Abstract
According to a coherentist position in philosophy of science, good theories cohere with the available data and one theory is better than another if it coheres better with the available data. This paper examines that relationship with a special focus on probabilistic measures of coherence. In a first step, it is shown that existing coherence measures satisfy a number of reasonable adequacy constraints for the comparison of two rival scientific theories. In a second step, the virtue of a coherentist position in philosophy of science is considered. More specifically, it is assessed whether coherence implies verisimilitude in the sense that a higher degree of coherence between theory and evidence entails a higher degree of (estimated) truthlikeness. To this end, it is demonstrated that there is an intimate relationship in this sense if we explicate coherence by means of the so-called overlap-measure.
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- 1.
In this paper I will use the terms “verisimilitude” and “truthlikeness” as synonyms.
- 2.
The first philosopher to propose a definition of verisimilitude that suffered from a similar drawback is Karl Popper. Miller (1974) and Tichý (1974) independently proved that Popper’s original definition Popper (1963, 1972) suffered from a logical flaw so that on his account a false theory can never be closer to the truth than another true or false theory. For a comprehensive survey of subsequent developments see Oddie 2008, Niiniluoto 2011 and Zwart 2001.
- 3.
See Sect. 4 for details.
- 4.
- 5.
The general case is slightly more intricate: Fitelson’s (2004) account as well as Douven & Meijs’ recipe are applicable to sets of propositions. As such they take into account the mutual degree of confirmation for each pair of non-empty, disjunct subsets of the given set under consideration.
- 6.
- 7.
Douven and Meijs (2007) consider further confirmation measures to be fed into their recipe, namely the ratio-measure (Keynes 1921; Horwich 1982) and the (log-) likelihood measure (Good 1984; Zalabardo 2009) of confirmation. However, \(\mathcal{S}_{d}\) is their favorite explication of coherence. Accordingly, since an evaluation of all possible coherence measures is beyond the scope of the present paper, we restrict our attention to \(\mathcal{S}_{d}\) and neglect all others.
- 8.
This measure has independently been proposed by Schippers and Siebel (2012, Reassessing probabilistic measures of coherence, Unpublished manuscript).
- 9.
This latter property, sometimes called the “Bar-Hillel-Carnap semantic paradox” (Floridi 2004, p. 198), might seem curious at first sight. However, (iii) follows naturally from the assumption that t’s information content is related to the amount of state descriptions precluded by t. A tautology precludes no state description at all and is consequently assigned the lowest possible degree of informativity. A contradiction on the other hand precludes every possible state description and is accordingly maximally informative. In this sense, Bar-Hillel and Carnap (1953) state that “a false sentence which happens to say much is thereby highly informative in our sense. […] A self-contradictory sentence asserts too much; it is too informative to be true” (p. 229).
- 10.
- 11.
- 12.
This is similar to the famous child’s play objection against content-definitions of verisimilitude (cf. Tichý 1974).
- 13.
Both in the case of explanation and in the case of predictive success to be considered below we assume simple cases of deductive explanation/success. This is not to say that these are the only relevant such cases; however, in the given context the focus is exclusively on these salient cases.
- 14.
The proofs only require elementary probability theory and straightforward arithmetic manipulations. For the proofs pertaining to measure \(\mathcal{O}\) see Zamora-Bonilla 1996. There, \(\Pr (e)^{-1} \times \mathcal{O}(t,e)\) is proposed as a probabilistic measure of verisimilitude. For further discussions see also Zamora-Bonilla 2002, 2013.
- 15.
Let \(\Gamma \) and \(\Sigma \) be two sets, then the symmetric difference, \(\Gamma \Delta \Sigma \), is defined as follows: \(\Gamma \Delta \Sigma = (\Gamma \setminus \Sigma ) \cup (\Gamma \setminus \Sigma )\).
- 16.
Cf. Schurz and Weingartner 2010 for similar characterizations.
- 17.
It is well-known that in propositional logics each formula can be translated into an equivalent formula in conjunctive normal form.
- 18.
The reason for focusing on this measure is that it performed well in the evaluation in Table 1. An extensive survey of different approaches is beyond the scope of the present paper.
- 19.
- 20.
- 21.
These logical probability distributions have famously been discussed by Carnap (1962).
- 22.
The same holds if we instead choose the firmness-based measure \(\mathcal{S}_{f}\). However, a detailed investigation of the other measures is beyond the scope of the present paper. The proof of Theorem 5.1 is given in the appendix.
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Appendix: A Proof of Theorem 5.1
Appendix: A Proof of Theorem 5.1
Let \(e_{\Pr }\) be a distribution of probabilities \(\Pr (c\vert e)\) for each constituent \(c \in \mathcal{C}\). The degree of coherence between c and an arbitrary c-theory \(t = p_{i_{1}} \wedge \ldots \wedge p_{i_{k}}\) is given by the following formula:
It can easily be shown that the following identity holds:
Now for conjunctive theories, both \(\Pr (\bigwedge t_{c}^{+}\vert c)\) and \(\Pr (\bigwedge t_{c}^{-}\vert \overline{c})\) are equal to 1. Hence, Eq. 2 reduces to the following formula:
Exploiting Bayes’ theorem, we get (remember that \(\Pr\) is the logical probability \(\Pr ^{{\ast}}\)):
Hence, \(\mathcal{O}(t,c,\overline{c})\) is strictly increasing in | t c + | and strictly decreasing in | t c − | for each constituent c. The same obviously holds for Vs(t, c). Therefore, for each pair of c-theories \(t,t^{{\prime}}\) and each constituent the following equivalence holds:
Hence, (5), (6) and the definition of Vs together entail that
Thus, we get the desired result that if \(\mathbb{E}(\mathcal{O}(t,e)) > \mathbb{E}(\mathcal{O}(t,e))\), then \(\mathrm{EVs}(t,e) >\mathrm{ EVs}(t^{{\prime}},e)\) (and also vice versa). □
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Schippers, M. (2015). Coherence and (Likeness to) Truth. In: Mäki, U., Votsis, I., Ruphy, S., Schurz, G. (eds) Recent Developments in the Philosophy of Science: EPSA13 Helsinki. European Studies in Philosophy of Science, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-23015-3_1
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