Abstract
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 settheoretic 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.
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Notes
Another methodological approach for evaluating coherence measures is based on test cases, i.e. paradigmatic situations featuring clearcut intuitions regarding the coherence ordering of sets of propositions. For a systematic evaluation of a great number of test cases see Koscholke (2013).
A proposition \(\varphi \) is called contingent iff it is neither tautologous nor contradictory; under a regular probability, \(0 < \Pr (\varphi )< 1\) for any contingent \(\varphi \).
Further measures are considered in the sequel of the paper.
The distinction between absolute confirmation and incremental confirmation is akin to Carnap’s (1962) distinction between measures of firmness and measures of increase in firmness.
More precisely, there are probability distributions such that \(\Pr (\varphi _1\wedge \ldots \wedge \varphi _n)=\prod _{i\le n}\Pr (\varphi _i)\) and \(\Pr (\varphi _{i_1}\wedge \ldots \wedge \varphi _{i_k})\not =\prod _{j\le k}\Pr (\varphi _{i_j})\) for some set \(\{i_1,\ldots ,i_k\}\subsetneq \{1,\ldots ,n\}\).
Bovens and Olsson concede that “one could argue that coherence is an ambiguous notion. One can think of coherence as a measure of agreement or as a measure of striking agreement. Coherence is sensitive to the specificity of the information on the latter but not on the former notion” (2000, p. 688f). As argued in the sequel of the paper, I do partly agree with respect to the ambiguity of the notion of coherence. However, part of the argument above against criticism of the equivalencecondition also applies to the view of coherence as striking agreement (cf. Schippers 2014c).
For simplicity we assume that \(\Pr (\varphi \psi )=0\) whenever \(\psi \) is logically inconsistent. For a discussion of this and related approaches to measure the coherence of inconsistent sets of propositions see Schippers and Siebel (2014).
By now, there is neither a proof to the effect that \(\mathcal{O}\) and \(\mathcal{O}^{*}\) satisfy (Agreement) for all \(n\ge 3\) nor any known counterexample. Indeed, I have tried to find such an example by means of two computer programs that allow to search for probability distributions that violate a given number of constraints. The one is Branden Fitelson’s PrSAT (see Fitelson 2008), the other one has recently been written by Jakob Koscholke. This seems to constitute at least some evidence that both measures might satisfy (Agreement).
A proof is given in Appendix 1.
A proof is given in Appendix 2.
\(\Gamma \) is a maximally consistent subset of \({\Delta }\) iff (i) \(\Gamma \subseteq {\Delta }\), (ii) \(\Gamma \) is consistent, and (iii) there is no other consistent subset \(\Gamma '\) such that \(\Gamma \subsetneq \Gamma '\subseteq {\Delta }\).
This assertion has been corroborated by a simulation study. In a sample of one million probability distributions over a triple of propositions, in each distribution such that \(\mathcal{RO}\) exceeds 0.7, the coherence value assigned by \(\mathcal{F}\) was always larger than the one assigned by \(\mathcal{RO}\).
This is in line with Glass’ intended reading of his test case (personal communication).
Given the distinction between MIC and MACmeasures of coherence, it is not initially clear why MACcoherence measures ought to comply with intuitions in the Tweety example. Might it be the case that what drives our intuition in this example is solely due to the relevancepart of MICmeasures? I think that this is not the case. What makes the pair of propositions incoherent is simply the fact that the probability that Tweety is a bird given that it is a grounddweller is nearly nil (and vice versa). In a similar vein, Siebel (2005) argues as follows: “Why does the proposition ‘Tweety is a bird’ fit ‘Tweety has wings’ much better than ‘Tweety cannot fly’? Because, one might argue, the probability that Tweety has wings, given that it is a bird, strongly exceeds the probability that Tweety cannot fly, given that it is a bird” (2005, p. 335). Thus, in Siebel’s argument there is no resort to unconditional probabilities whatsoever. Furthermore, Meijs’ (2006) discussion of the Tweety example presupposes that MACmeasures of coherence, like his \(\mathcal{O}^{*}\), can reasonably be applied to this example, although he himself argues for the existence of different classes of coherence measures [see his dissertation (2005)]. Thus, I conclude that in the absence of any persuasive argument to the contrary, MACmeasures of coherence ought to comply with intuitions in the Tweetyexample. I thank an anonymous reviewer for pressing me to be clearer on this point.
