Open Journal of Philosophy
2013. Vol.3, No.2, 329-336
Published Online May 2013 in SciRes (
Copyright © 2013 SciRes. 329
Irrationality Re-Examined: A Few Comments on
the Conjunction Fallacy
Michael Aristidou
Mathematics and Natural Sciences, American University of Kuwait, Safat, Kuwait
Received December 3rd, 2012; revised January 5th, 2013; accepted January 19th, 2013
Copyright © 2013 Michael Aristidou. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
In this paper, I argue that the probability model used to infer irrationality for the subjects in the famous
Linda problem is not appropriate, and I suggest different approaches based on fuzzy reasoning models.
My line of argument is two-fold: 1) If the term “probability” is understood properly (mathematically),
then the experimenters used the wrong model. 2) If the term “probability” is understood casually (non-
mathematically), then alternative models perhaps should be used to justify the subjects’ responses. The
objective is to experiment with new ways of looking at irrationality and raise a discussion regarding the
relation between irrationality, reasoning errors and logical models that are used as frameworks to study
Keywords: Probability; Rationality; Irrationality; Fuzzy; Conjunction; Fallacy
To examine how humans reason, psychologists have con-
ducted a series of experiments, the results of which demonstrate
a high degree of deviation in “rationality” from the so called
normative standards. Some of these experiments are: 1) The
Wason’s Card Selection Task1 (Wason and Johnson-Laird,
1972) and 2) Linda’s Conjunction Fallacy (Tversky & Kahne-
man, 1982). The former was reformulated by Oaksford and
Chater (1998) using Probability Theory and ultimately the “er-
ror” in judgment was justified, and further it was actually
claimed that it was a “good” error. Here, I will be concerned
with the latter, and I will argue that although the conjunction
experiment is quite useful in giving an insight into human cog-
nitive capacities, it is not sufficient to establish irrationality on
behalf of humans. It is good enough perhaps to cast a doubt on
the claim that humans are a priori, or mostly, rational (David-
son, 1984: pp. 183-198, 1982a: pp. 302-303). Not to be accused
that I operate in the context of an exclusivi st true/false dile mma,
that only considers extreme cases (either fully rational or com-
pletely irrational), I claim from the beginning that surely ra-
tionality admits of degrees, and humans display a high degree
of rationality on many occasions, and low degree on many oth-
1In this experiment, subjects have to pick those cards out of four that makes
a certain rule true or false. The four cards on the side that the subjects can
see have a “4”, “7”, “A” and “D” written on them. The rule is “If a card has
a vowel on the one side then it has an even number on the other.” On the
question of which cards’ other side one has to see in order to falsify the rule,
around 90% of the subjects responded incorrectly “A” and “4” where the
right answer is “A” and “7”, This controversial experiment revealed some
flaw in the deductive capacities of humans, but it surprised also researchers
much more when a more concrete version of i t ( Robertson, 1999: pp. 69-70),
dealing with everyday matters dramatically increased the correct responses
that people gave. For example, subjects did far better when they were asked
to play the role of a policeman that is trying to find whether the law of under
aged drinking has been violated. The subject has to go in a bar, where at one
of the tables four people are sitting. The first person is 15 years old, the
second is 45 years old, the third is drinking lemonade and the fourth person
is drinking beer. The rule is “If a
erson is under 18 years old, then she is
not allowed to drink alcohol.” Note that this task is nothing but the Wason’s
selection task stated above, where “lemonade” corresponds to “4”,“beer” to
“7”, “A” to “15” and “45” to “D.” On the question of what one has to do in
order to check th e rule, mos t of t he subj ects foun d the ri ght answ er which i s
check the 15-year old, and check the beer to see by who is being drunk. The
reason why subjects found the concrete version of Wason’s Task easier is,
accordin g to Roberston , that “certai n contexts made the selectio n task much
easier because the task fitted in with people’s previously constructed sche-
mas about those contexts”(Roberston, 1999: p. 71). People are more familiar
with permission/obligation contexts, therefore it was easier for them find the
correct answer, namely “15” and “beer”,since the reasoning that generates
those answers is clo ser to the permission/obligation schema.
The Conjunction Fallacy
If the probabilistic approach was effective on selection tasks,
such as Wason’s Task, it does not necessarily imply that the
same probabilistic techniques might work in other reasoning
tasks too. Proponents of the probabilistic approach think per-
haps that probabilistic reasoning is at the heart of human rea-
soning (carried away by its success on the Wason’s Task), but,
as Tversky and Kahneman (1982) revealed, this is not the case.
People tend to violate standard probability rules in many cir-
cumstances and “irrationalities” are apparent, even when adopt-
ing a probabilistic framework to explain reasoning.
