Correcting credences with chances

Abstract

Lewis’s Principal Principle is widely recognized as a rationality constraint that our credences should satisfy throughout our epistemic life. In practice, however, our credences often fail to satisfy this principle because of our various epistemic limitations. Facing such violations, we should correct our credences in accordance with this principle. In this paper, I will formulate a way of correcting our credences, which will be called the Adams Correcting Rules and then show that such a rule yields non-commutativity between conditionalizing and correcting. With the help of the notion of ‘accuracy’, then, I attempt to provide a vindication of the Adams Correcting Rule and show how we can respond to the non-commutativity in question.

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Fig. 1

Notes

  1. 1.

    A caveat needs to be stated from the start. This paper concerns ways of correcting our credences. Someone may think that this kind of correction leads us to be committed to doxastic voluntarism. Indeed, readers can find several voluntaristic expressions like ‘adopt the credence’ in this paper. However, this paper is not committed to such voluntarism. My argument in this paper may be regarded as criticizing or evaluating, rather than blaming, agents whose credences evolve in violation of the relevant epistemic norms. I should thank to Alan Hájek and Jaemin Jung for helping me make this point clear.

  2. 2.

    The unit set \(\{w\}\) is a proposition, but I will use interchangeably ‘\(\{w\}\)’ and w if there is no danger of confusion. When A is a proposition,‘\(\lnot A\)’ refers to the complement of A with respect to \(\mathcal {W}\)—namely, the negation of A. Similarly, when A and B are propositions, the conjunction ‘AB’ and the disjunction ‘\(A\vee B\)’, respectively, refer to the intersection and union of A and B.

  3. 3.

    Indeed, there may be many philosophical debates on what it is for an epistemic principle or rule to be a requirement of rationality. However, as the purpose of this paper is concerned, it is not necessary to discuss such a problem. Rather, I will assume that there are some reasons that the Principal Principle should be regarded as a requirement of rationality, and such reasons do not undermine my discussion that follows. A similar assumption goes with other principles appearing in this paper. In what follows, I will formulate a decision-theoretic principle that is related to credence correction. See Sects. 5.2 and 5.3. Regarding such a principle, I assume a similar thing to the rationality of the Principal Principle. I should thank an anonymous referee for making this point clear.

  4. 4.

    Someone may think that this formulation cannot be applied to the world that has no beginning—an infinite past. I agree. However, this feature has little to do with the discussion that follows. Indeed, if there is any proposition characterizing chance functions that one should respect, then my discussion will go through.

  5. 5.

    This argument for regularity is found in Lewis (1980). However, Hájek (2012) points out that there are various problems that beset regularity. I agree. So, regularity is not regarded, in this paper, as a requirement of rationality.

  6. 6.

    There are other attempts to overcome the bug. For example, see Roberts (2001) and Ismael (2008).

  7. 7.

    The expression ‘Self-esteem’ is intended to express that each chance function is certain that it is the only one true chance function. Pettigrew (2013) calls this feature of chance functions ‘Immodesty’. I do not use this name since I will use it in another context.

  8. 8.

    For example, see Hall (2004). In our context, the chance functions are analytic-experts who satisfy Self-esteem. Using Hall’s terminology, our chance functions are analytic-experts who know they meet the (analytic) expert conditions.

  9. 9.

    Suppose that an initial credence function C satisfies PP0 and is conditionalized on total evidence E. Then, it holds that, for any \(A\subseteq \mathcal {W}\) and \(U_{i}\in \mathcal {U}\), \(C_{E}(A|U_{i})=C(A|U_{i}E)=C(AE|U_{i})/C(E|U_{i})=ch_{i}(AE)/ch_{i}(E)=ch_{i}(A|E).\) Thus, \(C_{E}\) satisfies PP+.

  10. 10.

    In our context, all and only chance-free credence functions violate the Principal Principle, and all and only chance-fed credence functions satisfy the principle. In this paper, however, I am not committed to the claim that nothing but the Principal Principle properly governs the relationship between chances and credences. As shown in “Appendix”, the main arguments in this paper will go through even if the impact of chances is fed into the chance-free credence function via the New Principle. Thus, the terminology ‘chance-fed’ (and ‘chance-free’) may be applied to a credence function that satisfies the New Principle if the principle is accepted as a plausible chance–credence norm.

  11. 11.

    Note that it is assumed that \(ch_{i}\) is a probability function. Thus, the following two propositions are equivalent to each other: (i) For any \(A\subseteq \mathcal {W}\) and \(U_{i}\in \mathcal {U}\), \(C_{CH}(A|U_{i})=ch_{i}(A)\), if defined ; (ii) For any \(w\in \mathcal {W}\) and \(U_{i}\in \mathcal {U}\), \(C_{CH}(w|U_{i})=ch_{i}(w)\), if defined.

