Abstract
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.
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Notes
This restriction has been called “Rescher’s principle” (Olsson 2005, p. 17) and basically amounts to the fact that coherence is a property that propositions cannot have in isolation but only in groups of at least two propositions (cf. Rescher 1973, p. 32). For exceptions to Rescher’s principle in discussions on probabilistic coherence measures see Akiba’s (2000) and Fitelson’s (2003) discussions on self-coherence.
Probabilistic measures of coherence have been discussed critically by Bovens and Hartmann (2003), Moretti and Akiba (2007), Olsson (2005), Olsson and Schubert (2007), Siebel (2005) and Siebel and Wolff (2008). For an overview of the measures and their structural properties see Schippers (2014a, b, 2015), for an overview of their performance in a collection of test cases see Koscholke (2015). The relative overlap measure has received special attention in the literature due to its high degree of truth-conduciveness as shown by Angere (2007, 2008) and its strong performance in inferences to the best explanation as presented by Glass (2012).
Nonetheless, it is worth noticing that \(\mathcal{O}'\) has some flaws in the Tweety case. What if we had not received the information about Tweety being a penguin, i.e. \(x_3\), but rather \(\lnot x_3\), i.e. the information that Tweety is not a penguin? This proposition does not establish any inferential connections to \(x_1\) or \(x_2\). Instead, it seems even to decrease coherence because given \(\lnot x_3\), one possible explanation for why \(x_1\) and \(x_2\) might be the case, our last resort of making sense of \(x_1\) and \(x_2\) in some sense, vanishes. Hence, the extended set \(\{x_1,x_2,\lnot x_3\}\) should be less coherent than \(\{x_1,x_2\}\). Quite surprisingly, the Glass–Olsson naive overlap measure satisfies this intuition since \(\mathscr {O}(\{x_1,x_2,\lnot x_3\})=0\). Meijs’ refined measure does not because \(\mathscr {O}'(\{x_1,x_2,\lnot x_3\})= 0.247\). However, one should be careful with this modified case since it involves a negated proposition. Negations are known to be difficult to handle in intuitive judgements (cf. Deutsch et al. 2009).
Notice that the arguments of probabilistic coherence measures are usually not assumed to be closed under (classical) logical consequence.
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Acknowledgments
We would like to thank two anonymous reviewers whose comments helped us to improve this paper. 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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Koscholke, J., Schippers, M. Against relative overlap measures of coherence. Synthese 193, 2805–2814 (2016). https://doi.org/10.1007/s11229-015-0887-x
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DOI: https://doi.org/10.1007/s11229-015-0887-x