## 1. Coherentism Versus Foundationalism



A central problem in epistemology is to explain when we are justified in holding
a proposition to be true. It is not at all evident what epistemic
justification is, and classical accounts of that notion have turned
out to be severely problematic. A tradition inspired by Descartes holds that justified beliefs are those that are either self-evidently true or deduced from self-evident truths. But, as is often argued, little of what we
take ourselves to justifiably believe satisfies these austere
conditions: many of our apparently justified beliefs, it is commonly
thought, are neither based on self-evident truths nor derivable in a
strict logical sense from other things we believe in. Thus, the
Cartesian rationalist picture of justification seems far too
restrictive. Similar problems hound empiricist attempts to ground all
our knowledge in the allegedly indubitable data of the senses.
Depending on how they are understood, sense data are either not
indubitable or else not informative enough to justify a sufficient
portion of our purported knowledge. The exact characterization of
foundationalism is a somewhat contentious issue. There is another form
of foundationalism according to which some beliefs have some
non-doxastic source of epistemic support that requires no support of
its own. This support can be defeasible and it can require
supplementation to be strong enough for knowledge. This sort of
non-doxastic support would terminate the regress of justification. To
do so it may not have to appeal to self-evidence, indubitability or
certainty. Such foundationalist views vary on the source of the
non-doxastic support, how strong the support is on its own, and what
role in justification coherence plays, if any. Some critics of this
position have questioned the intelligibility of the non-doxastic
support relation. Thus, Davidson (1986) complains that advocates have
been unable to explain the relation between experience and belief that
allows the first to justify the second.



The difficulties pertaining to both rationalism and empiricism about
justification have led many epistemologists to think that there must
be something fundamentally wrong with the way in which the debate has
been framed, prompting their rejection of the foundationalist
justificatory structure underlying rationalism and empiricism alike.
Rather than conceiving the structure of our knowledge on the model of
Euclidean geometry, with its basic axioms and derived theorems, these
epistemologists favor a holistic picture of justification which does
not distinguish between basic or foundational and non-basic or derived
beliefs, treating rather all our beliefs as equal members of a
"web of belief" (Quine and Ullian 1970, cf. Neurath
1983/1932 and Sosa 1980).



The mere rejection of foundationalism is not itself an alternative
theory because it leaves us with no positive account of justification,
save a suggestive metaphor about webs of belief. A more substantial
contrasting proposal is that what justifies our beliefs is ultimately
the way in which they hang together or dovetail so as to produce a
coherent set. As Davidson puts it, "[w]hat distinguishes a
coherence theory is simply the claim that nothing can count as a
reason for a belief except another belief" (Davidson, 1986). The
fact that our beliefs cohere can establish their truth, even though
each individual belief may lack justification entirely if considered
in splendid isolation, or so it is thought. Following C. I. Lewis
(1946), some proponents think of this situation as analogous to how
agreeing testimonies in court can lead to a verdict although each
testimony by itself would be insufficient for that purpose.



There is a serious objection that any coherence theory of
justification or knowledge must immediately face. It is called the
*isolation objection*: how can the mere fact that a system is
coherent, if the latter is understood as a purely system-internal
matter, provide any guidance whatsoever to truth and reality? Since
a coherence theory, in its basic form, does not assign any essential role to experience, there is
little reason to think that a coherent system of belief will
accurately reflect the external world. A variation on this theme is
presented by the equally notorious *alternative systems
objection*. For each coherent system of beliefs there exist,
conceivably, other systems that are equally coherent yet incompatible
with the first system. If coherence is sufficient for justification,
then all these incompatible systems would be justifiably held. But this
observation, of course, thoroughly undermines any claim suggesting
that coherence is indicative of truth.



As we shall see, most, if not all, influential coherence theorists try
to avoid these traditional objections by assigning some beliefs that
are close to experience a special role, whether they are called
"supposed facts asserted" (Lewis, 1946),
"truth-candidates" (Rescher, 1973), "cognitively
spontaneous beliefs" (BonJour, 1985) or something else.
Depending on how this special role is construed, these theories are often
classified as versions of *weak
foundationalism*. An advocate of
weak foundationalism typically holds that while coherence is incapable
of justifying beliefs from scratch, it can provide justification for
beliefs that already have some initial, perhaps minuscule, degree of
warrant, e.g., for observational beliefs.



A fair number of distinguished contemporary philosophers have declared
that they advocate a coherence theory of justification. Apart from
this superficial fact, these theories often address some rather
diverse issues loosely united by the fact that they in one way or the
other take a holistic approach to the justification of beliefs. Here
are some of the problems and questions that have attracted the
attention of coherence theorists (cf. Bender, 1989):


* How can a regress of justification be avoided?
* How can we gain knowledge given that our information sources
(senses, testimony etc) are not reliable?
* How can we know anything at all given that we do not even know
whether our own beliefs or memories are reliable?
* Given a set of beliefs and a new piece of information (typically
an observation), when is a person justified in accepting that
information?
* What should a person believe if confronted with a possibly
inconsistent set of data?



The fact that these separate, though related, issues are not always
clearly distinguished presents a challenge to the reader of the
relevant literature.



Although the regress problem is not a central contemporary issue, it
is helpful to explain coherence theories as responses to the problem.
This will also serve to illustrate some challenges that a coherence
theory faces. We will then turn to the concept of coherence itself as
that concept is traditionally conceived. Unfortunately, not all
prominent authors associated with the coherence theory use the term
coherence in this traditional sense, and the section that follows is
devoted to such non-standard coherence theories. The arguably most
systematic and prolific discussion of the coherence theory of
justification has focused on the relationship between coherence and
probability. The rest of this entry will be devoted to this
development, which took off in the mid-1990s inspired by seminal work
by C. I. Lewis (1946). The development has given us precise and
sophisticated definitions of coherence as well as detailed studies of
the relationship between coherence and truth (probability),
culminating in some potentially disturbing impossibility results that
shed doubt on the possibility of defining coherence in a way that
makes it indicative of truth. More precise descriptions of key
entailments of these results, and ways to address the worries that
they raise, will be discussed in later sections of this entry.


## 2. The Regress Problem



On the traditional justified true belief account of knowledge, a
person cannot be said to know that a proposition \(p\) is true
without having good reasons for believing that \(p\) is true. If
Lucy knows that she will pass tomorrow's exam, she must have good
reasons for thinking that this is so. Consider now Lucy's reasons.
They will presumably consist of other beliefs she has, e.g., beliefs
about how well she did earlier, about how well she has prepared, and
so on. For Lucy to know that she will pass the exam, these other
beliefs, upon which the first belief rests, must also be things that
Lucy knows. Knowledge, after all, cannot be based on something less
than knowledge, i.e., on ignorance (cf. Rescher 1979, 76). Since the
reasons are themselves things that Lucy knows, those reasons must in
turn be based on reasons, and so on. Thus, any knowledge claim
requires a never-ending chain, or "regress", of reasons
for reasons. This seems strange, or even impossible, because it
involves reference to an infinite number of beliefs. But most of us
think that knowledge is possible.



What is the coherentist's response to the regress? The coherentist can
be understood as proposing that nothing prevents the regress from
proceeding in a circle. Thus, \(A\) can be a reason for \(B\)
which is a reason for \(C\) which is a reason for \(A\). If this
is acceptable, then what we have is a chain of reasons that is
never-ending but which does not involve an infinite number of beliefs.
It is never-ending in the sense that for each belief in the chain
there is a reason for that belief also in the chain. Yet there is an
immediate problem with this response due to the fact that
justificatory circles are usually thought to be vicious ones. If
someone claims \(C\) and is asked why she believes it, she may
reply that her reason is \(B\). If asked why she believes \(B\),
she may assert \(A\). But if prompted to justify her belief in
\(A\), she is not allowed to refer back to \(C\) which in the
present justificatory context is still in doubt. If she did justify
\(A\) in terms of \(C\) nonetheless, her move would lack any
justificatory force whatsoever.