  12. 12.

    Adams Conditionalization is named by Bradley (2005). A similar discussion is also found in Wagner (2003). There are some attempts to respond to several difficulties of Bayesian epistemology by means of this kind of Conditionalization. For example, Douven and Romeijn (2011) suggest a solution for the so-called Judy Benjamin Problem using Adams Conditionalization.

  13. 13.

    To understand the way of generalizing Adams Conditionalization, let me consider the following equation:

    $$\begin{aligned} C(A)=\sum _{E_{i}\in \mathcal {E}}C(F)C(E_{i}|F)C(A|E_{i}F)+C(A\lnot F), \end{aligned}$$

    which follows from the probability calculus. It is noteworthy that, according to Adams Conditionalization, the new credence function \(C_{\mathcal {E}|F}\) is obtained by replacing \(C(E_{i}|F)\) in the above equation with \(C_{\mathcal {E}|F}(E_{i}|F)\). Now, consider the following equation, which follows from the probability calculus:

    $$\begin{aligned} C(A)=\sum _{E_{i}\in \mathcal {E}}C(F_{1})C(E_{i}|F_{1})C(A|E_{i}F_{1})+\sum _{E_{i}\in \mathcal {E}}C(F_{2})C(E_{i}|F_{2})C(A|E_{i}F_{2})+\cdots . \end{aligned}$$

    Then, we can obtain a generalized version of Adams Conditionalization by replacing \(C(E_{i}|F_{j})\) with \(C_{\mathcal {E}|\mathcal {F}}(E_{i}|F_{j})\) for any \(E_{i}\) and \(F_{j}\).

  14. 14.

    The derivation of ACR0 from PP0 and GAC is given in “Appendix”. Similarly, ACR+, which is formulated below, is derived from PP+ and GAC. This derivation is also given in “Appendix”.

  15. 15.

    See, for example, Jeffrey (2004), Lange (2000), and Wagner (2003). This point can be made in a different way. One’s credences should be sensitive to one’s total evidence. The order in which one had the relevant experiences is part of one’s total evidence. If the order is changed, then one changes the total evidence and so does his credences. Therefore, such non-commutativity is not problematic.

  16. 16.

    As noted in footnote 15, the order in which we receive the relevant pieces of information may be taken as a part of our total evidence. By the same token, it may also be said that \(C_{CH,E}\) has different total evidence from \(C_{E,CH}\). In what follows, I will assume that, in some relevant situations, we could decide when our credences are corrected. Under this assumption, the decision between \(C_{CH,E}\) and \(C_{E,CH}\) may be regarded as the decision between two different pieces of total evidence.

  17. 17.

    For the candidates, see Joyce (2009) and Pettigrew (2016a) for example.

  18. 18.

    Consider the example in the previous section. Suppose that the chance-free initial credence function C is corrected to a credence function \(C^{*}\) whose credence assignment is as follows: \(C^{*}(w_{1})=C^{*}(w_{2})=C^{*}(w_{3})=1/12\) and \(C^{*}(w_{4})=2C^{*}(w_{5})=2C^{*}(w_{6})=3/8\). Note that \(C^{*}(w_{i}|U_{1})=C^{*}(w_{i}|U_{1})=ch_{1}(w_{i}|E)\) and \(C^{*}(w_{i}|U_{2})=C^{*}(w_{i}|U_{2})=ch_{2}(w_{i}|E).\) That is, this function satisfies the Principal Principle. However, \(C^{*}\) is not equal to \(C_{CH}\), which is corrected by means of the Adams Correcting Rule, namely ACR0.

  19. 19.

    This line of response to a similar problem is also found in Pettigrew (2016a, pp. 199–200). I owe a debt of gratitude to an anonymous referee for helping me make this point clear.

  20. 20.

    The proof of this is basically the same as the proof of Proposition 2, which is given in “Appendix”.

  21. 21.

    Meacham (2016, pp. 451–452) regards a very similar example as a motivation for adopting what he calls Ur Prior Conditionalization. He formulates Ur Prior Conditionalization as follows: If a subject has ur-priors up and current evidence E, her credence cr should be \(cr(\cdot )=up(\cdot |E)\), if defined. Here, ‘ur-priors’ corresponds to our ‘initial credences’.

  22. 22.

    Some readers may think that there is another way of incorporating the two pieces of information into the credal system. Note that the Conditionalizing-first and Correcting-first Rules might be regarded as sequential updating rules. However, the two pieces of information might be simultaneously incorporated into the credal system. I do not rule out this possibility in this paper. Be that as it may, we do not have to consider separately such a way. This is because the following discussion shows that the Conditionalizing-first Rule has an epistemic merit that any other relevant rules do not have.