For example, people seem to conform to the conjunction fal-
lacy in probability judgments. The most famous example per-
haps is this:
Linda is 31 years old, single, outspoken and very bright.
She majored in Philosophy. As a student, she was deeply
conce rned with issue s of disc riminat ion and soci al justice,
and also participated in anti-nuclear demonstrations (Tver-
sky & Kahneman, 1982: p. 361).
The subjects were asked to state which is the most probable
from the following five statements:
1) Linda is a teacher in an e l e m e nt a ry school.
2) Linda is a bank teller.
3) Linda works in a bookstore and takes yoga classes.
4) Linda is active in the feminist movement.
5) Linda is a bank teller and is active in the feminist move-
The majority of the subjects chose 5) as more probable that
2), violating a fundamental law of probability: that the prob-
ability of a conjunction must be less (or equal) than the prob-
ability of its conjuncts. This, according to the experimenters, is
an “irrationality” case. But is it? One could say that the above
“irrationalities” are irrationalities with respect to the approach
one chooses to model human reasoning. Oaksford and Chater,
commending on Tversky and Kahneman’s results, argue that:
If this [mismatch of probability and rationality]2 is correct,
then the whole idea of rational models of cognition is mis-
guided: cognition simply is not rational (Oaksford & Chater,
1998: p. 18).
One could easily disagree with this. If human reasoning fails
to be modeled or formalized by probability theory, does this
make human reasoning irrational? Humans might seem “irra-
tional” with respect to probabilistic systems of inference, but
we have no reason to believe that these are the only systems
that can give an account of human reasoning. There can be
alternative systems that come closer to how humans reason and
explain many of the alleged irrationalities that resulted from the
experiments. For example, a Fuzzy Reasoning System3 could
be, in the words of Stich, a “pragmatically superior alternative”
to the probability system mentioned above. In other words,
there could be better systems of inference relative to a problem,
as some systems (given a problem) might be more general and
provide more explanatory frameworks. They could also take
into consideration the weight of some information and its rele-
vance. For example, words like “discrimination” and “social
justice”, included in the information about Linda, are words
that indicate relevance or irrelevance to choices 2) and 5), a fact
that probability theory ignores. Also, words like “very” and
“deeply” indicate weight, or a degree, that certain information
about Linda bear in relation to choices 2) and 5), a fact that
probability theory also ignores.
Stich’s Theory
Davidson’s Principle of Charity states that “one should in-
terpret an agent’s utterances in such a way that most of her
assertions turn out to be true and most of her inferences turn out
to be rational” (Davidson, 1984: pp. 183-198, 1982a, pp. 302-
303). In other words, when we are interpreting someone and we
find that his reasoning deviates too much from what people
usually hold as “rational”, then it is more likely, according to
Davidson, that we are interpreting the subject incorrectly, and
less likely that the subject is irrational (Davidson, 1982a: p.
Inference that is frequently irrational, according to Davidson,
is conceptually impossible. That is because inference—a proc-
ess that generates beliefs—must have high levels of rationality
and truth. But, how could one link rationality with beli efs? The
keyword is interpretation, and Davidson’s argument (Davidson,
1982a: pp. 302-303, 1982b: p. 327; 1984: pp. 195-198, p. 170)
revolves around the following idea: the meaning of a word
cannot be fixed if what the agent utters by that word is not in
accord with what he means by that word and what he believes
about that word.
According to Davidson, an agent is irrational if he somehow
generates beliefs that do not cohere, or are not consistent, with
the pattern of his beliefs. Davidson’s approach to rationality,
through interpretation, demands high degrees of consistency.
Hence, any approach that allows much inconsistency is not ac-
ceptable to Davidson. That is clear when he says that: “… in-
consistency breeds unintelligibility” (Davidson, 1982a: p. 303),
and when he says:
If we are intelligently to attribute attitudes and beliefs …,
then we are committed to finding in the pattern of behavior,
belief, and desire a large degree of rationality and consistency
(Davidson, 1980: p. 237).
Stich argues (Stich, 1990: pp. 15,17) that if Davidson’s posi-
tion is true then the following problems arise:
1) Our ability to explore irrationality empirically is under-
mined; hence we are losing a good insight into human cogni-
tion, since the interpreters are the ones most probably mistaken.
2) The claim that bad reasoning is conceptually impossible
leads to a normative theory of rationality of no practical impor-
tance, since the theory turns its back on the empirical results.
Stich thinks that Davidson is right to say that content and ra-
tional reasoning are linked, but wrong in the assumption that
there is only one type of rational reasoning. The latter assump-
tion is something that Stich opposes, since he claims that it is
not the case that there are no alternative systems of reasoning
that are all rational. This is something that Stich calls Norma-
tive Cognitive Pluralism.
