psychologists often describe human behavior in
terms of its departure from behavior that is thought to be logically
required by the application of the laws of probability to the
information given to the decision maker. Many of the studies of
individual choice under uncertainty carried out by Tversky and
Kahneman (see, e.g., Kahneman, Slovic, & Tversky, 1982; Kah-
neman & Tversky, 1996; Tversky & Kahneman, 1980, 1982) in
their influential work are based on the premise that such departures
from the standards of correct inference that are based on the laws
of probability can be accurately measured through the observation
of behavior. Implicitly, they also assume that, given a problem
statement, there are ascertainable and logically valid standards of
reasoning from which these departures can be accurately and
objectively measured.
This assumption may be unfounded. Nickerson (1996) has
pointed out that the use of the calculus of probabilities to arrive atdegrees of certainty about various propositions rests on assump-
tions about the scenarios to which this reasoning is applied. He has
instantiated these qualifications by reference to decision problems
that are thought to have “correct” answers. Frequently, these
assumptions are neither supplied as part of the problem statement
nor are they self-evident. Rather, they must be constructed by each
person faced with that particular decision problem. For example, if
we estimate at 0.33 the probability that a family’s two children are
both boys on the basis of an observation of the children’s father
holding one boy in his arms as he walks down the street and the
information that he has another child, we are implicitly assuming
that the two-child family in question is randomly drawn from the
set of families in which there is at least one boy, which also implies
the assumption that all fathers prefer walking with their sons over
walking with their daughters, because otherwise, families with two
sons are twice as likely to be represented as are families with one
son and one daughter when the draw is random.
In this article, we discuss ways in which the application of
probabilistic reasoning depends on the prior assumptions we make
about the phenomena whose outcomes we are trying to predict or
infer. We show that the application of probabilistic reasoning relies
on a commitment to an ontological framework that maps percep-
tions into representations or propositions. Such frameworks deter-
mine which events are identical and which are different and
therefore allow an observer to conditionalize his predictions about
Mihnea Moldoveanu, Rotman School of Management, University of
Toronto, Toronto, Ontario, Canada; Ellen Langer, Department of Psychol-
ogy, Harvard University.
We acknowledge the useful comments and suggestions of Raymond
Nickerson and Klaus Fiedler on an earlier version of this article.
Correspondence concerning this article should be addressed to Mihnea
Moldoveanu, Rotman School of Management, University of Toronto, 105
St. George Street, #555, Toronto, Ontario M5S 3E6, Canada. E-mail:
micamo@rotman.utoronto.ca
Psychological Review Copyright 2002 by the American Psychological Association, Inc.2002, Vol. 109, No. 2, 358 –375 0033-295X/02/$5.00 DOI: 10.1037//0033-295X.109.2.358
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the occurrence of an event on his observation of similar events in
the past.
The application of the calculus of probabilities for assigning
measures to propositions about the world requires that descriptions
of different events stand in a logical relationship of exclusion one
to the other. Events themselves cannot stand in relationships of
logical entailment or contradiction one to the other, only state-ments about events can (Sen, 1993). Moreover, events admit of
many alternative representations, each of which generates a space
of propositions whose degree of logical connectedness can be
assigned probability figures. Different world views may contribute
different sets of logically incompatible propositions that describe
the same underlying event space.
To use an example from Lakoff (1996), political conservatives
and political liberals in the United States see each others’ positions
on various social issues as mutually incompatible and therefore
internally incoherent. Conservatives see a contradiction in the
liberals’ support for women’s rights to destroy their fetuses
through abortion (which they see as an implicit endorsement for
killing children) and the liberals’ support for child care and neo-
natal care programs (which they see as an attempt to nurture the
lives of young children). How can liberals be permissive to the
killing of children and committed to enhancing the survivability of
children at the same time? Liberals see a contradiction in the
conservatives’ position against the right to obtain an abortion
(which they see as a commitment to saving lives) and the conser-
vatives’ position against neonatal care programs (which they see as
a retreat from that commitment). How can conservatives say that
they want to save lives if they are against supporting lifesaving
measures for newborns? Lakoff argues that the contradictions only
appear when liberals represent conservatives’ positions through
the lens of the dominant liberal metaphor (society as a family with
the state as a nurturing mother figure), and conservatives represent
liberals’ positions through the lens of the dominant conservativemetaphor (society as a family with the state as a strict father
figure). The contradictions disappear when the conservative posi-
tion is represented through the lens of the conservative metaphor:
Permitting free access to abortions is tantamount to encouraging
the killing of children, and providing state funds for neonatal care
represents an inducement to idleness on the part of the indigent
mothers who benefit from the program. Thus, whereas liberals
might treat the two propositions (prohibit abortions, cut funds for
neonatal care) as mutually exclusive, conservatives might treat
them as either logically independent or as logically connected.
We examine several cases in which a narrow understanding of
the application of the probability calculus to decision problems and
judgments has led to conclusions about people’s cognitive pro-
cesses that are too strong and sometimes unjustified. We show, for
instance, how alternative “correct” interpretations of the same
problem statement can lead to the patterns of responses that are
judged to be “incorrect” by behavioral decision theorists, and how
in some cases, alternative interpretations of the problem statement
can lead to patterns of reasoning that are superior to the normative
solutions that are advanced as “correct” in some situations. For
example, in some cases, abandoning a probabilistic approach to
modeling belief formation can lead to an increase in a person’s de
facto ability to predict or control the outcomes of a phenomenon.
This is because the search for applicable causal models of a
phenomenon—which often requires seeing that particular phenom-
enon from many different perspectives—is often at odds with the
careful construction of long-term statistics for a phenomenon,
which must be founded on an invariant set of assumptions that
establish the relevant reference classes.
If, for instance, we model a coin toss by the instantiation of a
discrete random variable with two possible values and unknown
distribution, then, to form an opinion about the outcome of aparticular coin toss, we should toss the coin a very large number of
times in controlled conditions and construct the long-term statistics
of the outcomes of these tosses. This tactic is predicated, however,
on the assumption that coin tosses are independent and that the
person tossing the coin has not gotten tired and listless by the
1000th toss and systematically modified something about the way
in which the coin is tossed. One alternative—which we explore
below—is to construct a more realistic causal model of the work-
ings of a chance device, incorporating the physical laws and
properties that determine the dynamics of the device to arrive at a
noisy but deterministic prediction of the outcome. We show that
the accuracy of this prediction can be improved with improve-
ments in the observer’s ability to estimate or control the parame-
ters of his or her model, and therefore that it is not a priori
unreasonable for a gambler to try to predict the outcomes of such
devices on the basis of mental models of how they operate.
Important questions about the appropriate use of epistemic
norms in the interpretation of experiments that are aimed at ex-
posing cognitive biases and fallacies have recently been put forth
(Oaksford & Chater, 1994; Stanovich, 1999). The argument we
make is different from both these and other critiques of the
experimental findings that relate to the psychology of judgment
(Birnbaum, 1983; MacDonald, 1986; Politzer & Noveck, 1991;
Schwarz, 1998) and from the global critiques (e.g., Cohen, 1981)
of the work of Tversky and Kahneman, (1980, 1982), which are
usually aiming to exculpate lay rationality from the errors that it
apparently commits relative to a normative calculus of belief. It isdifferent from the former set of writings because, rather than
critiquing a particular finding by offering an alternative explana-
tion, we propose a framework for the investigation of epistemic
rationality and competence that incorporates, as special cases,
most of the more “local” critiques, while offering both new alter-
native interpretations of well-known experimental effects (see
Discussion of Four Experimental Results in Modern Cognitive
Psychology) and a map of the cognitive and metacognitive pro-
cesses that may underlie the processes of judgment under uncer-
tainty. It is different from the latter set of writings. We not only
criticize the part of the work of Kahneman and Tversky (1980,
1982) that shows people fail to apply normative principles of
reasoning with incomplete information and do not merely provide
one alternative model of cognitive intuition aimed at explaining
errors relative to a norm, but we aim also to show how and why
people may—sometimes rationally, sometimes not—contravene
these principles. We do so by showing how several different
normative principles may be used to understand response patterns
as either rational or irrational, depending on the framework that is
used and the participant’s interpretation of the cognitive task at
hand.
We do not merely generate a list of possible alternative inter-
pretations of “erroneous” response patterns, but aim to provide a
framework for evaluating epistemic rationality that transcends the
bounds of Bayesian reasoning, inductive inference, and the clas-
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person says that the coin is more likely to turn up tails, one
assumes that this is because the run of seven tails does not
correspond to his or her intuition about a string of binary random
variables; the probability theorist once again provides the same
explanation.
Suppose, however, that the person has based the estimate of a
higher chance of tails on a fallibilist interpretation of the assump-tion that the coin is unbiased. That is, the person only provisionally
accepts the hypothesis and looks for evidence and arguments that
refute it. The hypothesis that the person is testing is not “this is a
fair coin,” relative to which a run of seven tails is uninformative,
but rather, the assumption of an unbiased coin is incorrect; the coin
is biased, relative to which the run of seven “tails” is informative.
Now suppose that the person is assuming that the experimenter
is trying to be informative and then may think that the a priori
conception of a probability is not the right one to use in this case:
Some credal probability is called for that is based on a critical
consideration of all of the information.
Then, suppose that the person who has based the estimate of a
higher chance of heads on the next toss on the basis of observing
a correlation between the way in which the coin is tossed (e.g.,
which side is up, how many times it flips in the air) and the
outcome of the coin toss. “The coin may be unbiased,” the person
reasons, “but the toss is not.” Once again, the person is assigning
a credal probability to the proposition that the next outcome will be
heads, which is based on a different set of assumptions from those
typically thought to be embodied in a problem statement about
coins.
