Hence, even if we start by following the Bayesian approach we sooner or

later face the task of re-evaluating of our prior (“a priori”) probabilities. How

are we to do so? There is a branch of studies called “probability dynamics” that

addresses this very problem. I am not proposing here to offer any criticisms

of the ¬ne print of these studies. However, they do not incorporate one of

the most important ways in which we in fact reevaluate those probability

distributions on which our rational judgments of credibility rest. The evidence

(information) that is the basis of our knowledge of prior probabilities does not

come to us as a gift from a Platonic heaven or from some inscrutable oracle.

It comes from different sources, which in a rational inquiry are known to the

inquirer.

But how does this analysis apply to the conjunction effect? What we have

to realize is that what in a Bayesian approach are the priors is the probability

distribution on the informational input the evaluator receives from different

sources. And one of the most common ways of evaluating these probabilities

(degrees of credibility) is by considering the source of the different items of

input. This is, of course, the ¬rst and foremost question we raise concerning the

testimony of human witnesses. Is the witness likely to be reliable or not? And

such reliability is not an a priori matter but has to be evaluated by reference to

other evidence plus, of course, some a priori knowledge. For instance, we may

look at the other answers by the same witness to see whether he or she is in

a position to provide information or compare the witness™s answers with facts

that we know (or assume to be known). In a simple case, if a witness™s testimony

contradicts known facts, or other testimony by the same witness, we know that

he or she is lying. Conversely, if the witness™s answers conform to what we are

entitled to expect on the basis of known facts, or con¬rms with the witness™s

other testimony, our estimation of that witness™s reliability is enhanced.

A Fallacious Fallacy? 215

There is nothing magical about such a posteriori judgments of reliability,

either. Whenever enough background information is available, they can, in

principle, be made in accordance with the usual laws of probability calculus.

For a simple example, let us assume that it is initially known that there are two

possibilities concerning a given source of information, both equally likely”

namely, that it always yields a true answer and that it yields a true answer with

the constant probability 1/10. Then, if we receive an answer that is known to

be true, the probability that that source of information always tells the truth

is raised from 0.5 to 0.8333. . . .

Now we come to the crux of the conjunction “fallacy.” The fact that answers

like T or (T & F) come from a different source of information does not

change the objective evidence E on the basis of which their credibility has

to be judged. But in the light of what has been said, it may affect the prior

probability distribution. For the sources of the information may affect the

prior probabilities P1 (T) and P2 (T & F) that are the basis of the condition-

alized probabilities P1 (T/E) and P2 ((T & F)/E). Even though the former are

known technically as “prior” probabilities, according to what was found they

can be affected by evidence, though not by conditionalization. And part of

that evidence is the two different messages that the two sources provide the

evaluator with. For when we are evaluating an answer A from a source of

information with an unknown but presumably constant propensity to give

true answers, that particular answer A itself constitutes part of the data on

the basis of which the reliability of its source, and hence its own reliability,

must be judged. Hence the probabilities assigned in the course of an actual

inquiry to two answers (messages) A, B from two different sources of infor-

mation may have to be judged by reference to different “prior” probability

distributions.

In particular, even when on any ¬xed evidence E and any one probability

distribution P, the two answers P(A/E) > P(B/E), the answer B may so much

increase the credibility of the second source of information by comparison

with the source of the ¬rst answer that after the answers have actually been

given, the credibility of B is greater than that of A. This can be the case even

though the rest of the available evidence is the same in the two cases.

In practice, it is hard to put numerical values to the probabilities in question,

but simple qualitative examples are easy to give. Indeed, Linda the bank teller

example can serve as a case in point.

Speaking more explicitly (and more generally), when the two messages

A and B come from different sources, they must be judged by reference to

different “prior” probability distributions Pa and Pb . Then it may very well be

the case that the following are all true:

Pa (A/E) > Pa (B /E)

Pb(A/E) > Pb(B /E)

Pa (A/E) < Pb(B /E)

Socratic Epistemology

216

There is nothing strange about the use of two different “prior” probability

distributions in the case of the two different sources of information. It is merely

an extension of the strategy recommended by Savage of building background

information that cannot be naturally expressed as speci¬c evidence E into

what in Bayesian scheme are the priors.

