## Abstract

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.

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## Notes

A different example is given by Bovens and Hartmann in their Tokyo murder case (cf. Bovens and Hartmann 2003, p. 39f.).

An analysis of the coherence measures to be introduced in the next section shows disagreement with respect to this test case (proof omitted).

In a recent anthology, Martin Blauw even speaks of a “contrastivist

*movement*” in philosophy (Blauw 2013, p. 1).Note that the case for the pair \(((\dagger _d),(\dagger _J))\) is so far unsettled.

\((\dagger _l)\) satisfies

**(QD)**provided that division by zero is equated with infinity. Alternatively, an ordinally equivalent measure put forward by Kemeny and Oppenheim (1952) can be used.Condition

**(D3)**is similar to the Bovens–Olsson condition that is well-known in the literature on measuring coherence (cf. Bovens and Olsson 2000, p. 688). In contrast to**(D3)**, the Bovens–Olsson condition considers*one*pair of propositions with respect to*two*different probability distributions where (**D3)**only pertains to cases where*two*different pairs of propositions are assessed with respect to*one*probability distribution. For an assessment of (a generalized form of) the Bovens–Olsson condition with respect to probabilistic measures of coherence see Schippers (2015).**(D3)**itself is discussed by Glass (2007) as a condition for ranking different explanations.There is a small caveat in this observation: so far the case of \((\dagger )_z\) is unsettled, i.e. I have neither been able to prove that it

*does*satisfy (**D3**) nor have I been able to find a counterexample using Branden Fitelson’s*PrSAT*(see Fitelson 2008).Based on the close relationship between the concepts of coherence and explanation, Siebel even argues in these papers for the impossibility of probabilistically measuring coherence. Starting from the observation that the concept of explanation cannot be reduced to probability, Siebel concludes that “if probabilistic accounts cannot cope with explanation, they will hardly be able to deal with coherence because, as BonJour (1985) and many others have pointed out, coherence is a function of explanation.” For a rebuttal see Roche and Schippers (2013).

Note that this inequality is the common core of various measures of explanatory power (Good 1960; McGrew 2003; Schupbach and Sprenger 2011; Crupi and Tentori 2012), that is, each of these measures \({\mathscr {E}}_{\Pr }(e,h)\) that quantify the degree of explanatory power that

*h*provides for*e*(given \(\Pr \)) satisfies the following principle for any contingent \(e,h_1,h_2\) and any regular probability \(\Pr \) (cf. Crupi and Tentori 2012):$$\begin{aligned} {\mathscr {E}}_{\Pr }(e,h_1)\gtreqless {\mathscr {E}}_{\Pr }(e,h_2) \quad \text { iff}\quad \Pr (e|h_1)\gtreqless \Pr (e|h_2) \end{aligned}$$Measures that satisfy \((\ddagger )\) are (among others)

*d*,*l*,*r*,*z*,*ku*, Shogenji’s justification measure*J*and Kemeny and Oppenheim’s (1952) measure*k*.Another interpretation of the conjunction fallacy is given by Shogenji (2012). There, Shogenji shows that conditions (3) and (4) also imply that the conjunctive hypothesis \(b\wedge f\) is more justified by

*e*than*b*, as measured by his justification measure*J*. Given that*J*also satisfies \((\ddagger )\), we also know that the degree justification-based coherence as measured by \({\mathscr {C}}_J\) is higher for the conjunctive hypothesis.

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## Acknowledgments

This work was supported by Grant SI 1731/1-1 to Mark Siebel from the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program “New Frameworks of Rationality” (SPP 1516).

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Schippers, M. Competing accounts of contrastive coherence.
*Synthese* **193, **3383–3395 (2016). https://doi.org/10.1007/s11229-015-0937-4

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DOI: https://doi.org/10.1007/s11229-015-0937-4

### Keywords

- Contrastivism
- Coherence
- Probability
- Confirmation
- Bayesian epistemology
- Conjunction fallacy