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The Problem of Coherence and Truth Redux

Abstract

In “What price coherence?” (Analysis 54:129–132, 1994), Klein and Warfield put forward a simple argument that triggered an extensive debate on the epistemic virtues of coherence. As is well-known, this debate yielded far-reaching impossibility results to the effect that coherence is not conducive to truth, even if construed in a ceteris paribus sense. A large part of the present paper is devoted to a re-evaluation of these results. As is argued, all explications of truth-conduciveness leave out an important aspect: while it might not be the case that coherence is truth-conducive, it might be conducive to verisimilitude or epistemic utility. Unfortunately, it is shown that the answer for both these issues must be in the negative, again. Furthermore, we shift the focus from sets of beliefs to particular beliefs: as is shown, neither is any of the extant probabilistic measures of coherence truth-conducive on the level of particular beliefs, nor does weakening these measures to quasi-orderings establish the link between coherence and truth for an important amount of measures. All in all, the results in this paper cast a serious doubt on the approach of establishing a link between coherence and truth. Finally, recent arguments that shift the focus from the relationship between coherence and truth to the one between coherence and confirmation are assessed.

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Fig. 1

Notes

  1. For a detailed analysis see Sect. 2.

  2. In what follows, reference to the probability function in \({\mathbf {Coh}}_{\Pr }\) is dropped whenever it is clear from the context.

  3. For refined weighting systems see Schupbach (2011).

  4. The confirmation measures in Table 1 are based on two different qualitative notions of confirmation, sometimes called incremental confirmation and absolute confirmation. The details of this distinction are not important in the present context.

  5. Here and in what follows we assume the following notational convention: if S and \(S'\) are sets of propositions, then \(\xi (S,S')\) denotes \(\xi (\bigwedge S,\bigwedge S')\), where \(\bigwedge S=\bigwedge _{A\in S}A\).

  6. For a proof of this and other observations see the Appendix. For a small subset of measures, this observation has already been proved by Meijs and Douven (2007).

  7. Cf. Bovens and Hartmann (2003), Fitelson (2003), Glass (2005), Koscholke (2015), Olsson (2002), Roche (2013), Schippers (2014c), Schupbach (2011), Siebel (2005), Siebel and Wolff (2008), and Wheeler (2009).

  8. Olsson (2001) also points out that Shogenji (1999) argument for why the total individual strength ought to be kept fixed is far from conclusive.

  9. See also Bovens and Olsson (2002) and Cross (1999).

  10. Cf. Olsson (2005a). Olsson’s earlier definition of truth-conduciveness (2002) differs insofar as also the probability distribution is not allowed to vary between both sets.

  11. See also Olsson (2005b).

  12. Furthermore, it is assumed that each witness i is either completely reliable (\(R_i\)) or completely unreliable (\(U_i\)), and that their reliability profile is incompletely known, i.e. \(\Pr (R_1)=\Pr (R_2)>0\) and \(\Pr (R_i)+\Pr (U_i)=1\).

  13. Like in our discussion of Olsson’s account of testimonial systems, \(\Pr ({\mathbf {S}})\) denotes the joint posterior probability of the information set conditional on the reports.

  14. For a similar remark see Olsson (2005b, p. 403).

  15. Note that this approach is more akin to Shogenji’s requirement of equal total individual strength.

  16. Cf. Meijs (2007).

  17. For exceptions see Akiba (2000) and Fitelson (2003). Fitelson even maintains that “intuitively, all propositions ‘cohere with themselves’ (maximally), except for necessary falsehoods” (2003, p. 198). This makes it even harder to image a possible application of the concept of coherence within the common accounts to verisimilitude.

  18. It is assumed that \(\Delta \models \Gamma \) iff for all \(A\in \Gamma \): \(\Delta \models A\).

  19. Among these are the ones proposed by Kuipers (1982), Oddie (1986), Schurz and Weingartner (1987, 2010), Brink and Heidema (1987) and Gemes (2007); cf. Cevolani et al. (2011).

  20. Another option is to investigate the relationship between coherence and estimated verisimilitude, where the latter basically is an expectation value for verisimilitude in the light of a certain set of relevant pieces of evidence (cf. Niiniluoto 1987, ch. 7). Schippers (2015b) investigates the relationship between coherence and estimated verisimilitude based on the idea that what we are to compare are not the theories’ degree of coherence and its degree of verisimilitude but whether a higher degree of coherence between a theory and the available evidence leads to a higher degree of estimated verisimilitude.

