Abstract
According to van Fraassen, inference to the best explanation (IBE) is incompatible with Bayesianism. To argue to the contrary, many philosophers have suggested hybrid models of scientific reasoning with both explanationist and probabilistic elements. This paper offers another such model with two novel features. First, its Bayesian component is imprecise. Second, the domain of credence functions can be extended.
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Notes
Recently, there have been intense debates on the cogency of imprecise Bayesianism. First, Elga (2010) argued that an agent a in an imprecise credal state \({\mathfrak {C}}\) will not accept a series of bets B although it is clearly beneficial to her. In response, Chandler (2014) pointed out that imprecise probabilists have discussed several different decision rules R, and for some \(r\in R\), if a in \({\mathfrak {C}}\) makes a decision in accordance with r, then a will accept B. For other objections and replies, see Bradley (2019). The goal of this paper is to show that IBE and imprecise Bayesianism can be combined into a coherent hybrid model on the assumption that each is defensible in its own right. Hence, imprecise Bayesianism is adopted here without offering any new defense.
For a technical reason, this paper adopts sentences, not propositions, as the objects of credence. See footnote 7.
Douven himself calls his updating model “IBE” but I will call it “extra boost view” to distinguish it from other probabilistic forms of IBE.
Admittedly, this formal model will remain obscure in some aspects until the correct theories of explanation and explanatory virtues are found. However, it is completely legitimate to develop a formal model before the related philosophical debates are finished. For example, the formal probability theory is a mature field in mathematics, but there is still an ongoing debate on the interpretation of probability among philosophers.
In this paper, “\(\alpha\) adequately explains \(\beta\),” “\(\alpha\) is an adequate explanation of \(\beta\),” and “\(\alpha\) provides (offers) an adequate explanation of \(\beta\)” are used interchangeably.
If the covering law account of explanation is correct, then any explanatory sentence X will be noncontingent and assigned the credence of 0 or 1, in which case the agent cannot change her credences by conditioning on X. In such a case, \(\models\) needs to be defined as follows: for any set \(\Gamma\) of \({\mathcal {L}}\)-sentences and any \({\mathcal {L}}\)-sentence X, \(\Gamma \models X\) iff there is no assignment v of truth-values to atomic and explanatory \({\mathcal {L}}\)-sentences such that for any \(Y\in \Gamma\), \(v(Y)=1\) but \(v(X)=0\). So, even if X is a noncontingent explanatory sentence, \(\not \models X\), \(\not \models \lnot X\), and possibly \(p(X)\in (0,1)\). See Garber (1983) pp. 109–118 for a similar theory of logical learning.
Some authors disagree to this view. For example, Oaksford and Chater (1998) point out that Bayesianism captures some features of actual human reasoning (e.g., non-monotonic evaluation of the truth values of conditionals) very well. For another, Harman (1965) argues that the normative force of enumerative induction derives from that of IBE. Still, Liption’s view is attractive because, first, it is difficult to explain why one ought to regard a virtuous explanation as more likely to be true (Lipton, 2004, pp. 142–147) and, second, there is empirical evidence that actual human inference often violates some principles of probability theory (e.g., conjunction fallacy) (Oaksford & Chater, 1998, p. 281).
If \({\mathfrak {F}}_{1}\not \subseteq {\mathfrak {F}}_{2}\), \(U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right)\) is undefined. According to Miranda (2008), a coherent lower prevision \({\underline{P}}\) defined over the set of gambles \({\mathfrak {K}}\) can be extended to a natural extension \({\underline{E}}\) for a bigger set of gambles \({\mathfrak {K}}^*\supset {\mathfrak {K}}\), or the least committal coherent lower prevision for \({\mathfrak {K}}^*\) that coincides with \({\underline{P}}\) concerning the supremum acceptable buying prices of the gambles in \({\mathfrak {K}}^*\) (pp. 638–40). Since \(U({\mathfrak {C}},{\mathfrak {F}}_1,{\mathfrak {F}}_2)\) is the least committal set of the extensions of \(p\in {\mathfrak {C}}\) for \({\mathfrak {F}}_2\supset {\mathfrak {F}}_1\), we can think as if \(U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\) is the natural extension of \({\mathfrak {C}}_1\) for \({\mathfrak {F}}_2\). See Walley (1991), especially Chap. 3, for more details about natural extension.
See Appendix A.