But how could one decide then that a system S is a rational
cognitive system? Stich mentions Goodman’s attempts to de-
scribe a procedure (or a test) that a system of inferential rules
should pass, in order to count as rational. But as Stich observes,
it is very difficult to discern the relation between rationality and
“appropriate” test, because we basically assume a priori that the
test itself is appropriate. Hence one needs a more realistic ac-
count of what makes a system rational. Perhaps a middle
ground could be found. Davidson recognizes high degrees of
consistency, but nevertheless speaks of “degrees” and not ab-
solute consistency. Also he alerts us on the conceptual difficul-
ties of inference being frequently irrational. Stich demands a
more realistic system of inference, which possibly considers
relevance of data, and is more explanatory on how actually
humans reason in praxis and not normatively. A fuzzy system
could be such a system. I do not claim that this is the only cor-
rect way to go, but based on the nature of the problem at hand
(Linda’s Problem, which contains fuzzy parameters) it seems
that it is at least m o re explanatory.
2My italics.
3By fuzzy reasoning system I mean broadly any system of inference that
takes into consideration relevance of data, imprecision, subjectivity, percep-
ased information, and is closer to how humans actually reason, and not
how they “ought” to reason. For more on Fuzzy Reasoning see Siler and
Buckley (2004). Harten (2008) in a Master’s thesis examines a variety ap-
roaches in solving the conjunction fallacy and he concludes that fuzzy
theory made progress on the issue but did not resolve it in all cases. Instead
oses a
Copyright © 2013 SciRes.
A Fuzzy Model Perhaps?
I mentioned previously that assuming a probabilistic way of
thinking, the “irrationality” appeared in Wason’s Cards Task
was in a way explained away. That is, humans seem quite ra-
tional if we ascribe to them a probabilistic way of thinking
rather than a propositional-logic way of thinking. But how
about the “irrationality” appearing in Tversky and Khaneman’s
Linda problem? Could that be set in a framework in which, if
not to ascribe rationality to the responders, at least explain their
error in a rational manner? It is true that the subjects erred
(probabilistically?), but could one at least, suggest a formal
model that would explain their reasoning? As Oaksford and
Chater did in the Wason Task, to some extent, one could justify
people’s choice of premise 5) over 2) in the Linda Problem by
considering a non - c l a ss i c a l a p p r oach to the probl e m .
It is important to clarify that the reason I suggest another
model is not because I want to render humans a-priori rational
or defend their judgment. It is simply because the (classical)
probability model that was used here seems to be completely
inappropriate for the problem at hand. Hertwig and Gigerenzer
(1999: pp. 276-278), tried to explain the fallacy indicating the
ambiguity of the word “probability”. I agree to a large extent,
but I actually also claim that if indeed by “probability” the sub-
jects understood the (classical) mathematical probability, as the
examiners clearly intended, then the fault still lies more with
the examiners as classical probability is not the right model to
formulate this particular problem. The reason is because a good
amount of data that refer to Linda are fuzzy data, including the
conjunctive statement 5).
For example, from the description of Linda, it is clear that
her being “outspoken”, “very bright”, and “deeply concerned
with discrimination and justice issues” are all fuzzy data. Also,
one of the conjuncts in statement 5), namely “active in the
feminist movement”, is also fuzzy, which makes the whole
statement 5) a fuzzy statement. So, it is not just that the data did
not entered the probability calculus in this problem, as Hertwig
and Gigerenzer (1999: p. 276) correctly pointed out, it is also
that it is the wrong probability model to account for those data.
The appropriate model for the problem is Fuzzy Probability
(Zadeh, 1968, 1984; Buckley, 2010) which extends classical
probability as it accounts also for fuzzy events. It is also sensi-
tive to the degrees of relevance and membership of certain ele-
ments in certain sets. I will not get into the mathematical tech-
nicalities, as it is beyond the scope of this paper, but it should
be noted that the conjunction law should also apply in the fuzzy
probability case (as the latter generalizes the classical case), and
the experimenter should be aware of that when using the ap-
propriate model.
So, the only problem, as we said, is that fuzzy probability is
supposed to generalize, not to reject classical probability, so
any rule true in the later (such as P(A and B) < P(A)) must be
also true in the former. Hence, in fuzzy probability Pf(A and B)
< Pf(A) also applies. It seems that the experimenters might be
correct after all. But, it also seems that they were unaware of
the following three cases: 1) P(A and B) < P(A) where A, B
crisp, 2) P(A and B) < P(A) where A, B fuzzy, 3) Pf(A and B) <
Pf(A) where A, B fuzzy. Randomness is distinguished from
fuzziness as a form of uncertainly (Kosko, 1990). Obviously,
the experimenters used case 1) but this is not the right case
(considering the problem) despite of the fact that all three cases
render the same result. So, it is possible that humans are sus-
ceptible to the conjunction fallacy, probability errors, or as
Gould (1992: p. 469) said that “our minds are not built to work
by the rules of probability”, as the numerous experiments show4.