This latter verdict might be considered somewhat controversial. The overlap measures \(\mathcal{O}\) and \(\mathcal{O}^{*}\) do not have a clearcut threshold separating coherent and incoherent sets of propositions. To some extent, they only allow for ordinal rankings of higher or lower degrees of coherence. However, given that all assigned degrees of coherence are ‘close’ to the minimum possible degree of coherence, zero, it seems reasonable to infer that both sets are judged incoherent. On the other hand, the posterior probability \(\Pr (\varphi \psi )\), considered as a measure of the absolute degree of confirmation that \(\psi \) provides for \(\varphi \) clearly suggests \(.5\) as the demarcation value above which a positive degree of confirmation is assigned. This value transfers to the corresponding coherence measure \(\mathcal{C}_f\).
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Acknowledgments
I owe special thanks to David H. Glass, Jakob Koscholke, Tomoji Shogenji, Mark Siebel and two anonymous reviewers for very helpful discussion or comments. This work was supported by Grant SI 1731/11 to Mark Siebel from the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program “New Frameworks of Rationality” (SPP 1516).
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Appendices
Appendix 1: Proof of Theorem 1
Theorem 1
There cannot be a probabilistic measure of coherence that satisfies both (Independence) and (Agreement).
Proof
Consider the following probability distributions over the set \(\{A_1, A_2\}\):
Straightforward calculations yield that \(A_1\) and \(A_2\) are probabilistically independent on both distributions \(\Pr \) and \(\Pr '\). Thus, a measure satisfying (Independence) will assign equal degrees of coherence to \(\{A_1,A_2\}\) on both probability distributions. On the other hand, \(\Pr (A_1A_2)=1/2>1/4=\Pr '(A_1A_2)\) and \(\Pr (A_2A_1)=1/2>1/4=\Pr '(A_2A_1)\). Hence, a measure satisfying (Agreement) will assign a higher degree of coherence to the set \(\{A_1,A_2\}\) on \(\Pr \) than on \(\Pr '\), i.e. it will assign different degrees of coherence to the set depending on which distribution is chosen. Thus, there cannot be a measure that satisfies both constraints. \(\square \)
Appendix 2: Proof of Theorem 2
Theorem 2
There are two maximally consistent subsets of the set of all constraints:

\(\Sigma _1\)={(Dependence), (Independence), (Equivalence), (Inconsistency)}

\(\Sigma _2\)={(Equivalence), (Inconsistency), (Agreement)}
Proof
In order to prove theorem 2, we have to show that (i) \(\Sigma _1\) and \(\Sigma _2\) are maximally consistent subsets of the set of all constraints and that (ii) there is no other maximally consistent subset.

(i)
Obviously, \(\Sigma _1\) is a subset of all considered constraints. Furthermore, the fact that the coherence measure \(\mathcal{C}_{\lambda }\) satisfies all constraints in \(\Sigma _1\) establishes the consistency of \(\Sigma _1\). The only constraint that is not a member of \(\Sigma _1\) is (Agreement). By theorem 1, we know that (Agreement) and (Independence) are logically inconsistent. Hence, there is no consistent superset \(\Sigma _1\subsetneq \Sigma '\) of constraints. Therefore, \(\Sigma _1\) is a maximally consistent subset of all considered constraints. Now consider \(\Sigma _2\): consistency is established by measure \(\mathcal{C}_f\), and the fact that (Independence) and (Agreement) are inconsistent shows that we cannot add (Independence) to \(\Sigma _2\) without loosing consistency. But the same holds for (Dependence) as there cannot be a measure that satisfies both (Dependence) and (Agreement). To see this, consider the following probability distributions:
Straightforward calculations yield the following probabilities: \(\Pr (A_1A_2)=1/2>1/4=\Pr (A_1)\) and \(\Pr (A_2A_1)=1/2>1/4=\Pr (A_2)\). On the other hand, \(\Pr '(A_1A_2)=399/655<655/912=\Pr '(A_1)\) and \(\Pr '(A_2A_1)=399/655<655/912=\Pr '(A_2)\). Thus, for each possible coherence measure \(\mathcal{C}\) satisfying (Dependence) with neutrality threshold \(\beta _\mathcal{C}\), \(\mathcal{C}_{\Pr }(A_1,A_2)>\beta _\mathcal{C}>\mathcal{C}_{\Pr '}(A_1,A_2)\). However, since \(\Pr (A_1A_2)<\Pr '(A_1A_2)\) and \(\Pr (A_2A_1)<\Pr '(A_2A_1)\), we get the opposite coherence ordering for each measure \(\mathcal{C}'\) satisfying (Agreement), i.e. \(\mathcal{C}'_{\Pr }(A_1,A_2)<\mathcal{C}'_{\Pr '}(A_1,A_2)\). Thus, there cannot be a coherence measure that satisfies (Agreement) and (Dependence). Hence, there is no consistent superset \(\Sigma '\) of \(\Sigma _2\), so that \(\Sigma _2\) is a maximally consistent subset of all considered constraints.