  21. This model-theoretic explication of the concept of a scientific theory relies heavily on Bovens and Hartmann (2003, pp. 53–55). Cf. Hartmann (2008). This representation of a theory accounts for the fact that propositions are usually not tested in isolation, but is on the other hand fine grained enough to allow for testing of proper parts of a given theory. Furthermore, by making allowance for overlapping sets it takes into consideration that some propositions in t (for example scientific laws) might play a prominent role in more than one model.

  22. What is here called the ‘inverse likelihood ratio’ is sometimes also simply called the likelihood ratio (cf. Howson and Urbach 2006, p. 21).

  23. Note that a constant inverse likelihood ratio is also stipulated in Bovens and Hartmann (2003) model. Furthermore, the likelihood-ratio is provably equivalent to the Bayes factor which is a popular measure of evidence in Bayesian statistics (cf. Kass and Raftery 1995). Furthermore, the results are independent of the choice of \(\overline{x}\). The only condition is that \(\overline{x}\) is a continuous and strictly decreasing function of x.

  24. The following example is due to Bovens and Hartmann (2003, p. 20). However, there are many more weight vectors featuring differences in \(a_0\) that nonetheless lead to similar negative results.

  25. However, mutatis mutandis, these considerations can analogously be extended to the case of subtracting/adding any finite number of beliefs.

  26. There is a small caveat: \({\mathcal {O}}\) trivially satisfies Definition 5.2, but this is only due to the fact that according to \({\mathcal {O}}\) it is impossible to increase coherence by adding any proposition whatsoever (proof omitted).

  27. Recently, Shogenji (2013) proposed to change the focus from questions of truth-conduciveness of coherence to the question whether coherence boosts the transmission of support from a set of pieces of evidence to a hypothesis. Basically, he argues that it can be shown that the less coherent a set of pieces of evidence is, the higher the support it transmits to the hypothesis under consideration, given a suitable amount of ceteris paribus conditions. However, his result suffers from some limitations: (i) all that the proof shows is that we can easily change (conditional and unconditional) probabilities of conjunctions to terms involving Shogenji’s coherence measure \({\mathcal {D}}\), viz., by replacing \(\Pr (\bigwedge _{i\le n}A_i)\) by \(\prod _{i\le n}\Pr (A_i)\cdot {\mathcal {D}}(A_1,\ldots ,A_n)\). The same applies to conditional probabilities like \(\Pr (\bigwedge _{i\le n}A_i|B)\); this, however, does not show that the degree of coherence as measured by \({\mathcal {D}}\) should be considered to have some impact on whatever quantity we measure, because the adequacy of \({\mathcal {D}}\) as a measure of coherence is not beyond reasonable doubt (cf. Schippers 2014c; Siebel 2005; Siebel and Wolff 2008). Wheeler (2009), instead, proposes to interpret \({\mathcal {D}}\) as a measure of correlation, and recently Brössel (2015) highlights the fact that \({\mathcal {D}}\) has been proposed by Keynes (1921) as a coefficient of dependence. Accordingly, some more argumentation seems wanting to conclude from the simple replacement of probabilistic terms to the impact of coherence. (ii) One of Shogenji’s arguments for why coherence has a negative impact on the transmission of support is based on the following equation:

    $$\begin{aligned} r(H,E_1\wedge \ldots \wedge E_n)=\prod _{i\le n} r(E_i,H)\cdot \frac{{\mathcal {D}}(E_1,\ldots ,E_n|H)}{{\mathcal {D}}(E_1,\ldots ,E_n)} \end{aligned}$$

    This equation is supposed to show that “other things being equal, the more coherent the pieces of evidence \(E_1,\ldots , E_n\) are, the less probabilistic support H receives from \(E_1,\ldots ,E_n\)” (Shogenji 2013, p. 2532, emphasis Shogenji’s). However, Shogenji’s result is limited to confirmation measures satisfying a number of “minimum requirements” that are not shared by all extant confirmation measures. Furthermore, in anticipation of the Sect. 6 the above formula can also be interpreted as saying that the higher the degree of focused coherence, the higher the transmission of support (see Sect. 6). All in all, I think that Shogenji’s argument deserves a detailed investigation that is, unfortunately, beyond the scope of the present paper.