What if there is more than one hypothesis competing with A? In that case, let B be the disjunction of all hypotheses \(B_{1},\ldots ,B_{n}\) competing with A. Then, the same argument will apply to the change in the agent’s credence in A.
See Appendix A.
See Appendix B.
I feel grateful to an anonymous reviewer for raising this objection.
By contrast, no extension of the domain occurs at \(t_{3}\). Hence, \({\mathfrak {F}}_{2}\) is the domain of \({\mathfrak {C}}_{3}\), too.
See Appendix B.
I feel grateful to an anonymous reviewer for pressing to address this important issue.
See Appendix B.
See Appendix C.
For simplicity, I ignore, in this section, the cases in which empirical evidence and the relevant explanatory fact are learned at different moments. Of course, my discussion in this section can be generalized for such a case in an obvious way.
This is also true in the cases of \(H_{0}\) and \(H_{10},\) but these results were determined analytically, not by a simulation. For details, see Table 1 in Douven (2013).
Proof: Consider any \(E_{w^*}\in {\mathcal {E}}_{k+1}.\) By definition, for any \(a\in {\mathcal {A}}_{k+1}\), for any updating rule \(\alpha ,\) \(\sum _{w\in W}c_{k}^{\left\langle \beta ,w^{*}\right\rangle }\left( w\right) u\left( a^{c_{k}^{\left\langle \beta ,w^{*}\right\rangle }},w\right) \ge \sum _{w\in W}c_{k}^{\left\langle \beta ,w^{*}\right\rangle }\left( w\right) u\left( a^{c_{k}^{\left\langle \alpha ,w^{*}\right\rangle }},w\right) .\) For any \(w\in W\), if \(E_{w^{*}}\) is true in w, then \(c_{k}^{\left\langle \beta ,w^{*}\right\rangle }\left( w\right) =\frac{c_{k}\left( w\right) }{c_{k}\left( E_{w^{*}}\right) }\) and if \(E_{w^{*}}\) is false in w, then \(c_{k}^{\left\langle \beta ,w^{*}\right\rangle }\left( w\right) =0\). In the former case: both \(E_{w}\) and \(E_{w^{*}}\) are true in w; since \({\mathcal {E}}_{k+1}\) is a partition, \(E_{w}=E_{w^{*}}\); hence, \(c_{k}^{\left\langle \beta ,w\right\rangle }=c_{k}^{\left\langle \beta ,w^{*}\right\rangle }\) and \(c_{k}^{\left\langle \alpha ,w\right\rangle }=c_{k}^{\left\langle \alpha ,w^{*}\right\rangle }.\) So, \(\sum _{E_{w^{*}}\in {\mathcal {E}}_{k+1}}\sum _{v_{w}\left( E_{w^{*}}\right) =1}c_{k}\left( w\right) u\left( a^{c_{k}^{\left\langle \beta ,w^{*}\right\rangle }},w\right) \ge \sum _{E_{w^{*}}\in {\mathcal {E}}_{k+1}}\sum _{v_{w}\left( E_{w^{*}}\right) =1}c_{k}\left( w\right) u\left( a^{c_{k}^{\left\langle \alpha ,w^{*}\right\rangle }},w\right) ,\) where \(v_{w}\left( \cdot \right)\) is the omniscient function at w. Done. Observe that we can replace “\(\ge\)” with “>” in this proof if, for some \(w\in W\), \(c_{k}\left( w\right) >0\) and \(u\left( a^{c_{k}^{\left\langle \beta ,w\right\rangle }},w\right) \ne u\left( a^{c_{k}^{\left\langle \alpha ,w\right\rangle }},w\right) .\) This proof is originally from Brown (1976).
That is, for any probability function p and any world \(w\in W\), \({\mathfrak {B}}\left( p,w\right) =_{df.}\sum _{A\in \text {dom}\left( p\right) }-\left( v_{w}\left( A\right) -p\left( A\right) \right) ^{2},\) where \(v_{w}\left( \cdot \right)\) is the omniscient function at w. Douven also discusses the log score accuracy measure, but I will not talk about it here.
Thanks to an anonymous reviewer for encouraging me to discuss this issue.
An anonymous reviewer points out that, despite these merits, many proponents of IBE will find this type of hybrid models (e.g., Lipton (2004) and Weisberg (2009)) to be unsatisfactory because, in those models, once the agent’s initial priors are determined by taking explanatory factors into consideration, such factors will play no substantial role in the subsequent updating process. I agree wholeheartedly.