Suppose, finally, that the experimenter reversed the problem and
gave the person a list of past coin tosses (e.g., 1023) of the same
coin in ideal conditions. The fraction of heads is found to be 5
1022, with an error of 1 in 100 million. Now the experimenter asks,
“What is the probability of heads on the next toss?” and the person
answers, “0.7,” having observed that most of the tosses on the last,set of tries came up heads. Is the person wrong? Not necessarily.
Even if the person has bought into the assumption that long-run
average frequencies amount to probabilities, it is never clear a
priori what probability can be inferred from the observation of a
finite set of observations. Thus, the actual probability of heads may
be 1/2 1/4 sin2 [n /(9 1022)] (Russell, 1940), and it is this
probability that the coin’s sequence of tosses is revealing. There is
no logically necessary relationship between the frequency of tosses
observed in a finite string of experiments and the probability of a
particular outcome occurring on the next toss.
Assumptions Underlying Applications
of Probabilistic ReasoningThe three different accounts of probabilities have in common a
calculus that the resulting probability measures must obey. Gen-
erally, we speak of particular propositions as making up the atomic
elements of a sample space or a space of possibilities. This is
consistent with the formulation of support theory (Rottenstreich &
Tversky, 1997), which we discuss below. The propositions making
up the sample space, which may be finite or infinite in number and
denoted by { H k }k 1
M , are assigned probability measures on the
interval [0, 1] that satisfy the following conditions:
1. Definition. A probability is defined by a function P : A 3 R
on a sample space of all possibilities, { H k }, such that A
and H A. A represents a finite subset of the universal set, made
up of individual propositions.
2. Range of P. P( H ) 0 for all H ; P() 0; P() 1; that
is, the probability of the null space is 0, and the probability of the
universal space is 1.
3. Independence. If H i and H j are logically independent for all
i and j, then P( H 1 H 2 H 3 . . . H M ) k 1
M
P( H k ); that is,the probability of the conjunction of a set of logically independent
propositions is equal to the product of the individual probabilities
of the propositions in question. If H i and H j are logically indepen-
dent, then P( H i H j) P( H i) P( H j), where represents
disjunction; that is, the probability of the proposition “either H i or
H j” is equal to the sum of the probabilities of H i and H j if the two
propositions are logically independent.
4. Monotonicity of P. If H i 3 H j, then P( H i) P( H j); that is,
if one proposition entails another, then the probability of the
former will be less than will be the probability of the latter.
5. Finite Subadditivity of P. P( H i H j) P( H i) P( H j); that
is, probability of the conjunction of two propositions (logically
independent or logically dependent) cannot exceed the sum of the
probabilities of the propositions themselves.
Let us examine the assumptions underlying our acceptance and
application of the laws of the probability calculus to particular
decision problems.
Bases for Accepting Probability Calculus as Normative
Proposition 1. Acceptance of the laws of the calculus of probabilities
is based on prior assumptions and preferences. These are usually
hidden and often unjustified.
Scientists that write about cognitive biases and fallacies assume
that conditions (1–5) are normative and rarely stop to ask, “Why
should degrees of belief satisfy these conditions?” Whereas the apriori interpretation of probabilities leads naturally to a probabi-
listic calculus as an extension of propositional logic, credal prob-
abilities and frequency-based probabilities are not a priori con-
strained by their definitions. Once we come up with a reason for
constraining degrees of belief to obey the calculus of probabilities
(de Finetti, 1937), we realize that our acceptance of these rules
must rest on some unverified assumptions. In particular, de Finetti
showed that, if a person’s degrees of belief about a set of hypoth-
eses do not conform to the probability calculus, then someone can
construct a Dutch book against that person, that is, can extract a
positive gain from that person with probability 1 by offering the
person a set of bets on various subsets of hypotheses (see Resnik,
1987, for a pedagogical exposition of the Dutch book argument).
Whether this is a good reason for accepting the probability calcu-lus as normative must depend on how likely we think it is that
someone can construct such a series of bets against us. If we think
this eventuality to have a negligible likelihood, then clearly we do
not have a compelling reason for adopting the calculus of proba-
bilities as normative.
Further Challenges: Justificationism Versus Fallibilism
A justificationist will seek reasons for accepting a particular
proposition and will choose among competing alternatives the
proposition for which there is the mostsupport (Albert, 1985;
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Lakatos, 1970). Someone who adopts the probability calculus as
normative is a justificationist who seeks to differentiate among
different propositions on the basis of the relative level of support
(or justification) that these propositions receive from evidence
statements. On the other hand, a fallibilist seeks reasons against a
proposition (Popper, 1992) and seeks to falsify a proposition by
looking for evidence that disconfirms that proposition. Only whensuch evidence has not been found, despite the person’s best efforts
aimed at producing it, is the proposition in question deemed
corroborated and accepted as provisionally true.
Fallibilists argue that the prior probability of any law-like uni-
versal generalization is 0, as follows: If L is a general law and ei,
1 i n are individual instantiations of L, then, P( L) limn3
P(e1
, e2
, . . . , en), because L entails an infinite number of instan-
tiations of ei (because it is universal). If ei is independent of e j for
any i and j, then P(e1, e2, . . . , en) P(e1) P(e2) . . . P(en)
by the independence axiom. If the probability measure P() is
regular, then P(ei) 1 for any i. Finally, if the sequence of
instantiations {ei} is exchangeable, then P(ei) P(e j) . . .
P(en).
Now, the conditions of independence, regularity, and exchange-
ability, taken together imply that
li mn3
P(e1, e2, · · · , en) limn3
P(e1) P(e2) · · · P(en)
limn3
[P(e1)]n 0
and, together with the condition that P( L) limn3 P(e
1, e2, . . . ,
en), that P( L) 0. Now, any hypothesis such as “here is a glass of
water,” will contain universal terms such as glass and water and
therefore will qualify as a universal statement of the kind Popper
(1992) has in mind. Therefore, those who are fallibilist can rea-
sonably decline to invest their beliefs with measures that satisfy
the probability calculus.
The Dependence of Judgments About Competence on
Judgments About Representations
Proposition 2. Objects and events—collections of sense data—do not
of themselves imply their own representations. There are many dif-
ferent representations of a particular collection of sense data, and only
in the context of these representations does it make sense to speak of
an application of the logic of partial belief.
As Wittgenstein (1953) pointed out, there is no unique and
self-evidently correct way to represent or propositionalize a par-
ticular perception. In fact, Anderson (1978) proved a theorem to
the extent that there exist multiple equivalent theories that specify
internal representations of objects and events that make the same
behavioral (thus empirically testable) predictions. To the point of
our critique, it is always possible to come up with an alternative
theory of internal processes by which people form judgments,
which predicts the same observed behavior as that recorded in
experimental tests of cognitive “competence,” but which supplies
a very different interpretation for the same experimental results
than that adduced by the experimental researchers. Inferences from
such studies thus may reflect as much about the researchers’ own
hidden (and thus untested) assumptions as about the subjects’
internal processes of forming judgments about empirical matters.
That the link between word and the object it represents is tenuous
can also be deduced from the collapse of the analytic-synthetic
distinction argued in Quine (1960) and from Wittgensteinian con-
siderations about the ambiguity that is inherent in establishing the
meaning of a word by “pointing” at the object or event to which
that word refers (Barnes, Bloor, & Henry, 1996), which cannot be
eliminated by a concatenation of acts of pointing on recursivelyfiner space-time scales.
Let us examine how the nonuniqueness of representation
plays out in our attempts to update our degrees of belief on
account of new information. The normative solution here is
provided by Bayes’s theorem, which states that if D is a new
piece of (propositional) information that is relevant to the truth
of various propositions { H k }, then, when learning it is the case
that D, we should update the probability of any one hypothesis
by the formula
P( H i D) P( D H i)P( H i)
j1
M P( D H j)P( H j)
, (1)
where D stands for an observation statement (datum), H i stands for
the hypothesis for which corroboration is sought, and H j are all of
the relevant hypotheses that D could corroborate.
At least two conditions must hold for the application of Equa-
tion (1) to be justified. First, for D to discriminate between two or
more competing hypotheses, the hypotheses in question must
logically exclude one another: They must be mutually exclusive
vis-a-vis D, in the sense that D cannot lend equal support to two or
more hypotheses. D must discriminate among the different { H j}.
Call this the discriminant condition. Second, the set of hypotheses
{ H j} must be exhaustive of all conceivable explanations. Call this
the completeness condition. Now let us critically examine these
two conditions.
Counterexample to the discriminant condition. Consider, first,
a counterexample to the discriminant condition: When only two
hypotheses are considered ( H 0 H, H
1 ˜ H ), an investigator will
be called on to accept or reject H on the basis of some observation-
statement D. He will calculate the probability that the null hypoth-
esis is true by,
P( H 0 D) P( D H 0)P( H 0)
P( D H 0)P( H 0) P( D H 1)P( H 1)(2)
and the probability that the null hypothesis is false is calculated by
P( H 1 D) P( D H 1)P( H 1)
P( D H 1)P( H 1) P( D H 1)P( H 1)
. (3)
When we apply Equations 2 and 3 to calculate a degree of belief,
which a hypothesis H 1 commands on the basis of an observation
statement that we hold to be true, we implicitly answer the parallel
questions “When can we say that an observation provides an
instantiation of H 1?” and “Which hypothesis is instantiated by an
observation?” If we could show, for example, that an observation
statement could provide support for two mutually contradictory
statements then we also could show that the result obtained by
applying Bayes’s theorem to a set of data will only have value or
make sense relative to one of several possible views of the world:
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There are at least some hypotheses that we cannot use the theorem
to adjudicate among.