  23. 23.

    The following argument is basically similar to Accuracy-centred Probabilists’ argument for a synchronic version of Conditionalization. For example, see Easwaran (2013), Pettigrew (2016a), Greaves and Wallace (2006).

  24. 24.

    Here, we should note that this rule (or function) takes as arguments only the evidence that will be received later. Thus, the rule in question has nothing to do with the credence function that the relevant agents have before receiving evidence. In other words, \(\mathbf {R}_{\mathcal {E}}\) says nothing about what credence function the agents should have before receiving evidence from \(\mathcal {E}\).

  25. 25.

    The proof of Proposition 6 is very similar to the proof of Proposition 5. In particular, both proofs rely on Immodesty and the fact that the credences in the ur-chance propositions remain the same when a credence function is corrected by means of the Adams Correcting Rule. The detailed proof is given in “Appendix”.

  26. 26.

    In order for MIPP and Proposition 6 to imply Plan Conditionalizing-first, MIPP needs to be slightly modified. Note that MIPP appearing in Sect. 5.2 is related to the expected inaccuracy of a particular credence-fed function. However, what is needed to imply Plan Conditionalizing-first is a principle about the expected inaccuracy of an updating rule. Of course, such a modified version can be readily formulated in a similar way of MIPP.

  27. 27.

    Then, is there any rational way of deciding what credence function we have between \(t'\) and \(t''\)? Let me consider the following argument. Suppose that, at time \(t'\), an agent, whose credence function C violates the Principal Principle, gets to know (i) and (ii) appearing in Plan Conditionalizing-first. Suppose also that she knows at \(t'\) that she always updates her credences by means of Conditionalization on evidence. At time \(t'\), she deliberates whether she corrects her credences immediately or puts off the correction until receiving evidence from \(\mathcal {E}\). Note that she knows at \(t'\) that she is a conditionalizer. So, she also knows at \(t'\) that if she corrects her credences immediately, then she will have one of \(C_{CH,E_{i}}\)s at time \(t''\). However, Proposition 6 says that, by the light of her current credence function C, this result does not have the minimal expected inaccuracy. Thus, it seems rational that the agent does not correct her credences immediately. In this paper, however, I am not committed to this conclusion. This is partially because of the assumption that the agent in question knows that she always updates her credences by means of Conditionalization. There may be some arguments for having such knowledge. However, I leave the matter open for further investigation. Be that as it may, I would like to emphasize here that what credence function we should have between \(t'\) and \(t''\) does not undermine the main point I wish to argue for here. This is because Proposition 6, which my main results heavily depend on, concerns only the credence function at \(t''\), and the proposition can still follow no matter what credence function we should have between \(t'\) and \(t''\). Many thanks to anonymous referees for encouraging me to make these points clear.

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Acknowledgements

Some earlier versions of this paper were presented at the 7th Asia-Pacific Conference on Philosophy of Science and the Quarterly Meeting of Korean Association for Logic. I am grateful to the audience of the conferences for their comments and feedback. Special thanks are due to Alan Hájek, Jaemin Jung, and Namjoong Kim for reading and commenting on the earlier draft. Moreover, I owe special debt to Il Kwon Lee and Sung Min Kim for correcting some grammatical mistakes. Of course, I am very grateful to several anonymous reviewers for their invaluable suggestions and comments.

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Correspondence to Ilho Park.

Appendix

Appendix

The following proofs and derivations have various conditional credences and chances. For presentational simplicity, I will assume in what follows that such conditional credences and chances are all well defined. I think this assumption yields no confusion.

A derivation of ACR0 from PP0 and GAC

Suppose that \(C_{CH}\) satisfies PP0, and that \(C_{CH}\) is updated from C by means of GAC. Then we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH}(A)&=\sum _{U_{i}\in \mathcal {U},w_{j}\in \mathcal {W}}C(U_{i})C(A|w_{j}U_{i})C_{CH}(w_{j}|U_{i})\\&=\sum _{U_{i}\in \mathcal {U},w_{j}\in \mathcal {W}}C(U_{i})C(A|w_{j}U_{i})ch_{i}(w_{j})\\&=\sum _{U_{i}\in \mathcal {U}}C(U_{i})\sum _{w_{j}\in \mathcal {W}}C(A|w_{j}U_{i})ch_{i}(w_{j})\\&=\sum _{U_{i}\in \mathcal {U}}C(U_{i})\sum _{w_{j}\in A}C(A|w_{j}U_{i})ch_{i}(w_{j})\\&\quad +\sum _{U_{i}\in \mathcal {U}}C(U_{i})\sum _{w_{j}\in \lnot A}C(A|w_{j}U_{i})ch_{i}(w_{j})\\&=\sum _{U_{i}\in \mathcal {U}}C(U_{i})\sum _{w_{j}\in A}ch_{i}(w_{j})=\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(A), \end{aligned}$$

as required.