Here and in the rest of this paper, \(O\left( \psi |\phi \right)\) itself is used as an interpretation-neutral notation for conditional obligation.
A related issue is what kind of implication is expressed by ‘\(\rightarrow\)’ but this does not matter for the purpose of this paper. Also, some wide-scopers regard ‘\(O(\cdot |\cdot )\)’ as a primitive dyadic operator. If you are one of them, read ‘\(O(\cdot \rightarrow \cdot )\)’ as such an operator.
Some authors call this rule “Restricted Factual Detachment,” reserving “Factual Detachment” for the following application of MP: \(\phi \rightarrow O\left( \psi \right) ,\phi \models _{SDL+}O\left( \psi \right) .\) For brevity, we will call the former “Factual Detachment” and the latter simply “MP.”
Recall: \(\phi\) is an adequate explanation of \(\psi\) iff, if \(\psi\) were true, then \(\phi\) would be a better and sufficiently good explanation of \(\psi\) than any rival hypothesis.
I owe this example to an anonymous reviewer.
In R&S’s original paper 2013, \(p(\cdot )\) was interpreted as the given agent’s actual prior credence function.
For an easier discussion, I modified Lange’s example in a few inessential aspects.
We denote \(\phi\) by “the probabilistic antecedent” of a conditional probability \(p(\psi |\phi )\).
This presupposes that real scientists are not logically omniscient, which is true.
R&S claim that, in this type of cases, it is the logical fact, not the explanatory fact, that confirms the target hypothesis. In Example 7, the logical fact that \(S\vdash _K M\), not the explanatory fact \(X_S \& \lnot X_C\), provides additional support for S. However, this claim is unconvincing. The logical fact that \(S\vdash _K M\) is an essential constituent of the explanatory fact \(X_S\). So, if one says that the additional confirmation of S originates from \(S\vdash _K M\), one has already said that it comes from \(X_S\).
This proof was constructed by modifying Skyrms (1987)’s proof of Theorem V, the converse Dutch book theorem for standard conditioning.
Each set of bets can be empty but since \({\mathfrak {D}}\) is a diachronic Dutch book, any two sets of bets traded at the same time cannot be both empty. For example, if \(\{B_1,\ldots ,B_j\}=\{B_{j+1},\ldots ,B\}=\emptyset \ne \{B_{k+1},\ldots ,B_n\}\), then \({\mathfrak {D}}\) would be a synchronic Dutch book at \(t_2\).
For our purpose, \(a'\) does not have to be in the same possible world as a’s.
One may wonder why we should discharge the assumption that there is a diachronic Dutch book against a rather than concluding that there cannot be an agent \(a'\) as described in this proof. However, the point of introducing \(a'\) into our discussion is that if such an agent \(a'\) exists and is free from any synchronic Dutch book at \(t_1\), then a must be free from any diachronic Dutch book \({\mathfrak {D}}\) concerning her credal transition from \(t_1\) to \(t_2\) because any such \({\mathfrak {D}}\) would be easily converted to a synchronic Dutch book that \(a'\) would accept at \(t_1\). Once we establish in this way the conclusion that a does not become vulnerable to a diachronic Dutch book by virtue of updating in accordance with EIB, it does not matter whether such an agent \(a'\) actually exists.
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Appendices
Appendix
A. Extended Priors’ Satisfaction of (4)
If any probability function \(p\left( \cdot \right)\) satisfies (1)–(3), then \(p\left( \cdot \right)\) also satisfies (4).
Corollary
For any \(p\in U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right)\) , \(p\left( \cdot \right)\) satisfies (4).