Goodman (1954) has given just such an example. He consid-
ers the proposition, “All emeralds are green,” which is corrob-
orated by “This emerald is green.” Next, he constructs the
predicate grue to describe an object observed before time t and
found to be green or observed after time t and found to be blue,where t lies in the future. The proposition, “All emeralds are
grue” is also corroborated by the discovery of a green emerald
before time t , but has the consequence that it predicts the
emerald will be blue after time t and contradicts “All emeralds
are green.” It is the function of inductive reasoning to come up
with well-supported statements about unobserved or not-yet-
observed events, but Goodman’s example implies that the prod-
uct of induction is contingent on the predicates that we use to
describe the world. Because these predicates emerge from a
particular world view, the outcome of the process of induction
is similarly world-view dependent.
One could object that there is something pathological about
the word grue, because it is the product of a transformation of
the predicate green, which induces a dependence of the predi-
cate on time. However, trying to bring this objection to some-
one who speaks grue–bleen language will likely bring the retort
that blue–green are illegitimate transformations of grue and
bleen: Something is green just in case it has been observed
before t and found to be grue and also observed after t and
found to be bleen.
The problem relating to the indeterminacy of inductive in-
ference can be detected in the literature on personality testing
and categorization (Edwards, Morrison, & Weissman, 1993)
using standardized scales such as the Minnesota Multiphasic
Personality Inventory (MMPI) and in the categorization of
mental illness according to the Diagnostic and Statistical Man-
ual of Mental Disorders (4th ed.; American Psychiatric Asso-ciation, 1994). In its original use, the MMPI was used as a
diagnostic tool (i.e., a person who was expected to be depressed
was also expected to score highly on the Depressed axis of the
test). Subsequent applications of the MMPI have evolved away
from “diagnostic” applications toward “profiling” or “config-
ural” applications, whereby the entire pattern of responses is
used to produce a profile of the individual taking the test (Butler
& Satz, 1995). Like grue, depressed is not an easily projectible
predicate. (It receives corroboration from observations that
correspond not only to typically depressed behavior, but also to
social introversion, psychastenia, and hypochondriasis.) Thus,
any particular observation of a behavior (or an answer to the
questionnaire) can be seen as supporting a characterization of
the person as depressed or, as, psychastenic, just like the
observation of a green emerald in the example above can
support the hypothesis that “emeralds are green” just as well as
it can support the hypothesis that “emeralds are grue.”
Counterexample to the completeness condition. Second, con-
sider a counterexample to the completeness condition (Jeffrey,
1965). You are called on to lay a bet on a coin that you have seen
flipped a trillion times. Most of the tosses—89% of them—have
yielded outcomes of heads, which, strangely, have occurred on all
and only those toss numbers that are composite. The next toss is
prime numbered. How would you bet?
If your hypothesis is “The coin is biased towards ‘heads’” and
your alternative hypothesis is “The coin is fair,” then the data seem
to support the main hypothesis, as long as you also believe in the
law of large numbers. If your hypothesis is, “The coin comes up
‘tails’ on prime-numbered tosses,” then the data also support it
resoundingly.
A Bayesian would advise us to look at the priors on the twopossible partitionings of the hypothesis space. The space of
functions supported on the number of observed tosses is infi-
nite; therefore, the prior probability should be very nearly zero,
and the probability that the pattern will continue into the future
will also be nearly zero. By contrast, the prior on the “fair coin”
assumption is 0.5—the coin is either fair or not. Thus, choosing
a particular kind of hypothesis also determines the space of
possible hypotheses and the prior probabilities in a Bayesian
experiment. The data as a whole, however, cannot be made
relevant to both kinds of hypotheses simultaneously: Choosing
one kind of data (correspondence between outcome and primal-
ity of toss number) commits a person to a particular kind of
hypothesis (concerning the type of pattern that is instantiated by
the sequence as a whole).
Why Support Theory Does Not Go Far Enough
The support theory elaborated by Tversky and Koehler (1994;
see also Rottenstreich & Tversky, 1997, for a succinct summary of
the theory) can be seen as a direct response to concerns such as
those embodied in Proposition 2. Support theory distinguishes
between events and representations of events and admits that the
same events can have more than one representation. It attempts to
give a reconstructed logic of judgment under uncertainty that
predicts the deviations of people’s answers from the “normative”
logic of belief. Support theory links particular observed violations
of the extensional logic of the probability calculus to the underly-ing syntax and semantics of the propositions that make up the
sample space. Implicit propositions, for example, are assigned
probability measures that are smaller than their explicit versions.
The theory can best be described as a cognitive framework for
applying the probability calculus to particular kinds of proposi-
tional sample spaces. It is not, however, a cognitive road map for
understanding deviations from the logic of the probability calculus
on the basis of alternative (nonprobabilistic) approaches to form-
ing judgments.
There are, then, critical aspects of tests of cognitive process and
competence that support theory does not address and can be
deduced from our analysis. First, it does not address the fact that
there are many different ways of combining elementary proposi-
tions to form testable hypotheses (i.e., different “logics of scien-
tific discovery” or epistemological commitments). Inductive infer-
ence may or may not be the model of choice for most subjects, and
epistemological commitment should be considered as a relevant
alternative explanation for response patterns that are considered
“erroneous” from the standpoint of an inductivist approach. Sec-
ond, it does not fully address the degrees of freedom one has in
interpreting a particular event (or a particular interpretation of an
event). Even though it recognizes that representations and inter-
pretations matter to behavior, it does not allow that we can “come
to see” a particular object as the instantiationof a particular repre-
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sentation as a result of being exposed to that representation. This
assumption needs to be tested. And at least one superficial exper-
iment in visual perception suggests that it is false. A collection of
marks on a piece of paper that appears to have been randomly
generated at first sight (Rock, 1984, p. 57) is immediately “seen
as” a man sitting on a bench when some cue (“man sitting” or
“bench scene”) is supplied that guides the attention of the observerto the salient features of the image. Although the apparently
random collection of marks cannot be used to discriminate be-
tween the hypotheses “he is sitting” and “he is standing,” the
apparently random collection in conjunction with the cue can
provide the requisite discrimination. “Observation” is not a theory-
independent process. Theory-dependent cues can color percep-
tion and shape the process by which pure sensory impression is
translated into linguistic expression, that is, into an observation
statement.
Although it recognizes the difference between language and
experience, support theory, in its current form, neither shows how
language and perception interact nor does it explicitly work
through the effects of different representations of the same deci-
sion predicament on the choices or judgments that a person might
make.
Language Dependence of Judgments About Cognitive
Rationality
Proposition 3. Objects and events cannot of themselves stand in
logical relations vis-a-vis each other, only statements about objects
and events can do so. The application of probabilistic reasoning to a
particular situation or problem must rest on an ontological framework
of object names, person names, event names, and relation names that
can be used to construct propositions whose truth value can be
measured relative to some data. Therefore, the end product of a
probabilistic reasoning process will depend on the ontological frame-
work used to interpret, understand, or represent the world.
Once we have identified an event by a proposition or a name, p,
then we can begin to refer to that event by its name. The statement
that asserts the occurrence of the event will stand in logical
contradiction to the statement that actively denies the occurrence
of the event and in a nondeterminate logical relation to its passive
negation. However, the mere occurrence of the event cannot stand
in a logical relation to the nonoccurrence of the event. The lan-
guage that we use to describe events plays a trick on us because it
does not explicitly signal that words are merely placeholders for
things; it therefore gives the impression that there exists a neces-
sary connection between events because there is a necessary con-
nection between the propositions that describe them. We some-
times take it as evident that if any relationships exist between event
names, they will also exist between the events to which the names
refer.
This is precisely where Anderson’s (1978) result becomes rel-
evant. Because even lay language contains at least some theory-
dependent terms (see Kuhn, 1970, who argued that there is no
theory-independent observation language), propositionalization—
or the application of lay language to a practical context—rests on
the application of a theory about the object or event that is
propositionalized. Anderson’s result states that there are multiple
possible theories that can correspond to the same observable event
or object, hence multiple possible propositionalizations of a par-
ticular event or object. This state of affairs can lead to a situation
in which the occurrence or nonoccurrence of an event depends on
the language that is used to describe the event. Hence, understand-
ing events as standing in a logical relation to one another can be
understood as an illusion arising from the idea that the link
between a proposition and an event is unique and objective, ratherthan nonunique and contingent. Indeed, the contingency of the
process of labeling events by using words is the critical insight
behind Quine’s critique (1960) of the Kantian distinction between
analytic statements (statements that are true in virtue of their
definitional meaning) and synthetic statements (statements whose
truth is contingent on some state of the world): One can always
imagine a world in which the meaning of some word that is found
in an analytic statement is such that it makes the statement false.
When we consider sentences that describe different events, we
can see that the events that are meant to be described by the
propositions p and q, such as “He went to the theater at time t ” and
“He had a cup of tea at time t ,” cannot stand in a relationship of
logical contradiction to each other, except relative to a backgroundsystem of assumptions that states that one cannot do both at the
same time. Therefore, the two events that are denoted by going to
the theater or having a cup of tea are mutually exclusive only
relative to a true a priori model, which says that the person in
question cannot carry out both tasks at the same time without
contradicting the premises or predictions of the model.
