*ABSTRACT: Suppose you have two witnesses who independently report a
certain observation. Laurence BonJour argues that the agreement of the
reports provides reason for thinking they are true, even if neither
witness has any credibility individually. By analogy, he says, the
coherence of several independent empirical beliefs provides reason for
thinking they are true, even if no given belief has any foundational
justification. I use the probability calculus to show that this central
argument for coherentism is mistaken: the agreement of the witnesses'
reports raises the probability that they are true only if an individual
report has some degree of initial credibility.*

In *The Structure of Empirical Knowledge*, Laurence BonJour argues that
coherence among a set of empirical beliefs can provide justification for
those beliefs, in the sense of rendering them likely to be true. He also
repudiates all forms of foundationalism for empirical beliefs, including
what he calls "weak foundationalism" (the weakest form of foundationalism
he can find). In the following, I will argue that coherence cannot
provide any justification for our beliefs in the manner BonJour suggests
*unless* some form of foundational justification is assumed. In other
words, the argument that BonJour gives in favor of the thesis that
coherence provides a kind of justification succeeds if and only if some
beliefs have (at least weak) foundational justification.

*1. The nature of observation:*

According to BonJour, when we make observations, what happens is
that we find ourselves spontaneously believing certain very detailed
propositions, in the absence of any process of inference nor any exercise
of volition or deliberation on our part. For example, I might find
myself spontaneously believing that there is a book of a certain size,
with a certain shade of red on the cover, and so on, sitting on a desk
of a certain size, shape, and color in front of me, etc.^{(1)} BonJour calls
these beliefs "cognitively spontaneous beliefs," and they are the heart
of his account of observation.

BonJour stresses that these cognitively spontaneous beliefs are not
to be construed as *foundational*, for although they are initially acquired
without the exercise of inference, they are not *justified* unless and
until we find reasons for them. If we do not have any reasons (distinct
from the beliefs themselves) for thinking our cognitively spontaneous
beliefs are likely to be true, then although we may continue to hold
them, we won't be epistemically justified in holding them.

*2. The argument for coherence justification:*

BonJour argues that if our system of beliefs can incorporate a wide range of these cognitively spontaneous beliefs and remain coherent over time, then there is reason to think that our cognitively spontaneous beliefs are generally caused by the sorts of facts that would make them true, and that our belief system in general is largely accurate. The essential reason for this is that on the assumption that our cognitively spontaneous beliefs don't reflect reality, it is very unlikely that they would happen to fit together into a highly coherent picture.

BonJour refers with approval to an instructive example from C.I.
Lewis. Suppose that we are interviewing a number of unreliable
witnesses. Each witness, we think, is quite likely to lie or misreport.
However, suppose we find that all or nearly all of the witnesses tell us
the same story. Assuming that the witnesses aren't collaborating, this
circumstance makes it highly probable that the story is correct, because
if the witnesses were fabricating their stories (and not in communication
with each other), it is highly improbable that they would have all
happened to agree.^{(2)} Thus, even if each witness has only a very low
degree
of credibility individually, two or three witnesses' testimony can
combine to provide a very high probability of the truth of the story.
Similarly, BonJour believes, even if our cognitively spontaneous
(observation) beliefs don't have any independent (foundational)
justification, the mutual agreement of a large number of them on the same
general 'story' can provide adequate justification for the truth of that
story.

*3. On foundationalism:*

BonJour distinguishes three varieties of foundationalism:

(a) *Strong foundationalism* holds that there are certain basic beliefs
that do not depend on inference for their justification, and that the
basic beliefs are absolutely certain, or indubitable, etc.

(b) *Moderate foundationalism* holds that there are basic beliefs that
possess a degree of justification independent of their support by other
beliefs, that this degree of justification is sufficient for knowledge,
but that it does not amount to absolute certainty.

(c) Finally, *weak foundationalism* holds that the basic beliefs only
possess a small degree of non-inferential justification which is
insufficient by itself for knowledge, and that we must rely on coherence
to amplify our justification up to the level where it would be sufficient
for knowledge.