(ii)
Assume that \(\Sigma \) is a maximally consistent subset of the set of all considered constraints, and assume furthermore that (Independence) is an element of \(\Sigma \). Than we can consistently add all constraints except (Agreement). Hence, by being maximally consistent, \(\Sigma \) must be identical to \(\Sigma _1\). The same holds if (Dependence) is assumed to be an element of \(\Sigma \). On the other hand, if we assume (Agreement) to be in \(\Sigma \), then we can consistently add (Equivalence) and (Inconsistency). Hence, both should be in \(\Sigma \), turning \(\Sigma \) identical to \(\Sigma _2\). Finally, if either (Equivalence) or (Inconsistency) are in \(\Sigma \), then by consistency, both of them should be in \(\Sigma \). Furthermore, we can either add (Independence) and (Dependence) yielding \(\Sigma _1\) or (Agreement), thereby yielding \(\Sigma _2\). Hence, \(\Sigma _1\) and \(\Sigma _2\) are the only maximally consistent subsets of considered constraints.
Appendix 3: Proofs for Table 3
We prove the satisfaction and dissatisfaction of considered constraints by cases:
(Independence) Being based on probabilistic measures of confirmation, each of the coherence measures \(\mathcal{C}_{d1}, \mathcal{C}_{d2}, \mathcal{C}_{r1}\) and \(\mathcal{C}_{r2}\) satisfies (Independence). This is due to the fact that each confirmation measure \(\mathfrak {s}\) assigns a neutral degree of coherence to probabilistically independent pairs of propositions. Accordingly, the assigned degree of coherence by measure \(\mathcal{C}_{\mathfrak {s}}\) will, again, be equal to the confirmation measures neutrality threshold \(\beta _{\mathfrak {s}}\). Furthermore, if a set \({\Delta }=\{\varphi _1, \ldots ,\varphi _n\}\) satisfies mutual independence, then \(\Pr (\varphi _1\wedge \ldots \wedge \varphi _n)=\prod _{i\le n}\Pr (\varphi _i)\). Hence, \(\mathcal{D}(S)=1\) and \(\mathcal{D}^{*}(S)=0\). Since both are the neutrality thresholds for these measures, both satisfy (Independence). On the other hand, it is not totally clear what should be considered the neutrality threshold for any of the measures \(\mathcal{O}, \mathcal{O}^{*}\) or \(\mathcal{C}_f\) (see footnote 18). However, it can be shown that whatever threshold is chosen, it is not the case that probabilistically independent sets of propositions are always assigned a coherence degree in line with this threshold. To see this, consider again the probability distributions given in the proof of theorem 1. The corresponding degrees of coherence are as follows: \(\mathcal{O}_{\Pr }(A_1,A_2)=\mathcal{O}^{*}_{\Pr }(A_1,A_2)=1/3\), \(\mathcal{C}_{f,\Pr }(A_1,A_2)=1/2\), and \(\mathcal{O}_{\Pr '}(A_1,A_2)=\mathcal{O}^{*}_{\Pr }(A_1,A_2)=1/7\), \(\mathcal{C}_{f,\Pr '}(A_1,A_2)=1/4\). Thus, whatever the chosen threshold, all these measures will violate (Independence).