But, the original Linda problem has not showed that yet, as it is
essentially a fuzzy problem and is using the wrong probability
calculus. It is worth noticing, by the way, that most variations
of the Linda problem in the literature that came to my attention
are crisp versions, or do not have the degree of fuzziness as the
original Linda proble m (Tversky & Kahneman, 19 82, 1983; Bar-
Hillel & Nater, 1993; Tentori et al., 2004).
One could get into the technicalities and certainly show that
indeed Pf(A and B) < Pf(A), where A, B fuzzy, but we will not
do it here, as it is beyond the scope of this paper. It seems also a
bit pointless, as even if it is true, what exactly would this fact
say on the issue of rationality? Would it necessarily imply that
humans are irrational? Leaving aside broader objections to the
notion of probability per se, the question still remains: How
exactly probability, crisp or fuzzy, re lates to human reasoning?
Is even the term “probability” the appropriate term to be ana-
lyzed here, in the example of Linda? Is this term what the sub-
jects understood and interpreted?
It is possible though, as Hertwig and Gigerenzer (1999: pp.
276-278) suggest, that mathematical probability was not what
was understood in the experiment, assuming that the relevance
maxim applied. Then, I would like to suggest the following
approach (“Fuzzy Membership/Relevance Interpretation”) as an
alternative to the mathematical interpretation of probability. It
is possible that the subjects interpreted “Linda most probably
is” as “Linda most likely belongs” (or “most likely member
of”)5, and if that is the case, then the following (fuzzy, but
non-probabilistic) scenario might be plausible:
Let us view Linda as a (fuzzy) set of properties, i.e. as the set
L = {Phil, Discr, Int, Outsp, …etc}, where Phil = majored in
Philosophy, Discr = concerned with discrimination issues and
social justice, etc. The crucial properties of “being a bank
teller” and “being a bank teller and active in the feminist
movement” denote them by B and BF, respectively. The point
is what is the relation of the B and BF to L. In other words,
qualitatively and quantitatively speaking, how, which, and how
many of the properties that characterize L relate to the B and
BF. Now, based on the relation “relevant things go with rele-
vant things”6 which seems consistent also with neuroscientific
evidence (Nielsen, 2003: pp. 118-119), it is more likely that BF
relates more to L, casually, on the basis that usually less Phi-
losophy majors become bank tellers, and usually more people
that are “deeply” concerned with issues of discrimination, etc,
are active membe rs of feminist movements. Do B and BF relate
to any of the properties of Linda? In virtue of “being active in
the feminist movement”, yes, to some degree BF does relate.
4Also, in Manktelow (2012: p. 245), even though he advocates the probabil-
istic paradigm, he states that “these formal systems ask too much of ordinary
5Or ignored the “probability” term altogether as irrelevant, and simply
looked for relevance (likelihood, membership, etc). After all, these assess-
ments about people in everyday life are not done by throwing dice.
6That’s a very realistic classification, since humans tend to classify things
that way. As Nielsen says: “For example, if the first three words (the as-
sumed facts of the consensus building process) are down the garden, con-
sensus building is then carried on the fourth (answer) region to yield an
answer token, which turns out to be (in our system), path. This four-word
sequence actually occurred in our training cor
us. If we then blank out the
token for the word path and repeat the thought process, the next word ob-
tained is lane. (Nielsen, 2003: pp. 118-119).
Copyright © 2013 SciRes. 331
On the other hand, B does not seem to relate much. This deter-
mination probably strikes one as too subjective, but, that is
exactly the point on why a fuzzy model (not necessarily fuzzy
probabilistic) might be better. It allows for subjective judgment
and accounts for perception-based information and fuzzy data.
The subjects in the experiment had exactly this to do: a sub-
jective judgment given fuzzy data. Therefore, one could claim
that there was some rational basis for people choosing state-
ment 5) over 2) in the Linda problem.
More precisely, and without getting too technical, one could
suggest the following model of fuzzy reasoning (generalized
modus ponens) which incorporates how humans tend to classify
things based on membership and relevancy:
If x is A, then y is B
x is A'
y is B'
where x, y are variables taking values from the universal sets X
and Y, A and A' are fuzzy sets in X, and B, B' are fuzzy sets on
The above is quite useful for modeling human common sense
reasoning, which is usually reasoning in fuzzy environments.
For example:
If a book is large, then it is expensive
Book x is fairly large
Book x is fairly expensive
Conclusion B’ is calculated for any y in Y by the formula:
 
maxmin,x ,
usually referred to as the compositional rule of inference (Klir
et al., 1997: p. 212), where I is an appropriate fuzzy implica-
tion7. For simplicity and economy in calculations, we usually
adopt the matrix f o r mat:
in order to compute the conclusion B', where “o” is the max-
min operation8.
