Coherence Theory

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  • MICHAEL WILLIAMs concepts of systematical and relational coherence

    Uwe Wiedemann [1]
    (University of Leipzig)

    Content

    1  Introduction
    2  What is a coherence problem?
    3  Systematic coherence as a measure of constraint satisfaction
    4   Some types of coherence problems
    4.1  Pure coherence problems
    4.2  Foundational and contextualistic coherence problems
    4.3  Discriminating coherence problems
    5  Relational coherence
    6  Results
    7  Literature

    1  Introduction

    Based on the concept of coherence problems as constraint satisfaction we consider WILLIAMS' systematic coherence as a measure of constraint satisfaction. We trace the systematic coherence back to the coherence and incoherence relations. That is opposed to WILLIAMS' thesis that the systematic coherence is fundamental. We will expand the theory of coherence problems defining a more adequate type of coherence problem. By discussing some types of coherence problems, we will see that there are coherence problems in foundationalism and contextualism, too. Moreover, we will claim that WILLIAMS is not necessarily right with his thesis that relational coherence is subordinated to systematic coherence.

    2  What is a coherence problem?

    It is useful to study the structure of coherence problems in general. In a first step we nearly follow the characterization of pure coherence problems by THAGARD and VERBEURGT. [2] However, we will use some generalizations and corrections of this type, which will not be discussed here.

    The common idea of our coherence theory is it to put all hypotheses and theses on the market in a set and to divide the elements of this set into accepted and rejected ones. [3] We call all these theses and hypotheses elements. In other coherence problems, we consider other elements, for example concepts, parts of images, goals, actions, and legislative acts.

    Furthermore, there are relations between the elements, called coherence or incoherence relations. Coherence relations include explanation, deduction, association, and so on, while incoherence relations include for instance inconsistency, competition, and incompatibility. [4]

    In coherence theories of epistemic justification the elements are propositions. Explanations and entailments are coherence relations. Inconsistencies and weaker relations, such as competition, are incoherence relations. It is clear, that explanatory connections are only a part of coherence-making connections. Lehrer has given other examples. [5]

    Not all coherence and incoherence relations have the same power. So is an inconsistency of two propositions stronger than a competition between them. Therefore, we use constraints constituted by the coherence and incoherence relations. If two or more elements hold a coherence relation, there is a positive constraint between them. If an incoherence relation holds between two or more elements, there is a negative constraint between them. [6]

    In a bisection of the set of propositions the coherence and incoherence relations can be satisfied. For example two inconsistent propositions are satisfied only if at least one of the propositions is not in the accepted set. Analogously we speak about satisfied constraints.

    The coherence problem consist of dividing a set of elements into accepted and rejected sets in a way that the constraints are satisfied as much as possible.

    Simultaneous satisfaction may be impossible. It is not necessary to ensure that all the constraints are simultaneously satisfied. In many cases the problem is overconstrained and admits no complete solution. Therefore, the coherence problem is a partial constraint satisfaction problem. [7]

    The structure of coherence problems is the stable doctrine of coherence theories WILLIAMS had missed. [8] Of course, how we get the constraints of a relation and what it means to satisfy a constraint is not unitary. WILLIAMS is right in another sense claiming that there is no stable doctrine that deserves to be called the coherence theory of justification, because it is not clear what coherence means in the context of coherence problems.

    Our general characterization of coherence problems directs the attention away from the relations between individual propositions to certain features of proposition-systems taken as a whole. Relations between particular propositions are important especially for their contribution to the coherence of some global view.

    3  Systematic coherence as a measure of constraint satisfaction

    I claim to consider WILLIAMS' systematic coherence as a measure of constraint satisfaction. So we can read THAGARD's "goodness-of-fit" and "harmony" [9] or SCHOCH's maximum of the coherence function [10] as systematic coherence. In this claim the systematic coherence is the sum of the constraints, which are satisfied. [11]

    THAGARD and VERBEURGT have pointed out that the harmony measure is of bounded worth, because it is very sensible with respect to the number of elements. It seems possible solving the problem by dividing this measure by the number of elements or constraints. Such measures would also be measures of systematic coherence. For our considerations, these measures are as good as the simple sum.

    Our understanding of systematic coherence shows that WILLIAMS is right, if he claims that systematic coherence implies radical holism. [12] We have traced systematic coherence back to coherence and incoherence relations. Therefore, we have seen that systematic coherence is not fundamental.