The coherentist may respond by denying that she ever intended to
suggest that circular reasoning is a legitimate dialectical strategy.
What she objects to is rather the assumption that justification should
at all proceed in a linear fashion whereby reasons are given for
reasons, and so on. This assumption of linearity presupposes that what
is, in a primary sense, justified are individual beliefs. This, says
the coherentist, is simply wrong: it is not individual beliefs that
are primarily justified, but entire belief systems. Particular beliefs
can also be justified but only in a secondary or derived sense, if
they form part of a justified belief system. This is a coherence
approach because what makes a belief system justified, on this view,
is precisely its coherence. A belief system is justified if it is
coherent to a sufficiently high degree. This, in essence, is Laurence
BonJour's 1985 solution to the regress problem.



This looks much more promising than the circularity theory. If
epistemic justification is holistic in this sense, then a central
assumption behind the regress is indeed false, and so the regress
never gets started. Even so, this holistic approach raises many new
questions to which the coherentist will need to respond. First of all,
we need to get clearer on what the concept of coherence involves as
that concept is applied to a belief system. This is the topic of the
next section. Second, the proposal that a singular belief is justified
merely in virtue of being a member of a justified totality can be
questioned because, plausibly, a belief can be a member of a
sufficiently coherent system without in any way adding to the
coherence of that system, e.g., if the belief is the only member which
does not quite fit in an otherwise strikingly coherent system. Surely,
a belief will have to *contribute* to the coherence of the
system in order to become justified by that system. A particular
belief needs, in other words, to *cohere with* the system of
which it is a member if that belief is to be considered justified. We
will turn to this issue in section 4, in connection with Keith
Lehrer's epistemological work. Finally, we have seen that most
coherence theories assign a special role to some beliefs that are
close to experience in order to avoid the isolation and alternative
systems objections. This fact raises the question of what status those
special beliefs have. Do they have to have some credibility in
themselves or can they be totally lacking therein? A particularly
clear debate on this topic is the Lewis-BonJour controversy over the
possibility of justification by coherence from scratch, which we will
examine more closely in section 5.


## 3. Traditional Accounts of Coherence



By a traditional account of coherence we will mean one which construes
coherence as a relation of mutual support or agreement among given
data (propositions, beliefs, memories, testimonies etc.). Early
characterizations were given by, among others, Brand Blanshard (1939)
and A. C. Ewing (1934). According to Ewing, a coherent set is
characterized partly by consistency and partly by the property that
every belief in the set follows logically from the others taken
together. Thus, a set such as \(\{A\_1,
A\_2, A\_1 \amp A\_2\}\),
if consistent, is highly coherent on this view because each element
follows by logical deduction from the rest in concert.



While Ewing's definition is admirably precise, it defines coherence
too narrowly. Few belief sets that occur naturally in everyday life
satisfy the austere second part of his definition: the requirement
that each element follow logically from the rest when combined.
Consider, for instance, the set consisting of propositions \(A,
B\) and \(C\), where




|  |  |
| --- | --- |
| \(A =\) | "John was at the crime scene at the time of the
robbery" |
| \(B =\) | "John owns a gun of the type used by the
robber" |
| \(C =\) | "John deposited a large sum of
money in his bank account the next day" |



This set is intuitively coherent, and yet it fails to satisfy Ewing's
second condition. The proposition \(A\), for instance, does not
follow logically from \(B\) and \(C\) taken together: that John
owns a gun of the relevant type and deposited money in his bank the
day after does not logically imply him being at the crime scene at the
time of the crime. Similarly, neither \(B\) nor \(C\) follows
from the rests of the propositions in the set by logic alone.



C. I. Lewis's definition of coherence, or "congruence" to
use his term, can be seen as a refinement and improvement of Ewing's
basic idea. As Lewis defines the term, a set of "supposed facts
asserted" is coherent (congruent) just in case every element in
the set is supported by all the other elements taken together, whereby
"support" is understood not in logical terms but in a
probabilistic sense. In other words, \(P\) supports \(Q\) if and
only if the probability of \(Q\) is raised on the assumption that
\(P\) is true. As is readily appreciated, Lewis's definition is
less restrictive than Ewing's: more sets will turn out to be coherent
on the former than on the latter. (There are some uninteresting
limiting cases for which this is not true. For instance, a set of
tautologies will be coherent in Ewing's but not in Lewis's sense.
There cases are not interesting because they are not significant parts
of anyone's actual body of beliefs.)



Let us return to the example with John. The proposition \(A\),
while not logically entailed by \(B\) and \(C\), is under normal
circumstances nevertheless supported by those propositions taken
together. If we assume that John owns the relevant type of gun and
deposited a large sum the next day, then this should raise the
probability that John did it and thereby also raise the probability
that he was at the crime scene when the robbery took place. Similarly,
one could hold that each of \(B\) and \(C\) is supported, in the
probabilistic sense, by the other elements of the set. If so, this set
is not only coherent in an intuitive sense but also coherent according
to Lewis's definition. Against Lewis's proposal one could hold that it
seems arbitrary to focus merely on the support single elements of a
set receive from the rest of the set (cf. Bovens and Olsson 2000). Why
not consider the support any subset, not just singletons, receives
from the rest?



Another influential proposal concerning how to define coherence
originates from Laurence BonJour (1985), whose account is considerably
more complex than earlier suggestions. Where Ewing and Lewis proposed
to define coherence in terms of one single concept--logical
consequence and probability, respectively--BonJour thinks that
coherence is a concept with a multitude of different aspects
corresponding to the following "coherence criteria"
(97-99):


1. A system of beliefs is coherent only if it is logically
consistent.
2. A system of beliefs is coherent in proportion to its degree of
probabilistic consistency.
3. The coherence of a system of beliefs is increased by the presence
of inferential connections between its component beliefs and increased
in proportion to the number and strength of such connections.
4. The coherence of a system of beliefs is diminished to the extent
to which it is divided into subsystems of beliefs which are relatively
unconnected to each other by inferential connections.
5. The coherence of a system of beliefs is decreased in proportion to
the presence of unexplained anomalies in the believed content of the
system.



A difficulty pertaining to theories of coherence that construe
coherence as a multidimensional concept is to specify how the
different dimensions are to be amalgamated so as to produce an overall
coherence judgment. It could well happen that one system \(S\) is
more coherent than another system \(T\) in one respect, whereas
\(T\) is more coherent than \(S\) in another. Perhaps \(S\)
contains more inferential connections than \(T\), but \(T\) is
less anomalous than \(S\). If so, which system is more coherent in
an overall sense? Bonjour's theory is largely silent on this
point.



BonJour's account also raises another general issue. The third
criterion stipulates that the degree of coherence increases with the
number of inferential connections between different parts of the
system. Now as a system grows larger the probability that there will
be relatively many inferentially connected beliefs is increased simply
because there are more possible connections to be made. Hence, one
could expect there to be a positive correlation between the size of a
system and the number of inferential connection between the beliefs
contained in the system. BonJour's third criterion, taken at face
value, entails therefore that a bigger system will generally have a
higher degree of coherence due to its sheer size. But this is at least
not obviously correct. A possible modified coherence criterion could
state that what is correlated with higher coherence is not the number
of inferential connections but rather the inferential *density*
of the system, where the latter is obtained by dividing the number of
inferential connections by the number of beliefs in the system.