Why Previous Critiques Need to Be Amplified
Dawes (1988) has discussed the problem of language depen-
dence in probability judgments (p. 80), but he advocates using
Venn diagram representations of sample spaces on which proba-
bility measures are defined as a sort of alternative to thinking inwords. Unfortunately, the elements of the sets that are represented
by Venn diagrams must themselves be individuated in language. If
these elements are events, then they can be individuated as changes
in the property of a substance at a time (Kim, 1976). The identity
of a substance or of a property, however, depends on the causal
powers of that substance or property, according to at least one
approach to individuation (Shoemaker, 1980); and talking of
causal powers gets us back to talking about cause-and-effect
relationships. Relying on Venn diagrams does not allow us to
escape the problem of language dependence.
In the absence of a rule for representing events and objects and
for declaring two events to be identical regardless of changes in
perspective or language used for description, we cannot claim a
noncontingent or theory-free status for statements about sequencesof events. Such statements are contingent on the representations
we choose for these events and, surely, on the strategy that we use
for individuating events. It is very much the business of individuals
who care about predicting the future to improve not only their skill
in applying the axioms of probability to given representations, but
also their craft in picking out representations of objects and events
that yield better predictions. We show here that many of the results
of behavioral decision theory that seem damning to the predictive
prowess of the man on the street can in fact be interpreted as
sophisticated applications of just this type of reasoning.
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Effect of Assumptions About Event Sequence on the
Construction of the Reference Class
Proposition 4. The application of a statistical interpretation of a
probability to the formation of a degree of belief about a future
state of the world rests on assumptions about the underlying
properties of the sequence of events to which the statistical de-scription refers.
The empirically observed frequency of a particular event type
can only be equated to the probability of the occurrence of that
event in the limit as the number of observations increases without
bound if the sequence of events in question has the property of
exchangeability (see Kreps, 1988, pp. 154, 158). The concept of
exchangeability is most easily explained for a two-outcome
“chance” device such as an unbiased coin. If a sequence of
outcomes is exchangeable, then the joint probability of any set
of M outcomes {l1
, . . . , l M } of tosses is equal to the joint proba-
bility of any other of the M ! permutations of the M outcomes, that
is,
P(“heads”, “tails”, “heads”)
P(“tails”,“heads”, “heads”) · · · . (4)
de Finetti’s (1937) exchangeability theorem shows that two people
that start out with different subjective probability estimates for a
particular outcome in an exchangeable sequence will converge to
the same estimate as the number of observations of the events in
the sequence increases without bound.
Judgments of exchangeability therefore determine when we
can adequately think about the frequency of events in a partic-
ular class as the probability of the occurrence of an event of that
class. However, judgments about the exchangeability of events
in a sequence depend on our representation of the events thatmake up that sequence. For example, the sequence of events
(taking off my shoes, putting on a night shirt, climbing into bed)
is not exchangeable because the joint probability of the se-
quence (climbing into bed, taking off my shoes, putting on a
night shirt) will not be equal to that of the sequence (taking off
my shoes, putting on a night shirt, climbing into bed). On the
other hand, one may be tempted to see the sequence (pronating
my hand, supinating my hand, pronating my hand) as exchange-
able. This sequence, however, may just be another representa-
tion of the sequence (taking off my shoes, putting on a night
shirt, climbing into bed) being made up of propositions that are
true given that the propositions in the original sequence are
true, because hand movements are associated with various body
movements. Judgments about exchangeability of a sequence of
events, therefore, seem to depend on the representation of the
sequence about which we are making the judgments of
exchangeability.
The application of probabilistic reasoning to everyday se-
quences of events will depend on the ways in which we choose
to represent those events. These descriptions are not in any way
implied by our unpropositionalized, sensory experiences of
those events, but rather are created by our minds. This line of
reasoning suggests that questions about whether the mind is an
analyses of variance (ANOVA) statistician, a Bayesian statis-
tician, or a Neyman-Pearson statistician (Gigerenzer et al.,
1989) rest on a set of assumptions about the ways in which
minds construe the problems that are meant to test for the
relevant cognitive processes. In the next section, we will give
several examples of alternative explanations of experimental
results from cognitive psychology that are aimed at determining
whether people’s reasoning processes conform to the acceptedmethodological standards of empirical scientists.
A Framework for the Investigation of Cognitive Processes
We have thus far established the dependence of conclusions
about people’s reasoning process on assumptions that we make
about their personal epistemologies, their subjective representation
of an event space, and the assumptions that they make about the
underlying sample space. Figure 1 summarizes these dependencies
in the form of a sequence of questions that an investigation of
cognitive processes involved in prediction, judgment, or explana-
tion should attempt to answer.
1. First, what is the epistemological approach that a person
uses? To fallibilists, a question about relative likelihood may be
meaningless, and therefore his/her answer will not illuminate the
way she thinks about problems that are supposed to illustrate her
failure to be a good justificationist.
2. Second, if the person is a justificationist, what approach does
he/she take to the representation of beliefs? Do probability mea-
sures capture all of the relevant information about a particular
context? In the absence of a conclusive argument one way or
another, this question remains open, and debate in epistemology
should inform arguments and experimental inquiries in psychol-
ogy, and vice-versa.
3. Third, for a probabilist, how does the person update his or her
subjective probabilities in light of new information? Is he/she a
Bayesian? What is the class of allowable update rules for Bayesianprobability measures? How are ambiguities about the assignment
of prior probabilities resolved? Once again, epistemological dis-
cussion and cognitive inquiry can inform each other to mutual
advantage.
4. Fourth, how does the person represent the decision, judg-
ment scenario? What are the relevant objects that make up
the problem statement as he/she sees it? What are the meta-
phors that structure the way in which he/she relates to the
problem? What is the relationship between his/her intuitive use
of natural language connectives (and, or) and the use of these
connectives in formal logic (of which probability theory is a
variant)? Here, research from socio- and psycholinguistics,
from the philosophy of language, and from analytic philosophy
can very usefully complement the cognitive study of epistemic
rationality.
5. Fifth, what does the interactive epistemology of the situation
look like? What are the person’s beliefs about the purpose, scope,
aim, goal of the experimenter, or the experimental design? What
are the person’s beliefs about the experimenter’s beliefs? about the
experimenter’s beliefs about his/her beliefs? and so forth. What is
the relevant depth of interactive epistemology in the situation at
hand? Here, research from social psychology, game theory, and
anthropology can give us important insights into the study of
individual cognitive processes, which may turn out to be success-
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ful adaptations to interpersonal situations whose complexity far
outweighs the complexity of the technological predicament that
individual cognition is supposed to have evolved to master (see
Bogdan, 1999).
Discussion of Four Experimental Results in Modern
Cognitive Psychology
We now apply our analysis of the assumptions that are embed-
ded in the application of probabilistic reasoning to decision and
judgment problems to experimental results that purport to reveal
“deficiencies” in man-on-the-street reasoning processes, relative to
commonly accepted standards of inference. Our analysis shows
that several alternative interpretations of these experimental results
are possible, and at least some of them are plausible explanations
for the recorded response patterns that are usually thought toexemplify “fallacious” reasoning.
The Conjunction Fallacy
Tversky and Kahneman (1982) gave participants in an ex-
periment the task to rank, in order of truth, values or personal
degrees of credibility, different statements that could be true of
a person of whom it is also true that “she is 31 years old, single,
outspoken, and very bright. She majored in philosophy. As a
student, she was deeply concerned with issues of discrimination
and social justice and also participated in antinuclear demon-
strations.” The statements ranged from “she is a bank teller” to
“she is a bank teller who is active in the feminist movement,”
to “she is a psychiatric social worker.” Respondents regularly
assigned higher truth values to the compound statement “she is
a bank teller who is active in the feminist movement” than to
the simple statement “she is a bank teller.” The authors used
these response patterns to infer that respondents’ reasoning
process seemed to violate the laws of probability, which require
that, if A logically implies B, then P( A) P( B). They write,
“like it or not, ‘ B’ cannot be more probable than ‘ A and B,’ and
a belief to the contrary is fallacious. Our problem is to retain
what is useful and valid in intuitive judgments, while correcting
the errors and biases to which it is prone” (p. 178). Below, we
provide several alternative explanations of the effect, based on
different (but plausible) interpretations of the experimental
task. All of them challenge the authors’ conclusion about an
error in the reasoning processes that is used by respondents, butdo so on different grounds, and point to different directions of
empirical research in which the “reconstructive” program of
Tversky and Kahneman (1980) can be pursued.
A Bayesian interpretation. The situation on which the partic-
ipants were invited to opine had either been instantiated or not. To
wit, the statement, “Linda is a bank teller who is active in the
feminist movement” is either overall true or overall false. There is
little value—in real settings—in getting one half of the sentence
right: Unlike graders of college exams or papers, Nature does not
give partial credit. Therefore, either one accepts “she is a bank
teller and she is active in the feminist movement” as an entire
Figure 1. Epistemological map for tracking cognitive processes through forced-choice experiments.
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sentence, or one rejects it as an entire sentence. The conjunction
and , as it appears in this sentence is, in this case, not construed as
an invitation to form the intersection of the two sets, bank tellers
and people who are active in the feminist movement , but rather as
one of logical connectedness between two propositions.
There is, then, a reasonable interpretation of the laws of prob-
ability that explains the choices that participants in the Tverskyand Kahneman experiment made. Participants chose rationally
(according to the rules of inductive logic that the application of
Bayes’ Theorem rests on) if they actually chose among statements
describing possible worlds, rather than among single statements
describing a single world. Each possible world has the property
that statements about it are globally true or globally false, their
complexity notwithstanding. We may not have enough information
to determine whether these statements are true; however, they can
never be partially true or probably true when that information has
been gathered: They are either true or false.
On this interpretation, the participants in the experiment sought
inductive support for various statements about Linda that could
have been true from among the statements about Linda that are
known to be true. To the extent that stereotypical images have and
give inductive support—which was not the purpose of the exper-
iment to challenge—the description of Linda as an outspoken
liberal arts major lends the greatest inductive support to the state-
ment about Linda that refers to her extracurricular involvements in
the feminist movement.