A derivation of ACR+ from PP+ and GAC

Suppose that \(C_{E,CH}\) satisfies PP+, and that \(C_{E,CH}\) is updated from \(C_{E}\) by means of GAC. Then we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{E,CH}(A)&=\sum _{U_{i}\in \mathcal {U},w_{j}\in \mathcal {W}}C_{E}(U_{i})C_{E}(A|w_{j}U_{i})C_{E,CH}(w_{j}|U_{i})\\&=\sum _{U_{i}\in \mathcal {U},w_{j}\in \mathcal {W}}C_{E}(U_{i})C_{E}(A|w_{j}U_{i})ch_{i}(w_{j}|E)\\&=\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})\sum _{w_{j}\in \mathcal {W}}C_{E}(A|w_{j}U_{i})ch_{i}(w_{j}|E)\\&=\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})\sum _{w_{j}\in A}C_{E}(A|w_{j}U_{i})ch_{i}(w_{j}|E)\\&\quad +\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})\sum _{w_{j}\in \lnot A}C_{E}(A|w_{j}U_{i})ch_{i}(w_{j}|E)\\&=\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})\sum _{w_{j}\in A}ch_{i}(w_{j}|E)=\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})ch_{i}(A|E), \end{aligned}$$

as required.

A proof of Proposition 1

Suppose that our credences are updated by means of Conditionalization, ACR0, and ACR+. It is straightforward that \(C_{E,CH}(w|U_{i})=ch_{i}(w|E)\) for any \(w\in \mathcal {W}\), and so \(C_{E,CH}(A|U_{i})=ch_{i}(A|E)\) for any \(A\subseteq \mathcal {W}\). Similarly, we have that \(C_{CH}(w|U_{i})=ch_{i}(w),\) and so that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH}(A|U_{i})=\sum _{w\in A}C_{CH}(w|U_{i})=\sum _{w\in A}ch_{i}(w)=ch_{i}(A). \end{aligned}$$

Then, Conditionalization and the above equation imply that, for any \(A\subseteq \mathcal {W}\) and \(U_{i}\in \mathcal {U}\),

$$\begin{aligned} C_{CH,E}(A|U_{i})=C_{CH}(A|U_{i}E)=\frac{C_{CH}(AE|U_{i})}{C_{CH}(A|U_{i})}=ch_{i}(A|E). \end{aligned}$$

Then, we have Proposition 1. Done.

A proof of Proposition 2

Note that \(U_{i}\subseteq \mathcal {W}\) for any \(U_{i}\in \mathcal {U}\). Thus, it is straightforward that (a) implies (b). Now, let me prove that (b) implies (a). For this purpose, let’s assume (b)—that is, \(C_{E,CH}(U_{i})=C_{CH,E}(U_{i})\) for any \(U_{i}\in \mathcal {U}\). From this assumption and Proposition 1, then, it follows that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{E,CH}(A)&=\sum _{U_{i}\in \mathcal {U}}C_{E,CH}(U_{i})C_{E,CH}(A|U_{i})=\sum _{U_{i}\in \mathcal {U}}C_{E,CH}(U_{i})ch_{i}(A|E)\\&=\sum _{U_{i}\in \mathcal {U}}C_{CH,E}(U_{i})ch_{i}(A|E)=\sum _{U_{i}\in \mathcal {U}}C_{CH,E}(U_{i})C_{CH,E}(A|U_{i})\\&=C_{CH,E}(A), \end{aligned}$$

as required.

A proof of Proposition 3

According to ACR+ and Conditionalization, we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{E,CH}(A)=\sum _{U_{i}\in \mathcal {U}}C(U_{i}|E)ch_{i}(A|E). \end{aligned}$$
(1)

Similarly, it follows from Conditionalization and ACR0 that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH,E}(A)&=C_{CH}(A|E)=\frac{C_{CH}(AE)}{C_{CH}(E)}\nonumber \\&=\frac{\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(E)}. \end{aligned}$$
(2)

