8 Bridges between Mainstream and Formal Epistemology || Coherence in Epistemology and Belief Revision

Coherence in Epistemology and Belief RevisionAuthor(s): Sven Ove HanssonSource: Philosophical Studies: An International Journal for Philosophy in the AnalyticTradition, Vol. 128, No. 1, 8 Bridges between Mainstream and Formal Epistemology (Mar.,2006), pp. 93-108Published by: SpringerStable URL: http://www.jstor.org/stable/4321715 .

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Philosophical Studies (2006) 128:93-108 ? Springer 2006 DOI 10.1007/sI1098-005-4058-7

SVEN OVE HANSSON

COHERENCE IN EPISTEMOLOGY AND BELIEF REVISION*

ABSTRACT. A general theory of coherence is proposed, in which systemic and relational coherence are shown to be interdefinable. When this theory is applied to sets of sentences, it turns out that logical closure obscures the distinctions that are needed for a meaningful analysis of coherence. It is concluded that references to "all beliefs" in coherentist phrases such as "all beliefs support each other" have to be modified so that merely derived beliefs are excluded. Therefore, in order to avoid absurd conclusions, coherentists have to accept a weak version of epistemic priority, that sorts out merely derived beliefs. Furthermore, it is shown that in belief revision theory, coherence cannot be adequately represented by logical closure, but has to be represented separately.

1. INTRODUCTION

The notion of coherence has mostly been discussed in relation to coherentist epistemology, but its area of application is much wider. Coherence has a role also in some epistemological systems commonly classified as foundationalist, such as that of Lewis (1946). It has important applications in other branches of philosophy, including moral philosophy. Coher- ence, or attempts to achieve it, is essential in John Rawls's ethical theory. Others have questioned the feasibility of a fully coherent ethical system, and claimed that there is a basic heterogeneity in the realm of values, distinguishing it from the realm of facts (Daya, 1960). Coherence is also an important property of systems of goals and plans. Hence, Millgram (2000) claims that coherent plans make more sense than coherent theory choice.

*Contribution to "Seven Bridges"



94 SVEN OVE HANSSON

The literature on coherence is large, but most of it is focused on specific application areas, and too little emphasis has been put on the more general structural issues. Furthermore, although semi-formal treatments are common, the advantages of a fully formalized system have seldom been made use of. In this paper, I intend to show that philosophically important insights can be obtained in this area by applying formal meth- ods, including those developed in belief revision theory. The need for precise formalizations was also pointed out by Bender (1989) in his plea for a "move from mere 'theory-sketch' to actual theory" in studies of coherence.

Section 2 introduces a general definition of coherence. In section 3, this theory is applied to sets of sentences, i.e. sets that have a logical structure. The major conclusions of this investigation are summarized in section 4.

2. A GENERAL THEORY OF COHERENCE

2. 1. Basics

I will assume that coherence is a property of sets, i.e. unor- dered classes of elements. In section 2, no structure is imposed on the elements of the sets in question. Intuitively speaking, coherence refers to some property that keeps the elements to- gether. In epistemology, coherence may be inferential, eviden- tial, explanatory, or probabilistic (Hansson and Olsson, 1999). In applications to sets of plans or goals, coherence may refer to the extent to which the achievement of some goals or plans facilitates the achievement of other goals or plans.

Coherence has often been treated as a categorical (all- or-nothing) notion, i.e. a set is assumed to be either (fully) coherent or not coherent at all. As was observed by Brendel (1999), a more general approach should include a gradational notion of coherence, one in which coherence comes in degrees. Given a gradational notion, it is a trivial matter to introduce a categorical notion by just inserting a limit on the scale of degrees of coherence, below which a set is counted as incoher- ent and above which it is counted as coherent. The gradational



COHERENCE IN EPISTEMOLOGY AND BELIEF REVISION 95

approach will be adopted here; for simplicity it will be assumed that degrees of coherence can be expressed as real numbers. (An alternative approach would be to introduce a binary relation "at least as coherent as". This approach is more general since the relation may be incomplete and intransitive. The numerical approach is chosen for reasons of simplicity.)