In regards to the Linda problem, we basically follow the
“guessing game” application9 in Tanaka (1996: p. 76) mod
ifying things slightly. Instead of implication I will generalize to
relevancy (they are both types of relations). In other words,
instead of whether Phil implies B and to what degree, for ex-
ample, I will check whether Phil is relevant to B and to what
degree. The way I will produce the relation RI (hereafter R) will
be empirically (i.e. experimentally), and I will only be con-
cerned with two of the crucial properties of Linda, namely
“Philosophy major” and “deeply concerned with discrimination
issues and social justice”. By “empirically”, I mean how sub-
jects would actually infer (not how they supposedly ought to
infer) based on their life knowledge and experience, and the
specific data given. I explain:
Let X = {B, BF}, where B = bank teller, BF = bank teller and
feminist, and Y = {Phil, Discr} where Phil = majored in Phi-
losophy, Discr = concerned with discrimination issues and so-
cial justice. Then, given the data and people’s prior knowledge
and experience we have the following relation R:
B 00.2
RBF 0.30.8
where each entry shows the degree of relevance between two
parameters and was determined10 practically as follows:
A group of 10 college students were asked, to their knowl-
edge, to correlate with a line and a percentage degree (degree of
relevancy) the following two columns:
BF Int
Outs p
where the right column clearly contains some of the properties
of Linda. The average rounded percentage (normalized to 1)
correlating Phil and Discr to B and BF are shown in R above.
Given the description of Linda, a fair assessment regarding
properties Phil and Discr would be:
7Say Lukasiewicz’s or Mandami’s implication, etc. But, as known, there is
no a priori justification on how to interpret an implication, or any logical
connective for that matter. At best, one tries to capture some basic intuition
of what a true or partially true proposition is, with respect to certain phe-
nomena or applications at hand. One could actually push the argument all
the way and say that experience would be the only arbitrator of truth to an
implication. For example, consider the implication: “If I am a Philosophy
major, then I am an active feminist.” The truth or partial truth of this could
only be assessed by experience. In other words if actually, if to our knowl-
edge, there are philosophy majors who are active feminists, to what degree
they are active, how Philosophy relates or not to feminism so we can put
weight to future assessme nts, etc. Probability here has little to te l l us .
8It works as follows:
 
aa bcab bd
aba b
ca dc cb dd
cdc d
 
 
 
 
 
9“Sally is nearsighted and colorblind. When she goes to the fruit market,
where fru its are placed on high shelves , she caanot s ee them very wel l. She
can only recognize the size and blurred shape of the fruits. Sally lived in
such a world for 20 years and now she has some knowledge about the fea-
tures of the fruits. For example, tangerines are round and relatively small.
When Sally says ‘quite round and quite large’, can you guess what fruit she
sees?” (the solution in T anaka, 1996 : p. 76).
10More rigorously, one could define a relevance relation
RX between
two sets X and Y (“X is relevant to Y”) to be the matrix whose entries are
given by the relevance operator
RX x
, where
is the
degree of relevance of xi to yj with respect to some attributes and based on
data. This is actually what in praxis the students in the experiment used
when asked to correlate with a percentage the two columns in p. 7, the an-
swers of which filled in the matrix entries for the relation R. This relevance
relation (neither reflexive, nor symmetric, nor transitive—the implication is
reflexive and transitive but not symmetric) is a slight generalization of the
usual equivalence and compatibility relation in fuzzy logic. Also, given the
above, one could also define a relevancy measure by:
, where
is the degree of relevance of yito
j, and |X|, |Y| are the cardinalities of X and Y. Notice that for the special
cases Y = {B} and X = {Phil, Discr}, and Y = {BF} and X = {Phil, Discr},
we have:
0.1 0.55
R X, which once again shows why the
subjects would conside r BF as more relevant to the description of Linda.
Copyright © 2013 SciRes.
as she did major in Philosophy (so degree of membership = 1)
and she is “deeply” concern on issues of discrimination (hence
high degree of relevancy). Therefore, the matrix formula for the
conclusion (or “guess”) gives:
 
10.9 0.30.8
0.3 0.8
C LoRo
 
i.e. , which says that the subjects ranked BF
higher than B, and saw it more possible given the data.
0.3 0.8C
What about the probabilities then? Our model does not re-
quire any. And even though the question posed to the Linda
problem contain the term “probability” the subjects do not, and
apparently did not, have to interpret it literally. The problem is
a real life problem that involves human experiences and
knowledge; it is not a problem to be decided by throwing dice.