First, let me prove that if E is a disjunction of some \(U_{i}\)s, then it holds that \(C_{E,CH}(A)=C_{CH,E}(A)\) for any \(A\subseteq \mathcal {W}\). Suppose that E is a disjunction of some \(U_{i}\)s. Let \(\mathcal {U}_{E}\) be the set of the disjuncts in question. Then, it holds that \(U_{i}\) implies E when \(U_{i}\in \mathcal {U}_{E}\), and \(U_{i}\) implies \(\lnot E\) otherwise. Moreover, \(ch_{i}(E)=1\) when \(U_{i}\in \mathcal {U}_{E}\), and \(ch_{i}(E)=0\) otherwise (Note that Self-esteem was assumed). From this, it follows that \(ch_{i}(AE)=ch_{i}(A|E)=ch_{i}(A)\) when \(U_{i}\in \mathcal {U}_{E}\), and \(ch_{i}(AE)=0\) otherwise. Then, we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH,E}(A)&=\frac{\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(E)}\\&=\frac{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})ch_{i}(AE)+\sum _{U_{i}\notin \mathcal {U}_{E}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})ch_{i}(E)+\sum _{U_{i}\notin \mathcal {U}_{E}}C(U_{i})ch_{i}(E)}\\&=\frac{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})}=\frac{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})ch_{i}(A|E)}{\sum _{U_{i}\in \mathcal {U}_{E}}C(U_{i})}\\&=\frac{\sum _{U_{i}\in \mathcal {U}}C(U_{i}E)ch_{i}(A|E)}{C(E)}=\sum _{U_{i}\in \mathcal {U}}C(U_{i}|E)ch_{i}(A|E)=C_{E,CH}(A), \end{aligned}$$

as required.

Now, let us prove that if there is a \(U_{i}\) such that \(E\subseteq U_{i}\), then it holds that \(C_{E,CH}(A)=C_{CH,E}(A)\) for any \(A\subseteq \mathcal {W}\). Suppose that there is an ur-chance proposition that is a subset of E. Let \(U_{k}\) be such a proposition. Then, it holds that \(ch_{i}(AE)=ch_{i}(E)=0\) when \(U_{i}\ne U_{k}\), and that \(C(U_{k}|E)=1\). Then, we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH,E}(A)&=\frac{\sum _{U_{i}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}}C(U_{i})ch_{i}(E)}\\&=\frac{\sum _{U_{i}=U_{k}}C(U_{i})ch_{i}(AE)+\sum _{U_{i}\ne U_{k}}C(U_{i})ch_{i}(AE)}{\sum _{U_{i}=U_{k}}C(U_{i})ch_{i}(E)+\sum _{U_{i}\ne U_{k}}C(U_{i})ch_{i}(E)}\\&=\frac{C(U_{k})ch_{k}(AE)}{C(U_{k})ch_{k}(E)}=ch_{k}(A|E);\\ C_{E,CH}(A)&=\sum _{U_{i}}C(U_{i}|E)ch_{i}(A|E)\\&=\sum _{U_{i}=U_{k}}C(U_{i}|E)ch_{i}(A|E)+\sum _{U_{i}\ne U_{k}}C(U_{i}|E)ch_{i}(AE)\\&=C(U_{k}|E)ch_{k}(A|E)=ch_{k}(A|E). \end{aligned}$$

Thus, it holds that \(C_{E,CH}(A)=C_{CH,E}(A)\) for any \(A\subseteq \mathcal {W}\). Done.

Several calculations related to Proposition 4 and Fig. 1

As assumed in the example related to Fig. 1, \(\mathcal {U}=\{U_{1},U_{2}\}\). Then, it follows from (1) and (2) in the proof of Proposition 3 that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH}(A)&=C(U_{1})ch_{1}(A)+C(U_{2})ch_{2}(A), \end{aligned}$$
(4a)
$$\begin{aligned} C_{CH,E}(A)&=\frac{C(U_{1})ch_{1}(AE)+C(U_{2})ch_{2}(AE)}{C(U_{1})ch_{1}(E)+C(U_{2})ch_{2}(E)}, \end{aligned}$$
(4b)
$$\begin{aligned} C_{E}(A)&=\frac{C(AE)}{C(E)},\,\text {and} \end{aligned}$$
(4c)
$$\begin{aligned} C_{E,CH}(A)&=C(U_{1}|E)ch_{1}(A|E)+C(U_{2}|E)ch_{2}(A|E). \end{aligned}$$
(4d)