## 4. Other Accounts of Coherence



We will return, in section 6, to the problem of defining the
traditional concept of coherence while addressing some of the concerns
that we have raised, e.g., concerning the relationship between
coherence and system size. The point of departure for the present
discussion, however, is the observation that several prominent
self-proclaimed coherentists construe the central concept, and to some
extent also its role in philosophical inquiry, in ways that depart
somewhat from the traditional view. Among them we find Nicolas
Rescher, Keith Lehrer and Paul Thagard.



Central in Rescher's account, as laid out in Rescher (1973), his most
influential book on the subject, is the notion of a truth-candidate. A
proposition is a truth-candidate if there is something that speaks in
its favor. Rescher's truth-candidates are related to Lewis's
"supposed facts asserted". In both cases, the propositions
of interest are *prima facie* rather than *bona fide*
truths. Although Rescher's 1973 book is entitled *A Coherence
Theory of Truth*, the purpose of Rescher's investigation is not to
investigate the possibility of *defining* truth in terms of
coherence but to find a truth *criterion*, which he understands
to be a systematic procedure for selecting from a set of conflicting
and even contradictory truth-candidates those elements which it is
rational to accept as *bona fide* truths. His solution amounts
to first identifying the maximal consistent subsets of the original
set, i.e., the subsets that are consistent but would become
inconsistent if extended by further elements of the original set, and
then choosing the most "plausible" among these subsets.
Plausibility is characterized in way that reveals no obvious relation
to the traditional concept of coherence. While the traditional concept
of coherence plays a role in the philosophical underpinning of
Rescher's theory, it does not figure essentially in the final product.
In a later book, Rescher develops a more traditional
"system-theoretic" view on coherence (Rescher 1979).



Keith Lehrer employs the concept of coherence in his definition of
justification, which in turn is a chief ingredient in his complex
definition of knowledge. According to Lehrer, a person is justified in
accepting a proposition just in case that proposition coheres with the
relevant part of her cognitive system. This is the relational concept
of coherence alluded to earlier. In Lehrer (1990), the relevant part
is the "acceptance system" of the person, consisting of
propositions to the effect that the subject accepts this and that.
Thus, "\(S\) accepts that \(A\)" would initially be
in \(S\)'s acceptance system, but not \(A\) itself. In later
works, Lehrer has emphasized the importance of coherence with a more
complex cognitive entity which he calls the "evaluation
system" (e.g., Lehrer 2000 and 2003).



The starting point of Lehrer's account of coherence is the fact that
we can think of all sorts of objections an imaginative critic may
raise to what a person accepts. These objections might be directly
incompatible with what that person accepts or they might threaten to
undermine her reliability in making assessments of the kind in
question. For instance, a critic might object to her claim that she
sees a tree by suggesting that she is merely hallucinating. That would
be an example of the first sort of objection. An example of the second
sort would be a case in which the critic replies that the person
cannot tell whether she is hallucinating or not. Coherence, and
(personal) justification, results when all objections have been
met.



Lehrer's concept of coherence does not seem to have much in common
with the traditional concept of mutual support. If one takes it as
essential that such a theory make use of a concept of systematic or
global coherence, then Lehrer's theory is not a coherence theory in
the traditional sense because, in Lehrer's view, "[c]oherence
... is not a global feature of the system" (1997, 31), nor
does it depend on global features of the system (31). A critic may
wonder what reasons there are for calling the relation of meeting
objections to a given claim relative to an evaluation system a
relation of *coherence*. Lehrer's answer seems to be that it is
a relation of "fitting together with", rather than, say, a
relation of "being inferable from": "[i]f it is more
reasonable for me to accept one of [several] conflicting claims than
the other on the basis of my acceptance system, then that claim fits
better or coheres better with my acceptance system" (116), and
so "[a] belief may be completely justified for a person because
of some relation of the belief to a system to which it belongs, the
way it coheres with the system, just as a nose may be beautiful
because of some relation of the nose to a face, the way it fits with
the face" (88). Olsson (1999) has objected to this view by
pointing out that it is difficult to understand what it means for a
belief to fit into a system unless the former does so in virtue of
adding to the global coherence of the latter.



Paul Thagard's theory is clearly influenced by the traditional concept
of coherence but the specific way in which the theory is developed
gives it a somewhat non-traditional flavor, in particular considering
its strong emphasis on explanatory relations between beliefs. Like
Rescher, Thagard takes the fundamental problem to be which elements of
a given set of typically conflicting claims that have the status of
*prima facie* truths to single out as acceptable. However,
where Rescher proposes to base the choice of acceptable truths on
considerations of plausibility, Thagard suggests the use of
explanatory coherence for that purpose.



According to Thagard, *prima facie* truths can cohere (fit
together) or "incohere" (resist fitting together). The
first type of relation includes relations of explanation and
deduction, whereas the second type includes various types of
incompatibility, such as logical inconsistency. If two propositions
cohere, this gives rise to a positive constraint. If they incohere,
the result is a negative constraint. A positive constraint between two
propositions can be satisfied either by accepting both or by rejecting
both. By contrast, satisfying a negative constraint means accepting
one proposition while rejecting the other. A "coherence
problem", as Thagard sees it, is one of dividing the initial set
of propositions into those that are accepted and those that are
rejected in such a way that most constraints are satisfied. Thagard
presents several different computational models for solving coherence
problems, including a model based on neural networks.



How acceptability depends on coherence, more precisely, is codified in
Thagard's "principles of explanatory coherence" (Thagard,
2000):



Principle E1 (Symmetry)
Explanatory coherence is a symmetric relation. That is, two
propositions \(A\) and \(B\) cohere with each other
equally.
Principle E2 (Explanation)

1. A hypothesis coheres with what it explains, which can either be
evidence or another hypothesis.
2. Hypotheses that together explain some other proposition cohere
with each other.
3. The more hypotheses it takes to explain something, the lower the
degree of coherence.

 
Principle E3 (Analogy)
Similar hypotheses that explain similar pieces of evidence
cohere.
Principle E4 (Data Priority)
Propositions that describe the results of observation have a
degree of acceptability on their own.
Principle E5 (Contradiction)
Contradictory propositions are incoherent with each other.
Principle E6 (Competition)
If \(A\) and \(B\) both explain a proposition, and if
\(A\) and \(B\) are not explanatorily connected, then \(A\)
and \(B\) are incoherent with each other \((A\) and \(B\) are
explanatorily connected if one explains the other or if together they
explain something).
Principle E7 (Acceptance)
The acceptability of a proposition in a system of propositions
depends on its coherence with them.


Principle E4 (Data Priority) reveals that Thagard's theory is not a
pure coherence theory, as it gives some epistemic priority to
observational beliefs, making it rather a form of weak
foundationalism, i.e., the view that some propositions have some
initial epistemic support apart from coherence. Moreover, Thagard's
theory is based on binary coherence/incoherence relations, i.e.,
relations holding between two propositions. His basic theory does not
handle incompatibilities that involve, in an essential way, more than
two propositions. But incompatibilities of that sort may very well
arise, as exemplified by the three propositions "Jane is taller
than Martha", "Martha is taller than Karen" and
"Karen is taller than Jane". Nevertheless, Thagard reports
the existence of computational methods for converting constraint
satisfaction problems whose constraints involve more than two elements
into problems that involve only binary constraints, concluding that
his characterization of coherence "suffices in principle for
dealing with more complex coherence problems with nonbinary
constraints" (Thagard 2000, 19). Thagard (2009) argues that
there is a connection between explanatory coherence and (approximate)
truth, where explaining consists in describing causal mechanisms.
Several other authors have advocated coherence theories that emphasize
the importance of explanatory relations. See, for example, Lycan
(1988, 2012) and, for a book-length defense of explanatory
coherentism, Poston (2014). Also related to Thagard's work is Susan Haack's so-called foundherentist theory, which draws on a proposed analogy between coherence justification (with foundationalist ingredients) and crossword puzzle-solving (Haack, 2009).