The question posed was not “what is more probable to have in
general or theoretically/mathematically speaking”, but “what is
more probable Linda is”, given the data. Have the data been
taken into consideration? And how were they interpreted, in a
Bayesian or a frequentist manner?11 And if they have, and sub-
jects failed the “test”, why not just simply say they made a
mistake, or they are bad in probabilities, just like millions of
perfectly rational people are bad in basic mathematics? There is
a difference on how we actually reason and how we ought to
reason. And the evidence that we somehow ought to reason
probabilistically is not quite convincing.
Again, the experimenters ignored the degrees of membership
of B and BF to L, i.e. to L’s “relevant” properties. Their sub-
jects though did not consider all properties of L as equivalent,
and it seems that they considered certain properties having
more relevance and weight to the presented state of affairs that
they were asked to make a choice. For example, what if the
choices were:
1) Linda is an alien.
2) Linda is an alien and is active in the feminist movement.
Most likely, from the description, she is not an alien, and we
do not know whether aliens care or not for women’s rights. But,
the description does point to some relevancy of Linda to femi-
nism. So the subjects could have thought that Linda sounds like
a feminist, and whether or not she is an alien is secondary. Be-
cause just being an alien doesn’t necessarily imply that she is a
feminist, but being an alien and a feminist certai nly implies that
she is a feminist. So, the crucial quality or property of Linda
being a feminist was “locked” first, to be consistent or relevant
to the description, and the rest of the properties are either ir-
relevant or are examined second. And, so what if in general
there are less aliens feminists than aliens? In this particular case,
the particular subject L is more likely to be a feminists whether
she is an alien or not, in virtue of her properties/qualities.
Some Objections
Before I turn into some possible objections, it is important to
once again clearly state what are the main points that I am ar-
guing for. First, I am criticizing the methodology of Tversky
and Kahneman, the way they arrive to their conclusions, and
what those conclusions entail for human rationality. Obviously,
there is a vast literature devoted to this discussion and I could
not comment or refer to all of it. It is important to note, how-
ever, in what my criticism differs from other criticisms of the
Linda case. It differs to the point that it indicates a mistake on
behalf of the experimenters: The classical probability model
used in the Linda problem is not the right model. One has to
acknowledge that at first, and then continue with any further
discussion. Would a fuzzy probability model render different
probability outcomes? No, but that is not my point of concern.
My point is not to find an alternative model which will judge
the subjects rational. My point is to not to judge them irrational
on the basis of models they do not know, and they are not obli-
gated to know. It is easy for me to give you an algebra problem,
of a level you have not cover, and because you failed to solve it
to judge you irrational. It is seems unfair. But, this is what the
mathematical interpretation of probability in the Linda problem
amounts to. Is correct responding to probability questions a
pre-requisite to rational thinking? This is what the experiment-
ers seem to assume, but I am not sure yet as to why anyone
should accept such a prerequisite.
Second, as I stated earlier in this paper, I think a non-classi-
cal reasoning model is, perhaps, a more promising model in
describing how humans reason. Perhaps … this is the key word
here. I do not currently have an alternative model, and I am not
aware of anyone who does. The argument behind exploring an
alternative non-classical model is based on the fact that the
current one, that is classical probability, failed. People do not
use probability calculus to make decisions; most do not even
know what that is. Even in matters that involve chance most do
not use probability to make decisions. Poker, for example, is a
good example. Furthermore, there is a big distinction between
chance and imprecision (two different types of uncertainty) and
the problem of Linda is not a clear cut chance problem. The
argument behind the suggestion for a fuzzy model is simply the
nature of the data in the description of Linda. It is only reason-
able to request that the probability of fuzzy data to be ac-
counted with the correct model, if one is interested in linking
further those results to rationality. Yet, my point still stands.
The subjects are not obligated to know probability, classical or
fuzzy, to count as rational. Unless, someone (the experimenters
in this case) makes an argument on why should it be so. Finally,
say that the people went through a quick probability seminar,
they were all taught the conjunction rule, and they applied it
correctly and responded as they should. Would they all count as
rational then? How so? Or would that just simply mean that
they know probability, applied it correctly, and solved a math
problem successfully?
Now, a few comments on some specific objections:
1) I mentioned earlier that humans might seem “irrational”
with respect to probabilistic systems of inference, but we have
no reason to believe that these are the only systems that can
give an account of human reasoning. One perhaps could object
as follows: Suppose that we changed the system, and hence
make better sense of peoples responses. Then, the fact that
people contradict basic rules of probability theory does seem
problematic to rationality.
11It is well known that in probability the two approaches do not agree. For a
discussion on t he different definitions of probability s ee Manktelow (2012:p
2-5). It is worth noticing that in the same book, that the author in more than
one occasion (p. xii, p.247) distinguishes it from binary logic, or calls it
non-binary, but it is not clear to me what he means, as probability is also
ased on binary logic. He also never mentions fuzziness or fuzzy probability
Fuzzy logic can be distinguished from binary logic, but even that on the
level of theorems is base d on binary logic. I disagree. I believe that the last statement begs the question.