(Dependence) If each pair of nonempty, disjoint subsets is mutually confirmatory/disconfirmatory, then each confirmation measure \(\mathfrak {s}\) will assign a degree of confirmation above/ below \(\beta _{\mathfrak {s}}\). Accordingly, \(\mathcal{C}_{d1}, \mathcal{C}_{d2}, \mathcal{C}_{r1}\) and \(\mathcal{C}_{r2}\) satisfy (Dependence). Furthermore, let \({\Delta }=\{\varphi _1,\ldots ,\varphi _n\}\) be a set of mutually confirmatory/disconfirmatory propositions. The deviation measure \(\mathcal{D}\) can equivalently be represented as follows (proof omitted):
Thus, given that \({\Delta }\) is mutually confirmatory/disconfirmatory, the assigned degree of coherence, i.e. \(\mathcal{D}({\Delta })\) will exceed/ undercut 1, and analogously, \(\mathcal{D}^{*}(S)\) will exceed/ undercut 0. On the other hand, assume that \(\mathcal{C}\) is either \(\mathcal{O}, \mathcal{O}^{*}\) of \(\mathcal{C}_f\). Now consider the following probability distribution:
Given probability distribution \(\Pr \), we calculate that \(\Pr (A_iA_j)=5/22>11/85=\Pr (A_i)\) for \(1\le i\not =j\le 2\), so that \(A_1\) and \(A_2\) are mutually confirmatory. However, \(\mathcal{O}(A_1,A_2)=\mathcal{O}^{*}(A_1,A_2)=5/39\approx 0.13\) and \(\mathcal{C}_f(A_1,A_2)=5/22\approx 0.23\). Thus, both sets are assigned incoherent (given that the threshold separating coherence and incoherence should be equal to or above \(1/2\)). On the other hand, on \(\Pr '\) we get: \(\Pr (A_iA_j)=27/29<29/31=\Pr (A_i)\) for \(1\le i\not = j\le 2\), so that \(A_1\) and \(A_2\) are mutually disconfirmatory. However, \(\mathcal{O}(A_1,A_2)=\mathcal{O}^{*}(A_1,A_2)=27/31\approx 0.87\) and \(\mathcal{C}_f(A_1,A_2)=27/29\approx 0.93\). Thus, both sets are assigned coherent.
(Equivalence) A proof for all considered measures except \(\mathcal{C}_{r2}, \mathcal{D}^{*}, \mathcal{O}^{*}\) and \(\mathcal{C}_f\) is given by Siebel and Wolff (2008). Now assume that \(\varphi _1\) and \(\varphi _2\) are logically equivalent, then
is simply undefined since \(\Pr (\varphi _1\lnot \varphi _2)=\Pr (\varphi _2\lnot \varphi _1)=0\). For \(\mathcal{D}^{*}\) we get the following result:
Thus, the assigned degree of coherence is sensitive to \(\varphi _1\)’s prior probability and a fortiori not maximal for every set of equivalent propositions. Next we turn to \(\mathcal{O}^{*}\). It is easy to see that \(\mathcal{O}^{*}\) satisfies (Equivalence) given that \(\mathcal{O}\) does since it is only an average of \(\mathcal{O}\) values. Eventually, given that \({\Delta }=\{\varphi _1,\ldots ,\varphi _n\}\) is a set of logically equivalent propositions, each posterior probability that enters the calculation for \(\mathcal{C}_f({\Delta })\) equals 1. Hence, \(\mathcal{C}_f({\Delta })\) is maximal for every set of logically equivalent propositions.