  28. Note that for some of the above coherence measures a difference between \({\mathbf {Coh}}(S|A)\) and \({\mathbf {Coh}}(S)\) might seem more appropriate. More generally, every function that is strictly monotonically increasing in \({\mathbf {Coh}}(S|A)\) and decreasing in \({\mathbf {Coh}}(S)\) could be chosen. We leave this issues for future research.

  29. More precisely, Wheeler and Scheines consider six confirmation measures among which are rl and k.

  30. See footnote 4.

  31. We dispense with a discussion of Wheeler and Scheines’ interesting ideas on coherence and causal structure. Although they provide us with very stimulating observations, all of them rest on interpreting \({\mathcal {D}}\) as a measure of coherence. Nonetheless, we grant that Wheeler and Scheines highlight a number of interesting connections between confirmation, causal structure and correlation, which is what \({\mathcal {D}}\) seems to measure. An in-depth analysis of these further results, however, must be postponed to another paper and can not be the focus of the present paper, which is solely concerned with probabilistic measures of coherence.

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Acknowledgments

I would like to thank Jakob Koscholke and the anonymous reviewers for providing me with valuable comments and suggestions that helped to improve the paper. This work was supported by Grant SI1731/-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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Appendices

Appendix 1: Proof of Observation 3.1

Consider the following probability distributions, where \(x=1-\Pr (A_1\vee A_2\vee A_3)\):

\(A_1\) \(A_2\) \(A_3\) Probability \(A_1\) \(A_2\) \(A_3\) Probability
T T T 1/5 F T T 11/65
T T F 2/41 F T F 7/44
T F T 3/37 F F T 1/18
T F F \(\frac{122786261713}{1248935336885}\) F F F x
\(A_1^{\prime }\) \(A_2^{\prime }\) \(A_3^{\prime }\) Probability \(A_1^{\prime }\) \(A_2^{\prime }\) \(A_3^{\prime }\) Probability
T T T 34/171 F T T 13/71
T T F 2/13 F T F 1/52
T F T 1/25 F F T 3/20
T F F \(\frac{15124126187369}{13521705438129525}\) F F F \(\hbox {x}^{\prime }\)

The table only gives a small sample of the complete distribution over six variables. However, note that \(\Pr (\bigwedge _{i\le 3}A_i)=0.200>0.199\approx \Pr (\bigwedge _{i\le 3}A_i')\) and \(\prod _{i\le 3}\Pr (A_i)=\prod _{i\le 3}\Pr (A_i')=0.125\). Straightforward calculations yield the following results:

  \({\mathcal {O}}\) \({\mathcal {D}}^{*}\) \({\mathcal {O}}^{*}\) \({\mathcal {C}}_d\) \({\mathcal {C}}_r\) \({\mathcal {C}}_s\) \({\mathcal {C}}_l\) \({\mathcal {C}}_k\) \({\mathcal {C}}_z\) \({\mathcal {C}}_f\)
\(\{A_1,A_2,A_3\}\) 0.246 1.292 0.381 0.119 1.276 0.212 1.792 0.246 0.228 0.572
\(\{A_1^{\prime },A_2^{\prime },A_3^{\prime }\}\) 0.266 1.367 0.425 0.125 1.281 0.230 2.176 0.263 0.247 0.587

All mentioned measures assign a lower degree of coherence to the set \(\{A_1,A_2,A_3\}\) which is in conflict with the requirements of Definition 3.2.

Appendix 2: Proof of Observation 5.1

For the vast majority of measures it is possible to show that Definition 5.1 is already violated for pairs of propositions. For these measures consider two sets of propositions \(\{A_1,A_2\}\) and \(\{A_1,A_3\}\) and the following probability distribution, where \(x=1-\Pr (A_1\vee A_2\vee A_3)\):

\(A_1\) \(A_2\) \(A_3\) Probability \(A_1\) \(A_2\) \(A_3\) Probability
T T T 7/62 F T T 5/51
T T F 14/65 F T F 5/52
T F T 6/73 F F T 1/182
T F F 1/220 F F F x

The coherence values for the relevant measures are given in the following table.