Bender (1989) introduced the terms "systemic" and "relational" coherence. Systemic coherence is the coherence of a belief system as a whole, whereas relational coherence holds between a part of such a system and the rest of the system. Both these notions have a tradition in epistemology. Quine's and Bonjour's coherentism refers to systemic coherence whereas the coherence discussed by Lehrer comes clo- ser to the relational variant. The distinction can easily be transferred to other subject-matter than knowledge, and it can therefore be included in a general theory of coherence.

Bender (1989) asked the important question whether systemic coherence holds in a set just in case there is relational coherence between each of its elements and the remaining members. We can generalize this question by asking: Can systemic coherence be defined in terms of relational coherence? The obvious converse question (already treated in Olsson 1999) is: can relational coherence be defined in terms of systemic coherence?

A further issue of interdefinability arises if we relate the systemic coherence of a set to that of its proper subsets: Given that we know the systemic coherence of each of the proper subsets of a set A (its "subsystemic coherence"), can we then infer the systemic coherence of A itself?

To treat these issues, we need to introduce two measures, one for systemic and one for relational coherence. For each set A, let s(A) be its systemic coherence, and for each pair (A,B) of two sets, let r(A,B) be the relational coherence of A with B.

Clearly, the addition of an element to a set can either increase or decrease its coherence. Therefore, systemic coherence will not be assumed to be monotonic or antimono- tonic, i.e. from A c B neither s(A) ? s(B) nor s(A) < s(B) can be concluded.' Relational coherence is assumed to be



96 SVEN OVE HANSSON

order-independent, i.e. r(A,B) = r(B,A) for all A and B. This generalizes Thagard's (1991) proposal, in a categorical con- text, that coherence is symmetric in the sense that A coheres with B if and only if B coheres with A.

2.2. Systemic and Subsystemic Coherence

There are good reasons why the systemic coherence of a set cannot in general be derived from the systemic coherence of its proper subsets. Consider the three elements:

(a) Roger is married. (b) Roger has been ordained. (c) Roger is a catholic.

Given reasonable background beliefs, the set {a,b,c} is much less coherent than any of its proper subsets. It is also possible to construct sets of four etc. elements in which the coherence of the whole set depends on features that are not present in any of its proper subsets. However, for larger numbers of elements these examples will have to be more con- trived, and the argument loses much of its practical relevance.

To make this precise, we can use an additive model in which for each set A, its systemic coherence s(A) is the sum of the (positive or negative) contributions to its coherence obtained from each of its proper subsets. We can introduce a function b that represents the uniquely contributed coherence. For each set A, let b(A) be the contribution to its coherence that is not obtained from any of the proper subsets of A. Hence, in general:

(1) s(B) = X{b(X)IX C B}

Note that b only distributes s(B) among the subsets of B; the introduction of b does not add any new assumption or property to the function s that represents systemic coherence. Indeed, s and b are interdefinable in finite domains; if we have the value of s(A) for every set A, then we can obtain b(A) for every set A by recursion on the number of elements, since for every A, b(A)=s(A)-Z {b(X) I XcA}.




We can assume that s(0) =0. Then b(0) =0. It is also reasonable, at least for some purposes, to assume that b({x}) =0 for all singletons {X}.2 It then follows that b({x,y}) = s({x,y}) for all sets {x,y} with exactly two elements.

In some applications it may be reasonable to assume that b(A) = 0 for all sets A above a certain size limit. For a simple example, let this size limit be 3, so that the degree of systemic coherence of any set is equal to the sum of the contributions of its subsets with at most three elements. Assuming for simplicity that singletons do not contribute to coherence, so that b({x,y}) = s({x,y}) for all x and y, this means for a set {a,b,c,d} of four elements that:

s({a, b, c, d}) =b({a, b, c}) + b({a, b, d}) + b({a, c, d}) + b({b, c, d}) + b({a, b}) + b({a, c}) + b({a, d}) + b({b, c}) + b({b, d}) + b({c, d})

=s({a, b, c}) + s({a, b, d}) + s({a, c, d}) + s({b, c, d}) - s({a, b}) - s({a, c}) - s({a, d}) - s({b, c}) -s({b, d}) -s({c, d}).