Thus, letting H 1
denote the hypothesis “Linda is a bank teller
who is active in the feminist movement,” H 0 denote the hypothesis
“Linda is not a bank teller who is active in the feminist move-
ment,” and D denote the description of Linda, we have, by the
application of Bayes’ Theorem, that
P( H 0 D) P( D H 0)P( H 0)
P( D H 0)P( H 0) P( D H 1)P( H 1) 1 P( H 0 D) P( H 1 D), (5)
if P( D H 1) P( D H 0), that is, if the stereotype of an outspoken
female philosophy major as a feminist activist has any evidentiary
support. Therefore, the probability that H 0
is true will be less than
the probability that H 0 is false, and H
1 will be accepted by a
reasonable inductivist, such as a Bayesian.
Now, let H 1 denote the hypothesis “Linda is a bank teller,” H 0denote the hypothesis “Linda is not a bank teller,” and D denote
the description of Linda given to the participants. In the absence of
some inductively supported stereotype of bank tellers, we have
P( H 0 D)
P( D H 0)P( H 0)
P( D H 0)P( H 0) P( D H 1)P( H 1)
1 P( H 0 D) P( H 1 D), (6)
that is, H 0
will be supported to the same extent as will H 1
. In this
case, there is no inductive justification for choosing “Linda is a
bank teller” over “Linda is not a bank teller” given D, the descrip-
tion of Linda. Thus, a straightforward application of inductivist
logic leads one to choose—correctly—the statement “Linda is a
bank teller who is active in the feminist movement” as more likely
to be true than is the statement “Linda is a bank teller.”
A Popperian interpretation. Popper (1992) has argued for an
approach to scientific knowledge in which there is no inductive
support for a statement. Taking as a point of departure Hume’s
argument that there is no logical basis for induction, Popper argues
that scientists should (a) seek information that could falsify their
theories rather than verify them and (b) choose from among
competing theories, those that have the greatest empirical content,
have received the most severe empirical tests, and have most
successfully passed them.If a theory is formed by the conjunction of two falsifiable
propositions, a and b, then it will have greater empirical content
than a theory that comprises a alone (Popper, 1992). Moreover, if
one of a and b has been tested against some observation-statement
d , then the theory made up of a and b will be preferable to a theory
made up of two untested empirical propositions, c and d . The
consequence of this argument is that one is usually advised to
choose, as most likely to be true, the a priori least likely proposi-
tion that has survived empirical testing because a priori the em-
pirical content of “a and b” will be greater than will the empirical
content of a or b alone, whereas the a priori probability of “a and
b” will be less than or equal to the probability of either a alone or
b alone. This negation of probabilism is consistent with Popper’s
insistence that the prior probabilities of lawlike universal general-
izations is zero (Gemes, 1997).
Let a represent “Linda is a bank teller” and b represent “Linda
is active in the feminist movement.” By a falsificationist account
of participants’ reasoning, the conjunction “a and b” will be
chosen over a because it has greater empirical content, and b has
already been “tested” against D, the description of Linda, than
against the proposition a alone. This interpretation of cognitive
processes underlying the “Linda” experiment has even more dra-
matic implications than does the former: The “intuitive scientist,”
so much maligned in socio-psychological studies of inference
(Gilovich, 1991; Nisbett & Ross, 1980), may be more of a scientist
by the Popperian account of science than are the scientists that
administer the tests of scientific competence (for a discussion of failures of the critical spirit of inquiry among psychologists, see
Greenwald, Leippe, Pratkanis, & Baumgardner, 1986).
A psycholinguistic interpretation. We usually assume that par-
ticipants parse the statement “Linda is a bank teller who is active
in the feminist movement” as a straight conjunction of the two
propositions, “Linda is a bank teller” and “Linda is active in the
feminist movement.” In first-order logic, the conjunction “a and b”
is identical to the conjunction “b and a.” In natural language,
however, this is hardly the case; indeed, asymmetry of conjunctive
sentences is singled out by Dawes (1988) as the reason why we
should be cautious of applying probability measures to “language-
dependent” representations. “I bought a machine gun and went to
the market” is not (usually) understood to be identical to “I went
to the market and bought a machine gun.”
In language, conjunction is asymmetric. Moreover, the fact of a
conjunction may change our interpretation of the terms in the
conjunction. “I bought a machine gun,” in the first case (wherein
it appears that I bought it to murder people at the market) is
different from “I bought a machine gun” in the second case
(wherein my intention is not apparent).
When we say, “Linda is a bank teller,” we understand her to
currently do the work of bank tellers. She is, therefore, part of the
set of currently active bank tellers. When we say, however, “Linda
is a bank teller who is active in the feminist movement,” we may
infer that she was trained as a bank teller or that she once worked
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as a bank teller, in addition to the possibility that she is currently
doing the work of a bank teller. Thus, the set {bank tellers1
} may
be a proper subset of the set {bank tellers2}, in which case, saying
that P ( Linda {bank tellers1}) P ( Linda {bank tellers2} and
Linda feminist group) may not be fallacious. In this interpreta-
tion, the experiment reveals that people may not use the rules of
first-order logic to parse natural language sentences, which ishardly a surprise to cognitive linguists, who have figured out that
logical form and grammatical structure are different (Hacking,
1984).
An interpersonal interpretation. Grice (1975) proposed that
conversations between people cannot be understood simply by
reference to the transcript of their conversation and to a dictionary
or thesaurus that translates words and phrases and parses gram-
matical structures. Rather, the meaning that one gives to a phrase
uttered in a conversation depends on one’s assumptions about the
intentions of the person uttering the sentence, which are them-
selves related in many ways to the immediate context of the
sentence. Grice proposed that people assume each other to be
cooperative and therefore try to interpret each other’s words to
make them informative and relevant to a particular topic.
If one assumes that the laws of probability are a priori dispos-
itive of the choice between the statements “Linda is a bank teller”
and “Linda is a bank teller who is active in the feminist move-
ment” as to their relative likelihood, then one must infer that the
description of Linda in the experimental materials is irrelevant. But
this contradicts Grice’s (1975) cooperation principle. To find it
relevant, participants must find an interpretation of the problem
that allows them to consider all of the information given by the
experimenter as relevant and informative. Choosing “Linda is a
bank teller who is active in the feminist movement” as more likely
to be true than is “Linda is a bank teller” is no more than a signal
that participants were trying to solve an interpersonal problem
vis-a-vis the experimenter, rather than the first-order problem thatthey were apparently resolving.
The work of Schwarz and his coworkers (Schwarz, 1998;
Schwarz & Bless, 1992; Schwarz, Strack, & Mai, 1991) and the
review of the subject by Hilton (1995) posit an explanation for
representativeness-based judgments that is similar to the Gricean
logic in the emphasis on the information impacted to a person
making a judgment by the context of the conversation in which
that judgment is asked for. The representativeness heuristic (Tver-
sky & Kahneman, 1982) relates to the propensity of people to
make judgments about the likelihood of the validity of a universal
proposition (“My life is going well”) on the basis of statements
about particular circumstances that are deemed to be “representa-
tive” of the reference class of the universal proposition (“My
marriage is going well”). In the experiment run by Schwarz,
Strack, and Mai, people from one group were first asked how
satisfied they were with their life in general and then asked how
satisfied they were with their marital situation. The researchers
found a correlation coefficient of .32 between the (coded) answers
to the two questions. In a second group, the order of the questions
was reversed, and the correlation coefficient increased to .67.
Schwarz (1998) offers a cognitive explanation for the effect:
“Presumably, answering the marital satisfaction question first ren-
dered information about one’s marriage highly accessible and this,
rather than other, information, was subsequently used in evaluating
one’s life as a whole” (p. 96). In the Linda example, what is prima
facie accessible is the congruence between the description of Linda
and the suggestion that she is active in the feminist movement; and
by the explanation offered by Schwarz, it is the accessibility of
some decision rule, rather than the presumed intent of the speaker,
that accounts for the conjunction bias. A more detailed account of
such “congruence effect” is examined in the next section.
An Interpretation Based on “Intuitive Probabilities”
Cohen (1977, 1979, 1981; MacDonald, 1986) developed a prob-
ability calculus based on his understanding of the use of words that
appeal to the concept of probability (i.e., “probable motive, prob-
able cause”) in English jurisprudence. The sentence “Mike prob-
ably killed her because he was jealous” asserts a strong claim to
knowing that he was jealous and a weak claim to knowing that he
killed her. The entire sentence is proven false by the discovery that
Jones killed her: “Probably” does not provide an alibi for “he was
jealous” in Cohen’s interpretation. More generally, P( A and B)
min(P( A), P( B)). Furthermore, according to Cohen’s model, if a
person believes that p is probably true or probably P, then she doesnot believe that p is probably false or probably not p. That is, if
P( A) P( A) then P( A) 0. Finally, the probability of a
hypothesis or statement about which we have no opinion is zero
in Cohen’s interpretation: Uncertainty is equivalent to logical
impossibility.
MacDonald (1986) offers the Linda problem as an example of
the application of Cohen’s exculpatory logic of belief as an alter-
native to the condemnationist approach of Tversky and Kahneman
(1980). He states, “In the intuitive probability model, subjects start
with no reason to believe any of the statements” (p. 20). He then
goes on to give a “Grice-like” account of their choices, writing, “in
natural language questions are always motivated, that is, they are
only asked when there is some reason to expect a positive answer”(p. 20). Leaving aside the question of whether the statement is true
in general, the added credence effect should color all of the
alternatives in a positive light. Moreover, using the static part of
Cohen’s theory, which says that P( A and B) min(P( A), P( B))
and parsing “Linda is a bank teller who is active in the feminist
movement” as “Linda is a bank teller (A) and Linda is active in the
feminist movement (B),” we have that P(“Linda is a bank teller
who is active in the feminist movement”) P(“Linda is a bank
teller”).
