Skip to main content

The Problem of Coherence and Truth Redux

Erkenntnis volume 81, pages 817–851 (2016)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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 r, l 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.

References

  • Akiba, K. (2000). Shogenji’s probabilistic measure of coherence is incoherent. Analysis, 60, 356–359.

    Article  Google Scholar 

  • BonJour, L. (1976). The coherence theory of empirical knowledge. Philosophical Studies, 30, 281–312.

    Article  Google Scholar 

  • BonJour, L. (1985). The structure of empirical knowledge. Cambridge, MA and London: Harvard University Press.

    Google Scholar 

  • BonJour, L. (1999). The dialectic of foundationalism and coherentism. In J. Greco & E. Sosa (Eds.), The blackwell guide to epistemology. Malden, MA: Blackwell.

  • Bovens, L., & Hartmann, S. (2003). Bayesian epistemology. Oxford and New York: Oxford University Press.

    Google Scholar 

  • Bovens, L., & Hartmann, S. (2005). Why there cannot be a single probabilistic measure of coherence. Erkenntnis, 63, 361–74.

    Article  Google Scholar 

  • Bovens, L., & Hartmann, S. (2006). An impossibility result for coherence rankings. Philosophical Studies, 128, 77–91.

    Article  Google Scholar 

  • Bovens, L., & Olsson, E. J. (2002). Believing more, risking less: On coherence, truth and non-trivial extensions. Erkenntnis, 57, 137–150.

    Article  Google Scholar 

  • Brössel, P. (2015). Keynes’s coefficient of dependence. Erkenntnis. doi:10.1007/s10670-014-9672-3.

  • Brink, C., & Heidema, J. (1987). A verisimilar ordering of theories phrased in a propositional language. British Journal for the Philosophy of Science, 38, 533–549.

    Article  Google Scholar 

  • Carnap, R. (1962). Logical foundations of probability. Chicago: University of Chicago Press.

    Google Scholar 

  • Cevolani, G., Crupi, V., & Festa, R. (2010). The whole truth about Linda: Probability, verisimilitude and a paradox of conjunction. In M. D’Agostino, et al. (Eds.), New essays in logic and philosophy of science (pp. 603–615). London: College Publications.

    Google Scholar 

  • Cevolani, G., Crupi, V., & Festa, R. (2011). Verisimilitude and belief change for conjunctive theories. Erkenntnis, 75, 183–202.

    Article  Google Scholar 

  • Cevolani, G., & Tambolo, L. (2013). Progress as approximation to the truth: A defence of the verisimilitudinarian approach. Erkenntnis, 78, 921–935.

    Article  Google Scholar 

  • Christensen, D. (1999). Measuring confirmation. Journal of Philosophy, 96, 437–461.

    Article  Google Scholar 

  • Cross, C. B. (1999). Coherence and truth-conducive justification. Analysis, 59, 186–s193.

    Article  Google Scholar 

  • Crupi, V., Chater, N., & Tentori, K. (2013). New axioms for probability and likelihood ratio measures. The British Journal for the Philosophy of Science, 64, 189–204.

    Article  Google Scholar 

  • Crupi, V., Tentori, K., & Gonzalez, M. (2007). On Bayesian measures of evidential support: Theoretical and empirical issues. Philosophy of Science, 74, 229–252.

    Article  Google Scholar 

  • Douven, I., & Meijs, W. (2007). Measuring coherence. Synthese, 156, 405–425.

    Article  Google Scholar 

  • Eells, E., & Fitelson, B. (2000). Measuring confimation and evidence. Journal of Philosophy, 97, 663–672.

    Article  Google Scholar 

  • Festa, R. (2012). For unto every one that hath shall be given. Matthew properties for incremental confirmation. Synthese, 184, 89–100.

    Article  Google Scholar 

  • Fitelson, B. (2003). A probabilistic theory of coherence. Analysis, 63, 194–199.

    Article  Google Scholar 

  • Fitelson, B. (2004). Two technical corrections to my coherence measure. http://www.fitelson.org/coherence2.

  • Gemes, K. (2007). Verisimilitude and content. Synthese, 154, 293–306.