How exactly does probability relate to rationality? One cannot
Copyright © 2013 SciRes. 333
simply refer to the Linda test. The fact that they failed it does
not render them irrational, as the link between probability and
rationality has not been established yet. Neither their success to
the Linda test would render the subject rational. When Giger-
enzer et al. claim that “we should not care about being irrational
in that sense”, this I interpret I as a quiet admission that prob-
ability does not have much to tell us about rationality. It is in-
deed not quite clear how introducing another logic one could
shed more light on the Linda problem. But, considering the fact
that the logic used is at least controversial, if not the wrong one,
then it is only reasonable for one to explore other venues.
2) When I spoke of “alternative systems that could give an
account of human reasoning”, one could quickly respond: What
does it mean to give an account of human reasoning”? Would
not this simply amount to specify the rules that we follow and
the calculus of those rules? If this is so, then such a project
would be simply descriptive, and therefore the rationality ques-
tion would still remain untouched.
Regarding the last statement, I believe the same applies for
probability theory. The model in the Linda experiment gave at
best a descriptive account, but has not been related to the ra-
tionality question in and of itself. Furthermore, specifying the
rules that we follow and the mechanics and unfolding of those
rules is one issue, but there are other issues too. For example,
those rules have to be empirical too, and have some connection
to how humans actually reason. Most humans are not aware of
the specified probabilistic rules, their calculus, etc, and they
certainly do not follow them even in matters of chance. So, I
am not sure how much it can inform us about how humans
reason and whether their reasoning is rational.
3) We noted that words like “very” and “deeply” indicate
weight, or a degree, that certain information about Linda bear in
relation to choices 2) and 5), a fact that probability theory also
ignores. Question: How does being very bright relate to
Linda being a bank teller or a feminist bank teller?
This is not to say that feminists are smarter or that if you are
smart you must be a feminist. It is more to show that being a
feminist adds another dimension to Linda’s qualities. Positive
or negative, the point is that she thought about it. To put it
bluntly, she is not simply counting money; she is concerned
with other issues too, and exposed herself, as she also majored
in Philosophy, to more interesting and difficult issues such as
human rights, etc. It could be that one is very bright but has no
other interests, but here it is clear that the description of Linda
points to the fact that she is involved in other topics, and has
opinion on matters. Such a person is perhaps more rare to find,
but it is not going to be decided by throwing dice.
4) I tried to emphasize and clarify that the reason I suggest
another model is not because I want to render people a-priori
rational or defend their judgment. It is simply because the
(classical) probability model that was used here seems to be
completely inappropriate for the problem at hand. One then
could ask: Inappropriate how and to whom? If people were
using the model correctly, then perhaps they would not have
failed the test this way. Why is it inappropriate for us to judge
them based on that model? Is the model still inappropriate if
people had been reminded of the relevant rules of probability
It is inappropriate tool to model the experiment. This is sim-
ply a mistake on behalf of the experimenters, and I am indicat-
ing it. It might not affect the conclusion of the experiment, but
the experimenters have the obligation to apply the right tools.
People have no obligations to use, or know, the probability
model at all, just because the experimenters included the word
“probability” in the question. Certain issues are not settled by
throwing dice, and the distinction between randomness and
fuzziness (imprecision) needs to be drawn. And probability,
even on everyday matters of chance, gives us an idea and a hint,
but we certainly do not know or explicitly compute rules of
probability in order to make a decision. And, say that the peo-
ple went through a quick probability seminar, as we mentioned
above, and they were all taught the conjunction rule, and they
applied it correctly and responded as they should. Would they
all count as rational then? Or would it just simply mean that
they know probability, applied it correctly, and solved a math
problem successfully?
5) I noted that “a good amount of data that refer to Linda are
fuzzy data, including the conjunctive statement 5).” One could
object: In what sense this data is fuzzy? People are receiving a
description that involves no numbers, so how is this fuzzy
data”? For example, Linda being outspoken”, “very bright”,
and deeply concerned with discrimination and justice issues
these are supposed to be fuzzy data”, but this is not quite clear
what this is supposed to mean.
These are fuzzy date in the sense that the sets of all “bright”
people, or “outspoken” people, are inherently fuzzy. The mem-
bership or not in that those sets is not a yes or no situation, but
there is a degree of membership in those sets for each case we
examine, which is determined subjectively. Now, these mem-
bership degrees in the corresponding fuzzy sets are supposed to
be taken into considerations when one consideration when one
understands probability in the mathematical sense and does the
calculations explicitly. Is this what the expectation of the ex-
perimenters was?