What is needed for an application of Cohen’s (1982) logic of
belief to the “Linda” problem is an extension of that logic to the
kinematics of the probability function: This is the problem of
updating probabilities in virtue of new information, which Bayes’
Theorem was meant to resolve. In this case, we can use Jeffrey’s(1965) updating rule as an intuitive alternative to Bayes’ rule: Let
p( A) p( A B) p( B) p( A / B) p( B) and P( A) p( A B) p( B)
p( A B) p( B) denote the probabilities that A is true before and
after learning some information about B, and p( B) and P( B) denote
the prior and posterior probabilities of B. Then P( A) p( A)
(P( B) p( B)) Rel( A, B), where Rel( A, B) is the relevance of A to
B and is given by Rel( A, B) p( A B) p( A B). If we find out
that B is true, then P( B) 1 and P( A) p( A) (1 p( B)) Rel( A,
B). Finally, if we assume, following Cohen, that in the absence of
any other information about Linda, p( B) 0, then P( A) p( A)
Rel( A, B).
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Now, look at the Linda problem again. Let B represent Linda’s
description (bright, outspoken, single, former philosophy major).
The relevance of the description to the statement, “Linda is a bank
teller who is active in the feminist movement” is greater than that
of the description of Linda to the other choosable statements.
Therefore the posterior probability of “Linda is a bank teller who
is active in the feminist movement” will be greater than that of other choosable statements. If prima facie credence of this state-
ment has been established by its inclusion as an alternative, then
this statement will be the one with the highest posterior intuitive
probability.
Base-Rate Neglect
Tversky and Kahneman (1980, 1982) and Kahneman and Tver-
sky (1996) have also argued that lay persons do not properly
incorporate base rates for the occurrence of a phenomenon in their
probability estimates of the occurrence of that phenomenon in the
future. They have argued this point on the basis of two experiments
(described and discussed below), and their results have been cri-
tiqued by Gigerenzer (Gigerenzer, 1993, 1994, 1996; Gigerenzer et
al., 1989) on the grounds that (a) people do take base rates into
account when they are allowed to perform their own draws from
the relevant population distributions, and (b) that base rates as
specified in Tversky and Kahneman’s experiments should not be
construed as properties of single events or propositions. Kahneman
and Tversky (1996), argue, contra Gigerenzer’s (1993) claim that
the mind is a frequency monitoring device, that experimental
evidence seems to contradict the hypothesis that people begin to
heed base rates when they themselves sample the relevant popu-
lation; they also argue, contra Gigerenzer’s (1996) normative
challenge to the interpretation of individual probabilities via fre-
quencies, that the “refusal to apply the concept of probability to
unique events is a position that has some following among statis-ticians, but it is not generally shared by the public” (Tversky &
Kahneman, 1996). Leaving aside the lurking contradiction in-
volved in making the public the arbiter of a norm that they use to
criticize members of that public, we will try to show that many
more possible interpretations of the base rate neglect studies than
those that have been adduced by Gigerenzer (1996) are possible in
view of the broader conception of reasoning under uncertainty that
we have developed here.
The “hit and run” experiment. Tversky and Kahneman (1980)
used the following problem to illustrate the ignorance of base rate
effects:
A cab was involved in a hit-and-run accident at night. Two cab
companies, the Green and the Blue, operate in the city. You are given
the following data: (a) 85% of the cabs in the city are Green and 15%
are Blue. (b) A witness identified the cab as a Blue cab. The court
tested his ability to identify cabs under the appropriate variability
conditions. When presented with a sample of cabs (one half of which
were Blue and one half of which were Green), the witness made
correct identifications in 80% of the cases and erred in 20% of the
cases. Question: What is the probability that the cab involved in the
accident was Blue rather than Green?
Tversky and Kahneman (1980) interpreted the median answer
P( Blue“ Blue”) 0.80—the probability that the cab was Blue,
given that the witness said it was Blue—as a sign that people
systematically ignore base rate information in probabilistic mod-
eling and decision-making problems. They argue that the “correct”
use of Bayes’s theorem in this case would give
P( Blue“ Blue”)
P( Blue)P(“ Blue” Blue)
P(Green)P(“ Blue”Green) P( Blue)P(“ Blue” Blue)
0.150.80
0.850.20 0.150.80 0.41, (7)
assuming, of course, that the witness did not exhibit any systematic
bias in the (80/20) pattern of errors toward Blue cabs or Green cabs
(i.e., that the witness was not more likely to err if the cab was Blue
than was the case if the cab was Green).
Suppose that a participant was thinking as follows:
The experimenter is giving me a typical overdetermined problem, that
is, one in which the answer depends on only a subset of the given
information. To get the right answer, I must determine the subset that
is relevant. Well, the subset that seems the most relevant is thecredibility of the witness—I mean, he saw it happen. Therefore only
the credibility of the witness should factor into the right answer. In
this case, the participant would report P(Blue/“Blue”) 0.80.
Now, suppose that a participant trusts the witness’s proficiency
in identifying a cab’s color in the dark and the base rate of Green
and Blue cabs in the city. Rationally, the witness should always
answer “Green” when asked about the color of a cab. Therefore,
the report of the witness is uninformative relative to the problem of
deciding on the color of the cab involved in the hit-and-run
accident, and the participant would report P( Blue“ Blue”) 0.15.
Birnbaum (1983) has shown that, if the participant assumes that
the witness is behaving like a Neyman-Pearson statistician when
approaching the color discrimination problem and that the witnessknows the base rates of Green and Blue cabs in the city and
chooses answers to minimize the overall probability of error, then
the normatively correct answer to the Tversky and Kahneman
(1980) problem can be as high as 0.82. Once again, working out
the assumptions that the participant makes about what the witness
knows (or, alternatively, about what the experimenter thinks is
understood or implicit in the problem statement) can provide
relevant alternative hypotheses of the experimental results, which
do not necessarily condemn the reasoning process of the partici-
pants on the basis of these results alone.
The “lawyer–engineer” experiment. Consider another exper-
iment that is meant to highlight base rate ignorance which uses the
following experimental problem (Tversky & Kahneman, 1982):
A panel of psychologists has interviewed and administered personal-
ity tests to 30 engineers and 70 lawyers, all of whom are successful in
their respective fields. On the basis of this information, thumbnail
descriptions of the 30 engineers and 70 lawyers have been written.
You will find in your forms 5 descriptions, chosen at random from the
100 available descriptions. For each description, please indicate your
probability that the person described is an engineer, on a scale of 1
to 100. A typical “thumbnail sketch” reads as follows: Jack is a
45-year-old man. He is married and has four children. He is generally
conservative, careful, and ambitious. He shows no interest in political
and social issues and spends most of his free time on his many
hobbies, which include home carpentry, sailing, and mathematical
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puzzles. The probability that Jack is one of the 30 engineers in the
sample of 100 is . Participants’ responses are judged to be mostly
invariant to changes in the base rates of engineers and lawyers in the
group.
There is much in this experiment that depends on what the
participants think that the experimenter thinks. For instance, theymay think that the sentence A panel of psychologists has inter-
viewed and administered personality tests to 30 engineers and 70
lawyers serves to establish the “informativeness” of the thumbnail
sketch of the person. Inasmuch as the participants see themselves
trying to solve a “trick question”— one whose clues are available
but not obvious—they might use this information to the detriment
of the “self-evident” answer that is based on the relative frequen-
cies of engineers and lawyers in the group. Call this the “trick-
question” approach to interactive reasoning.
More sophisticated participants (than the experimenter, in this
case) might think that the experimenter was trying to test their
understanding of the application of the concept of probability. In
particular, they think the experimenter thinks that a frequency canonly become a probability in the limit as N , the sample size,
increases without bounds. Whereas a frequency is the property of
an ensemble of observations or individuals, a probability is a
property of a single instance or event. The description of Jack
proceeds in terms of properties of Jack. Therefore, if probability is
to be interpreted as a property of the proposition “Jack is an
engineer,” then it cannot be the case that the frequency of engi-
neers in Jack’s group is relevant to the problem.
A somewhat contrived example may serve to illustrate the
possible disadvantages of reasoning normatively about base rates.
Suppose you are a young college basketball star, playing in the
NCAA championship final. Your team is down 81 to 79, and there
are 5 s of regular time remaining. You have the ball and are in the
three-point zone. Your record on three-pointers in past games is
atrocious, but you have made all of the three-point attempts to-
night. You have access to two kinds of information. One kind
consists of a propositionalized set of information about statistical
probabilities of making the three-pointer: frequencies of successful
three-pointers by yourself in the past, by yourself in critical situ-
ations, by your team in the past 10 years, by your team in the
past 20 years, by your team in this game, by yourself against the
two blockers you see in front of you, and so forth. Another kind
consists of very detailed—and possibly unpropositionalized—
knowledge of the specific context: that one blocker is leaning the
wrong way, that another is looking for you to make the two-point
shot, that a player from your team, who is just behind you, is a
good sprinter and can pick up the rebound from the three-pointattempt, to go along with the unpropositionalized “feel” of coor-
dinating your movements to guide the ball into the hoop, which
you seemed to have developed throughout the night. Are you
“objectively unreasonable” in trying for the three-pointer, contra
the overwhelming statistical evidence in favor of going for the
two-point attempt? We think not: Your rich knowledge of the
circumstance of time and place that relate to this particular shot
should be more relevant than should be the statistical information
that relies—implicitly—on only a subset of the information that
you have and on a particular “method” of interpreting that
information.