(Inconsistency) Let \(\varphi _1\) and \(\varphi _2\) be contingent propositions such that the set \(\{\varphi _1, \varphi _2\}\) is inconsistent. Then we get the following result for \(\mathcal{C}_{d1}\):
Thus, the assigned degree of coherence is sensitive to the propositions’ prior probabilities, and so \(\mathcal{C}_{d1}\) violates (Independence). Now let us turn to \(\mathcal{C}_{d2}\). By Bayes’ theorem we calculate:
Hence, \(\mathcal{C}_{d2}\) is also sensitive to prior probabilities. Next we consider the ratiobased measures \(\mathcal{C}_{r1}\) and \(\mathcal{C}_{r2}\). Let \({\Delta }=\{\varphi _1,\ldots ,\varphi _n\}\) be a set of contingent propositions such that all subsets with at least two propositions are unsatisfiable. Then we get:
These latter terms are undefined for all \(S'\) with at least two propositions (since in this case \(\Pr (S')=\Pr (S'S'')=0\) and therefore we would have to divide by \(0\)). Now we turn to the \(\mathcal{D}\) and \(\mathcal{O}\) and their refined versions \(\mathcal{D}^{*}\) and \(\mathcal{O}^{*}\). To do so, let again \({\Delta }=\{\varphi _1,\ldots ,\varphi _n\}\) be a set of contingent propositions such that all nonsingleton subsets of \({\Delta }\) are unsatisfiable. Then
On the other hand, \(\mathcal{D}^{*}\) is not even defined in these cases. However, if we stipulate that \(log(0)=\infty \), then \(\mathcal{D}^{*}\) passes the inconsistency test. \(\mathcal{O}^{*}(S)\) is an average of \(\mathcal{O}\) values for all subsets with at least two propositions. Since it is assumed that all those subsets are unsatisfiable, we get
Finally, we consider the firmnessbased coherence measure \(\mathcal{C}_f\):
Since all posterior probabilities are equal to zero (remember that we stipulated that \(\Pr (\varphi \psi )=0\) for all inconsistent \(\psi \)), the assigned degree of coherence is equal to the minimum possible degree \(0\).
(Agreement) By Theorem 1 we know that there cannot be a probabilistic measure of coherence that satisfies both (Independence) and (Agreement). Thus, given that \(\mathcal{C}_{d1}, \mathcal{C}_{d2}, \mathcal{C}_{r1}, \mathcal{C}_{r2}\), \(\mathcal{D}\) and \(\mathcal{D}^{*}\) satisfy the former, they necessarily have to violate the latter. The case for \(\mathcal{O}\) and \(\mathcal{O}^{*}\) is so far unsettled, i.e. we were neither able to prove that these measures satisfy (Agreement) nor to find a counterexample.
Next, given that \(\mathcal{C}_f\) is an average of posterior probabilities, it must necessarily be the case that if all relevant posterior probabilities for \(\Pr \) exceed their counterparts on \(\Pr '\), then the assigned degree of \(\mathcal{C}_f\)coherence on \(\Pr \) exceeds the one on \(\Pr '\).
Appendix 4: Proofs for Table 6
Being based on probabilistic measures of confirmation, each of the coherence measures \(\mathcal{C}_{z}, \mathcal{C}_{\lambda }\) satisfies (Independence) and (Dependence). Now let \({\Delta }=\{\varphi _1,\ldots ,\varphi _n\}\) be a set of equivalent propositions, then given that \(\Pr ({\Delta }'{\Delta }'')=1\ge \Pr ({\Delta '})\) we get:
and accordingly
Thus, both measures satisfy (Equivalence). Now we turn to the inconsistencyconstraint. First, consider \(z\). If \(\Pr ({\Delta }'{\Delta }'')=0<\Pr ({\Delta }')\), then
On the other hand, if \(\Delta '\models \bot \), then \({\Delta }'\models \lnot {\Delta }''\) and therefore by means of the underlying case distinction we get \(z({\Delta }',{\Delta }'')=1\). Thus, \(\mathcal{C}_z\) satisfies (Inconsistency). Finally consider \(\lambda \). If \(\Pr ({\Delta }'{\Delta }'')=0<\Pr ({\Delta }')\), then
On the other hand, if \(\Delta '\models \bot \), then \({\Delta }'\models \lnot {\Delta }''\) and therefore we get \(\lambda ({\Delta }',{\Delta }'')=1\) by definition. Thus, \(\mathcal{C}_{\lambda }\) satisfies (Inconsistency). Finally, given the inconsistency of (Independence) and (Agreement), we conclude that \(\mathcal{C}_z\) and \(\mathcal{C}_{\lambda }\) violate (Agreement).
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Schippers, M. Probabilistic measures of coherence: from adequacy constraints towards pluralism. Synthese 191, 3821–3845 (2014). https://doi.org/10.1007/s1122901405017
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DOI: https://doi.org/10.1007/s1122901405017