6) Earlier we mentioned that “based on the relation ‘relevant
things go with relevant things’ which seems consistent also
with neuroscientific evidence, it is more likely that y2 relates
more to L, casually on the basis that usually less Philosophy
majors become bank tellers, and usually more people that are
‘deeply’ concerned with issues of discrimination, etc, are active
members of feminist movements.” One could say: What neuro-
scientific evidence have to do with the question at issue? Also,
where are the claims about philosophy majors becoming bank
tellers or not coming from?
I am simply saying that humans tend to classify and group
things based on relevancy. For example, if I tell you “ice-
cream” you could reply back to me some relevant things that
your mind related to it by experience. Such as “cone”, “cold”,
“cup”, etc. You probably would not replied “calculator” or
“fish” to me. And that is what I think happened with the Linda
case. The subjects most I think saw more relevancy between
“feminism” and “philosophy major” than between “bank teller”
and “philosophy major”. If this is backed by experience, which
I think it does, then the experimenters should take a gallop and
actually count whether bank tellers or feminist bank tellers had
philosophy degrees, or took philosophy classes, or have broader,
or casual, philosophical concerns in general, etc.
7) Finally, I mentioned that “the question posed was not
‘what is more probable to have in general, but what is more
probable Linda is’, given the data.” One could say: Well, this is
not the question posed to the subjects after all.
Yes, but why was the description given to the subjects then?
Was it given to mislead them? Was not given to them so they
could weigh, and take into consideration, the information about
Copyright © 2013 SciRes.
Linda and compute the probability? Because if not, and the
description was irrelevant, then we basically test the subjects on
whether they were aware of a certain math rule, namely the
conjunction rule, and because of the lack of that knowledge we
rendered them irrational. I do not believe that was reasonable,
or fair.
Concluding Remarks
Davidson argues that inference that is frequently irrational is
conceptually impossible since inference (a process that gener-
ates beliefs) must have high levels of rationality and truth.
Therefore, humans must be rational. Davidson’s argument is a
plausible argument if one relaxes the demands for consistency.
That wi ll make it more pragmatic and bring it closer to Stich’s
position and experimental results. But then, one needs a new
(more general) framework and model to study rationality.
Just as Oaksford and Chater have suggested that committing
the fallacy in Wason’s Task was, actually, the rational thing to
do, other authors (such as Dulany & Hilton, 1991, Slugoski &
Wilson, 1998) claim that committing the fallacy in the conjunc-
tion problem was a rational response. On the other hand, other
authors (such as Stanovich & West, 1998, 2000) conducted
studies which showed that subjects that committed the conjunc-
tion fallacy also scored lower (in general) in SAT tests than
subjects that did not commit the fallacy. This fact is an impor-
tant empirical piece of evidence that comes in the discussion,
although it is still not clear how it translates in terms of ration-
ality. Also, it is uncertain what SAT has to do with rationality
anyway, considering the extensive criticism12 against it on be-
ing an unfair, and biased test, and irrelevant to measuring rea-
soning skills. Perhaps if we tested the subjects in chess playing
the ones who committed the conjunction fallacy might have
been better chess players than the ones who avoided the fallacy.
But that is not the point. Because whatever the test is, it still
falls under Goodman’s project to describe a test that tests ra-
tionality, which in turn falls into Stich’s criticism who says that
it is very difficult to establish a relation between rationality and
“appropriate” test, since we always assume a priori that the test
itself is appropriate.
In conclusion, a non-classical reasoning model is, perhaps, a
more promising model in describing how humans reason. It is
not the first time that an alternative to probabilistic models has
been suggested. Cohen (1979) suggested a Baconian model of
judgment as an alternative to Bayesian ones. Perhaps, a fuzzy
reasoning model (not necessarily fuzzy probabilistic) could be a
better way to model humans’ natural way of communicating
and reasoning. As Hilton says:
Understanding conversational inference may help clarify the
question as to which normative model is appropriate in a given
situation (Hilton, 1995: p. 266).
Indeed, since an important aspect of the conversational in-
ferential model is the effect of conventional and conversational
implicatures in which fuzzy words such as “intelligent”, “ac-
tive” and “few” are frequently used (see Hilton, 1995: p. 265),
then we need a model that is appropriate to deal with such
vague concepts and respects implicatures.
Whether there was an error on the part of the subjects (in re-
sponding the way they did to a probability problem) or an error
on the part of the experimenters (in testing subjects on prob-
abilities to account for rationality), the lesson is that alternative
models of reasoning perhaps need to be considered. Models that
are not blind to relevance or imprecision. The nature of the
testing, perhaps, needs also to be inverted. The objective should
not be how well human reasoning approximates a reasoning
model, but how well a reasoning model approximates human
I would like to thank Andrei Zavaliy for his comments on an
earlier draft as well as the anonymous referee for his comments
and productive criticism.
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