BonJour makes it clear that he wishes to repudiate all forms of
foundationalism, including weak foundationalism. In the above argument
for coherence justification, we are not to suppose the conclusion is
merely that coherence can amplify the level of justification that our
observation beliefs start out with; rather, BonJour holds, our
observation beliefs start out with *no* initial degree of justification
whatever, and coherence alone provides justification for them:

[Lewis'] example shows quite convincingly that no antecedent
degree of warrant or credibility is required. For as long as
we are confident that the reports of the various witnesses are
genuinely independent of each other, a high enough degree of
coherence among them will eventually dictate the hypothesis of
truth telling.... And by the same token, so long as
apparently cognitively spontaneous beliefs are genuinely
independent of each other, their agreement will eventually
generate credibility, without the need for any initial degree
of warrant.^{(3)}

It is this claim that I aim to refute below: if the witnesses lack any degree of independent credibility, then their mutual agreement will never generate any credibility for their story; and if our cognitively spontaneous beliefs do not possess any degree of foundational justification, then no amount of coherence will ever generate any justification for them either.

BonJour's argument for coherence justification is straightforwardly probabilistic. The natural way to test both Lewis' claim about his own case and BonJour's claims about the case is to specify the assumptions of the case (sufficiently to generate the relevant set of probabilities) and then calculate the relevant prior and conditional probabilities.

For the sake of simplicity, let us suppose that there are two
witnesses, Alice and Bert, and that they are reporting on the value of
a certain variable, *x*. *x* can take on a number of different values --
suppose that there are *n* possible values that *x* can assume, and that one
of these possible values is *x* = 2. Assume that each witness is equally
credible and that Alice's level of credibility is such that the
probability that Alice will report the value of *x* correctly is *r*. Now
let "*X*", "*A*", and "*B*" stand for the following propositions:

X= The value ofxis 2.

A= Alice reports that the value ofxis 2.

B= Bert reports that the value ofxis 2.

Now, the claim that the concurrence of the two witnesses' testimony
confirms that what they attest to is true amounts at least to this: that
the probability of *X* given *A* and *B* is higher than the probability of
*X*.
What we need now is a general determination of P(*X*|*A*&*B*). We
make the
following assumptions:

(1) The probability that Alice reports the value of

xcorrectly = the probability that Bert reports correctly =r. (As discussed above.)

(2) There arenpossible values ofx, each equally likely to be the actual value.

(3) If Alice or Bert reports incorrectly then they select randomly from then- 1 incorrect values which to report.

(4) Alice's and Bert's answers are entirely independent of each other. (As stipulated by Lewis and BonJour.)

(5) The chances of Alice or Bert reporting incorrectly are independent of what the true value ofxis. (Assumed for the sake of simplicity.)

We now set out to calculate P(*X*|*A*&*B*) in terms of *n*
and* r*. From Bayes'
Theorem,

We next determine each of the values on the right hand side of Eq. 1. From assumption (2),

So

Also,

whence, since *A* and *B* are independent (assumption 2; we may assume they
are probabilistically independent relative to *X*), we get

From assumptions (1) and (5), we know that P(*A*|*X*) =
P(*B*|*X*) = *r*, so

Next,

Probabilistic independence again dictates that this is equivalent to

and symmetry considerations dictate that P(*A*|¬*X*) =
P(*B*|¬*X*), since Alice
and Bert are equally reliable (or unreliable) and we have given no
different information about either of them than we have about the other.
Therefore,

Now, given that the value of *x* is something other than 2, Alice has a
probability of 1 - *r* of reporting incorrectly (assumptions (1) & (5)).
Assuming Alice does report incorrectly, then, we have said, she selects
randomly from the *n* - 1 incorrect values of *x* to report. Since 2 is one
of those values, there is a chance of 1 in* n* - 1 that Alice would report
*x* = 2. Therefore,

and plugging this into the previous equation,

Finally, plugging equations 2, 3, 4, and 5 into Eq. 1, we obtain,

which thankfully simplifies to the following equation:

Now, equation 6 is interesting because it exhibits the likelihood
that *X* is true given that both witnesses have attested that *X* is true as
a function of two variables: first, the degree of reliability, *r*, of the
witnesses, and second, the number, *n*, of possible values of *x*. This
enables us to vary assumptions about* r* and* n* and see what happens to the
probability we're interested in. It also enables us to test both Lewis'
and BonJour's assertions about the hypothetical example in question.