  \({\mathcal {O}}\) \({\mathcal {D}}^{*}\) \({\mathcal {O}}^{*}\) \({\mathcal {C}}_d\) \({\mathcal {C}}_s\) \({\mathcal {C}}_l\) \({\mathcal {C}}_k\) \({\mathcal {C}}_z\) \({\mathcal {C}}_f\)
\(\{A_1,A_2\}\) 0.539 1.574 0.539 0.241 0.453 2.921 0.480 0.463 0.710
\(\{A_1,A_3\}\) 0.376 1.514 0.376 0.205 0.316 2.370 0.402 0.326 0.562

Given that \(\Pr (A_1|A_2)\approx 0.628<0.653\approx \Pr (A_1|A_3)\), all these measures are not truth-conducive in the sense of Definition 5.1. Note that in order to prove the analogous result for the two missing coherence measures \({\mathcal {D}}\) and \({\mathcal {C}}_{r}\), we have to consider at least one set of propositions with more than two elements. This is because for the considered pairs of sets, \(\Pr (A_1|A_2)<\Pr (A_1|A_3)\) already entails \({\mathcal {D}}(A_1,A_2)<{\mathcal {D}}(A_1,A_3)\) and the same holds for \({\mathcal {C}}_r\).

Therefore, let \(S=\{A_1,A_2\}\) and \(S'=\{A_1,A_2,A_3\}\) and consider the following probability distribution, where \(x=1-\Pr (A_1\vee A_2\vee A_3)\):

\(A_1\) \(A_2\) \(A_3\) Probability \(A_1\) \(A_2\) \(A_3\) Probability
T T T 12/49 F T T 12/37
T T F 2/25 F T F 3/44
T F T 9/73 F F T 1/72
T F F 6/53 F F F x

Against the background of this distribution we get the desired result that even though \(A_1\)’s posterior probability given \(A_2\) (approx. 0.453) exceeds its posterior probability given both \(A_2\) and \(A_3\) (approx. 0.430), we have

$$\begin{aligned} {\mathcal {D}}(S) \approx 0.807&< 0.861 \approx {\mathcal {D}}(S'')\\ {\mathcal {C}}_{r}(S) \approx 0.807&< 0.936 \approx {\mathcal {C}}_r(S'') \end{aligned}$$

Hence, these measures are not truth-conducive in the sense of Definition 5.1, too.

Appendix 3: Proof of Observation 5.2

Consider again the former probability distribution:

\(A_1\) \(A_2\) \(A_3\) Probability \(A_1\) \(A_2\) \(A_3\) Probability
T T T 12/49 F T T 12/37
T T F 2/25 F T F 3/44
T F T 9/73 F F T 1/72
T F F 6/53 F F F x

We can easily extend the calculated coherence values to all considered measures. The following table contains additional values for all measures but \({\mathcal {O}}\).

  \({\mathcal {D}}^{*}\) \({\mathcal {O}}^{*}\) \({\mathcal {C}}_d\) \({\mathcal {C}}_s\) \({\mathcal {C}}_l\) \({\mathcal {C}}_k\) \({\mathcal {C}}_z\) \({\mathcal {C}}_f\)
\(\{A_1,A_2\}\) 0.807 0.341 −0.124 −0.350 0.594 −0.256 −0.193 0.516
\(\{A_1,A_2,A_3\}\) 0.930 0.417 −0.036 −0.082 0.940 −0.064 −0.028 0.565

As the table shows, these measures agree with \({\mathcal {D}}\) and \({\mathcal {C}}_r\) in that the extended set \(\{A_1,A_2,A_3\}\) is more coherent than its subset \(\{A_1,A_2\}\). Taking into account that nonetheless \(A_1^{\prime }\) posterior probability is lower for this extended set, this result shows that all considered measures (except \({\mathcal {O}}\)) are not truth-conducive in the sense of Definition 5.2.