## 5. Justification by Coherence from Scratch



The arguably most significant development of the coherence theory in
recent years has been the revival of C. I. Lewis's work and the
research program he inspired by translating parts of the coherence
theory into the language of probability. The kind of coherence in
question should be distinguished from a probability function being
coherent in the sense of conforming to the axioms of the probability
calculus. The theory of coherence that we are concerned with here is
an application of such coherent probability functions to model
coherence as mutual support, agreement etc. Thus "probabilistic
coherence" means something else than it does in standard
Bayesian theories. The probabilistic translations of coherence theory
has made it possible to define concepts and prove results with
mathematical precision. It has also led to increased transferability
of concepts and results across fields, e.g., between coherence theory
and confirmation theory as it is studied in philosophy of science. As
a result, the study of coherence has developed into an
interdisciplinary research program with connections to philosophy of
science, cognitive psychology, artificial intelligence and philosophy
of law. The rest of this article will be devoted to this recent
transformation of the subject.



To introduce Lewis's view on the role of coherence, consider the
following famous passage on "relatively unreliable witnesses who
independently tell the same story" from his 1946 book:



> 
> For any one of these reports, taken singly, the extent to which it
> confirms what is reported may be slight. And antecedently, the
> probability of what is reported may also be small. But congruence of
> the reports establishes a high probability of what they agree upon, by
> principles of probability determination which are familiar: on any
> other hypothesis than that of truth-telling, this agreement is highly
> unlikely; the story any one false witness might tell being one out of
> so very large a number of equally possible choices. (It is comparable
> to the improbability that successive drawings of one marble out of a
> very large number will each result in the one white marble in the
> lot.) And the one hypothesis which itself is congruent with this
> agreement becomes thereby commensurably well established. (346)
> 



While Lewis allows that individual reports need not be very credible
considered in isolation for coherence to have a positive effect, he is
firmly committed to the view that their credibility must not be nil.
He writes, in his discussion of reports from memory, that "[i]f
... there were no initial presumption attaching to the mnemically
presented ... then no extent of congruity with other such items
would give rise to any eventual credibility" (357). In other
words, if the beliefs in a set have no initial credibility, then no
justification will ensue from observing the coherence of that set.
Thus, Lewis is advocating weak foundationalism rather than a pure
coherence theory.



In apparent agreement with Lewis, Laurence BonJour (1985, 148) writes:
"[a]s 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 as the only available explanation of their
agreement." However, BonJour proceeds to reject Lewis's point
about the need for positive antecedent credibility: "[w]hat
Lewis does not see, however, is that his own [witness] example shows
quite convincingly that no antecedent degree of warrant or credibility
is required" (148). BonJour is here apparently denouncing
Lewis's claim that coherence will not have any confidence boosting
power unless the sources are initially somewhat credible. BonJour is
proposing that coherence can play this role even if there is no
antecedent degree of warrant, so long as the witnesses are delivering
their reports independently.



Several authors have objected to this claim of BonJour's, arguing that
coherence does not have any effect on the probability of the report
contents if the independent reports lack individual credibility. The
first argument to that effect was given by Michael Huemer (1997). A
more general proof in the same vein is presented in Olsson (2002).
What follows is a sketch of the latter argument for the special case
of two testimonies, couched essentially in the terminology of Huemer
(2011). In the following, all probabilities are assumed to lie
strictly between 0 and 1.



Let \(E\_1\) be the proposition that the first witness
reports that \(A\), and let \(E\_2\) be the proposition
that the second witness reports that \(A\). Consider the following
conditions:



**Conditional Independence**  

\(P(E\_2 \mid E\_1, A) = P(E\_2 \mid A)\)
  

\(P(E\_2 \mid E\_1,\neg A) = P(E\_2 \mid \neg A)\)




**Nonfoundationalism**  

\(P(A \mid E\_1) = P(A)\)
  

\(P(A \mid E\_2) = P(A)\)




**Coherence Justification**  

\(P(A \mid E\_1, E\_2) \gt P(A)\)




Conditional independence is intended to capture the idea that the
testimonies are independent in the sense that there is no direct
influence between the testimonies. The probability of a testimony is
influenced only by the fact it reports on, meaning that once that fact
is given, this "screens off" any probabilistic influence
between the individual testimonies making them irrelevant to each
other. Nonfoundationalism states that neither testimony confers any
justification upon \(A\) by itself: assuming merely that one single
witness has testified that \(A\) has no effect on the probability
of \(A\). Finally, Coherence Justification states that testimonies,
when combined, do provide justification for \(A\).



The debate between Lewis and BonJour can be reconstructed as a debate
over the joint consistency of these three conditions. BonJour is
claiming that the conditions are jointly consistent, and that
Coherence Justification follows from Conditional Independence even in
the context of Nonfoundationalism, whereas Lewis is rejecting these
claims. Olsson (2002) established that if the dispute is couched in
these terms, then Lewis was provably right. From Conditional
Independence and Nonfoundationalism it follows, via Bayes's theorem,
that



\[
P(A \mid E\_1, E\_2) = P(A)
\]


so that combining collectively independent but individually useless
testimonies, however coherent, fails to give rise to anything useful.
(As noted in Olsson, 2005, section 3.5, the matter is somewhat
complicated by the fact that Lewis adopted a notion of independence
that is weaker than Conditional Independence. Ironically, Lewis's
weaker notion turns out to be compatible with the combination of
Nonfoundationalism and Coherence Justification.)



Nonfoundationalism should be contrasted with the following
condition:



**Weak Foundationalism**  

\(P(A \mid E\_1) \gt P(A)\)
  

\(P(A \mid E\_2) \gt P(A)\)




Weak Foundationalism does not by itself entail Coherence
Justification: it is common knowledge in probability theory that even
if two pieces of evidence each support a given conclusion, that
support may disappear, or even turn into disconfirmation, if they are
combined. However, in the context of Conditional Independence, Weak
Foundationalism does imply Coherence Justification. Indeed, the
combined testimonies will, in this case, confer more support upon the
conclusion than the testimonies did individually. As confirmed by
James Van Cleve (2011), the conclusions supported by these
considerations are that coherence can boost justification or
credibility that is already there without being able to create such
justification or credibility from scratch.



