Abstract
I develop a probabilistic account of coherence, and argue that at least in certain respects it is preferable to (at least some of) the main extant probabilistic accounts of coherence: (i) Igor Douven and Wouter Meijs’s account, (ii) Branden Fitelson’s account, (iii) Erik Olsson’s account, and (iv) Tomoji Shogenji’s account. Further, I relate the account to an important, but little discussed, problem for standard varieties of coherentism, viz., the “Problem of Justified Inconsistent Beliefs.”
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- 1.
There are similar questions about justified (or rational) belief-revision.
- 2.
The third question (i.e., the question of whether coherence is truth-conducive), in particular, has been widely discussed of recent. See Angere (2007, 2008), Bovens and Hartmann (2003a, b, 2005, 2006), Bovens and Olsson (2000, 2002), Cross (1999), Huemer (1997, 2007, 2011), Klein and Warfield (1994, 1996), Meijs and Douven (2007), Merricks (1995), Olsson (2001, 2002, 2005a, b), Olsson and Shogenji (2004), Roche (2010, 2012), Schubert and Olsson (2012), Schupbach (2008), Shogenji (1999, 2005, 2007, Shogenji forthcoming), van Cleve (2005, 2011), and Wheeler (2009, 2012).
- 3.
The account is not meant to accurately describe how in fact ordinary people make coherence evaluations. Clarification as to what the account is meant to do is given below in Sect. 3.2.1.
- 4.
- 5.
- 6.
Olsson holds, at least, that (CO) has some initial appeal.
- 7.
- 8.
- 9.
Here and throughout the paper I assume that only finite sets of two or more claims can have a degree of coherence or incoherence. Cf. Akiba (2000).
- 10.
This case is adapted from BonJour (1985, 95–96).
- 11.
Also important, I believe, is the extent to which an account of coherence is fruitful in that it leads to results concerning, say, the reliability of witnesses in a witness scenario, the transmission of confirmation, and so on. For relevant discussion, see Dietrich and Moretti (2005) and Schubert (2012). I do not have the space to evaluate (CDM), (CF), (CO), (CR), and (CS) in terms of fruitfulness. Fitelson (2003, 194) gives a desideratum (referred to as “(1)”) on which, inter alia, an adequate account of coherence should imply that a set S = {p 1,..., p n } is maximally coherent if p 1,..., p n are logically equivalent (and S is satisfiable). I would be happy to accept this part of the desideratum; (CR), like (CF), implies that a set S = {p 1,..., p n } is maximally coherent if p 1,..., p n are logically equivalent (and S is satisfiable). But I would not be happy to accept the desideratum as a whole. See Sect. 3.4.2.1 below.
- 12.
It is not trivial that (CDM), (CF), (CO), (CR), and (CS) all imply (a). Not all probabilistic accounts of coherence imply (a). Some probabilistic accounts of coherence are purely qualitative. See Douven and Meijs (2007, Sect. 2), where five such accounts are developed and compared. See, also, Lewis (1946, 338); there the term “congruence” is used in place of the term “coherence.”
- 13.
By this standard, which admittedly is somewhat vague, each of (CDM), (CF), (CO), (CR), and (CS) is a probabilistic account of coherence. A more stringent standard could be employed. Perhaps then not all of (CDM), (CF), (CO), (CR), and (CS) would be probabilistic accounts of coherence. But, of course, nothing of importance hinges on how the various accounts are categorized.
- 14.
- 15.
See Douven and Meijs (2007, 407).
- 16.
By “explanatory facts” I mean to include facts about the explanatory virtues, for example, simplicity. How can at least certain of the probabilistic facts in a given case be determined by (in part) the explanatory facts in the case? Imagine a case where h 1 and h 2 are scientific hypotheses, and h 1 is preferable to h 2 in terms of simplicity. One might hold that \( \Pr \left( {{h_1}} \right)>\Pr \left( {{h_2}} \right) \), and that this is owing in part to the fact that h 1 is preferable to h 2 in terms of simplicity. Or suppose (adapting a case from Okasha 2000, 702–703) Smith is in some distress, where e describes Smith’s symptoms. Suppose h 1 is the claim “Smith has pulled a muscle,” and h 2 is the claim “Smith has torn a ligament.” Suppose, given background information, e is better explained by h 1 than by h 2 in that e would be expected if h 1 were true but not if h 2 were true. Then, the idea goes, \( \Pr \left( {e|{h_1}} \right)>\Pr \left( {e|{h_2}} \right) \). The issues here, however, are many and difficult, and the relevant literature is vast. See, for starters, Day and Kincaid (1994), Douven (1999, 2011, Sect. 4), Harman (1970), Huemer (2009a, b), Iranzo (2008), Lipton (2001, 2004, Chaps. 7), Lombrozo (2007), McGrew (2003), Niiniluoto (1999, 2004), Okasha (2000), Psillos (2004, 2007), Salmon (1970, 1990, 2001a, b), Sober (2002), Swinburne (1973, Chaps. 7), Tregear (2004), van Fraassen (1989, Chaps. 7, Sect. 4), and Weisberg (2009).