    Supposing we substitute BARTELBORTH's condition of incoherence degree by his conditions of incoherence, we can characterize his concept of systematic coherence by the following properties:

    X is the more coherent,

    (a) the more inferential relations (logical and explanatory) connect the propositions in X [condition of the degree of connection]
    (b) the better the explanations in X are [condition of explanatory power]
    (c) the fewer inconsistencies exist in X (including probabilistic ones) [inconsistency condition]
    (d) the fewer subsystems X contains, which are weakly connected [subsystem condition]
    (e) the fewer explanatory anomalies appear in X [anomaly condition]
    (f) the fewer competitive explanations exist in X [competition condition]
    (g) the more stable X was in the past [stability condition]. [13]

    Our definition of systematic coherence as a measure of constraint satisfaction fulfils the conditions of the degree of connection, of explanatory power, the inconsistency condition, the subsystem condition and the competition condition in a nice way. However, the anomaly condition and the stability condition have insufficiently been realized so far. We have to discuss how belief revision works for integrating the stability condition. How the anomaly condition works in our framework is not clear.

    4  Some types of coherence problems

    4.1 Pure coherence problems

    We now have to discuss more precisely, what it means by saying that a constraint is satisfied.

    THAGARD and VERBEURGT outlined the following characterization of constraint satisfaction (slightly generalized): [14]

    (PS1) A positive constraint between two or more elements can be satisfied by either accepting all or rejecting all of these elements.
    (NS1) A negative constraint between two or more elements can be satisfied only by accepting at least one element and by rejecting another one.
    The authors call coherence problems with these constraint satisfaction properties pure coherence problems.

    There is however a weak point in this description: The sets of accepted and rejected elements are interchangeable without any loss in constraint satisfaction. That means, if we consider the coherence problem of justification as pure coherence problem, then we can interchange the sets of justified and rejected elements and the elements in the new accepted set are justified, too. Of course, that is a refutation of the thesis that the coherence problem of justification is a pure coherence problem.

    SCHOCH uses de facto another satisfying condition for negative constraints than it is described by the condition for pure coherence problems. We can reconstruct his condition in the following way:

    (NS2) A negative constraint between two or more elements can be satisfied by rejecting at least one element.

    Here a negative constraint is satisfied, if all its elements are rejected contrary to the satisfaction of negative constraints we discussed before. This condition is more intuitive than that of THAGARD and VERBEURGT.

    This condition blocks the interchangeability of the sets of accepted and rejected elements.

    Similar acts the condition of positive constraint satisfaction (PS2), which SCHOCH uses implicitly, too.

    (PS2) A positive constraint between two or more elements can be satisfied by accepting all of these elements.

    This condition is also more intuitive than that of THAGARD and VERBEURGT.

    SCHOCH's constraint satifaction conditions show that we can get a coherence theory of justification without any faith of evidence. We can solve the problem of interchangeability only by reason of explanatory and inconsistency relations. SCHOCH's constraint conditions are opposed to the many system objection.

    4.2  Foundational and contextualistic coherence problems

    We can also add evidence conditions to block the interchangeability of the accepted and rejected elements. One type of coherence problem we get in this way is called foundational coherence problem. THAGARD and VERBEURGT wrote about this type:

    "A foundational coherence problem selects a set of favored elements for acceptance as self-justified." [15]

    Let me give a more precise description of this property. The conditions are almost the same as those of pure coherence problems, but additionally we have the following:

    (EE) There is a nonempty subset of the set of elements, called the set of evidence.
    (ES) An evident element is satisfied if this element is accepted.
    Including evidences we get another formulation of the coherence problem:

    (FCP) The foundational coherence problem consists of dividing a set of elements into accepted and rejected sets in a way that all evidences are satisfied, and the constraints are additionally being satisfied as much as possible.

    We can see that there is not a dichotomy between foundationalism and coherentism. There are coherence problems within the framework of formal foundationalism. [16] Supposing we have evidences of any kind, we have to solve the problem to obtain the other theses and hypotheses we can accept.

    In a similar way we can describe the coherence problem in the framework of contextualism. In this case, we select a set of favored elements for acceptance as context-justified.

    That means, the foundational coherence problems and contextualistic coherence problems have the same structure.

    4.3  Discriminating coherence problems

    In foundational coherence problems the evidences are a dogma. In this type of coherence problems, they are necessarily accepted. That is not very plausible. In our scientific work, we sometimes reject observations, for instance. The notion of discriminating coherence problems is an answer to this circumstance.