We examine Lewis' claim first. Lewis' fairly modest claim seems to
have been just this: that you can get a high degree of confirmation as
a result of the agreement of the witnesses even when each witness has
only a low degree of credibility considered by himself. As an
interpretation of low credibility, let's suppose that Alice and Bert each
only tell the truth half of the time. And let's suppose that *x* has ten
possible values. The values of *x* could be any of a number of things, of
course -- possible colors of a car that they claim to have seen; possible
last digits in an address; etc. Plugging the values *n* = 10 and *r* = 1/2
into equation (6) gives the result (omitting the arithmetic),

Since the antecedent probability of *X* would have been .1, this result of
a 90% probability after Alice and Bert both attest that *X* is fairly
impressive. It looks as if Lewis is correct, which, for precisely the
reason he gave, is not terribly surprising: if *X* were not the case, it
would be very unlikely that *A* and *B* would both happen agree that *X*.

But this logic simply does not apply in the sort of case BonJour
would envision: that is, if there really is no antecedent reason for
expecting either of the witnesses to get things right, then the
likelihood of the witnesses both reporting *X* given that *X* was the right
answer would be no better than the likelihood of their both reporting *X*
given that *X* was one of the wrong answers.

To make the issue more precise, we must formulate BonJour's claim
in terms of a probability relation. BonJour says that even if neither
witness has any credibility at all, enough coherence will eventually
render a belief in *X* justified. What is the relevant notion of
'credibility'? The interest of the example, of course, is for the
analogy that can be drawn to observational beliefs and the degree of
initial credibility that a foundationalist would attribute to such
beliefs. Now, the natural interpretation of the claim, made by the weak
foundationalist, that a given cognitively spontaneous belief carries with
it *some* (possibly very weak) degree of *prima facie* justification is this:
my having a cognitively spontaneous belief that* H* raises the probability
of* H*. That is, the probability of* H* given that I have the cognitively
spontaneous belief that* H* is greater than the probability of* H*.
Likewise, in the case of the two witnesses, the statement that Alice has
some degree of initial credibility amounts to this: when Alice says that
*X*, *the probability of *X* goes up*.

Given this interpretation, we have a well-defined probabilistic
question: if we let *r* be sufficiently low so that P(*X*|*A*) = P(*X*)
(i.e.
Alice has no credibility), can we still have P(*X*|*A*&*B*) >
P(*X*)? Notice
that the only way Alice's testimony will fail by itself to raise the
probability of *X* is if Alice's likelihood of being correct is *no better
than chance* -- that is, if *r* = 1/*n*.

In other words: we must set P(*X*|*A*) = P(*X*). P(*X*) =
1/*n*. P(*X*|*A*) is
equal to the probability that Alice is correct given that Alice asserts
that *x* = 2. This is the same as the probability of her being correct if
she reports any other value of *x*, which = *r*. So we must set *r* =
1/*n*.
Plugging this into Eq. 6, we get

which is exactly the same as the antecedent probability of *X*, P(*X*).
Thus, we see that if *X* receives no confirmation at all from either *A* or
*B* individually, then *X* receives no confirmation at all from *A* and
*B*
together. If neither witness has any independent credibility, then the
correspondence of the two witnesses' testimony provides no reason at all
for thinking that what they report is true, and this holds regardless of
how many different stories they could have told -- that is, no matter what
the value of *n* is, the probability that the witnesses' story is true will
remain equal to 1/*n*.

This result is, intuitively, not very surprising either. It is,
indeed, highly counter-intuitive that one can manufacture confirmation
for a hypothesis merely by combining a sufficient number of pieces of
evidence that are, individually, *completely irrelevant* to the hypothesis.
And what is perhaps the central anti-coherentist intuition is closely
related: one cannot obtain *justification* for a proposition merely through
its agreement with a sufficient number of* completely unjustified* other
propositions.