- 17.
There are alternative generalizations of the thesis that whether a two-member set of claims is coherent hinges on whether the claims are positively probabilistically relevant to each other. One is developed by Jonah Schupbach (2011). I do not have the space to examine this account (or any of the other alternative generalizations). But what I say in Sects. 3.4.1.1, 3.4.1.2, and 3.4.1.3 about (CS) can also be said mutatis mutandis about the account developed by Schupbach.
- 18.
- 19.
It might be better to define S so that it is an ordered set, and to define S* so that S’ is a nonempty subsequence of S. See Schubert (2012, 311–312).
- 20.
If S’ has just one member, then ∧S’ is simply that member. Likewise with respect to S”.
- 21.
- 22.
That C X(S) is the mean of \( \sum\nolimits_{i=1}^{n-1 } {\left( {\begin{array}{*{20}{c}} n \\ i \\ \end{array}} \right)\left( {{2^{n-i }}-1} \right)} \) confirmation values follows, ultimately, from the fact that S ** has exactly \( \sum\nolimits_{i=1}^{n-1 } {\left( {\begin{array}{*{20}{c}} n \\ i \\ \end{array}} \right)\left( {{2^{n-i }}-1} \right)} \) members. See Douven and Meijs (2007, 412, n. 15).
- 23.
The result that C X(S) is the mean of \( {3^n}\hbox{--} {2^n}^{+1 }+1 \) confirmation values is due essentially to Kyle Kloster (to whom I am grateful).
- 24.
Douven and Meijs (2007, Sects. 2 and 3) give a compelling defense of this idea. This defense, though, can be strengthened a bit. Douven and Meijs fail to prove “Conjecture 2.1” (2007, 408). The thesis in question—viz., that “one-any + partition coherence” does not entail “any-any coherence”—is true, as Douven and Meijs conjecture, and can be proven.
- 25.
Why cannot C DM(S) = 1? C DM(S) = 1 only if each of the various confirmation values equals 1. But, none of the various confirmation values equals 1. This is because d(h, e) cannot equal 1. Suppose Pr(h) = 0. Then, assuming Pr(h | e) is defined, Pr(h | e) = 0, hence \( \Pr \left( {h|e} \right)-\Pr (h)=0 \). Suppose, instead, Pr(h) > 0. Then, even if Pr(h | e) = 1, it follows that \( \Pr \left( {h|e} \right)-\Pr (h)<1 \). Why cannot C DM(S) = −1? C DM(S) = −1 only if each of the various confirmation values equals −1. But, none of the various confirmation values equals −1. This is because d(h, e) cannot equal −1. Suppose Pr(h) = 1. Then, assuming Pr(h | e) is defined, Pr(h | e) = 1, hence \( \Pr \left( {h|e} \right)-\Pr (h)=0 \). Suppose, instead, Pr(h) < 1. Then, even if Pr(h | e) = 0, it follows that \( \Pr \left( {h|e} \right)-\Pr (h)>-1 \).
- 26.
Douven and Meijs (2007, 411, n. 14) make a point to this effect.
- 27.
I am assuming, here and throughout the paper, that if a claim p is necessarily true, then, on any (admissible) probability function, Pr(p) = 1, and that if a claim p is necessarily false, then, on any (admissible) probability function, Pr(p) = 0. In assuming this I am glossing over some difficult issues in epistemology, philosophy of language, and philosophy of mind. I leave it for further investigation how best to treat these issues and whether the best treatment would require substantive changes to the main points of this paper. For relevant discussion, see Chalmers (2011), Douven and Meijs (2007, Sect. 3.5.1), Garber (1983), and Swinburne (1973, Chap. 4).
- 28.