There are various ways to save the coherence theory from this
probabilistic attack. The most radical strategy would be to dismiss
the probabilistic framework as altogether unsuitable for coherentism.
Independent reasons for this response can be found in Thagard's work
(e.g., Thagard 2000 and 2005). A less radical approach would be to
refrain from any blanket dismissal of probability theory in this
context but reject one of the premises of the troublesome proof. This
is the strategy recently taken by Huemer, who now considers his 1997
probabilistic refutation of coherentism to be mistaken (Huemer 2011,
39, footnote 6). While he thinks that Coherentist Justification
correctly captures a minimal sense of coherentism, he reports
dissatisfaction with both Conditional Independence and
Nonfoundationalism (his term for the latter is "Strong
Nonfoundationalism"). Huemer now thinks independence, in the
intuitive sense, is better captured by the condition
\(P(E\_2 \mid E\_1, A) \gt P(E\_2 \mid E\_1, \neg A)\).
Moreover, he takes the condition \(P(A \mid E\_1, \neg E\_2) = P(A)\),
or "Weak Nonfoundationalism" in his terminology, to be a
more suitable explication of nonfoundationalist intuitions than the
condition \(P(A \mid E\_1) = P(A)\). He goes on to show that they are jointly
consistent with Coherentist Justification: there are probability
distributions satisfying all three conditions. Thus the immediate
threat to coherentism presented by the observed inconsistency of the
three original conditions has been neutralized, even though a critic
might point out that the defense is weak since it has not been shown
that Coherence Justification follows from the two new conditions.



Whatever merits Huemer's new conditions might have, their standing in
the literature is hardly comparable to that of the original
conditions. Conditional Independence, for instance, is an extremely
powerful and intuitive concept which has been put to fruitful use in
many areas in philosophy and computer science, the most spectacular
example being the theory of Bayesian networks (Pearl, 1985).
Similarly, the Nonfoundationalist condition is still the most widely
used--and many would say most natural--way of stating, in
the language of probability theory, that a testimony fails to support
that which is testified. Thus, it would seem that coherentism is saved
at the price of disconnecting it from the way in which probability
theory is standardly applied. Roche (2010) criticizes
Nonfoundationalism from another perspective. In his view, a close
reading of BonJour reveals that the latter requires only that the
witness reports lack individual credibility in the sense that
\(P(A \mid E\_i) = 0.5\) and not in the
sense of \(P(A \mid E\_i) = P(A)\), which is the condition we called
Nonfoundationalism. Since the former does not entail the latter,
coherentists, to the extent that they follow BonJour, need not worry
about the joint inconsistency of Conditional Independence,
Nonfoundationalism and Coherence Justification. Still, this account of
what it means to lack initial credibility is non-standard if taken as
a general characterization, and it may in the end be more charitable
to interpret BonJour as not having subscribed to it. For an
elaboration of this point the reader is referred to Olsson (2005, 65),
footnote 4. In later works, BonJour has gradually retracted from his
original coherentist position (e.g., BonJour 1989 and 1999).


## 6. Probabilistic Measures of Coherence



We recall that Lewis's defined coherence, or congruence, not for any
old set of proposition but rather for a set of supposed facts
asserted. One way to capture this idea is in terms of the notion of a
testimonial system introduced in Olsson (2005). A testimonial system
\(S\) is a set
\(\{\langle E\_1,A\_1\rangle ,\ldots ,\langle E\_n,A\_n\rangle \}\)
where \(E\_i\) is a report to the effect that
\(A\_i\) is true. We will say that
\(A\_i\) is the *content* of report
\(E\_i\). The content of a testimonial system
\(S = \{\langle E\_1,A\_1\rangle ,
\ldots ,\langle E\_n,A\_n\rangle \}\)
is the ordered set of report contents
\(\langle A\_1 ,\ldots ,A\_n\rangle\).
By the degree of coherence \(C(S)\) of such a testimonial
system we will mean the degree of coherence *of its content*.
Bovens and Hartmann (2003) proposed a similar representation of
supposed facts asserted in terms of ordered sets.



To illustrate these concepts, consider a case in which all witnesses
report exactly the same thing, e.g., that John was at the crime scene.
That would be a paradigm case of a (highly) coherent set of reports.
Now contrast this situation with one in which only one witness reports
this. That would be a situation which would intuitively not qualify as
coherent. Indeed, it does not even seem meaningful to apply the
concept of coherence to a case of just one report (except in the
trivial sense in which everything coheres with itself). Letting
\(A\) be the proposition "John was at the crime scene",
and \(E\_1 ,\ldots ,E\_n\) the
corresponding reports, this intuitive difference can be represented as
the difference between two testimonial systems: \(S = \{\langle E\_1,A\rangle ,\ldots ,\langle E\_n ,A\rangle \}\) and
\(S' = \{\langle E\_1,A\rangle \}\). If, by
contrast, the entities to which coherence applies are represented as
simple unstructured sets, the sets of testimonies in question would be
given the same formal representation in terms of the set having
\(A\) as its sole member.



By a (probabilistic) coherence measure, as defined for ordered sets of
propositions, we shall mean any numerical measure
\(C(A\_1 ,\ldots ,A\_n)\)
defined solely in terms of the probability of
\(A\_1 ,\ldots ,A\_n\) (and their
Boolean combinations) and standard arithmetical operations (Olsson,
2002). This definition makes the degree of coherence of a set of
witness reports a function of the probability of the report contents
(and their Boolean combinations). Huemer (2011, 45) refers to this
consequence as the Content Determination Thesis. We will return to the
status of this thesis in section 8, in connection with the recent
impossibility results for coherence. A reasonable constraint on any
coherence measure is that the degree of coherence of an ordered set
should be independent of the particular way in which the content
propositions are listed. Thus,
\(C(\langle A\_1,A\_2 , \ldots ,A\_n\rangle) = C(\langle B\_1,B\_2 , \ldots ,B\_n\rangle)\) whenever
\(\langle B\_1,B\_2 , \ldots ,B\_n\rangle\) is a permutation of
\(\langle A\_1,A\_2 , \ldots ,A\_n\rangle\). This is a formal way of stating
that all propositions in the relevant set should be treated as
epistemic equals. All measures that will be discussed below satisfy
this condition.



Our starting point will be an attempt to identify the degree of
coherence of a set with its joint probability:



\[
C\_0 (A,B) = P(A\wedge B)
\]


However, it is easily seen that this is not a plausible proposal.
Consider the following two cases. Case 1: Two witnesses point out the
same person as the perpetrator, John, say. Case 2: One witness states
that John or James did it, and the other witness that John or Mary did
it. Since the joint probability is the same in both cases, equaling
the probability that John did it, they yield the same degree of
coherence as measured by \(C\_0\). And yet, the reports in
the first case are more coherent from a presystematic standpoint
because the witnesses are in complete agreement.



One way of handling this example would be to define coherence as relative overlap, in the following sense (Glass 2002, Olsson 2002):



\[
C\_1 (A,B) = \frac{P(A\wedge B)}
 {P(A\vee B)}
\]


\(C\_1 (A,B)\), which also takes on values
between 0 and 1, measures how much of the total probability mass
assigned to either \(A\) or \(B\) falls into their intersection.
The degree of coherence is 0 if and only if
\(P(A\wedge B) = 0\), i.e., just in case \(A\) and
\(B\) do not overlap at all, and it is 1 if and only if
\(P(A\wedge B) = P(A\vee B)\), i.e., just in case
\(A\) and \(B\) coincide. The measure is straightforwardly
generalizable:



\[
C\_1 (A\_1 ,\ldots A\_n) =
 \frac{P(A\_1\wedge \ldots \wedge A\_n)}
 {P(A\_1\vee \ldots \vee A\_n)}
\]


This measure assigns the same coherence value, namely 1, to all cases
of total agreement, regardless of the number of witnesses that are
involved. Against this it may be objected that agreement among the
many is more coherent than agreement among the few, an intuition that
can be accounted for by the following alternative measure introduced
by Shogenji (1999):



\[
C\_2 (A,B) = \frac{P(A \mid B)}{P(A)}
 = \frac{P(A\wedge B)}{P(A)\times P(B)}
\]


or, as Shogenji proposes to generalize it,



\[
C\_2 (A\_1 ,\ldots ,A\_n) = 
 \frac{P(A\_1\wedge \ldots \wedge A\_n)}{P(A\_1)\times \ldots \times P(A\_n)}
\]


It is easy to see that this measure is sensitive, in the way we
suggested, to the number of reports in cases of total agreement:
\(n\) agreeing reports correspond to a coherence value of
\(\bfrac{1}{P(A)^{n-1}}\), meaning that as
\(n\) approaches infinity, so does the degree of coherence. Like
the other measures, \(C\_2 (A,B)\) equals 0
if and only if \(A\) and \(B\) do not overlap. An alternative
generalization of the Shogenji measure is presented in Shupbach
(2011). However, whatever its philosophical merits, Schupbach's
proposal is considerably more complex than Shogenji's original
suggestion. Akiba (2000) and Moretti and Akiba (2007) raise a number
of worries for the Shogenji measure and for probabilistic measures of
coherence generally but they seem to be predicated on the assumption
that the concept of coherence is interestingly applicable to unordered
sets of propositions, an assumption that we found reason to question
above.