    THAGARD und VERBEURGT wrote:

    "A discriminating coherence problem favors a set of elements but their acceptance still depends on their coherence with all the other elements." [17]

    The conditions are the same as in foundational coherence problems, but we replace (FCP) by the following:

    (DCP) The discriminating coherence problem consists of dividing a set of elements into accepted and rejected sets in a way that both the evidences and the constraints are satisfied as much as possible. [18]

    Not all evidences have to be satisfied, but as much as possible with respect to satisfying the constraints, too.

    Discriminating coherence problems are as foundational coherence problems examples of formal foundationalism. Again, we can consider discriminating coherence problems within the framework of contextualism by interpreting the favored elements as context-justified.

    Using SCHOCH's satisfaction conditions and the framework of discriminating coherence problems, we get the following structure of the coherence problem of justification:

    1. Elements are propositions.
    2. Coherence and incoherence relations can exist between elements. Coherence relations include explanation and deduction. Incoherence relations include inconsistency and competition.
    3. If a coherence relation exists between two or more elements, than there is a positive constraint between them. If an incoherence relation exists between two or more elements, than there is a negative constraint between them.
    4. Elements are to be divided into accepted or rejected ones.
    5. A negative constraint between two or more elements can be satisfied by rejecting at least one element.
    6. A positive constraint between two or more elements can be satisfied by accepting all of these elements.
    7. There is a nonempty subset of the set of elements, called the set of evidence.
    8. The coherence problem consists of dividing a set of elements into accepted and rejected sets in a way that both the evidences and the constraints are satisfied as much as possible.
    9. The elements of the accepted set are called justified.
    10. The measure of the constraint and evidence satisfaction is called systematic coherence.
    5  Relational coherence

    Similar to OLSSON [19] we make a distinction between two kinds of belief changes:

    (1) changes necassary to move to a state of systematic coherence, and
    (2) changes necassary to incorporate new input.

    OLSSON calls the first kind of change consolidation. [20] The aim of consolidation is to maximize constraint satisfaction.

    The new input can be a new theory or a single proposition (hypothesis, evidence). For the analysis of the first kind of new input we will use the notion of the coherence of a subset. [21] For the second kind we will use WILLIAMS notion of relational coherence.

    WILLIAMS claims that relational coherence is subordinated to systematic coherence. [22]

    THAGARD and VERBEURGT are in accordance with the idea that it is possible to trace the coherence of a particular element back to the systematic coherence.

    "It would be desirable to define, within our abstract model of coherence as constraint satisfaction, a measure of the degree of coherence of a particular element or of a subset of elements, but it is not clear how to do so. Such coherence is highly non-linear, since the coherence of an element depends on the coherence of all the elements that constraint it, including elements with which it competes. The coherence of a set of elements is not simply the sum of the weights of the constraints satisfied by accepting them, but depends also on the comperative degree of constraint satisfaction of other elements that negatively constrain them." [23]

    However, it is not necessary to subordinate relational coherence to systematic coherence. It is a needless restriction of our space of research.

    We can identify, for instance, acceptability and relational coherence. It is also thinkable to subordinate relational coherence to acceptance in other ways.

    BARTELBORTH has another idea, but with the same result. He only uses the coherence relation to characterize the relational coherence. He formulates two conditions, the abduction condition and the embedding condition.

    The abduction condition says:

    The more p coheres with the system X the more propositions exist in X, which are explained or infered by p, and the better the explanations are.

    The embedding condition says:

    The more p coheres with the system X the more often can p be deduced or explained from the propositions in X, and the better the explanations are. [24]

    BARTELBORTH gets a concept of relational coherence that fulfils most of the expected properties of relational coherence.

    In his analysis of relational coherence contradictions, competitions and evidences play no part at all. In an analogous way to BARTELBORTH's definition of relational coherence, we can trace the relational incoherence back to incoherence relations.

    To subordinate the concept of relational coherence to the concept of acceptance, acceptability or coherence and incoherence relation are ways without subordinating relational coherence to systematic coherence and therefore this subordination is not necessary.

    6  Results

    We considered WILLIAMS' systematic coherence as a measure of constraint satisfaction. We traced the systematic coherence back to the coherence and incoherence relations, and saw in this way that systematic coherence is not fundamental. By discussing some types of coherence problems we got a new one and saw that there are coherence problems in foundationalism and contextualism, too. Moreover, we saw that relational coherence is not necessarily subordinated to systematic coherence. However, it is eventually the best way to get relational coherence.