Note that \(E\equiv w_{1}\vee w_{2}\vee w_{4}\vee w_{5}\), \(U_{1}\equiv w_{1}\vee w_{2}\vee w_{3}\), and \(U_{2}\equiv w_{4}\vee w_{5}\vee w_{6}\). With the help of (4a)–(4d), the chance assignments of \(ch'\) and \(ch^{*}\), and the initial credence assignment of C, we have that, for any \(A\subseteq \mathcal {W}\),

$$\begin{aligned} C_{CH}(A)&=\frac{1}{2}ch_{1}(A)+\frac{1}{2}ch_{2}(A),\\ C_{CH,E}(A)&=\frac{12}{17}\left( ch_{1}(AE)+ch_{2}(AE)\right) ,\\ C_{E}(A)&=\frac{10}{7}C(AE),\,\text {and}\\ C_{E,CH}(A)&=\frac{9}{14}ch_{1}(AE)+\frac{16}{21}ch_{2}(AE). \end{aligned}$$

Now, we can derive the probability assignments of \(C_{CH}\), \(C_{CH,E}\), \(C_{E}\), and \(C_{E,CH}\). For example,

$$\begin{aligned} C_{CH}(w_{1})&=\frac{1}{2}ch_{1}(w_{1})+\frac{1}{2}ch_{2}(w_{1})=\frac{1}{2}ch_{1}(w_{1})=\frac{1}{6};\\ C_{CH,E}(w_{1})&=\frac{12}{17}\left( ch_{1}(w_{1}E)+ch_{2}(w_{1}E)\right) =\frac{12}{17}ch_{1}(w_{1})=\frac{4}{17};\\ C_{E}(w_{1})&=\frac{10}{7}C(w_{1}E)=\frac{10}{7}C(w_{1})=\frac{1}{7};\,\text {and}\\ C_{E,CH}(w_{1})&=\frac{9}{14}ch_{1}(w_{1}E)+\frac{16}{21}ch_{2}(w_{1}E)=\frac{9}{14}ch_{1}(w_{1})=\frac{3}{14}. \end{aligned}$$

These results conform with the probability assignments in Fig. 1.

Proofs of Propositions 5 and 6

Let me start with noting that it follows from GAC and the probability calculus that: for any \(F_{k}\in \mathcal {F}\),

$$\begin{aligned} C_{\mathcal {E}|\mathcal {F}}(F_{k})&=\sum _{E_{i}\in \mathcal {E},F_{j}\in \mathcal {F}}C(F_{j})C{}_{\mathcal {E}|\mathcal {F}}(E_{i}|F_{j})C(F_{k}|E_{i}F_{j})\\&=\sum _{E_{i}\in \mathcal {E},F_{j}=F_{k}}C(F_{j})C{}_{\mathcal {E}|\mathcal {F}}(E_{i}|F_{j})C(F_{k}|E_{i}F_{j})\\&\quad +\sum _{E_{i}\in \mathcal {E},F_{j}\ne F_{k}}C(F_{j})C{}_{\mathcal {E}|\mathcal {F}}(E_{i}|F_{j})C(F_{k}|E_{i}F_{j})\\&=\sum _{E_{i}\in \mathcal {E}}C(F_{k})C{}_{\mathcal {E}|\mathcal {F}}(E_{i}|F_{k})=C(F_{k}). \end{aligned}$$

That is, when some experience directly changes the relevant conditional credences, and so C is updated to \(C_{\mathcal {E}|\mathcal {F}}\) by means of GAC, the credences in the conditioning propositions, namely \(F_{i}\)s, remain the same. As explained above, the correction of \(C_{E}\) to \(C_{E,CH}\) by means of ACR+ can be regarded as a belief updating by means of GAC, in which the conditioning propositions are the ur-chance propositions \(U_{i}\)s. So, the credences in \(U_{i}\)s remain the same through the correction in question. More formally, it holds that: for any \(E\subseteq \mathcal {W}\) and \(U_{i}\in \mathcal {U}\),

$$\begin{aligned} C_{E,CH}(U_{i})=C_{E}(U_{i})=C(U_{i}|E). \end{aligned}$$
(A)

Now, with this mathematical feature of the Adams Correcting Rule in hand, we can prove Propositions 5 and 6.

A proof of Proposition 5

Let \(\mathbb {R}\) be a set of updating rules for a chance-free credence function \(C_{E}\). All members of \(\mathbb {R}\) correct \(C_{E}\) to a chance-fed function. Let \(\mathbf {R}\) be the Adams Correcting Rule, i.e., ACR+. Then, it holds that \(C_{E}^{\mathbf {R}}=C_{E,CH}\). Moreover, let \(\mathbf {R}^{*}\) be a correcting rule in \(\mathbb {R}\). Then, Immodesty implies that,

$$\begin{aligned} EI_{C_{E}^{\mathbf {R}}}[C_{E}^{\mathbf {R}},\mathcal {U}]\le EI_{C_{E}^{\mathbf {R}}}[C_{E}^{\mathbf {R}^{*}},\mathcal {U}], \end{aligned}$$
(5a)

