Michael Schippers

Probabilistic measures of coherence: : from adequacy constraints towards pluralism

The debate on probabilistic measures of coherence flourishes for about 15 years now. Initiated by papers that have been published around the turn of the millennium, many different proposals have since then been put forward. This contribution is partly devoted to a reassessment of extant coherence measures. Focusing on a small number of reasonable adequacy constraints I show that (i) there can be no coherence measure that satisfies all constraints, and that (ii) subsets of these adequacy constraints motivate two different classes of coherence measures. These classes do not coincide with the common distinction between coherence as mutual support and coherence as relative set-theoretic overlap. Finally, I put forward arguments to the effect that for each such class of coherence measures there is an outstanding measure that outperforms all other extant proposals. One of these measures has recently been put forward in the literature, while the other one is based on a novel probabilistic measure of confirmation.

Towards a Grammar of Bayesian Coherentism

One of the integral parts of Bayesian coherentism is the view that the relation of ‘being no less coherent than’ is fully determined by the probabilistic features of the sets of propositions to be ordered. In the last one and a half decades, a variety of probabilistic measures of coherence have been put forward. However, there is large disagreement as to which of these measures best captures the pre-theoretic notion of coherence. This paper contributes to the debate on coherence measures by considering three classes of adequacy constraints. Various independence and dependence relations between the members of each class will be taken into account in order to reveal the ‘grammar’ of probabilistic coherence measures. Afterwards, existing proposals are examined with respect to this list of desiderata. Given that for purely mathematical reasons there can be no measure that satisfies all constraints, the grammar allows the coherentist to articulate an informed pluralist stance as regards probabilistic measures of coherence.

INCOHERENCE AND INCONSISTENCY

This paper scrutinizes the relationship between inconsistency and incoherence with a special focus on probabilistic measures of coherence. As is shown, while the majority of extant coherence measures face problems regarding the assessment of inconsistent sets of propositions, it is possible to adapt the measures in order to improve their performance. Furthermore, different intuitions regarding the degree of incoherence of inconsistent sets of propositions are surveyed and assessed with respect to extant measures. In this context, a refined approach to measuring coherence is introduced. As is argued, by means of this approach one can account for the diverging coherence intuitions regarding inconsistent sets independently of the discussion on the adequacy of different probabilistic explications of coherence. The last part of the paper is devoted to the question of whether there is a covariation between degrees of inconsistency and degrees of incoherence in the sense that the higher the degree of inconsistency of a set of propositions, the higher its degree of incoherence. Focusing on two straightforward measures of the degree of inconsistency, this latter question is answered in the negative.

Against relative overlap measures of coherence

Coherence is the property of propositions hanging or fitting together. Intuitively, adding a proposition to a set of propositions should be compatible with either increasing or decreasing the set’s degree of coherence. In this paper we show that probabilistic coherence measures based on relative overlap are in conflict with this intuitive verdict. More precisely, we prove that (i) according to the naive overlap measure it is impossible to increase a set’s degree of coherence by adding propositions and that (ii) according to the refined overlap measure no set’s degree of coherence exceeds the degree of coherence of its maximally coherent subset. We also show that this result carries over to all other subset-sensitive refinements of the naive overlap measure. As both results stand in sharp contrast to elementary coherence intuitions, we conclude that extant relative overlap measures of coherence are inadequate.

STRUCTURAL PROPERTIES OF QUALITATIVE AND QUANTITATIVE ACCOUNTS TO COHERENCE

This paper evaluates four different qualitative (probabilistic) accounts to coherence with a focus on structural properties (symmetries, asymmetries, and transitivity). It is shown that while coherence is not transitive on any of these accounts, there are screening-off conditions that render coherence transitive. In a second step, an array of quantitative (probabilistic) accounts to coherence is considered. The upshot is that extant measures differ considerably with respect to a number of symmetry constraints.

Competing accounts of contrastive coherence

The proposition that Tweety is a bird coheres better with the proposition that Tweety has wings than with the proposition that Tweety cannot fly. This relationship of contrastive coherence is the focus of the present paper. Based on recent work in formal epistemology we consider various possibilities to model this relationship by means of probability theory. In a second step we consider different applications of these models. Among others, we offer a coherentist interpretation of the conjunction fallacy.

Coherence and (Likeness to) Truth

According to a coherentist position in philosophy of science, good theories cohere with the available data and one theory is better than another if it coheres better with the available data. This paper examines that relationship with a special focus on probabilistic measures of coherence. In a first step, it is shown that existing coherence measures satisfy a number of reasonable adequacy constraints for the comparison of two rival scientific theories. In a second step, the virtue of a coherentist position in philosophy of science is considered. More specifically, it is assessed whether coherence implies verisimilitude in the sense that a higher degree of coherence between theory and evidence entails a higher degree of (estimated) truthlikeness. To this end, it is demonstrated that there is an intimate relationship in this sense if we explicate coherence by means of the so-called overlap-measure.

Genuine Confirmation and Tacking by Conjunction

Tacking by conjunction is a deep problem for Bayesian confirmation theory. It is based on the insight that to each hypothesis h that is confirmed by a piece of evidence e one can ‘tack’ an irrelevant hypothesis h′ so that h∧h′ is also confirmed by e. This seems counter-intuitive. Existing Bayesian solution proposals try to soften the negative impact of this result by showing that although h∧h′ is confirmed by e, it is so only to a lower degree. In this article we outline some problems of these proposals and develop an alternative solution based on a new concept of confirmation that we call genuine confirmation. After pointing out that genuine confirmation is a necessary condition for cumulative confirmation we apply this notion to the tacking by conjunction problem. We consider both the question of what happens when irrelevant hypotheses are added to a hypothesis h that is confirmed by e as well as the question of what happens when h is disconfirmed. The upshot of our discussion will be that genuine confirmation provides a robust solution for each of the different perspectives. 1 Introduction 2 Tacking by Conjunction: Existing Solution Proposals 3 Genuine Confirmation   3.1 Content elements and content parts   3.2 Qualitative genuine confirmation   3.3 Quantitative genuine confirmation 4 Tacking by Conjunction: The Case of Confirmation 5 Tacking by Conjunction: The Case of Disconfirmation 6 Tacking by Conjunction: Adding Multiple Hypotheses 7 Conclusion Appendix  1 Introduction 2 Tacking by Conjunction: Existing Solution Proposals 3 Genuine Confirmation   3.1 Content elements and content parts   3.2 Qualitative genuine confirmation   3.3 Quantitative genuine confirmation   3.1 Content elements and content parts   3.2 Qualitative genuine confirmation   3.3 Quantitative genuine confirmation 4 Tacking by Conjunction: The Case of Confirmation 5 Tacking by Conjunction: The Case of Disconfirmation 6 Tacking by Conjunction: Adding Multiple Hypotheses 7 Conclusion Appendix 

Genuine Coherence as Mutual Confirmation Between Content Elements

The concepts of coherence and confirmation are closely intertwined: according to a prominent proposal coherence is nothing but mutual confirmation. Accordingly, it should come as no surprise that both are confronted with similar problems. As regards Bayesian confirmation measures these are illustrated by the problem of tacking by conjunction. On the other hand, Bayesian coherence measures face the problem of belief individuation. In this paper we want to outline the benefit of an approach to coherence and confirmation based on content elements. It will be shown that the resulting concepts, called genuine coherence and genuine confirmation, can be used in order to solve the two mentioned problems. In a final section we present some results on degrees of genuine coherence and genuine confirmation.