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Deference and Uniqueness

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Abstract

Deference principles are principles that describe when, and to what extent, it’s rational to defer to others. Recently, some authors have used such principles to argue for Evidential Uniqueness, the claim that for every batch of evidence, there’s a unique doxastic state that it’s permissible for subjects with that total evidence to have. This paper has two aims. The first aim is to assess these deference-based arguments for Evidential Uniqueness. I’ll show that these arguments only work given a particular kind of deference principle, and I’ll argue that there are reasons to reject these kinds of principles. The second aim of this paper is to spell out what a plausible generalized deference principle looks like. I’ll start by offering a principled rationale for taking deference to constrain rational belief. Then I’ll flesh out the kind of deference principle suggested by this rationale. Finally, I’ll show that this principle is both more plausible and more general than the principles used in the deference-based arguments for Evidential Uniqueness.

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Notes

  1. For example, see White (2005) and Matheson (2011).

  2. For some replies, see Ballantyne and Coffman (2011), Kelly (2013), Schoenfield (2013), Meacham (2013), Peels and Booth (2014) and Titelbaum and Kopec (2016).

  3. The arguments I’ll consider, offered by Levinstein (2015) and Greco and Hedden (2016), appeal to a kind of deference principle in order to provide an argument for Evidential Uniqueness. Dogramaci and Horowitz (2016) have also presented an argument for Evidential Uniqueness that appeals (in part) to considerations involving deference. But Dogramaci and Horowitz’s argument is very different from the arguments offered by Levinstein and Greco and Hedden. While Levinstein and Greco and Hedden and offer theoretical arguments driven by particular deference principles, Dogramaci and Horowitz offer an empirical argument driven by a particular application of inference to the best explanation. Thus while Dogramaci and Horowitz’s argument deserves careful discussion, I won’t engage with it here, since the issues it raises are largely tangential to the kind of argument I’ll be focusing on.

  4. A function is typically defined as only having one output, whereas a permission function can intuitively have multiple outputs (if multiple things are permissible). So, strictly speaking, one might characterize the permission function as a multivalued function. Alternatively, one can take the permission function to be an ordinary function whose output is a set containing all of the permissible options.

  5. Thus given Evidential Inputs, the epistemic permission function can always be characterized as a function whose only input is a subject’s total evidence at a time.

  6. There’s a large literature on alternative ways of understanding deontic conditionals; for some recent discussions and further references, see McNamara (2010), vonFintel (2011) and Kratzer (2012).

  7. Greco and Hedden (2016) formulate Deference as follows (pp. 372–373): “\(\ldots\)agents ought to satisfy the following conditional: Deference: If agent \(S_1\) judges that \(S_2\)’s belief that P is rational and that \(S_1\) does not have relevant evidence that \(S_2\) lacks, then \(S_1\) defers to \(S_2\)’s belief that P.”

    A couple comments. First, placing the ought/obligatory operator outside of the statement of Deference, as Greco and Hedden do, obscures some features of Deference’s normative structure that will turn out to be important. So I’ve moved this operator into the statement of the principle. Second, by “rational” Greco and Hedden mean “rationally permissible” (indeed, if they didn’t the principle wouldn’t apply to the permissive cases Greco and Hedden focus on). Since this will also turn out to be important, I’ve made this explicit in my statement of the principle. Third, in order to ward off some possible misunderstandings of the principle and nearby variants of it in the discussion to come, I’ve made the location of the quantifiers over subjects and propositions explicit in my statement of the principle. Finally, to cut down on notation and indices, I’ve replaced references to \(S_1\) with “you” and \(S_2\) with “S”. This is merely to make the principle (and my discussion of it) a bit more readable; the claims Deference (and its relatives) make about about “you” should be understood as claims that apply to all subjects.

  8. Greco and Hedden (2016, p. 373).

  9. Greco and Hedden (2016) present the argument as follows:

    “combining Permissivism with Deference\(\ldots\) threatens to yield inconsistent deferential commitments. \(\ldots\) In a nutshell, if two agents have the same total evidence but different beliefs about whether P, then you cannot defer to each’s belief about whether P on pain of inconsistency. Let’s take this a bit more slowly. Suppose you know that one agent has credence n in P while another has credence m in P, and that they have the same total evidence. Then, you cannot defer to each one’s credence in P; you cannot simultaneously adopt credence n in P and credence m in P, where \(n \ne m\). Hence, if judging that an agent’s credence is rational involves a commitment to deferring to that agent’s credence, then you cannot judge each agent’s credence in P to be rational.” (p. 373)

  10. Of course, what’s problematic is not merely that we’ve formulated Deference using two different deontic operators. Since the permission and obligation operators are inter-definable, it’s trivial to reformulate any principle with two deontic operators so that it employs only permission or obligation operators. What’s problematic is that the amount of latitude in what the principle demands of the subject (in cases of deference) doesn’t line up with the amount of latitude that the subject believes to obtain (in these cases of deference).

  11. Deference applies to both cases in which (a) you believe S is in the same epistemic situation (has the same evidence), and (b) cases in which S is in a better epistemic situation (has strictly more evidence). But I’ve restricted Deference (prudential) to just apply to the analog of the (a) cases—cases in which (you believe) you’re both in the same prudential situation. I’ve done this because, in the context of prudential assessments, it’s difficult to see what kind of fact would put one subject in a better prudential situation than another. In any case, it doesn’t matter; this special case is enough to show how a norm like Deference which employs mismatching deontic operators is implausible in prudential contexts.