where equality holds if and only if \(C_{E}^{\mathbf {R}}(U_{i})=C_{E}^{\mathbf {R}^{*}}(U_{i})\) for any \(U_{i}\in \mathcal {U}\). Note that \(C_{E}^{\mathbf {R}}=C_{E,CH}\). Thus, it follows from (A) that

$$\begin{aligned} EI_{C_{E}^{\mathbf {R}}}[C_{E}^{\mathbf {R}},\mathcal {U}]&=\sum _{U_{j}\in \mathcal {U}}C_{E,CH}(U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E}^{\mathbf {R}},V_{j})\nonumber \\&=\sum _{U_{j}\in \mathcal {U}}C_{E}(U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E}^{\mathbf {R}},V_{j})=EI_{C_{E}}[C_{E}^{\mathbf {R}},\mathcal {U}]. \end{aligned}$$
(5b)

Similarly, we also have that:

$$\begin{aligned} EI_{C_{E}^{\mathbf {R}}}[C_{E}^{\mathbf {R}^{*}},\mathcal {U}]&=EI_{C_{E}}[C_{E}^{\mathbf {R}^{*}},\mathcal {U}]. \end{aligned}$$
(5c)

Now, (5a), (5b), and (5c) jointly imply that

$$\begin{aligned} EI_{C_{E}}[C_{E}^{\mathbf {R}},\mathcal {U}]\le EI_{C_{E}}[C_{E}^{\mathbf {R}^{*}},\mathcal {U}], \end{aligned}$$

where equality holds if and only if \(C_{E}^{\mathbf {R}}(U_{i})=C_{E}^{\mathbf {R}^{*}}(U_{i})\) for any \(U_{i}\in \mathcal {U}\). In a similar way to the proof of Proposition 2, on the other hand, we can prove that the following two propositions are equivalent to each other: (i) \(C_{E}^{\mathbf {R}}(U_{i})=C_{E}^{\mathbf {R}^{*}}(U_{i})\) for any \(U_{i}\in \mathcal {U}\); (ii) \(C_{E}^{\mathbf {R}}=C_{E}^{\mathbf {R}^{*}}\). Finally, we have that

$$\begin{aligned} EI_{C_{E}}[C_{E}^{\mathbf {R}},\mathcal {U}]\le EI_{C_{E}}[C_{E}^{\mathbf {R}^{*}},\mathcal {U}], \end{aligned}$$

where equality holds if and only if \(C_{E}^{\mathbf {R}}=C_{E}^{\mathbf {R}^{*}}\). Done.

A proof of Proposition 6

We can prove Proposition 6 in the very similar way to the proof of Proposition 5. Let \(\mathbb {R}_{\mathcal {E}}\) be a set of updating rules on \(\mathcal {E}\) for a chance-free credence function C. All members of \(\mathbb {R}_{\mathcal {E}}\) correct C to a chance-fed function such that \(C_{E_{i}}^{\mathbf {R}}\) satisfies the Principal Principle and assigns \(E_{i}\) to 1, for any \(\mathbf {R}_{\mathcal {E}}\in \mathbb {R}_{\mathcal {E}}\) and \(E_{i}\in \mathcal {E}\). Let \(\mathbf {R}_{\mathcal {E}}\) be the Conditionalizing-first Rule on \(\mathcal {E}\) for C such that \(C_{E_{i}}^{\mathbf {R}}=C_{E_{i},CH}\) for any \(E_{i}\in \mathcal {E}\). Lastly, let \(\mathbf {R}_{\mathcal {E}}^{*}\) be an updating rule in \(\mathbb {R}_{\mathcal {E}}\). Then, Immodesty implies that: for any \(E_{i}\in \mathcal {E}\),

$$\begin{aligned} EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}},\mathcal {U}]\le EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}^{*}},\mathcal {U}], \end{aligned}$$
(6a)

where equality holds if and only if \(C_{E_{_{i}}}^{\mathbf {R}}(U_{j})=C_{E_{i}}^{\mathbf {R}^{*}}(U_{j})\) for any \(U_{j}\in \mathcal {U}\), which is equivalent to \(C_{E_{i}}^{\mathbf {R}}=C_{E_{i}}^{\mathbf {R}^{*}}\). Note that \(C_{E_{i}}^{\mathbf {R}}=C_{E_{i},CH}\) for any \(E_{i}\in \mathcal {E}\). Similar to (5b) and (5c), it follows from (A) that, for any \(E_{i}\in \mathcal {E}\),

$$\begin{aligned} EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}},\mathcal {U}]&=\sum _{U_{j}\in \mathcal {U}}C_{E_{i},CH}(U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E_{i}}^{\mathbf {R}},V_{j})\nonumber \\&=\sum _{U_{j}\in \mathcal {U}}C_{E_{i}}(U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E}^{\mathbf {R}},V_{j})\nonumber \\&=\sum _{U_{j}\in \mathcal {U}}C(U_{j}|E_{i})\mathfrak {I}_{\mathcal {U}}(C_{E}^{\mathbf {R}},V_{j}). \end{aligned}$$
(6b)