Similarly, for a set {a,b,c,d,e} with five elements we obtain

s({a, b, c, d, e}) = b({a, b, c}) + b({a, b, d}) + b({a, b, e}) +b({a,c,d}) +b({a,c,e}) +b({a,d,e}) + b({b, c, d}) + b({b, c, e}) + b({b, d, e}) +b({c,d,e}) +b({a,b}) +Fb({a,c}) + b({a, d}) + b({a, e}) + b({b, c}) + b({b, d}) + b({b, e}) + b({c, d}) + b({c, e}) + b({d, e})

= s({a,b, c}) +is({a,b,d}) +s({a,b,e}) + s({a, c, d) + s({a, c, e}) + s({a, d, e}) + s({b, c, d}) + s({b, c, e}) + s({b, d, e}) + s({c, d, e}) - 2s({a, b}) - 2s({a, c}) - 2s({a, d}) - 2s({a, e}) - 2s({b, c}) - 2s({b, d}) - 2s({b, e}) -2s({c, d}) - 2s({c, e}) - 2s{d, e}).

In the literature on epistemic coherence, several authors have emphasized that a coherent set of beliefs should not contain



98 SVEN OVE HANSSON

"isolated" subsystems that are unconnected with the rest of the set. (For references, see Hansson and Olsson (1999). See also Spohn (1999).) In the present gradational approach, two mutually exclusive sets A and B can be defined as disconnected if b(X) =0 for all X such that AnX ? 0 ? BnX. Such disconnectedness leads to a drastic reduction in the number of possible contributions to systemic coherence. If A and B have four elements each, then the condition that b(X)=O if AnX ? 0 ? BnX reduces the maximal number of subsets X of AUB with b(X) ? 0 from 247 to 22. This example confirms the high relevance of disconnected subsets for overall systemic coherence.

2.3. Systemic and Relational Coherence

A definition of relational coherence in terms of systemic coherence was proposed by Olsson (1999) in a categorical framework. According to that proposal, A is coherent with B if and only if the combination of A and B (for sets: A U B) is coherent. This proposal can be transferred to our gradational framework through the formula r(A,B)=s(A U B). However, cases can easily be found in which this proposal is implausible. Consider the following four statements about bat species:

(a) The Noctilio leporinus has longer legs, toes and claws than other bats.

(b) The Noctilio leporinus flies low over coastal water, catch- ing surface crustacea with its legs.

(c) The incisors of the Desmodus rotundus are razor-edged, whereas its cheek teeth are degenerate.

(d) The Desmodus rotundus ("true vampire") lives entirely on fresh blood that it obtains by making incisions in the skin of living animals.

Clearly, {a,c} coheres more with {b,d} than {a,b} coheres with {c,d}.




A more promising alternative is to define the relational coherence of two mutually exclusive sets A and B as the increase in total coherence obtained by joining them, thus:

(2) r(A, B) = s(A U B)-s(A)--s(B) (for A n B = 0) To see why the formula is only applicable when A and B are mutually exclusive, consider the extreme opposite case when A = B. The formula would then reduce to r(A,A) = -s(A), making relational self-coherence inversely correlated to systemic coherence, which is absurd.

With this definition, systemic coherence of finite sets turns out to be derivable from relational coherence by recursion on the number of elements, using the following two formulas that both follow from (2):

(3) s(B U {a}) = r({a}, B) + s({a}) + s(B) (for a * B)

(4) r({a}, {b}) = s({a, b})3

Combining (1) and (2) we obtain a definition of r in terms of b, according to which the coherence between two mutually exclusive sets is equal to the sum of the coherence contributed by each subset of their union that connects them:

r(A, B) = E{b(X)IX c p(A U B) \ (p(A) U pc(B))} whenever AnB = 0

Hence, with reasonable definitions, systemic, contributed, and relational coherence are all interdefinable. This result is prob- lematic for philosophical views that favour either systemic or relational coherence to the exclusion of the other.4

3. COHERENCE OF SETS OF SENTENCES

3.1. Basics

In this section, we are going to add the further condition that the sets under study consist of sentences. This assumption increases the epistemological relevance of our deliberations, since beliefs can be represented by sentences. Admittedly,



100 SVEN OVE HANSSON

actual beliefs do not necessarily have the structure of sentences in a language. However, although sentences do not capture all aspects of beliefs, they provide the best available general-purpose representation of beliefs. (Hansson, 2003)