\(C\_1\) and \(C\_2\) can also be contrasted
with regard to their sensitivity to the specificity of the
propositions involved. Consider two cases. The first case involves two
witnesses both claiming that John committed the crime. The second case
involves two witnesses both making the weaker disjunctive claim that
John, Paul or Mary committed the crime. Which pair of witnesses are
delivering the more coherent set? One way to reason is as follows.
Since both cases involve fully agreeing testimonies, the degree of
coherence should be the same. This is also the result we get if we
apply \(C\_1\). But one could maintain instead that since
the first two witnesses agree on something more specific--a
particular individual's guilt--the degree of coherence should be
higher. This is what we get if we apply \(C\_2\). In an
attempt at reconciliation, Olsson (2002) suggested that
\(C\_1\) and \(C\_2\) may capture two
different concepts of coherence. While \(C\_1\) measures
the degree of agreement of a set, \(C\_2\) is more
plausible as a measure of how *striking* the agreement is.



Recently, however, Koscholke and Schippers (2015) have noted a
counterintuitive feature of \(C\_1\): that it is impossible to increase
a set's degree of coherence by adding further propositions to
the set. They also show that according to a more sophisticated
relative overlap measure which avoids this issue no set's degree
of coherence exceeds the degree of coherence of its maximally coherent
subset. On the other hand, they also prove in a later article, in
collaboration with Stegman, that there is a further relative overlap
measure which avoids both these problems and which therefore in their
view re-establishes relative overlap as a candidate for a proper
formalization of coherence (Koscholke, Schippers and Stegman,
2019).



A further much discussed measure is that proposed in Fitelson (2003).
It is based on the intuition that the degree of coherence of a set
\(E\) should be "a quantitative, probabilistic generalization
of the (deductive) logical coherence of \(E\)" (ibid., 194).
Fitelson takes it to be a consequence of this idea that a maximum
(constant) degree of coherence is attained if the propositions in
\(E\) are all logically equivalent (and consistent). This is in
accordance with \(C\_1\) but not with
\(C\_2\), which as we saw is sensitive to the specificity
(prior probability) of the propositions involved. Fitelson, who
approached the subject from the standpoint of confirmation theory,
proposed a complex coherence measure based on Kemeny and Oppenheim's
(1952) measure of factual support. A further innovative idea is that
Fitelson extends this measure to take into account support relations
holding between *all* subsets in the set \(E\), whereas
Lewis, we recall, only considered the support relation holding between
one element and the rest. The degree of coherence of a set, finally,
is defined as the mean support among the subsets of \(E\). An
alleged counterexample to this measure can be found in Siebel (2004)
and criticisms and proposed amendments in Meijs (2006). The reader may
wish to consult Bovens and Hartmann (2003), Douven and Meijs (2007),
Roche (2013a) and Shippers (2014a) for further coherence measures and
how they fare in relation to test cases in the literature, and
Koscholke and Jekel (2017) for an empirical study of coherence
assessments drawing on similar examples. The latter study indicates
that the measures by Douven and Meijs and by Roche are more in line
with intuitive judgement than other established measures. Some recent
works have focused on applying coherence measures to inconsistent
sets, e.g., Schippers (2014b) and Schippers and Siebel (2015).



It is fair to say that coherence theorists have yet to reach anything
like consensus on how best to define coherence in probabilistic terms.
Nevertheless, the debate so far has given rise to a much more
fine-grained understanding of what the options are and what
consequences they have. What is more, some quite surprising
conclusions can be drawn even with this issue largely unresolved: all
we need to assume in order to prove that no coherence measure can be
truth conducive, in a sense to be explained, is that those measures
respect the Content Determination Thesis.


## 7. Truth Conduciveness: the *Analysis* Debate



Peter Klein and Ted Warfield's 1994 paper in *Analysis*
initiated a lively and instructive debate on the relationship between
coherence and probability (e.g., Klein and Warfield 1994 and 1996,
Merricks 1995, Shogenji 1999, Cross 1999, Akiba 2000, Olsson 2001,
Fitelson 2003 and Siebel 2004). According to Klein and Warfield, just
because one set of beliefs is more coherent than another set, this
does not mean that the first set is more likely to be true. On the
contrary, a higher degree of coherence can, so they claimed, be
associated with a lower probability of the whole set. The idea behind
their reasoning is simple: We can often raise the coherence of an
informational set by adding more information that explains the
information already in the set. But as more genuinely new information
is added, the probability that all the elements of the set are true is
correspondingly diminished. This, Klein and Warfield wrote, follows
from the well-known inverse relationship between probability and
informational content. They concluded that coherence is not truth
conducive.



Much in the spirit of C. I. Lewis, Klein and Warfield illustrated
their argument referring to a detective story (the so-called
"Dunnit example"). It turns out that this example is
unnecessarily complex and that the main point can be illustrated by
reference to a simpler case (borrowed from computer science where it
is used to exemplify the concept of non-monotonic inference). Suppose
that you are told by one source, Jane, that Tweety is a bird and by
another source, Carl, that Tweety cannot fly. The resulting
information set \(S = \langle\)"Tweety is a bird",
"Tweety cannot fly"\(\rangle\) is not particularly coherent
from an intuitive standpoint. Nor is it coherent from the point of
view of Lewis's definition: assuming one of the items true decreases
the probability of the other. At this point, it would be reasonable to
conjecture that either Jane or Carl is not telling the truth. However,
upon consulting a further source, Rick, we receive the information
that Tweety is a penguin. The new set \(S' = \langle\)"Tweety is a bird", "Tweety cannot fly",
"Tweety is a penguin"\(\rangle\) is surely more coherent than
\(S\). In explaining the previous anomaly, the information supplied
by Rick contributes to the explanatory coherence of the set.



The new enlarged set \(S'\) is more coherent than the
original smaller set \(S\). And yet \(S\), being less
informative, is more probable than \(S'\): the conjunction of
all the propositions in \(S\) is more probable than the conjunction
of all the propositions in \(S'\). Hence, more coherence does
not necessarily imply higher likelihood of truth in the sense of
higher joint probability. Klein and Warfield seem to be right:
coherence is not truth conducive.