By the same token, it also holds that: for any \(E_{i}\in \mathcal {E}\),

$$\begin{aligned} EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}^{*}},\mathcal {U}]&=\sum _{U_{j}\in \mathcal {U}}C(U_{j}|E_{i})\mathfrak {I}_{\mathcal {U}}(C_{E}^{\mathbf {R}^{*}},V_{j}). \end{aligned}$$
(6c)

Then, (6b) and the relevant definitions imply that:

$$\begin{aligned} EI_{C}[\mathbf {R}_{\mathcal {E}},\mathcal {U}]&=\sum _{E_{i}\in \mathcal {E},U_{j}\in \mathcal {U}}C(E_{i}U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E_{i}}^{\mathbf {R}},V_{j})\nonumber \\&=\sum _{E_{i}\in \mathcal {E}}\left( \sum _{U_{j}\in \mathcal {U}}C(E_{i}U_{j})\mathfrak {I}_{\mathcal {U}}(C_{E_{i}}^{\mathbf {R}},V_{j})\right) \nonumber \\&=\sum _{E_{i}\in \mathcal {E}}C(E_{i})\left( \sum _{U_{j}\in \mathcal {U}}C(U_{j}|E_{i})\mathfrak {I}_{\mathcal {U}}(C_{E_{i}}^{\mathbf {R}},V_{j})\right) \nonumber \\&=\sum _{E_{i}\in \mathcal {E}}C(E_{i})EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}},\mathcal {U}]. \end{aligned}$$
(6d)

Similarly, it follows from (6c) and the relevant definition that:

$$\begin{aligned} EI_{C}[\mathbf {R}_{\mathcal {E}}^{*},\mathcal {U}]=\sum _{E_{i}\in \mathcal {E}}C(E_{i})EI_{C_{E_{i}}^{\mathbf {R}}}[C_{E_{i}}^{\mathbf {R}^{*}},\mathcal {U}]. \end{aligned}$$
(6e)

Now, (6a), (6d), and (6e) imply that:

$$\begin{aligned} EI_{C}[\mathbf {R}_{\mathcal {E}},\mathcal {U}]\le EI_{C}[\mathbf {R}_{\mathcal {E}}^{*},\mathcal {U}], \end{aligned}$$

where equality holds if and only if \(\mathbf {R}_{\mathcal {E}}=\mathbf {R}_{\mathcal {E}}^{*}\). Done.

Correcting chance-free credences in accordance with the new principle

As mentioned in Sect. 2, the original version of the Principal Principle suffers from the Big Bad Bug—however, the New Principle does not. That said, my discussions go through even if we accept the New Principle rather than the original version. In particular, we can derive the new versions of ACR0 and ACR+, which require us to correct in accordance with the New Principle. Let C and \(C_{E}\) be chance-free credence functions, and \(C_{CH}\) and \(C_{E,CH}\) be chance-fed credence functions that are updated from C and \(C_{E}\), respectively, in accordance with the New Principle. Then, we can formulate such versions, as follows:

  • New ACR0: \(C_{CH}(A)=\sum _{U_{i}\in \mathcal {U}}C(U_{i})ch_{i}(A|U_{i})\), for any \(A\subseteq \mathcal {W}\).

  • New ACR+: \(C_{E,CH}(A)=\sum _{U_{i}\in \mathcal {U}}C_{E}(U_{i})ch_{i}(A|EU_{i})\), for any \(A\subseteq \mathcal {W}\).

Note that these versions follow, in a similar way to ACR0 and ACR+, from GAC and the New Principle. In particular, we can derive these versions by replacing the unconditional chance function \(ch_{i}(\cdot )\) in ACR0 and ACR+ with the conditional chance function \(ch_{i}(\cdot |U_{i})\). In a similar way, moreover, the Propositions corresponding to Propositions 16 also follow from New ACR0 and New ACR+. Therefore, we can say that my discussions do not depend on which version of the Principal Principle we accept.

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Park, I. Correcting credences with chances. Synthese 198, 509–536 (2021). https://doi.org/10.1007/s11229-018-02037-0

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Keywords

  • Chances
  • Credences
  • Correction
  • Conditionalization
  • The Principal Principle
  • Accuracy-centred probabilism
  • The Adams Correcting Rule