Proof
Consider any probability function \(p\left( \cdot \right)\). Suppose (1)–(3) and show (4). For the supposition to hold, \(p\left( A|\lnot X_{A} \& X_{B} \& E\right)\) must be defined. Hence, \(p\left( \lnot X_{A} \& X_{B}|E\right) \ge p\left(\neg X_{A} \& X_{B} \& E\right) >0.\) For similar reasons, \(p\left( \lnot X_{A} \& \lnot X_{B}|E\right) ,\) \(p\left( X_{A} \& X_{B}|E\right) >0\). Thus, \(p\left( X_{A} \& \lnot X_{B}|E\right) =1-[p(X_{A} \& X_{B}|E)+p(\lnot X_{A} \& X_{B}\) \(|E)+p(\lnot X_{A} \& \lnot X_{B}|E)]<1.\) By supposition,
By the law of total probability, \(p\left( A|E\right)\) is the sum of the terms on the right-hand side, and clearly, \(p\left( A|E \& X_{A} \& \lnot X_{B}\right)\) is that of the terms on the left-hand side. Therefore, \(p\left( A|E \& X_{A} \& \lnot X_{B}\right) >p\left( A|E\right)\). Done. \(\square\)
B. Posterior Credence from Imprecise Prior Credal State
For any \(A\in {\mathfrak {F}}_{1}\), if \({\mathfrak {C}}_{2}\left( A\right) =\left( U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right) \right)\) \(\left( A|E \& X_{A} \& \lnot X_{B}\right)\), then \({\mathfrak {C}}_{2}\left( A\right) =\left( \underline{{\mathfrak {C}}_{1}}\left( A|E\right) ,1\right]\).
Corollary
For any \(A\in {\mathfrak {F}}_{1}\), if \({\mathfrak {C}}_{2}\left( A\right) =\left( U\left( c_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right) \right)\) \(\left( A|E \& X_{A} \& \lnot X_{B}\right)\), \({\mathfrak {C}}_{2}\left( A\right) =\left( c_{1}\left( A|E\right) ,1\right]\).
Proof
Consider any \(A\in {\mathfrak {F}}_{1}\). Suppose \({\mathfrak {C}}_{2}\left( A\right) =(U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right) )(A|E \& X_{A} \& \lnot X_{B})\), and show \({\mathfrak {C}}_{2}\left( A\right) =\left( \underline{{\mathfrak {C}}_{1}}\left( A|E\right) ,1\right]\). It suffices to show \((U({\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}))(A|E \& X_{A} \& \lnot X_{B})=\left( \underline{{\mathfrak {C}}_{1}}\left( A|E\right) ,1\right]\). (\(\subseteq\)) By Appendix A, \(U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right) \left( A|E \& X_{A} \& \lnot X_{B}\right) \subseteq \left( \underline{{\mathfrak {C}}_{1}}\left( A|E\right) ,1\right]\). (\(\supseteq\)) Consider any \(x\in \left( \underline{{\mathfrak {C}}_{1}}\left( A|E\right) ,1\right]\). By definition, there is \(p\in {\mathfrak {C}}_{1}\) such that \(p\left( A|E\right) <x\le 1\). Let \(f:{\mathbb {R}}_{\ge 0} \rightarrow \left( 0,1\right]\) be \(y\mapsto \frac{p\left( A \& E\right) }{p\left( A \& E\right) +y}\) and r be \(\frac{\left( 1-x\right) p\left( A \& E\right) }{xp\left( \lnot A \& E\right) }\). So \(f\left( p\left( \lnot A \& E\right) \right) =p\left( A|E\right) <x=f\left( r\cdot p\left( \lnot A \& E\right) \right) \le 1=f\left( 0\right) .\) Since \(f\left( \cdot \right)\) is a strictly decreasing monotonic continuous function, \(0\le r<1\). Hence, we can construct an extension \(q\left( \cdot \right)\) of \(p\left( \cdot \right)\) for \({\mathfrak {F}}_{2}\) such that
\(\begin{aligned} q(A \& E \& X_{A} \& \lnot X_{B})&= q(A \& E \& \lnot X_{A} \& \lnot X_{B}) =\\ q(A \& E \& X_{A} \& X_{B})&= q(A \& E \& \lnot X_{A} \& X_{B}) = \frac{p\left( A \& E\right) }{4}, \end{aligned}\)
\(\begin{aligned} q(A \& \lnot E \& X_{A} \& \lnot X_{B})&= q(A \& \lnot E \& \lnot X_{A} \& \lnot X_{B}) =\\ q(A \& \lnot E \& X_{A} \& X_{B})&= q(A \& \lnot E \& \lnot X_{A} \& X_{B}) = \frac{p\left( A \& \lnot E\right) }{4}, \end{aligned}\)