Appendix 4: Proof of Observation 5.3

First of all, we can utilize the following probability distribution in order to show that the orderings induced by Definition 5.6 are not truth-conducive for all confirmation measures but f:

\(A_1\) \(A_2\) \(A_3\) Probability \(A_1\) \(A_2\) \(A_3\) Probability
T T T 1/30 F T T 5/44
T T F 4/59 F T F 10/61
T F T 1/153 F F T 53/105
T F F 6/55 F F F x

The confirmation values for the relevant measures that are currently of interest are given in the following table:

Confirmation d r s l k z
\({\xi (A_1,A_2)}\) 0.050 1.232 0.081 1.317 0.137 0.064
\({\xi (A_1,A_2|A_3)}\) 0.166 3.745 0.214 4.550 0.640 0.177
\({\xi (A_2,A_1)}\) 0.088 1.232 0.112 1.435 0.179 0.141
\({\xi (A_2,A_1|A_3)}\) 0.613 3.745 0.652 17.743 0.893 0.789

As the table indicates, all considered confirmation measures agree in that the there is a larger degree of confirmation between \(A_1\) and \(A_2\) when \(A_3\) is taken for granted. This, however, is in sharp contrast with the relevant conditional probabilities: as was mentioned before, \(A_1\)’s conditional probability given \(A_2\) exceeds its conditional probability given \(A_2\) and \(A_3\). Accordingly, these measures are not truth-conducive in the sense of Definition 5.3.

The latter distribution also suffices to show that the refined deviation measure \({\mathcal {D}}^{*}\) is not truth-conducive in this sense. This is due to the fact that

$$\begin{aligned} {\mathcal {D}}^{*}(A_1,A_2) \approx 1.232 < 3.745 \approx {\mathcal {D}}^{*}(A_1,A_2|A_3) \end{aligned}$$

To show that the refined overlap measure \({\mathcal {O}}^{*}\) is not truth-conducive in the sense of Definition 5.3 we utilize the following distribution involving four propositions with \(x=1-\Pr (A_1\vee A_2\vee A_3\vee A_4)\):

\(A_1\) \(A_2\) \(A_3\) \(A_4\) Probability \(A_1\) \(A_2\) \(A_3\) \(A_4\) Probability
T T T T 1/25 F T T T 1/43
T T T F 1/23 F T T F 1/36
T T F T 3/56 F T F T 1/44
T T F F 5/69 F T F F 2/39
T F T T 4/39 F F T T 1/59
T F T F 1/57 F F T F 6/71
T F F T 1/56 F F F T 2/53
T F F F 27/94 F F F F x

According to the refined overlap measure all non-singleton subsets of \(\{A_2,A_3,A_4\}\) are assigned a higher degree of coherence conditional on \(A_1\). However, \(A_2\)’s conditional probability given \(A_1\), \(A_3\) and \(A_4\) is lower than its conditional probability given only \(A_3\) and \(A_4\). Hence, \({\mathcal {O}}^{*}\) also violates Definition 5.3.

Now we turn to the remaining confirmation measure f. In order to show that this measure is truth-conducive in the sense of Definition 5.6, note that for each pair \((S',S'')\in [S]\) the following claim holds by definition:

(\(\dagger _f\)):

If \(f(S',S''|A)>f(S',S'')\), then \(\Pr (S'|S'',A)>\Pr (S'|S'')\).

Hence, let \(S'=\{x\}\) for some \(x\in S\) and \(S''=S\setminus \{x\}\), then the fact that \(f(S',S''|A)>f(S',S'')\) by definition together with \((\dagger _f)\) entails the desired claim.

Appendix 5: Proof of Observation 6.2

If \({\mathcal {D}}({\mathbf {E}},H)>1\) for some set \({\mathbf {E}}=\{E_1,\ldots ,E_n\}\), then we get the following derivation:

$$\begin{aligned} {\mathcal {D}}({\mathbf {E}},H)>1&\Rightarrow \frac{\Pr (H|E_1,\ldots ,E_n)\Pr (H)^{n-1}}{\Pr (H|E_1)\cdot \ldots \cdot \Pr (H|E_n)}>1\\&\Rightarrow \Pr (H|E_1,\ldots ,E_n)>\underbrace{\frac{\Pr (H|E_1)}{\Pr (H)}}_{>1} \cdot \ldots \cdot \underbrace{\frac{\Pr (H|E_{n-1})}{\Pr (H)}}_{>1}\cdot \Pr (H|E_n)\\&\Rightarrow \Pr (H|E_1,\ldots ,E_n)>\Pr (H|E_n)\\&\Rightarrow \Pr (H|E_1,\ldots ,E_n)>\Pr (H) \end{aligned}$$