An immediate consequence of sentential representation is that logical relationships enter the scene. This they do in essentially two ways. First, coherence can be based at least in part on logical implication. The two sets {p,q} and {p q, pAq} are logically equivalent, and it would be strange to deny that they have a high degree of relational coherence. On the other hand, it would be too demanding to require that epistemic coherence always be based on logical inference. Weaker relationships such as "increases the probability of' or "makes it more justified to believe that" can provide the basis for coherence. However, these relations typically have logical implication as a limiting case. Therefore, we can assume that the addition to a set of sentences of some sentence that it implies should not decrease coherence, or in other words:

(6) If B F- a, then s(B U {a}) > s(B)

or more generally:

(7) If A C Cn(B) then s(B U A) > s(B) (logical addition)

The other way in which logical relationships have impact on coherence is the negative way: with inconsistency comes inco- herence. As was noted by Olsson (1999) in a categorical framework, that coherent sets be consistent "is the only thing that researchers tend to agree upon when it comes to the qualitative nature of coherence". One way to express this intuition in a gradational account is the following:

(8) If A is consistent and B inconsistent, then s(A) > s(B).

In what follows, we will be concerned with sets of sentences that represent belief states. The basic assumption is that for each belief state K there is a belief-representing set K of sentences, such that the degree of coherence of K is adequately represented by the degree of coherence of K, as measured with a systemic coherence measure s.




As was noted in section 2. 1, systemic coherence is not monotonic. It follows that there should be cases in which the removal of a sentence from a belief-representing set increases the degree of systemic coherence:

There is some belief-representing set K and some (9) set A such that K\A is a belief-representing set

and that s(K\A)>s(K). (non-monotonicity).

3.2. Logical Closure

It is generally assumed in belief revision theory that all belief- representing sets are logically closed, i.e.:

If K is a belief-representing set, then K = Cn(K) (10) (logical closure)

Similarly, we should expect the resulting new set after a removal to be logically closed. However, this gives rise to the following formal result:

Observation 1. Let 0 ? A c K= Cn(K) and K\A = Cn (K\A). Then Cn (A) = K.

Proof. Excluding trivial limiting cases, let a E A and b E K\A. It follows from the logical closure of K\A and a * K\A that a b * K\A, hence a b E A, hence b E Cn(A). Since this holds for all b E K\A we have K\A c Cn(A), hence Cn(A) = K.

This means that if the both the original set of beliefs and the set resulting from the removal of A are logically closed, then the removed unit A logically implies the original set. This is implausible, and gives us reason to give up the requirement of logical closure. However, before doing that we should consider alternatives. Perhaps, if we retain logical closure for the original set of beliefs, but give it up for the set remaining after the removal, then the removed unit (A in the above observation) can have a more plausible structure than in Observation 1. This amounts to adding a structural requirement on A to the non-monotonicity postulate. There




are two obvious ways to do this. We can require A to be a singleton, and thus let the postulate refer to the removal of a single belief, or we can require that A be logically closed, thus letting the units of coherence have the same structure as sets of belief-representing sentences. This amounts to the following two versions of the postulate:

There is some belief-representing set K and some a such that s(K \ {a}) > s(K) . (non-monotonicity 1).

There is some belief-representing set K and some (12) A such that A = Cn(A) c K and s(K\A) > s(K).

(non-monotonicity 2).

However, neither of these two variants of the postulate is compatible with the other postulates given above:

Observation 2. The following conditions are incompatible.

If A c Cn(B) then s(BUA) ? s(B) (logical addition) If K is a belief-representing set, then K= Cn(K) (logical closure) There is some belief-representing set K and some a such that s(K\{a}) > s(K). (non-monotonicity 1).

Proof. Excluding a trivial case, we have a E K. It follows from logical closure that there is some b E K\\{a} and that avb, av-,b E K\{a}. Hence K\{a} H- a, so that K c Cn(K\{a}). Logical addition then yields s(K) ? s(K\{a}), contrary to non-monotonicity 1.

Observation 3. The following conditions are incompatible:

If A c Cn(B) then s(BUA) ? s(B) (logical addition) If K is a belief-representing set, then K=Cn(K) (logical closure) There is some belief-representing set K and some A such that A = Cn(A) c K and s(K\A) > s(K). (non-monotonicity 2).