This point can be established in two other ways. First, observe that:
$$ {C_{\mathrm{S}}}\left( {{S_6}} \right)=\frac{{\Pr \left( {f\wedge \sim\ f} \right)}}{{\Pr (f)\times \left( {\Pr \sim\ f} \right)}}=\frac{{\Pr \left( {\sim\ f} \right)\times \Pr \left( {f|\sim\ f} \right)}}{{\Pr (f)\times \Pr \left( {\sim\ f} \right)}} = \frac{{\Pr \left( {f|\sim\ f} \right)}}{{\Pr (f)}}. $$Since the numerator in \( \frac{{\Pr \left( {f|\sim\ f} \right)}}{{\Pr (f)}} \) is undefined, C S(S 6) is undefined. Second, observe that:
$$ {C_{\mathrm{S}}}\left( {{S_6}} \right)=\frac{{\Pr \left( {f\wedge \sim\ f} \right)}}{{\Pr (f)\times \Pr \left( {\sim\ f} \right)}}=\frac{{\Pr (f)\times \Pr \left( {\sim\ f\left| f \right.} \right)}}{{\Pr (f)\times \Pr \left( {\sim\ f} \right)}} = \frac{{\Pr \left( {\sim\ f\left| f \right.} \right)}}{{\Pr \left( {\sim\ f} \right)}}. $$Since the denominator in \( \frac{{\Pr \left( {\sim\ f\left| f \right.} \right)}}{{\Pr \left( {\sim\ f} \right)}} \) equals 0, C S(S 6) is undefined.
- 29.
Douven and Meijs (2007, Sect. 3.5.1) raise a problem for (CDM), and for certain other accounts of coherence, and give two proposals for solving the problem. Each proposal has the result that the only sets that should be considered when evaluating (CDM) are sets consisting of pairwise logically independent claims. This result entails that, since S 6 is not a set consisting of pairwise logically independent claims, S 6 should not be considered when evaluating (CDM). It seems clear, though, that S 6 has a coherence value, in fact a very low if not maximally low coherence value. Moreover, it seems clear that many sets consisting of pairwise logically dependent claims have coherence values. So for the purposes of this paper I shall assume that the two proposals given by Douven and Meijs should be rejected and that the problem raised by Douven and Meijs for (CDM), and for the other accounts in question, can be adequately answered without appeal to those proposals. See Huemer (2011, 46–47) and Schubert (2012, 311–312).
- 30.
A slight variant of this proposal with respect to (CO) is given in Glass (2005, 384, n. 7).
- 31.
38 = 6,561.
- 32.
There are variants of the case where (CDM) is silent on the larger set but (CS) is not. Suppose S 9* is just like S 8 except that S 9* includes two additional claims, p n+1 and p n+2, where each claim has a nonextreme probability and the one claim entails the falsity of the other claim. (CDM) is silent on S 9*; \( \Pr \left( {{p_n}_{+1 } \wedge {p_n}_{+2 }} \right)=0 \), thus \( \Pr \left( {{p_1}|{p_n}_{+1 } \wedge {p_n}_{+2 }} \right) \) is undefined, thus \( d\left( {{p_1},{p_n}_{+1 } \wedge {p_n}_{+2 }} \right) \) is undefined, thus C DM(S 9*) is undefined. (CS), by contrast, is not silent on S 9*. C S(S 9*) = 0 and so (CS) implies that S 9* is maximally incoherent. This, it seems, is the wrong result. S 9* has a lesser coherence value than S 8. But, since (by hypothesis) certain of the claims in S 9* (namely, p 1,..., p n ) hang together in the requisite sense, S 9* is not maximally incoherent.
- 33.
In Sect. 3.5, below, I consider the question of whether C R(S 9) can have a coherence value greater than 0.5.
- 34.
Two claims are subcontraries just in case the falsity of the one claim entails the truth of the other.
- 35.
Meijs (2006, 237) gives an argument along these lines.
- 36.
In fact, I prefer the more general idea that for any set with two or more members maximal coherence is a matter of pairwise mutual entailment (where for each pair of claims in the set the one claim entails the other claim, and each claim has a probability greater than 0) and maximal incoherence is a matter of pairwise incompatibility (where for each pair of claims in the set the one claim entails the negation of the other claim).
- 37.
There are at least two further reasons for preferring the idea that maximal coherence is a matter of mutual entailment and maximal incoherence is a matter of incompatibility. First, if coherence is a matter of positive probabilistic relevance, it follows that no set of necessary truths is coherent. Hence no set of mathematical necessities is coherent, and no set of philosophical necessities is coherent (for example, no set of logical necessities is coherent), and so on. If, instead, maximal coherence is a matter of mutual entailment and maximal incoherence is a matter of incompatibility, it follows that all sets of necessary truths are coherent, indeed, maximally coherent. Second, if coherence is a matter of positive probabilistic relevance, it follows that there can be sets S and S* such that the claims in S are mutually entailing, the claims in S* are mutually entailing, and yet, because the prior probabilities of the claims in S are lower than the prior probabilities of the claims in S*, S has a greater coherence value than S*. If, instead, maximal coherence is a matter of mutual entailment and maximal incoherence is a matter of incompatibility, it follows that if the claims in S are mutually entailing and the claims in S* are mutually entailing, then, regardless of the prior probabilities of the claims in S and S*, S and S* have the same coherence value. For helpful discussion of (CF), (CS), (CO), and the issue of “prior-dependence,” see Glass (2005). See, also, Fitelson (2003, Sect. 2) and Siebel and Wolff (2008).