\(\begin{aligned} q(\lnot A \& \lnot E \& X_{A} \& \lnot X_{B})&= q(\lnot A \& \lnot E \& \lnot X_{A} \& \lnot X_{B}) =\\ q(\lnot A \& \lnot E \& X_{A} \& X_{B})&= q(\lnot A \& \lnot E \& \lnot X_{A} \& X_{B}) = \frac{p\left( \lnot A \& \lnot E\right) }{4}, \end{aligned}\)
and
\(\begin{aligned} q(\lnot A \& E \& X_{A} \& \lnot X_{B})&= \frac{r\cdot p\left( \lnot A \& E\right) }{4},\\ q(\lnot A \& E \& \lnot X_{A} \& \lnot X_{B})&= q(\lnot A \& E \& X_{A} \& X_{B})&=\\ q(\lnot A \& E \& \lnot X_{A} \& X_{B})&= \frac{p\left( \lnot A \& E\right) +e}{4}, \end{aligned}\)
where \(e=\frac{1-r}{3}p\left( \lnot A \& E\right) >0\). Since \(q(A|E \& X_{A} \& \lnot X_{B})=f(r\cdot p(\lnot A \& E))>f(p(\lnot A \& E)+e)=q(A|E \& \lnot X_{A} \& \lnot X_{B})=q(A|E \& X_{A} \& X_{B})=q(A|E \& \lnot X_{A} \& X_{B}),\) \(q\left( \cdot \right)\) satisfies (1)–(3). By definition, \(q\in U\left( {\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}\right)\). Therefore, \(x=q(A|E \& X_{A} \& \lnot X_{B})\in (U({\mathfrak {C}}_{1},{\mathfrak {F}}_{1},{\mathfrak {F}}_{2}))(A|E \& X_{A} \& \lnot X_{B})\). Done. \(\square\)
C. Conditional Converse Dutch Book Theorem for EIB
If no agent in a possibly imprecise credal state, consisting of (a) probability function(s), is vulnerable to synchronic Dutch books, then any such agent will be immune to diachronic Dutch books against EIB.
Proof
Footnote 38Suppose that no agent in a possibly imprecise credal state is vulnerable to synchronic Dutch books. Consider an agent a who updates \({\mathfrak {C}}_1\) into \(\{p(\cdot |E \& X_A \& \lnot X_B)|p\in U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\}\), in accordance with EIB (\({\mathfrak {F}}_1\subsetneq {\mathfrak {F}}_2\)). Assume, for reductio, that a’s credal transition from \(t_1\) to \(t_2\) is vulenrable to a diachronic Dutch book \({\mathfrak {D}}\). So, \({\mathfrak {D}}\) is a sure loss strategy consisting of the following bets:
\(B_1,\ldots ,B_j\), which a cunning bookie b offers to buy at \(t_1\) at the prices of \(\$r_1,\ldots ,\$r_j\),
\(B_{j+1},\ldots ,B_k\), which b offers to sell at \(t_1\) at the prices of \(\$r_{j+1},\ldots ,\$r_k\),
\(B_{k+1},\ldots ,B_l\), which b offers to buy at \(t_2\) at the prices of \(\$r_{k+1},\ldots ,\$r_l\) if \(E \& X_{A} \& \lnot X_{B}\) is true, and,
\(B_{l+1},\ldots ,B_n\), which b offers to sell at \(t_2\) at the prices of \(\$r_{l+1},\ldots ,\$r_n\) if \(E \& X_{A} \& \lnot X_{B}\) is true,Footnote 39
such that the gain and loss conditions of \(B_1,\ldots ,B_k\) are defined using the sentences from \({\mathfrak {F}}_1\) and those of \(B_{k+1},\ldots ,B_n\) are defined using the sentences from \({\mathfrak {F}}_2\). Consider another agent \(a'\) such that \(a'\)’s credal state at \(t_1\) is \(U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\) and \(a'\) receives the same evidence E and learns the same explanatory fact \(X_A \& \lnot X_B\) at \(t_2\) as a does.Footnote 40 Since no domain extension occurs for her, \(a'\) updates \(U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\) by conditioning on \(E \& X_A \& \lnot X_B\) at \(t_2\). Consequently, \(a'\)’s credal state at \(t_2\) is \(\{p(\cdot |E \& X_A \& \lnot X_B)|p\in U(c_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\}\). Observe that (i) \({\mathfrak {C}}_1=\{p\upharpoonright \mathfrak {F}_1 |p\in U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\}\), (ii) a and \(a'\) have the same credal state at \(t_2\), and for each of a and \(a'\), (iii) her acceptable selling prices at \(t_1\) of \(B_1,\ldots ,B_j\) and acceptable buying prices at \(t_1\) of \(B_{j+1},\ldots ,B_k\) are determined by her credences at \(t_1\) in the members of \({\mathfrak {F}}_1\), and (iv) her acceptable selling prices at \(t_2\) of \(B_{k+1},\ldots ,B_l\) and acceptable buying prices at \(t_2\) of \(B_{l+1},\ldots ,B_n\) are determined by her credences at \(t_2\) in the members of \({\mathfrak {F}}_2\). Since a is vulnerable to \({\mathfrak {D}}\), a will accept all of b’s offers in \({\mathfrak {D}}\). By (i) and (iii), \(a'\) sells \(B_1,\ldots ,B_j\) and buys \(B_{j+1},\ldots ,B_k\) at \(t_1\) and by (ii) and (iv), \(a'\) sells \(B_{k+1},\ldots ,B_l\) and buys \(B_{l+1},\ldots ,B_n\) at \(t_2\). Thus, \(a'\) is also vulnerable to \({\mathfrak {D}}\). As Skyrms (1987) points out, \(B_{k+1},\ldots ,B_n\) can be converted to conditional bets \(B_{k+1}^C,\ldots ,B_n^C\) that return the same payoffs as those from \(B_{k+1},\ldots ,B_n\) in all possible cases and will be cancelled if E or \(X_A \& \lnot X_B\) is false (p. 16). So we can construct a synchronic Dutch book \({\mathfrak {S}}\) consisting of
\(B_1,\ldots ,B_j\) and \(B_{k+1}^C,\ldots ,B_l^C\), which b offers to buy at \(t_1\) at the prices of \(r_1,\ldots ,r_j\) and \(r_{k+1},\ldots ,r_l\), and
\(B_{j+1},\ldots ,B_k\) and \(B_{l+1}^C,\ldots ,B_n^C\), which b offers to sell at \(t_1\), at the prices of \(r_{j+1},\ldots ,r_k\) and \(r_{l+1},\ldots ,r_n\).
Since \({\mathfrak {D}}\) and \({\mathfrak {S}}\) consist of bets that return the same payoffs in all possible cases and take effect in the same conditions, \({\mathfrak {S}}\) is a sure loss strategy, too. The remaining job is to show that \(a'\) will accept all of b’s offers in \({\mathfrak {S}}\). It was already argued that \(a'\) sells \(B_1,\ldots ,B_j\) and buys \(B_{j+1},\ldots ,B_k\) at \(t_1\). Since \(a'\) is a conditionalizer, (v) \(r_{k+1},\ldots ,r_l\) are \(a'\)’s acceptable selling prices for \(B^C_{k+1},\ldots ,B^C_l\) at \(t_1\) iff \(r_{k+1},\ldots ,r_l\) are \(a'\)’s acceptable selling prices for \(B_{k+1},\ldots ,B_l\) at \(t_2\). Similarly, (vi) \(r_{l+1},\ldots ,r_n\) are \(a'\)’s acceptable buying prices for \(B^C_{l+1},\ldots ,B^C_n\) at \(t_1\) iff \(r_{l+1},\ldots ,r_n\) are \(a'\)’s acceptable buying prices for \(B_{l+1},\ldots ,B_n\) at \(t_2\). Since \(a'\) is vulnerable to \({\mathfrak {D}}\), the right-hand sides of (v) and (vi) are true. Thus, \(a'\) will accept all of b’s offers regarding \(B^C_{k+1},\ldots ,B^C_n\). In sum, \(a'\) will accept all of b’s offers in \({\mathfrak {S}}\) at \(t_1\). So, \(a'\) is vulnerable to \({\mathfrak {S}}\) at \(t_1\). This contradicts the supposition because \(U({\mathfrak {C}}_1,{\mathfrak {F}}_1,{\mathfrak {F}}_2)\) is \(a'\)’s possibly imprecise credal state at \(t_1\). By reductio, there is no diachronic Dutch book against EIB.Footnote 41 Done. \(\square\)
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Kim, N. Imprecise Bayesianism and Inference to the Best Explanation. Found Sci 28, 755–781 (2023). https://doi.org/10.1007/s10699-022-09841-5
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DOI: https://doi.org/10.1007/s10699-022-09841-5