The False Incorporation of Narrative Postevent
Information
Loftus and her collaborators (1979; Loftus, Miller, & Burns,
1978) have argued that people are likely to incorporate misleading
information about events they have observed firsthand or to cor-
rupt or impair first-hand memories of these events, after thisinformation is given to them in a narrative recounting the events.
The essence of the alleged effect is the following: You see a man
climbing into a sports utility vehicle and speeding away. It turns
out that he was fleeing the scene of a murder. You are called as a
witness in his trial, after the victim’s left hand is found in his
kitchen cabinet. The lawyer for the prosecution asks you to de-
scribe the pick-up truck in which the man climbed, which another
witness described as blue. If you are subject to the effect in
question, you proceed to give a vivid account of a blue pick-up
truck, even though what you saw was a black sports utility vehicle.
Investigating the way we propositionalize perceptions and thus get
from experience to text or narrative can help once again unpack the
experimental effect and perhaps even discover that there is no
overt substitution of narrative memory for visual memory.
Loftus and Hoffman (1989) acknowledge, in response to the
criticism of McCloskey and Zaragoza (1985) and Tversky and
Tuchin (1989), that there are several mechanisms that could lead to
the “memory impairment effect”: (a) apparent replacement of the
visual memory with the narrative proposition because there was no
specific visual memory in the first place; (b) mistrust of one’s own
memory in the face of a narrative account that came from an
authoritative source; (c) “guessing” in the face of a lack of memory
about either the narrative account or the visual memory; and (d)
direct replacement of the visual memory by a memory of the
narrative.
McCloskey and Zaragoza (1985) argued against Loftus’s (1979)
conclusions by claiming to have ruled out (a) and (d) by askingparticipants to choose between reports that recounted the true
events and reports that recounted events that were neither observed
nor narrated and by reporting unimpaired ability to discriminate
between the true version of the events and the alternative. The
findings of Tversky and Tuchin (1989) and Belli (1989) seem to
rule out (c), on the basis of findings that participants were more
likely to choose the planted narrative account over both the real
account and over an account that was neither observed nor planted.
Accordingly, Loftus and Hoffman (1989) modified the theme of
their argument by replacing the concept of memory impairment
with that of misinformation acceptance. This modification is sig-
nificant, as it places the memory impairment effect in the same
group of effects that are obtained by people who study credulity
and gullibility about statements simpliciter (Gilbert, 1991), which,
moreover, can be explained by a Gricean approach to interpersonal
communication.
Our analysis above that concerns the difference between obser-
vations and (propositionalized) reports of observations suggests
that different people can propositionalize the same set of observa-
tions in different ways. Moreover, unpropositionalized accounts
may not be directly compared in memory with propositionalized
accounts. When we engage in answering written questions, we
engage with language. Therefore, we may be more likely to be
swayed in our answers by propositionalized accounts of the inci-
dent than by unpropositionalized accounts. As we wrote above,
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only two propositions can stand in a relationship of logical con-
tradiction to one another; an observation—a collection of sense
data—cannot contradict or support a proposition about that obser-
vation, except after the observation has been propositionalized.
If we follow through this argument, we will pay close attention
to the steps that require a transition from-unpropositionalized
observations to propositions. Thus, in the original Loftus (1979)study, people are required to propositionalize their beliefs about
the events they witnessed after receiving the propositionalized
misleading information. In this case, we would predict that propo-
sitionalized information they already have will influence their
reports in the direction of that information—as Loftus indeed
finds. Seen in this light, the McCloskey and Zaragoza (1985)
studies test for people’s ability to correctly propositionalize their
observations when asked to do so because they are not given the
option to report having “seen” the misleading account of the
events. Tversky and Tuchin’s (1989) studies show that proposi-
tionalized postevent information is indeed persuasive, but that
people retain the ability to propositionalize first-hand information.
Two remarks are in order: First, students who are presumably
used to taking reading comprehension tests are likely to treat the
experiment as a test of their ability to remember a statement after
they have read it. The statement “You have seen a can of 7UP” is
far easier to compare with the forced-choice answers “You have
seen a can of Coke/7UP/Sunkist” than is the unpropositionalized
set of observations. Moreover, students are likely to treat the menu
of the forced-choice experiment as itself informative: They will
simply look up on the menu a word that they have heard come up
in the past. Because all of the studies are forced choice, however,
the “menu dependence” of the response pattern cannot be mea-
sured. Second, none of the studies show the effect of misleading
propositionalized accounts of the sequence of events on the par-
ticipants’ unpropositionalized memory of those events. Notably,
none of the studies use visual tests of people’s memories; that is,none ask people to choose, after being textually misinformed,
between the actual sequence of events they observed and an edited
sequence of events that fits the propositionalized account.
The Gambler’s Fallacy
The “gambler’s fallacy” relates to the propensity of people to
predicate their actions on the supposition of sequential dependen-
cies in sequences of events that are “really” independent, such as
those that are supposedly produced by chance devices, such as
coins, basketball games, or roulette wheels (Gilovich, 1991; Gilo-
vich, Vallone, & Tversky, 1985). Many basketball fans, for in-
stance, make predictions about the likelihood of a successful shot
by extrapolating from the record of success and failure on the
previous shot attempts of the player whose behavior they are trying
to predict. Experimental participants’ betting behavior seems to
ignore information about the randomness of the phenomenon that
they are called to bet on. In such cases, the operative claim is that
people “falsely” perceive sequential dependencies where in fact
there are none to be found, hence “the gambler’s fallacy.”
However, one may observe the same “fallacious” judgments
such as those coming out of purely extrapolative reasoning (naive
induction) as those that might come out of judgment processes
predicated on a causal model of the underlying phenomenon,
including a chance device. If one has a model of behavior that
predicts sequential dependence between consecutive shots on the
basis of a set of well-corroborated assumptions about the psycho-
logical and physiological characteristics of a particular player, then
one might also ostensibly make a set of judgments that seem to
instantiate the “gambler’s fallacy” without necessarily committing
the “gambler’s fallacy,” which relates to the purely extrapolative
judgment process that was described above. The reason for thisindeterminacy is that, as in our previous examples, the “gambler’s
fallacy” refers to an inferred psychological process for arriving at
a judgment, whose inferential basis is a pattern of observable
behavior that can be interpreted to lend support to several alter-
native hypotheses about the underlying processes that are used to
arrive at the same observable behavior. It is not clear, therefore, if
the “gambler’s fallacy” findings exhibit the instantiation of a
naively extrapolative approach to prediction, if they exhibit a
failure to seek and incorporate refuting evidence for a causal
model of player behavior, or indeed if they exhibit a rational
attempt to test a detailed hypothesis about the behavior of a
particular player, in a situation where the cost of “erring” is not
very high.
It would be exculpatory of the “gambler’s fallacy” argument
if it turned out that some processes that are used to generate
sequences of events about which subjects are to make predic-
tions are random in some fundamental way, if, for instance, it
turned out that there is no causal model that could in principle
account for these sequences of events. However, most “chance”
devices that are meant to produce “random” sequences of
events, such as a roulette wheel, can be modeled deterministi-
cally. A mechanical model of the wheel (Figure 2; Keller, 1986)
posits an angle (t ) between a point on the wheel and the
position of the pointer on the wheel and a torque that opposes
the motion of the wheel, such that the differential equation of
motion of the wheel is
d 2 (t )
dt 2 ; 0 t t 0, (8)
with the initial and final conditions given by
0 0;d (0)
dt 0.
Figure 2. Mechanical model variables for a spin of a roulette wheel.
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The integrated equation of motion of the wheel is given by
(t ) t t 2
2.
Because at t t 0
, t 0 0, we have that t
0 / and
(t 0) 2
2 .
Because is taken modulo 2 , an initial velocity n will lead to
a final angle 2n , which is related to n via
n [2 ( 2n )]1/2.
If the person who is spinning the wheel can measure or estimate
the initial velocity and the opposing torque of the wheel with
accuracies of and , respectively, and knows the right number
will be hit if the wheel stops at a final angle of /2, then we
can describe the dynamics of the error in the experiment by the
equation
n
2 [2( /2)( /2 2n )]1/2.
How confident can one be of getting the wheel to stop in a
neighborhood of of size , given that one has estimates of orders
and of n and , respectively? Suppose 2 . Then, we have
the reduced error equation
2
2
4
2 /2.
The range of , the right-hand side of this equation, is
2
2
4
2
2
2
4
2 .
The degree of confidence in the prediction of the outcome of the
roll of the wheel should be determined by the ratio / , that is,
P( /2 (t 0) /2) max min
.
Now, assume that can exceed 2 . If we can estimate the torque
on the wheel perfectly and need to “hit” a precise angle (t 0)
2n , then the distance between n and n1 is a decreasing-
function of n, that is,
n1 n 2
n
[1 (n2)].
As n increases, the margin or error in the estimate of n that is
needed to hit a particular will decrease, and it will take increas-
ingly accurate estimates of n to make estimates of of a given
accuracy: This may explain why roulette wheels in casinos are
spun at such high initial velocities.
We are not claiming that all lay persons actually do engage in
such sophisticated modeling of chance devices (although some
may). What we are claiming is (a) that the “deluded gambler” may
engage in some model-building for the chance device or phenom-
enon that he is trying to predict and (b) that building a determin-
istic model of a “chance” device is not epistemically unreasonable.