But, as will soon be clear, this conclusion is premature. As a
preliminary, let us state Klein and Warfield's argument more formally
using the following abbreviations:




|  |  |
| --- | --- |
| \(A\_1 =\) | "Tweety is a bird." |
| \(A\_2 =\) | "Tweety cannot fly." |
| \(A\_3 =\) | "Tweety is a penguin." |



The first information set \(S\) consists of \(A\_1\)
and \(A\_2\). The second, more coherent set
\(S'\) contains, in addition, \(A\_3\). We let
\(C\) denote the degree of coherence, intuitively understood. What
we have then is:



\[
C(A\_1,A\_2) \lt C(A\_1,A\_2,A\_3).
\]


As we saw, due to the greater informational content of the larger set,
its probability is lower than that of the smaller set:



\[
P(A\_1,A\_2,A\_3) \lt P(A\_1,A\_2).
\]


Yet behind this seemingly impeccable piece of reasoning lurks a
serious difficulty. As we saw, it is part of the example that we are
supposed to know also that *Jane reports* that Tweety is a
bird, that *Carl reports* that Tweety cannot fly and that
*Rick reports* that Tweety is a penguin. Let:




|  |  |
| --- | --- |
| \(E\_1 =\) | "Jane reports that Tweety is a bird" |
| \(E\_2 =\) | "Carl reports that Tweety cannot fly" |
| \(E\_3 =\) | "Rick reports that Tweety is a penguin" |



The well-known principle of total evidence now dictates that all
relevant evidence should be taken into consideration when computing
probabilities. Since it cannot be excluded at the outset that the
evidence represented by
\(E\_1\)-\(E\_3\) may be relevant to the
probability of the information sets \(S\) and \(S'\), the
probability of the smaller set is not
\(P(A\_1,A\_2)\) but rather
\(P(A\_1,A\_2 \mid E\_1,E\_2)\). Similarly, the probability
of the larger set is not
\(P(A\_1,A\_2,A\_3)\)
but rather
\(P(A\_1,A\_2,A\_3 
\mid E\_1, E\_2,
E\_3)\).



Bovens and Olsson (2002) raised the question whether, given this
revised understanding of the probability of a set of reported
propositions, it would still follow that extended sets are no more
probable than the sets they extend. Referring to our Tweety example,
would it still hold that



\[
P(A\_1,A\_2,A\_3 \mid E\_1,E\_2,E\_3) \lt P(A\_1,A\_2 \mid E\_1,E\_2)?
\]


Bovens and Olsson demonstrated that the answer to the general question
is in the negative by giving an example of a more coherent extended
set that is also more probable, on the revised understanding of what
this means, than the original smaller set. Klein and Warfield's
reasoning is based on an problematic understanding of the joint
probability of a set of reported propositions. In the end, they have
not shown that coherence is not truth conducive. 



Let us say that a measure \(C\) of coherence is *propositionally
truth conducive* if and only if the following holds: 



if 
 \(C(A\_1 ,\ldots ,A\_n) \gt C(B\_1 ,\ldots ,B\_m)\), then
  

 \(P(A\_1\wedge \ldots \wedge A\_n) \gt P(B\_1\wedge \ldots \wedge B\_m)\).





One lesson emerging from the *Analysis* debate is
that this way of construing truth conduciveness should be replaced by
a notion of truth conduciveness where the relevant probabilities take
all relevant evidence into account, whatever that evidence may be
(beliefs, testimonies etc.). For example, a coherence measure \(C\)
is *doxastically truth conducive* (for a subject \(S)\) if
and only if:



if \(C(A\_1 ,\ldots ,A\_n) \gt C(B\_1 ,\ldots ,B\_m)\), then   

\(P(A\_1\wedge \ldots \wedge A\_n \mid \mathrm{Bel}\_S A\_1,\ldots,\mathrm{Bel}\_{S}A\_n)
\gt\) \(P(B\_1\wedge \ldots \wedge B\_m \mid \mathrm{Bel}\_S B\_1,\ldots,\mathrm{Bel}\_{S}B\_m)\),




where \(\mathrm{Bel}\_S A\) abbreviates "\(S\) believes
that \(A\)". In other words, a measure of coherence is
doxastically truth conducive just in case a more coherent set of
*believed* propositions is jointly more probable than a less
coherent set of *believed* propositions. This the how we will
understand the probability (likelihood of truth) of a set in the
following.


## 8. Impossibility Results



The impossibility results for coherence draw on all three
debates summarized above: the Lewis-BonJour controversy, the debate
over probabilistic measures of coherence and also the dispute in
*Analysis* regarding truth conduciveness. Before we can discuss
the results we need to make one further observation. Given the
conclusion of the Lewis-BonJour dispute, it is a reasonable
expectation that no coherence measure is truth conducive, in the
relevant conditional sense, unless the reports (beliefs, memories
etc.) in question are individually credible and collectively
independent. But assuming this is not sufficient for coherence to
stand a reasonable chance of being truth conducive. We must also
require that when we compare two different sets of reports, we do so
while keeping the degree of individual credibility fixed. Otherwise we
could have a situation in which one set of report contents is more
coherent than another set but still fails to give rise to a higher
likelihood of truth simply because the reporters delivering the
propositions in the less coherent set are individually more reliable.
Thus, truth conduciveness must be understood in a *ceteris
paribus* sense. The question of interest, then, is whether more
coherence implies a higher probability (given independence and
individual credibility) *everything else being equal*. We are
now finally in a position to state the impossibility theorems. What
they show is that no measure of coherence is truth conducive even in a
weak *ceteris paribus* sense, under the favorable conditions of
(conditional) independence and individual credibility.



The first result of this nature was presented by Bovens and Hartmann
(2003). Their definition of truth conduciveness deviates slightly from
the standard account given above. As they define it, a measure
\(C\) is truth conducive if and only if, for all sets \(S\) and
\(S'\), if \(S\) is at least as coherent as
\(S'\) according to \(C\), then \(S\) is at least as
likely to be true as \(S'\) *ceteris paribus* and
given independence and individual credibility. Very roughly, their
proof has the following structure: They show that there are sets
\(S\) and \(S'\), each containing three
propositions, such that which set is more likely to be true will
depend on the level at which the individual credibility (reliability)
is held fixed. Thus for lower degrees of reliability, one set, say
\(S\), will be more probable than the other set, \(S'\);
for higher degrees of reliability, the situation will be reversed. One
can now find a counterexample to the truth conduciveness of any
measure \(C\) through a strategic choice of the level at which the
reliability is held fixed. Suppose for instance that, according to
\(C\), the set \(S\) is more coherent than the set
\(S'\). In order to construct a counterexample to \(C\)'s
truth conduciveness, we set the reliability to a value for which
\(S'\) will be more probable than \(S\). If, on the other
hand, \(C\) makes \(S'\) more coherent than \(S\), we
fix the reliability to a level at which \(S\) will be the more
probable set. For the details, see Bovens and Hartmann (2003, section
1.4).



Olsson defines truth conduciveness in the standard fashion. His
impossibility theorem is based on the following alternative proof
strategy (Olsson 2005, appendix B): Consider a situation of two
witnesses both reporting that \(A\), represented by
\(S = \langle A, A\rangle\). Take a measure
\(C\) of coherence that is informative with respect to \(S\), in
the sense that it does not assign the same degree of coherence to
\(S\) regardless of which probability assignment is used. This
means that the measure is non-trivial in the situation in question.
Take two assignments \(P\) and \(P'\) of probabilities to
the propositions in \(S\) that give rise to different coherence
values. Olsson shows that a counter example to the truth conduciveness
of \(C\) can be constructed through a strategic choice of the
probability of reliability. If \(P\) makes \(S\) more coherent
than does \(P'\) according to \(C\), we fix the
probability of reliability in such a way that \(S\) comes out as
more probable on \(P'\) than on \(P\). If, on the other
hand, \(P'\) makes \(S\) more coherent, then we choose a
value for the probability of reliability so that \(P\) makes
\(S\) more probable. It follows that no coherence measure is both
truth conducive and informative.