    7  Literature

    Bartelborth, Thomas: Begründungsstrategien. Ein Weg durch die analytische Erkenntnistheorie. Berlin 1996

    Bartelborth, Thomas: Coherence and Explanations. (forthcoming in Erkenntnis)

    Freuder, Eugene C./Wallace, Richard J.: Partial constraint satisfaction. Artificial Intelligence 58 (1992), 21 - 70

    Lehrer, Keith: Knowledge. Oxford 1978

    Lehrer, Keith: Theory of Knowledge. Boulder, Colo./London 1990

    Lehrer, Keith: Self-Trust. A Study of Reason, Knowledge, and Autonomy, Oxford 1997

    Olsson, Erik J.: Making Beliefs Coherent. Journal of Logic, Language, and Information 7 (1998), 143 - 163

    Schoch, Daniel: A Fuzzy Measure for Explanatory Coherence (forthcoming)

    Thagard, Paul: Conceptual Revolutions. Princeton, N. J. 1992

    Thagard, Paul/Verbeurgt, Karsten: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 1 - 24

    Williams, Michael: Unnatural Doubts. Epistemological Realism and the Basis of Scepticism. Princeton, N. J. 1996

    Footnotes

    [1] The author is supported by the graduate college "Knowledge representation" from the Deutsche Forschungsgemeinschaft (DFG).
    [2] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 16
    [3] That is nearly the way in the sciences. There are many ideas on the market and scientists evaluate them.
    [4] If the coherence or incoherence relations are binary, we get Thagard's notion of pairwise coherence or pairwise incoherence. (Thagard, P.: Conceptual Revolutions. Princeton, N. J. 1992, 66)
    [5] Lehrer, K.: Theory of Knowledge. Boulder, Colo./London 1990, 105f.
    [6] In SCHOCH's theory is explanation not only a coherence relation, but also an incoherence relation. If P1 and P2 explain P3, then there is in SCHOCH's model de facto a negative constraint between P1, P2 and ~P3 (Schoch, D.: A Fuzzy Measure for Explanatory Coherence (forthcoming)).
    [7] Freuder, E. C./Wallace, R. J.: Partial constraint satisfaction. Artificial Intelligence 58 (1992), 21 - 70
    [8] Williams, M.: Unnatural Doubts. Epistemological Realism and the Basis of Scepticism. Princeton, N. J. 1996, 268
    [9] vgl. Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 13
    [10] Schoch, D.: A Fuzzy Measure for Explanatory Coherence (forthcoming)
    [11] Strictly speaking THAGARD and SCHOCH implicitly subtract the number of negative constraints from the total sum of the constraints (not only the constraints which are satisfied, but also those that are not satisfied). That is not essential with respect to dividing the set of propositions into an accepted and a rejected set.
    [12] Williams, M.: Unnatural Doubts. Epistemological Realism and the Basis of Scepticism. Princeton, N. J. 1996, 279
    [13] Bartelborth, T.: Begründungsstrategien. Ein Weg durch die analytische Erkenntnistheorie. Berlin 1996, 193
    [14] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 3
    [15] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 16
    [16] I follow WILLIAMS' distinction between formal and substantive foundationalism (Williams, M.: Unnatural Doubts. Epistemological Realism and the Basis of Scepticism. Princeton, N. J. 1996, 114).
    [17] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 16
    [18] THAGARD's model ECHO is an attempt to solve discriminating coherence problems in some special cases. (about ECHO see for instance: Thagard, P.: Conceptual Revolutions. Princeton, N. J. 1992, 70ff.)
    [19] Olsson, E. J.: Making Beliefs Coherent. Journal of Logic, Language, and Information 7 (1998), 143
    [20] Olsson, E. J.: Making Beliefs Coherent. Journal of Logic, Language, and Information 7 (1998), 143
    [21] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 13
    [22] Williams, M.: Unnatural Doubts. Epistemological Realism and the Basis of Scepticism. Princeton, N. J. 1996, 279
    [23] Thagard, P./Verbeurgt, K.: Coherence as Constraint Satisfaction. Cognitive Science, 22 (1998), 14
    [24] Bartelborth, T.: Begründungsstrategien. Ein Weg durch die analytische Erkenntnistheorie. Berlin 1996, 193; Bartelborth, Thomas: Coherence and Explanations. (forthcoming in Erkenntnis)