This latter fact means that \(\xi (H,{\mathbf {E}})>\theta \) for all relevance-sensitive \(\xi \). \(\square \)

Appendix 6: Proof of Observation 6.4

Keeping in mind that by assumption \(\Pr (H|E_2)=\Pr (H|E_3)\), we get

$$\begin{aligned} {\mathcal {D}}({\mathbf {E}},H)>{\mathcal {D}}({\mathbf {E}}^{\prime },H)&\Leftrightarrow \frac{\Pr (H|E_1,E_2)\cdot \Pr (H)}{\Pr (H|E_1)\Pr (H|E_2)} >\frac{\Pr (H|E_1,E_3)\cdot \Pr (H)}{\Pr (H|E_1)\Pr (H|E_3)}\\&\Leftrightarrow \Pr (H|E_1,E_2)>\Pr (H|E_1,E_3) \end{aligned}$$

Thus, if \(\xi \) satisfies (FPI), then \(\Pr (H|E_1,E_2)>\Pr (H|E_1,E_3)\) entails that \(\xi (H,{\mathbf {E}})>\xi (H,{\mathbf {E}}^{\prime })\). \(\square \)

Appendix 7: Proof of Observation 6.5

H \(E_1\) \(E_2\) \(E_2\) Probability H \(E_1\) \(E_2\) \(E_2\) Probability
T T T T 1/495 F T T T 1/33
T T T F 3/46 F T T F 1/47
T T F T 6/41 F T F T 1/11
T T F F 1/41 F T F F 401/182172
T F T T 1663/39606 F F T T 1/68
T F T F 5099/39606 F F T F 4621/58938
T F F T 1/21 F F F T 1/114
T F F F 1/177 F F F F x

Straightforward calculations yield the following results: \(\Pr (H|E_1)\approx .622>.462\approx \Pr (H)>.363\approx \Pr (H|\lnot E_1)\) and \(\Pr (H|E_i)=\Pr (H|E_j)\) as well as \(\Pr (H|\lnot E_i)=\Pr (H|\lnot E_j)\) for all \(1\le i,\,j\le 3\). Hence, \({\mathbf {E}}\cup {\mathbf {E}}^{\prime }\) is an equal positive evidence set for H. Furthermore, \({\mathcal {D}}({\mathbf {E}},H)\approx .676>.657\approx {\mathcal {D}}({\mathbf {E}}^{\prime },H)\); however, \(n(H,{\mathbf {E}})\approx .050<.096\approx n(H,{\mathbf {E}}^{\prime })\), \(m(H,{\mathbf {E}})\approx .027< .052\approx m(H,{\mathbf {E}}^{\prime })\) and \(s(H,{\mathbf {E}})\approx .118<.121\approx s(H,{\mathbf {E}}^{\prime })\).

Appendix 8: Proof of Observation 6.6

H \(E_1\) \(E_2\) \(E_2\) Probability H \(E_1\) \(E_2\) \(E_2\) Probability
T T T T 1/12 F T T T 1/31
T T T F 1/205 F T T F 1/306
T T F T 1/1406 F T F T 1/651
T T F F 1/150 F T F F 431/12852
T F T T 3293/817950 F F T T 1/918
T F T F 6763/2017610 F F T F 4513/132804
T F F T 1/133 F F F T 1/28
T F F F 20/67 F F F F x

Given this probability distribution we calculate: \(\Pr (H|E_1)\approx .575>.409\approx \Pr (H)>.376\approx \Pr (H|\lnot E_1)\) and \(\Pr (H|E_i)=\Pr (H|E_j)\) as well as \(\Pr (H|\lnot E_i)=\Pr (H|\lnot E_j)\) for all \(1\le i,\,j\le 3\). Hence, \({\mathbf {E}}\cup {\mathbf {E}}^{\prime }\) is an equal positive evidence set for H. Furthermore, \({\mathcal {O}}({\mathbf {E}},H)\approx 1.444>1.428\approx {\mathcal {O}}({\mathbf {E}}^{\prime },H)\); however, \(\Pr (H|{\mathbf {E}})\approx .7129<.7132\approx \Pr (H|{\mathbf {E}}^{\prime })\) and therefore all (FPI)-measures will agree in that \(\xi (H,{\mathbf {E}})<\xi (H,{\mathbf {E}}^{\prime })\).