Proof. Since A c K, there is some b E K\A. Let a E A. Then due to the logical closure of A, a b E K\A, hence a E Cn(K\A). Thus A c Cn(K\A). Logical addition yields that s((K\A) U A) ? s(K\A), i.e. s(K) ? s(K\A), contrary to non- monotonicity 2.

The above three observations give us reason to give up the idea that belief-representing sets of sentences should be logically closed. It does not seem possible to apply a measure of systemic coherence to logically closed sets of sentences in a way that reflects the effects of logical relations on coherence. (A similar conclusion was reached for a categorical notion of coherence in Hansson and Olsson (1999).) The reason for this can be seen from the simple example of a person believing in two logically independent sentences p and q and their logical consequences. If we move from using {p,q} to Cn({p,q}) as a belief-representing set, then we add a large number of "connecting sentences" such as p&q, p q, p A q, r -* p&q, -,r -+ p&q, etc. These "logical additions" to the belief- representing set tend to add so much coherence that distinctions are lost.

It is important to note that this argument is not directed against the view that the logical consequences of beliefs are themselves beliefs. Someone who believes in p and q is also committed to believe in p&q, p q, p A q, etc. Therefore, logical closure of the total set of beliefs held by an agent is still a useful idealization (but a total set of beliefs should be dis- tinguished from a belief-representing set of sentences in the sense introduced in section 3.1.). What is at stake is instead the assumption that all beliefs are on an equal footing with respect to contributions to the coherence of the state of beliefs. If I believe that r and that q, I also believe that r&q, but it does not follow that the latter belief has in all respects the same standing as r and q. In particular, the fact that it implies r and q does not necessarily mean that it contributes to their justification or to the coherence of the belief state. For a concrete example, since I believe that Paris is the site of the French foreign ministry (p) I also believe that either




Paris or Quito is the site of the French foreign ministry (pVq), but this latter belief does not have an independent standing; it stands or fall with the former. Such merely derived beliefs do not seem to contribute to the coherence of the belief state.

3.3. Consequences for Epistemic Coherentism and for Belief Revision

These formal results do not invalidate the basic coherentist ideas, but they nevertheless have consequences for coherentism. The view that "all beliefs" are capable of contributing to the justification (etc.) of other beliefs has to be modified so that "mere logical consequences" are excluded. Coherentists have to recognize that there are two categories of beliefs, such that the beliefs in one of these categories (the merely derived beliefs) depend entirely on those in the second category. However, this second category of beliefs does not correspond to the basic beliefs referred to in foundationalist epistemology. Our formal analysis give us no reason to main- tain that they have to be based on experience. Indeed, as far as these results are concerned they need not have any other justification than the coherence that holds among them.

In belief revision theory it has mostly been taken for gran- ted that models in which the belief state is represented by a logically closed set of sentences (such as the AGM model, see Alchourron et al. (1985)) correspond to coherentist epistemology, whereas models employing "belief bases" that are not logically closed represent foundationalism. (Gardenfors, 1990; see further references in Hansson and Olsson (1999).) How- ever, as should be clear from the above, the logical relationships among the elements of a logically closed set do not adequately represent epistemic coherence. A representation of coherence is something that has to be added.

An explicit representation of coherence was introduced into a belief revision framework in by Olsson (1997, 1998). In (Hansson, 2000a) such a representation was employed in a study of contractions, i.e. changes in which beliefs are




removed but no new beliefs are added. Coherence was treated as categorical; hence it was assumed that belief states are either coherent or non-coherent, with no intermediate degrees. Clearly, if the current belief state is coherent it does not follow that all subsets of the current set of beliefs represent the beliefs held in some coherent belief state that can be reached through contraction.5 For instance, I presently believe that there is a radio transmitter operating on the surface of Mars (r). I also believe that a spaceship from Earth has brought such a device to Mars (s). There are various ways in which my belief state may be contracted, i.e. changed so that my set of beliefs is reduced to a subset of the original set of beliefs. It holds for all coherent belief states that I can arrive at through contraction that if r is retained, then so is s. (There may be ways in which I can be brought to coherently believe in r but not in s, but these involve the acquisition of some new belief such as that extraterrestrial beings have sent instruments to Mars.)