Indeed, the well-known “familiarity bias” (Heath & Tversky,
1991), in which individuals prefer to bet on lotteries that are drawn
on events into which they have some insight and for which they
supply their own, subjective probabilities (outcomes of certain
football games in the case of football fans) over betting on eventswith objectively supplied probabilities that are identical to the
probabilities supplied by the bettor, may be due to the presence of
a causal model that predicts a particular event in the first case and
its absence in the second case (“random draws” are random pre-
cisely because of the absence of a causal model for the process that
is generating the individual draw; Moldoveanu, 2000a, 2000b).
Building and testing causal models for individual events is not
in itself irrational. Some apparently “deluded” gambler may be
reasoning like a Popperian scientist, who builds models of local-
ized phenomena and tests them by making predictions on the basis
of those models. In the laboratory, where the phenomena one is
trying to predict have been designed to be the outcomes of sto-
chastic processes, this approach to prediction may seem unreason-
able. In the real world, however, where the successful prediction of
individual events depends on the incorporation of the information
that is relevant to the particular event in a competent model of the
underlying phenomenon, the “naive” lay person may be signifi-
cantly more successful than would be the scientist who relies on
the metaphor of the “random draw” from a known probability
distribution. Of course, a person’s model of the stochastic process
that is used in the laboratory to reveal his purported naivete may be
simple minded. In the case of the “hot-hand” phenomenon, the
model may simply be the psychological insight that a string of
successful field goal attempts gives a player a higher level of
confidence in his skill, and that higher confidence correlates pos-
itively with a higher “hit” rate. We should not, however, confuse
the naivete of the model with the naivete of the epistemic approachto prediction.
Summary: Levels of Analysis in the Understanding of
“Illusions and Biases” Literature
Table 1 ties together the ideas in the first half of the article to the
experimental results discussed in the second half. For each of the
experimental results cited, the table shows the mechanism for
generating relevant alternative interpretations and the relevant
individual-level variables and constructs that should be taken into
consideration in a thoughtful analysis of the results.
The critical analysis that we have put forth focuses attention on
several levels at which any experimental result can be “unpacked”
and extended: linguistic (syntactic and semantic), epistemological
(metacognitive), and cognitive. On the linguistic level, we can ask
whether the grammatical structure of the sentences that are used by
experimenters as subject material are understood by participants in
the same way in which they are meant by the experimenters.
Statements about cognitive competence rest on the judging of
grammatical form to be the same as logical structure (Hacking,
1984). As we saw in the “Linda” example, if “linguistic” conjunc-
tion is asymmetric, whereas logical conjunction is symmetric, we
can judge people as incompetent users of the probability calculus
when in fact we are insensitive to subtle differences between
grammatical form and logical form. Further, different interpreta-
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tions of the questions and of the interpersonal context in which the
questions were asked can lead to response patterns that are rea-
sonable given the problem that the subject was really trying to
solve, but unreasonable given the problem the experimenter
thought the subject was trying to solve. Here again, we stand the
danger of confusing our misinterpretation with subjects’ cognitive
errors.
On the epistemological level, we can ask what alternative (and
normatively valid) epistemologies can lead to response patterns
that are often judged to be “fallacious” by the standards of induc-
tive inference and Bayesian kinematics of the probability function.As we saw in several examples, different epistemological commit-
ments can lead to response patterns on cognitive tasks that are
“fallacious” by the standards of inductive inference, but are quite
reasonable given some other epistemological commitment. Thus,
we would benefit from asking (and pursuing experimental studies
to answer) the question, “What kind of epistemologists can we
expect to find among our experimental subjects?”
On the cognitive level, even after we have ascertained that
participants are “really” inductivists and that probability measures
are sound ways in which to represent their degrees of belief, there
are still different probability measure structures (Cohen, 1982) and
kinematics (Jeffrey, 1965) that can give us “normative” explana-
tory accounts of observed “deviations” from the Bayesian logic of belief. Once again, experimental investigations aimed at validating
these alternative accounts of the dynamics of belief can go a long
way to serving the “reconstructive” project that we have advocated
in this article.
Concluding Comments
As Nickerson (1996) has pointed out, the tasks that participants
in experiments involving choice under uncertainty are asked to
carry out are—on close inspection—considerably more ambigu-
ous than they seem to be at first sight. Making “correct” judgments
or choices from the information that was given to the participants
depends on an unspoken set of assumptions about that information,
which may or may not be justified. It is not clear, therefore, that
deviations from such “correct” answers instantiate cognitive “er-
rors”; they may alternatively instantiate divergences in the set of
assumptions that people make about the decision scenario. If the
latter is true, then the decision scenarios that are associated with
the “cognitive biases” literature are unspecified, and conclusions
about the “incompetence” of decision makers vis-a-vis the axioms
of calculus of probability may be premature.
We argued in this article a version of the argument that “data
underconstrains theory” that is particularized to the specific field
of cognitive proclivities that surround judgment under uncertainty;
specifically, that (a) the choice of an epistemological approach to
judgment formation and (b) the choice of the proper model for
predicting an event or forming a belief about a proposition with
yet-unknown truth value are not self-evidently implied by the way
in which a problem is formulated. We gave examples from the vast
literature that purports to document cognitive “biases” and “illu-
sions,” wherein the assumption of a single, self-evidently correct
interpretation of a decision problem is crucial to reaching the
conclusion that people are “poor intuitive scientists.” It is impor-
tant to differentiate our focus from the focus—no less impor-
tant—on the relationship between the cognitive schemata and
algorithms that people use to solve a problem and the format inwhich information about the problem is presented to them (Gig-
erenzer & Hoffrage, 1995). In that case, the “data” that is to
function as evidentiary basis for the solution of the problem is
itself a source of information, not only as “input” to the “solution
algorithm” that an individual mind uses, but also as a discriminator
among different alternative solution algorithms that one can use.
To the point, presenting data in frequency formats rather than in
probability formats seems to induce participants to use Bayesian
solution algorithms for updating beliefs without the need for
training in the logic of Bayesian probability kinematics. Although
quite different from our focus, this link between data and choice of
Table 1
Summary of Alternative Epistemic Approaches to Forced-Choice Problems Found in the Experimental Literature and of the
Alternative Bases They Provide for Answering Forced-Choice Problems
Approach Conjunction fallacy Base-rate ignorance Memory distortion effect “Gambler’s fallacy”
Personal epistemological
commitment
Popperian interpretation of choice
problem that leads to adifferent “logic of decision”
Prior model of physical device
that influences interpretationof subsequent informationabout the device
Interpretation of interpersonal logic of the situation
Gricean interpretation based oncooperation principle; menu of choices seen as informativeand relevant
Gricean logic that governsunderstanding of propositionalinformation; menu of choices seen asinformative
Interpretation of probability measure
Intuitive approach to probabilitymeasure
“Classical” interpretationof probability measure
Interpretation of probability kinematics
Bayes/Jeffrey approach toupdating probabilities
Observations versuspropositionalrepresentations of
observations
Mismatch between visualinformation andpropositional account of
visual information
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theory used to interpret or process the data will, we expect, require
a similarly detailed analysis to the one we have provided regarding
the multiplicity of possible—and possibly normative—theories for
understanding a problem statement.
Our article focuses on several sources of epistemic indetermi-
nacy in the formulation of decision problems that are used in tests
of cognitive processes: First, how is an outcome space repre-sented? As Rottenstreich and Tversky (1997) argue, the proper
“objects” that populate a probabilistic state-space are propositions
about events. Simply put, to use the calculus of probability, we
must first propositionalize an event space. We showed that differ-
ent propositionalizations of an event space can have different
implications for our use of the probability calculus. Second, what
problem is the participant trying to solve when answering ques-
tions in forced-choice experiments? If the participant is trying to
outguess the experimenter or trying to read the experimenter’s
intentions into the problem statement, then the experimental results
will reveal different features of cognition than if we assume that
the participant is reacting to the problem as it is understood by the
experimenter. Third, what epistemic approach is taken by theparticipant? We showed that, depending on the hypotheses that
were tested and the technique of drawing inferences that was used,
participants can arrive at different response patterns and can, on
the basis of normatively sound reasoning, arrive at what are
portrayed as normatively incorrect judgments. We know on a
priori reasoning grounds that several different cognitive process
theories can be used to explain the same observed sequence of
empirical observations on forced-choice tasks that require some
predictive judgment. What we have done is to give a map of the
alternative models of judgment formation that one can use to
explain findings that stem from research on cognitive biases and
fallacies and to argue that many of these models can be used to
interpret those same findings as indicative of cognitive compe-
tence, rather than of incompetence.
Of particular significance is the proposition that people’s judg-
ments may reveal a particular commitment to the process of
modeling (trial and error) that is a cornerstone of Popperian
fallibilistic reasoning, rather than be an outright failure to follow
the extensional logic of probability theory that is attributable either
to the correct application of probabilistic logic to alternative rep-
resentations of the underlying cognitive task (Rottenstreich &
Tversky, 1997) or to the incorrect applications of the logic of
probability to the intended representation of the cognitive task
(Cohen, 1981). Rather, the logic-in-use that experiment partici-
pants follow may not be fashioned on the probabilistic mode at all
and may reveal different epistemological—or metacognitive—
approaches to the task at hand that are valid in their own frame of reference (e.g., falsificationism). The art of forming beliefs about
individual events may have little to do with long-run average
statistics of past events of a particular type or, indeed, with any of
the alternative interpretations of “probability” that we outlined in
this article. Such statistics are bound to be different for different
representations of the same event, thus presaging a choice between
alternative representations of the event space. Perhaps it is time to
reconsider our indictment of the lay person’s “psychology of
prediction” and to take seriously the reconstructive program of
searching for valid psychological mechanisms for the experimental
findings in this burgeoning literature.
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