Appendix 9: Proof of Observation 6.7

The proof of Observation 6.7 utilizes Lemma 6.1:

$$\begin{aligned} {\mathcal {C}}_f({\mathbf {E}},H)&>{\mathcal {C}}_f({\mathbf {E}}^{\prime },H)\\&\Leftrightarrow \frac{\Pr (E_1|E_2, H)+\Pr (E_2|E_1,H)}{\Pr (E_1|E_2)+\Pr (E_2|E_1)} >\frac{\Pr (E_1|E_3, H)+\Pr (E_3|E_1,H)}{\Pr (E_1|E_3)+\Pr (E_3|E_1)}\\&\Leftrightarrow \Pr (H|E_1,E_2)\cdot \frac{\sum _{i=1,2}\Pr (H\wedge E_i)^{-1}}{\sum _{i=1,2}\Pr (E_i)^{-1}} > \Pr (H|E_1,E_3)\cdot \frac{\sum _{j=1,3}\Pr (H\wedge E_j)^{-1}}{\sum _{j=1,3}\Pr (E_j)^{-1}}\\&\Leftrightarrow \frac{ \Pr (H|E_1,E_2)}{ \Pr (H|E_1,E_3)} > \frac{\sum _{i=1,2}\Pr (H\wedge E_i)^{-1}}{\sum _{i=1,2}\Pr (E_i)^{-1}}\cdot \frac{\sum _{j=1,3} \Pr (E_j)^{-1}}{\sum _{j=1,3}\Pr (H\wedge E_j)^{-1}}\\&\Leftrightarrow \frac{ \Pr (H|E_1,E_2)}{ \Pr (H|E_1,E_3)} > 1 \quad\quad\quad ({\text {Lemma}}\; 6.1) \end{aligned}$$

This completes the proof of Observation 6.7.

Appendix 10: Proof of Lemma 6.1

If \({\mathbf {E}}\cup {\mathbf {E}}^{\prime }\) is an equal positive evidence set for H, then (i) \(\Pr (H|E_i)>\Pr (H)>\Pr (H|\lnot E_i)\) and (ii) \(\Pr (H|\pm E_i)=\Pr (H|\pm E_j)\) for all \(1\le i,\,j\le 3\). Now we get:

$$\begin{aligned} \Pr (H)&= \Pr (H\wedge E_i) + \Pr (H\wedge \lnot E_i)\\&= \Pr (H|E_i)\cdot \Pr (E_i) + \Pr (H|\lnot E_i)\cdot \Pr (\lnot E_i)\\&\mathop {=}\limits ^{\text {(ii)}} \Pr (H|E_j)\cdot \Pr (E_i) + \Pr (H|\lnot E_j) \cdot \Pr (\lnot E_i) \end{aligned}$$

and also

$$\begin{aligned} \Pr (H)&= \Pr (H|E_j)\cdot \Pr (E_j) + \Pr (H|\lnot E_j)\cdot \Pr (\lnot E_j) \end{aligned}$$

Hence we get

$$\begin{aligned} \Pr (H|E_j)\cdot \Pr (E_i) + \Pr (H|\lnot E_j)\cdot \Pr (\lnot E_i) = \Pr (H|E_j)\cdot \Pr (E_j) + \Pr (H|\lnot E_j)\cdot \Pr (\lnot E_j) \end{aligned}$$

and thus

$$\begin{aligned} \Pr (H|E_j)\cdot (\Pr (E_i)-\Pr (E_j)) = \Pr (H|\lnot E_j)\cdot (\Pr (\lnot E_j) -\Pr (\lnot E_i)) \end{aligned}$$

from which we conclude that either \(\Pr (H|E_i)=\Pr (H|\lnot E_i)\) in contradiction to (i) or \(\Pr (E_i)=\Pr (E_j)\). With this latter identity and \(\Pr (H|E_i)=\Pr (H|E_j)\) we conclude that also \(\Pr (H\wedge \pm E_i)=\Pr (H\wedge \pm E_j)\).

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Schippers, M. The Problem of Coherence and Truth Redux. Erkenn 81, 817–851 (2016). https://doi.org/10.1007/s10670-015-9771-9

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