- 38.
Certain of the accounts, even if inadequate as accounts of coherence, can be useful nonetheless, for example, in contexts of confirmation. See Dietrich and Moretti (2005), for discussion of (CF), (CO), and (CS) and the transmission of confirmation.
- 39.
For discussion of the “regress problem” and foundationalism, social contextualism, infinitism, and coherentism, and for references, see Cling (2008). It might be best to allow for varieties of coherentism on which some justification is noninferential, and thus on which it is not required for justification that (CCI) be satisfied. See Lycan (2012) and Poston (2012).
- 40.
A circular chain of implication should not be confused with a circular chain of justification. Coherentists (of the sort I have in mind) deny that justification is transferred between beliefs. Coherentists hold that justification is holistic: Beliefs are justified together when the requisite conditions are satisfied. For further discussion of this and related issues, see Roche (2012b).
- 41.
For discussion of varieties of coherentism of this sort, and for references, see Roche (2012b).
- 42.
(B) should be understood so that whether S’s belief system is consistent is determined by whether the set of claims believed by S is consistent.
- 43.
(C), like (A), is fully general and so applies to all of S’s beliefs.
- 44.
There is a second version of the Problem of Justified Inconsistent Beliefs pertaining to lottery-style cases. See Kvanvig (2012) for an explanation of the problem and an attempted solution. Eric Senseman (2010), a former undergraduate student of mine (at Texas Christian University), considers a variant of (CR) and how that variant relates to the problem.
- 45.
If \( \wedge S^{\prime} \) involves p n+1, \( \wedge S^{\prime\prime}\vDash \sim\ \wedge S^{\prime} \). If \( \wedge S^{\prime\prime} \) involves p n+1, \( \wedge S^{\prime\prime}\vDash \sim\wedge S^{\prime} \).
- 46.
Recall (from Sect. 3.3.3) that where S = {p 1,..., p n }, C X(S) is the mean of \( {3^n}\hbox{--} {2^n}^{+1 }+1 \) confirmation values. Thus where S = {p 1,..., p n , p n+1}, C X(S) is the mean of \( {3^n}^{+1}\hbox{--} {2^n}^{+2 }+1 \) confirmation values.
- 47.
This follows from the fact that (i) \( \frac{{{3^n}-{2^{n+1 }}+1}}{{{3^{n+1 }}-{2^{n+2 }}+1}}<\frac{1}{3} \) when n = 2, (ii) \( \frac{{{3^n}-{2^{n+1 }}+1}}{{{3^{n+1 }}-{2^{n+2 }}+1}} \) is a strictly increasing function of n (given the constraint that 2 ≤n< ∞ where n \( \in \mathbb{N} \)), and (iii) \( {{\rm lim}_{n\rightarrow\infty}} \frac{{{3^n}-{2^{n+1 }}+1}}{{{3^{n+1 }}-{2^{n+2 }}+1}}=\frac{1}{3}. \)
- 48.
One possibility would be to understand probability so that Pr(p) can be greater than 0 even if p is a necessary falsehood (and so that Pr(p) can be less than 1 even if p is a necessary truth). For relevant discussion, see Chalmers (2011), Douven and Meijs (2007, Sect. 5.1), Garber (1983), and Swinburne (1973, Chap. 4). See, also, Kvanvig (2012, Sect. 2) and Lycan (1996, Sect. VII; 2012, Sect. 7).
- 49.
- 50.
- 51.
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Acknowledgments
I wish to thank Michał Araszkiewicz, Kyle Kloster, Michael Roche, Eric Senseman, and the participants in the Artificial Intelligence, Coherence and Judicial Reasoning Workshop at ICAIL 2011 (the 13th International Conference on Artificial Intelligence and Law) for helpful comments on or discussion of ancestors of this paper. Further, I wish to thank Igor Douven, Kyle Kloster, and Eric Senseman for helpful correspondence on some of the issues discussed in the paper and related issues.
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Roche, W. (2013). Coherence and Probability: A Probabilistic Account of Coherence. In: Araszkiewicz, M., Šavelka, J. (eds) Coherence: Insights from Philosophy, Jurisprudence and Artificial Intelligence. Law and Philosophy Library, vol 107. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6110-0_3
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