    Article  Google Scholar 

  • Glass, D. H. (2002). Coherence, explanation, and Bayesian networks. In M. ONeill, et al. (Eds.), AICS 2002, LNAI 2464 (pp. 177–182), Berlin.

  • Glass, D. H. (2005). Problems with priors in probabilistic measures of coherence. Erkenntnis, 63, 375–385.

    Article  Google Scholar 

  • Good, I. J. (1984). The best explicatum for weight of evidence. Journal of Statistical Computation and Simulation, 19, 294–299.

    Article  Google Scholar 

  • Harman, G. (1986). Change in view. Cambridge, MA: MIT Press.

    Google Scholar 

  • Hartmann, S. (2008). Modeling in philosophy of science. In M. Frauchiger & W. K. Essler (Eds.), Representation, evidence, and justification: Themes from Suppes (Lauener Library of Analytical Philosophy, vol. 1) (pp. 95–121). Frankfurt: Ontos Verlag.

    Google Scholar 

  • Hempel, C. G. (1960). Inductive inconsistencies. Synthese, 12, 439–469.

    Article  Google Scholar 

  • Howson, C., & Urbach, P. (2006). Scientific reasoning. The Bayesian approach (3rd ed.). Chicago: Open Court.

    Google Scholar 

  • Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.

    Article  Google Scholar 

  • Keynes, J. M. (1921). A treatise on probability. London: Macmillan.

    Google Scholar 

  • Kemeny, J., & Oppenheim, P. (1952). Degrees of factual support. Philosophy of Science, 19, 307–324.

    Article  Google Scholar 

  • Klein, P., & Warfield, T. A. (1994). What price coherence? Analysis, 54, 129–132.

    Article  Google Scholar 

  • Koscholke, J. (2015). Last measure standing. Evaluating test cases for probabilistic coherence measures. Erkenntnis. doi:10.1007/s10670-015-9734-1.

  • Kuipers, T. A. F. (1982). Approaching descriptive and theoretical truth. Erkenntnis, 18, 343–378.

    Article  Google Scholar 

  • Levi, I. (1967). Gambling with truth. New York: A. A. Knopf.

    Google Scholar 

  • Meijs, W. (2005). Probabilistic measures of coherence. PhD thesis, Erasmus University, Rotterdam.

  • Meijs, W. (2006). Coherence as generalized logical equivalence. Erkenntnis, 64, 231–252.

    Article  Google Scholar 

  • Meijs, W. (2007). A corrective to Bovens and Hartmann’s measure of coherence. Philosophical Studies, 133, 151–80.

    Article  Google Scholar 

  • Meijs, W., & Douven, I. (2007). On the alleged impossibility of coherence. Synthese, 157, 347–360.

    Article  Google Scholar 

  • Merricks, T. (1995). On behalf of the coherentist. Analysis, 55, 306–309.

    Article  Google Scholar 

  • Miller, D. (1974). Popper’s qualitative theory of verisimilitude. British Journal for the Philosophy of Science, 25, 166–177.

    Article  Google Scholar 

  • Mortimer, H. (1988). The logic of induction. Paramus: Prentice Hall.

    Google Scholar 

  • Niiniluoto, I. (1987). Truthlikeness. Dordrecht: Reidel.

    Book  Google Scholar 

  • Nozick, R. (1981). Philosophical Explanations. Oxford: Clarendon.

    Google Scholar 

  • Oddie, G. (1986). Likeness to truth. Dordrecht: Reidel.

    Book  Google Scholar 

  • Olsson, E. J. (2001). Why coherence is not truth-conducive. Analysis, 61, 236–241.

    Article  Google Scholar 

  • Olsson, E. J. (2002). What is the problem of coherence and truth? The Journal of Philosophy, 99, 246–272.

    Article  Google Scholar 

  • Olsson, E. J. (2005a). Against coherence. Truth, probability, and justification. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Olsson, E. J. (2005b). The impossibility of coherence. Erkenntnis, 63, 387–412.

    Article  Google Scholar 

  • Olsson, E. J., & Schubert, S. (2007). Reliability conducive measures of coherence. Synthese, 157, 297–308.