These observations provide a basis for a solution to the conjunction para-

dox. What happens in the Linda case is that the two propositions T and (T & F)

are thought of as answers by two different sources of information to the

question: What is Linda doing now? The background information does not

tell us much about whether T is likely to be true or not. If anything, the story

about Linda makes it unlikely that she should be a bank teller, of all things.

In contrast, the information that F (i.e., that Linda is active in the feminist

movement) is made likely by the background information. Hence it enhances

the credibility of the second witness, for the reasons adumbrated earlier. For

this reason, the answer (T & F) coming from a different source of information

might have to be assigned rationally a higher degree of credibility than the

answer T, coming from a different witness. There need not be anything falla-

cious or otherwise irrational, such as the conjunction effect.

This diagnosis of the conjunction effect has a number of implications. For

instance, the conjunction effect takes place only if the added conjunct is apt to

enhance the credibility of the source of information in question. If F were the

statement “Linda is a stamp collector,” the conjunction effect would be dis-

tinctly weakened. Furthermore, the conjunction effect becomes smaller when

it is not natural to think of the two statements as being made by two dif-

ferent speakers with different degrees of reliability and/or access to relevant

information.

Nothing in the line of thought I have followed turns on using a notion of

probability different from the normal epistemic probability. Hence the solu-

tion proposed here is different from a suggestion by Gerd Gigerenzer (1991).

He attributes the alleged paradox to a confusion between epistemic and statis-

tical probability. However, there is a connection. When the statistical character

of probability judgments is made explicit, it becomes unlikely that the proposi-

tions whose probabilities are at issue are expressed by statements made by two

different witnesses. Hence it is not surprising that changing Linda-type exam-

ples into ones in which the statistical character of the probability judgments

in question is emphasized can appreciably reduce the conjunction effect, as

Gigerenzer has shown.

For instance, Tversky and Kahneman pointed out as early as in 1983 that

the situation is quite different if one changes the operative question in the

Linda case to the following:

There are 100 persons who ¬t the description above (i.e., Linda™s). How

many of them are

(a) bank tellers

(b) bank tellers and active in the feminist movement

A Fallacious Fallacy? 217

This drastically reduces the conjunction effect. Gigerenzer interprets this to

mean that the conjunction effect results from the use of a single-case epistemic

probability concept rather than a frequency concept. The diagnosis proposed

in this chapter suggests a less drastic explanation. If we put the question in

frequentist terms, we destroy the tacitly accepted scenario in which (a) and

(b) are answers by two different witness-like sources of information. As was

seen, this should be enough to greatly reduce the conjunction effect, which is

what Tversky and Kahneman found.

Apart from this indirect in¬‚uence, nothing in the arguments presented

here turn on a contrast between different kinds of probability”for example,

between single-case probabilities and probabilities that can be interpreted as

frequencies.

It is also worth reiterating that my diagnosis does not depend on any assump-

tions concerning whatever connections there might be between the two wit-

nesses and the subject matter of T and (T & F). What is at issue is merely the

different testimonies, which by de¬nition have only informational and not, for

example, causal connections with the facts of the case.

In a nutshell, I might sum up my diagnosis of the conjunctive fallacy by say-

ing that given suitable collateral evidence, certain messages are self-con¬rming

in that they tend to increase our con¬dence in their source. There is noth-

ing paradoxical in this kind of self-con¬rming effect. What makes the Linda

case unusual is that the extra information that makes (T & F) more prob-

able is introduced by the very assertion of this proposition by the second

presumed witness. This assertion is therefore, in a certain sense and to some

degree, self-certifying. Such a self-certi¬cation might at ¬rst sight seem para-