There are some further subtle differences between the two results.
First, Olsson's theorem is proved against the backdrop of a
*dynamic* (or, in the language of Bovens and Hartmann, 2003,
*endogenous*) model of reliability: the assessment of witness
reliability, which in this model is represented as a probability of
reliability, may change as we obtain more reports. Bovens and
Hartmann's detailed proof assumes a non-dynamic (exogenous) model of
reliability, although they indicate that the result carries over to
the dynamic (endogenous) case. Second, there is a difference in the
way the *ceteris paribus* condition is understood. Olsson fixes
the initial probability of reliability, but allows the prior
probability of the report contents to vary. Bovens and Hartmann fix
not only the reliability but also the prior probability of the report
contents.



These impossibility results give rise to a thought-provoking paradox.
It can hardly be doubted that we trust and rely on coherence reasoning
when judging the believability of information, in everyday life and in
science (see Harris and Hahn, 2009, for an experimental study in a
Bayesian setting). But how can this be when in fact coherence is not
truth conducive? Since the impossibility results were published a
number of studies have been dedicated to the resolution of this
paradox (see Meijs and Douven, 2007, for an overview of some possible
moves). These studies can be divided into two camps. Researchers in
the first camp accept the conclusion that the impossibility results
show that coherence is not truth conducive. They add, however, that
this does not prevent coherence from being valuable and important in
other ways. Researchers in the other camp do not accept the conclusion
that the impossibility results show that coherence is not truth
conducive because they think that at least one premise used in proving
the results is doubtful.



Let us start with responses from the first camp. Dietrich and Moretti
(2005) show that coherence in the sense of the Olsson measure is
linked to the practice of indirect confirmation of scientific
hypotheses. That measure turns out to be, in the terminology of
Moretti (2007), "confirmation conducive". Glass (2007)
argues, similarly, that coherence can provide the key to a precise
account of inference to the best explanation, the main idea being to
use a coherence measure for ranking competing hypotheses in terms of
their coherence with a given piece of evidence. Furthermore, Olsson
and Schubert (2007) observe that, while coherence falls short of being
truth conducive, it can still be "reliability conducive",
i.e., more coherence, according to some measures, entails a higher
probability that the sources are reliable, at least in a paradigmatic
case (cf. Schubert 2012a, 2011). Nevertheless, Schubert has proved an impossibility theorem to the effect that no coherence
measure is reliability conducive in general (Schubert 2012b). For yet
another example, Angere (2007, 2008) argues, based on computer
simulations, that the fact that coherence fails to be truth conducive,
in the above sense, does not prevent it from being connected with
truth in a weaker, defeasible sense. In fact, almost all coherence
measures that have an independent standing in the literature satisfy
the condition that *most* cases of higher coherence are also
cases of higher probability, although they do so to different degrees. Moreover, Roche (2013b) has demonstrated that assuming a set to
be coherent implies an increase in the probability of truth of any of
its elements. This is a weak form of truth-conduciveness, and Roche is
right to point out that it should not give the coherentist much
comfort. It has also been noted that coherence plays an important
negative role in our thinking. If our beliefs show signs of
incoherence, this is often a good reason for contemplating a revision.
See chapter 10 in Olsson (2005) for an elaboration of this point.



As for the other approach to the impossibility results (questioning
the premises used in their derivation), we have already seen that
Huemer (2007, 2011), in connection with the Lewis-BonJour dispute, has
expressed doubts regarding the standard way of formalizing
independence in terms of conditional probability. It should come as no
surprise that he objects to the impossibility results (ibid.) on the
same grounds. In his 2011 article, Huemer even questions the Content
Determination Thesis, which plays a pivotal role in the derivation of
the results, for reasons that we have to leave aside here.



All these things can be consistently questioned. But the question is:
at what cost? We have already seen that there are strong systematic
reasons for explicating independence in terms of conditional
independence in the standard way. Furthermore, the Content
Determination Thesis is deeply entrenched in just about all work on
coherence that takes agreeing witnesses to be the prototypical case.
Giving up Content Determination would mean purging the coherence
theory of one of its clearest and most distinctive pre-systematic
intuitions: that coherence is a property at the level of report
*contents*. The worry is that coherentism is saved at the cost
of robbing it of almost all its significance, as Ewing put it almost a
century ago in response to a similar worry (Ewing 1934, 246).



These concerns do not obviously carry over to another dialectical
move: questioning the *ceteris paribus* conditions employed in
the impossibility results, i.e., the conditions that determine what to
hold fixed as the degree of coherence is varied. This line of
criticism has been taken up by several authors, including Douven and
Meijs (2007), Schupbach (2008) and Huemer (2011), and it may well be
the internally least problematic strategy to explore for those who are
inclined to challenge the premises upon which the impossibility
results are based. It should be borne in mind, though, that the
tendency to offer ever stronger *ceteris paribus* conditions
may in the end be self-defeating. As more things are held fixed, it
becomes easier for a coherence measure to be truth conducive. Hence,
researchers pursuing this line of defense ultimately run the risk of
trivializing the debate by making coherence truth conducive by
definition (cf. Schubert 2012b).



There are some attempts to explain or come to grips with the
impossibility results that do not easily fit into the two camps
identified above or represent a combination of ideas from both. For an
example of the latter, Wheeler (2012; see also Wheeler and Scheines,
2013) both suggest focusing on reliability conduciveness as opposed to
truth conduciveness (camp 1) and question the assumptions, primarily
independence but also the Content Determination Thesis, used in the
derivation of the impossibility results (camp 2). Shogenji (2007,
2013) and McGraw (2016) are other complex and insightful attempts to
deepen the Bayesian analysis and diagnose of those results. Specifically, Lydia McGrew argues for a shift in focus from the coherence of contents of reports to coherence of the evidence with an hypothesis \(H\) (which need not coincide with the conjunction of the report contents). If the conjunction of the members of a set of evidence supports some hypothesis \(H\), and if all members of that set are playing a role in that epistemic effect, then that set of evidence is, McGrew shows, indeed confirmatory of \(H\) and in that sense truth-conducive for \(H\) (McGrew, 2016). McGrew offers several proposals for how to spell out "coherence with \(H\)".


## 9. Conclusions



The coherence theory of justification represents an initially
suggestive solution to some deeply rooted problems of epistemology.
Perhaps most significantly, it suggests a way of thinking about
epistemic justification as arising in a "web of belief".
As such, it competes with, and could potentially replace, the
historically dominating, but increasingly disreputable,
foundationalist picture of knowledge as resting on a secure base of
indubitable fact. Coherentism may also be more promising than
alternative foundationalist views with their reliance on non-doxastic
support. Unfortunately, coherence theorists have generally struggled
to provide the details necessary for their theory to advance beyond
the metaphorical stage, something which has not gone unnoticed by
their critics. Following the seminal work of C. I. Lewis, contemporary
scholars have taken on that challenge with considerable success in
terms of clarity and established results, although a fair number of
the latter are to the coherentist's disadvantage. Some results support
a weak foundationalist theory according to which coherence can boost
credibility that is already there, without creating it from scratch.
However, on the face of it, the impossibility results negatively
affect this less radical form of coherence theory as well. It is often
observed that while it is relatively easy to put forward a convincing
theory in the outline, the ultimate test for any philosophical
endeavor is whether the product will survive detailed specification
(the devil is in the details, and so on). What the recent developments
in this area have shown, if nothing else, is that this is very much
true for the coherence theory of epistemic justification.