    Article  Google Scholar 

  • Pearl, J. (2000). Causality: Models, reasoning, and inference. New York: Cambridge University Press.

    Google Scholar 

  • Popper, K. R. (1963). Conjectures and refutations. London: Routledge.

    Google Scholar 

  • Popper, K. R. (1968). The logic of scientific discovery. London: Routledge.

    Google Scholar 

  • Roche, W. (2013). Coherence and probability. A probabilistic account of coherence. In M. Araszkiewicz & J. Savelka (Eds.), Coherence: Insights from philosophy, jurisprudence and artificial intelligence (pp. 59–91). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Schippers, M. (2014a). Probabilistic measures of coherence. From adequacy constraints towards pluralism. Synthese, 191, 3821–3845.

    Article  Google Scholar 

  • Schippers, M. (2014b). Structural properties of qualitative and quantitative accounts to coherence. The Review of Symbolic Logic, 7, 579–598.

    Article  Google Scholar 

  • Schippers, M. (2014c). Coherence, striking agreement, and reliability. On a putative vindication of the Shogenji measure. Synthese, 191, 3661–3684.

    Article  Google Scholar 

  • Schippers, M. (2014d). On the impossibility of measuring coherence. Manuscript.

  • Schippers, M. (2015a). Towards a grammar of Bayesian coherentism. Studia Logica, 103, 955–984.

    Article  Google Scholar 

  • Schippers, M. (2015b). Coherence and (likeness to) truth. In Mäki, Ruphy, Schurz & Votsis (Eds.), Recent developments in the philosophy of science: EPSA13 Helsinki.

  • Schubert, S. (2012a). Is coherence conducive to reliability? Synthese, 187, 607–621.

    Article  Google Scholar 

  • Schubert, S. (2012b). Coherence reasoning and reliability: A defense of the Shogenji measure. Synthese, 187, 305–319.

    Article  Google Scholar 

  • Schupbach, J. N. (2008). On the alleged impossibility of Bayesian coherentism. Philosophical Studies, 41, 323–331.

    Article  Google Scholar 

  • Schupbach, J. N. (2011). New hope for Shogenji’s coherence measure. The British Journal for the Philosophy of Science, 62, 125–142.

    Article  Google Scholar 

  • Schurz, G., & Weingartner, P. (1987). Verisimilitude defined by relevant consequence elements. In T. Kuipers (Ed.), What is closer-to-the-truth? (pp. 47–77). Amsterdam: Rodopi.

    Google Scholar 

  • Schurz, G., & Weingartner, P. (2010). Zwart and Franssen’s impossibility theorem holds for possible-world-accounts but not for consequence-accounts to verisimilitude. Synthese, 172, 415–436.

    Article  Google Scholar 

  • Shogenji, T. (1999). Is coherence truth conducive? Analysis, 59, 338–345.

    Article  Google Scholar 

  • Shogenji, T. (2013). The degree of epistemic justification and the conjunction fallacy. Synthese, 184, 29–48.

    Article  Google Scholar 

  • Siebel, M. (2005). Against probabilistic measures of coherence. Erkenntnis, 63, 335–360.

    Article  Google Scholar 

  • Siebel, M., & Wolff, W. (2008). Equivalent testimonies as a touchstone for coherence measures. Synthese, 161, 167–182.

    Article  Google Scholar 

  • Tichý, P. (1974). On Popper’s definition of verisimilitude. The British Journal for the Philosophy of Science, 25, 155–160.

    Article  Google Scholar 

  • Wheeler, G. (2009). Focused correlation and confirmation. British Journal for the Philosophy of Science, 60, 70–100.

    Article  Google Scholar 

  • Wheeler, G. (2012). Explaining the limits of Olsson’s impossibility result. The Southern Journal of Philosophy, 50, 136–150.

    Article  Google Scholar 

  • Wheeler, G., & Scheines, R. (2013). Coherence and confirmation through causation. Mind, 122, 135–170.

    Article  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Oldenburg, 26111, Oldenburg, Germany

    Michael Schippers

Authors
  1. Michael Schippers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Schippers.

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 \(\Pr (H\wedge \pm E_i)=\Pr (H\wedge \pm E_j)\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10670-015-9771-9