Based on this reasoning, we can postulate that there is a set C consisting of those subsets of the current set of beliefs, K, that represent a coherent belief state. K is assumed to be an element of C. Note that the elements of C are logically closed; as was observed above the idealization that the total set of beliefs is logically closed does not have to be given up. The central result obtained in (Hansson, 2000a) was based on the further assumptions that C is finite and all its elements finite- based, and that it forms a semi-lattice for K (i.e Cn(0) E C, K E C, and if X, Y E C then Cn(XU Y) E C). Under these conditions, a contraction operator on K such that K - a E C for all a can be obtained as the closure of a partial meet contraction on a finite belief base for B, i.e. for all a, K- a = Cn(B-a) where B is a set such that Cn(B) = K and - is an operator of partial meet contraction for B. This is "coherentist contraction" in the sense that the outcome of contracting a set K of beliefs by a non-tautologous sentence a is always a coherent subset of K that does not imply a. The fact that coherentist contraction brings us to an operation that can be con- structed with a belief base shows again that the conventional




association between belief bases and foundationalist epistemology is misleading.

With this approach, the dynamic behaviour of the belief state determines which beliefs are basic. This solves one of the major problems with belief bases, namely that it is diffi- cult to determine which are the basic sentences. It should be emphasized, however, that the approach to coherence in (Hansson, 2000a) is too simplistic; a gradational representation of coherence should be useful here as well.

4. CONCLUSION

The formalization of philosophical concepts can provide us with new insights that have implications for informal philosophy. (Hansson, 2000b) The results reported here have two major such implications. (1) Systemic and relational accounts of coherence have been treated in informal epistemology as competitors. We have shown that systemic and relational coherence are interdefinable. Therefore, they should be treated as alternative formulations of essentially the same con- cept. (2) Epistemic coherentism has been interpreted as meaning that "all beliefs support each other". We have seen that this interpretation is untenable. In order to avoid absurd conclusions, coherentists have to accept a weak version of epistemic priority, that sorts out merely derived beliefs.

NOTES

1 However, a plausible stability property should be noted that is some- what related to monotonicity: If it holds for a set B that s(A) < s(B) for all A c B, then B is stable against coherence-driven contraction. 2 Cf. Rescher's (1973, p. 32) view that coherence is a property of sets with at least two elements. 3 This holds under the assumption that singletons make zero contribution to overall systemic coherence. If singletons make non-zero contribution, then recursion is still possible provided that r({a},{a}) = s({a}). 4 See Olsson (1999) for a critical discussion of Lehrer's renunciation of systemic coherence. 5 Note again the distinction between (1) the set of all beliefs held in a




certain belief state, and (2) a set representing that belief state, and such that the interrelations among its elements correspond to the relations of coherence in that belief state. In section 3.2 we saw that a set satisfying (2) cannot be logically closed. In the rest of this section we will be concerned with logically closed sets that satisfy (1).

Acknowledgements

I would like to thank Erik J Olsson for valuable comments on an earlier version of this paper.

REFERENCES

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Bender, J.W. (1989): 'Coherence, Justification, and Knowledge: The Current Debate', in J.W. Bender (ed.), The Current State of the Coherence Theory (pp. 1-14), Dordrecht: Kluwer.

Brendel, E. (1999): 'Coherence Theory of Knowledge: A Gradational Account', Erkenntnis 50, 293-307.

Daya, K. (1960): 'Types of Coherence', Philosophical Quarterly 10, 193-204. Gardenfors, P. (1990): 'The Dynamics of Belief Systems: Foundations vs.

Coherence Theories', Revue Internationale de Philosophie 44, 24-46. Hansson, S.O. (2000a): 'Coherentist Contraction', Journal of Philosophical

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Open Court Publishing Co. Millgram, E. (2000): 'Coherence: The Price of the Ticket', Journal of

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Spohn, W. (1999): 'Two Coherence Principles', Erkenntnis 50, 155-175. Thagard, P. (1991): 'The Dinosaur Debate: Explanatory Coherence and the

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Department of Philosophy and the History of Technology Royal Institute of Technology (KTH) Teknikringen 78B, 100 44 Stockholm Sweden E-mail. soh@,infra.kth.se



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8 Bridges between Mainstream and Formal Epistemology || Coherence in Epistemology and Belief Revision