So where did BonJour's reasoning go wrong? After all, isn't it true
that the concurrence of the witnesses' testimony constitutes a surprising
coincidence, which requires some explanation? Well, surprising it may
be, but the problem is just that under the assumption that the witnesses
each have zero independent credibility, the hypothesis that the story
they give is the truth is *not* an explanation of the coincidence. If
there is no antecedent presumption that the witnesses are more likely to
report the true value of *x* than to report any other value, the hypothesis
that *x* = 2 is no sort of explanation of the fact of coincidence, since
it would not render it any more likely that the witnesses would report
that *x* = 2 (i.e., P(*A*&*B*|*X*) is no greater than
P(*A*&*B*|¬*X*)). You might just
as well propose the hypothesis that *X* is *false* as an explanation of why
both witnesses agree that *X* -- for that would make it just as likely that
they would both assert *X*.

Perhaps it would be said that I'm taking an overly simplified view
of explanation -- as if *A*'s explaining *B* just amounts to *A*'s raising the
probability of *B*. It might be claimed that to explain something is,
rather, to cite a causal mechanism, or perhaps that to explain a
collection of phenomena is to give a theory that *unifies* them in a
relatively simple way. My response to this is that while such enriched
notions of explanation may well be correct, they will not suit BonJour's
purposes here. The task that BonJour undertakes, which the argument
under discussion is supposed to serve, is to provide an *a priori* argument
for the conclusion that a belief that belongs to a coherent set of
beliefs is likely to be true. He would have done this if he could argue
that the truth of a certain set of propositions 'explains' why we believe
them in the sense of rendering it much more probable that we would
believe them (for if* H* raises the probability of* E*, then* E* raises the
probability of* H*, according to Bayes' Theorem). But if that isn't what
he's claiming, then the issue of explanation seems to have nothing to do
with the probabilistic argument.

Our first conclusion is that weak foundationalism should be taken seriously as an epistemological position. BonJour questions the plausibility of weak foundationalism:

The basic idea is that an initially low degree of
justification can somehow be magnified or amplified by
coherence, to a degree adequate for knowledge. But how is
this magnification or amplification supposed to work? How can
coherence, not itself an independent source of justification
on a foundationalist view, justify the rejection of some
initially credible beliefs and enhance the justification of
others.^{(4)}

We have seen the answers to these questions above. In the Lewisian
example, we saw that even if P(*X*|*A*) is only 1/2, the justification for
*X* reaches 90% with corroboration from a single additional witness. The
coherence-amplification effect would obviously be even more pronounced
if we assumed a larger range of values for *x* -- which, not incidentally,
would be more typical of real examples (what's the number of
distinguishable things I could be visually perceiving at the moment?) --
and corroboration from a larger number of witnesses. The amplification
occurs precisely for the reason Lewis states. As long as the truth of
*X* makes it somewhat more likely that each source reports *X* (in terms of
the example, as long as *r* > 1/*n*), repeated mutual corroboration will
strongly favor the hypothesis of truth. We saw that this does not imply
that coherence is an independent source of justification, and that in
fact the amplificatory effect of coherence is parasitic on the initial
foundational justification. Without that there would be, so to speak,
nothing to amplify.

Our second conclusion is that the coherence theory of justification
has been seriously undermined. The thrust of BonJour's severe criticisms
of foundationalism in *The Structure of Empirical Knowledge* is this: any
theory of justification must explain why the beliefs it identifies as
"justified" are likely to be true, and we have no satisfactory
explanation of why foundational beliefs are likely to be true.
Furthermore, BonJour considers the most fundamental objection to
coherentism to be the objection that there's no apparent connection
between having an internally coherent belief system and having *true*
beliefs. His bid to show that coherent beliefs are likely to be true is
thus the cornerstone of BonJour's theory. The probabilistic argument
that we've considered is the sole argument he provides for this crucial
point, so the failure of that leaves us with no reason to accept the
coherence theory of justification.^{(5)}

1. *The Structure of Empirical Knowledge* (Cambridge: Harvard University Press, 1985), p. 117.

2. C.I. Lewis, *An Analysis of Knowledge and Valuation* (La Salle, Ill: Open Court, 1962), p. 346. Discussed in
BonJour, pp. 147-8.

3. BonJour, p. 148.

4. p. 29.

5. I would like to thank Barry Loewer and Mike Hardy for their comments on the manuscript and for checking my math.