  12. In Sect. 3 of their paper, Greco & Hedden (2016) discuss the analog of Unique Outputs in prudential contexts, and defend the viability of such a principle. Such a principle seems difficult to maintain in the kind of coin toss case described in the text. Let the prudential permission function be a function which takes a subject’s credences, utilities and available options as inputs, and yields the permissible subset of those options as outputs. Then the following two claims:

    1. (1)

      Expected Utility Maximization: an option is prudentially permissible iff it maximizes expected utility.

    2. (2)

      Existence of Symmetric Cases: there are cases in which more than one option maximizes expected utility (such as the coin toss case described above). [END LIST] entail that there are cases in which multiple options are prudentially permissible. Thus adopting the prudential analog of Unique Outputs with respect to options requires rejecting either (1) or (2). Neither choice seems attractive.

    What would Greco and Hedden say about the coin toss case? In correspondence, Greco and Hedden have suggested a couple potential replies. First, they might grant that the prudential analog of Unique Outputs with respect to options is false, but maintain that a prudential analog of Unique Outputs with respect to preferences is true. And they might claim that permissions regarding preferences are more fundamental than permissions regarding options. Second, they might accept the prudential analog of Unique Outputs with respect to options and reject (1), by holding that in symmetric cases like the coin toss case neither option is prudentially permissible.

  13. There’s a large literature on this topic; for a classic account of the irrationality of akrasia, see Davidson (1970). For some recent defenses of the irrationality of akrasia in the epistemic case, see Horowitz (2014) and Greco (2014). For a broad discussion of these issues in prudential and moral cases, and further references, see Stroud (2014).

  14. These two principles are wide scope, but one might take the narrow scope versions of these principles—that if you believe X is permissible/obligatory for you, then X is permissible/obligatory for you—to also be prima facie plausible to those sympathetic to subjectivist accounts. Plausible or not, such principles cannot be endorsed by proponents of Evidential Uniqueness. For these narrow scope principles are incompatible with Evidential Uniqueness—they entail that two subjects with the same evidence but different beliefs can be obligated/permitted to believe different things. Since I want to motivate a deference principle in a manner that’s neutral with respect to Evidential Uniqueness, I won’t appeal to principles like this.

  15. For the derivation, see the appendix. One might wonder why an appendix is needed, since General Deference (Same Inputs) might seem to follow trivially from the anti-akratic principles given above. But while the derivation of the narrow scope version of General Deference (Same Inputs) from narrow scope versions of the anti-akratic principles is straightforward, the derivation of the actual (wide scope) version of General Deference (Same Inputs) from the actual (wide scope) versions of the anti-akratic principles take a bit more time.

  16. I.e., believe that the inputs of f are the same for you and S.

  17. For examples of such views, see Stich (1993), Fantl and McGrath (2009) and Podgorski (2016).

  18. I’ve followed Greco and Hedden here in formulating General Deference in terms of belief instead of more fine-grained doxastic attitudes, like credence. And if we restrict our attention to the epistemic realm, it’s natural to extend General Deference so that it takes these more fine-grained attitudes into account, allowing it to model cases in which (say) you’re uncertain about what S’s normative situation is, or uncertain about what it’s permissible/obligatory for S to believe in a situation. But it’s hard to provide this kind of quantitative formulation of General Deference if we want it apply to other normative realms as well (e.g., prudential, moral, etc.) as I do. The reason is that in these other normative realms, the objects of deference (e.g., acts) aren’t fine-grained in the way that credences are. And when the objects of deference aren’t fine-grained, there’s no easy way to turn fine-grained differences in your credence about (say) what S’s normative situation is, or what it’s permissible/obligatory for S to believe in a situation, into fine-grained differences in the verdicts the principle delivers. (That said, I talk more about the bearing of this discussion on quantitative epistemic deference principles in Sect. 5.)

  19. The prescriptions of General Deference will depend, in part, on which inputs one takes to be ranked inputs, and what one takes the partial ordering associated with these ranked inputs to be. And while the answers to these questions are relatively straightforward in some cases (e.g., it’s natural to take evidence to be a ranked input with a partial ordering determined by entailment), there are other cases in which there are substantive questions to be settled, questions that will impact the prescriptions General Deference makes.

    For example, consider a view which takes background beliefs to be an input, and holds that only some background beliefs are permissible. One stance, given such a view, is that background beliefs are ranked inputs, and that all permissible background beliefs are ranked higher than all impermissible background beliefs. A second stance is that background beliefs are ranked inputs, but only some particular permissible background beliefs are ranked higher than some particular impermissible background beliefs. (For example, if some impermissible background beliefs B are almost rational—if one added a certain belief they would be permissible—then one might take the permissible background beliefs you get by adding that belief to B to be higher ranked than B. But for other permissible background beliefs, such as ones that assign radically different beliefs than B, there might be no ranking between them and B.) A third stance is that background beliefs are unranked inputs, because while some of them (e.g., the permissible ones) are better in some respects than some of the others (e.g., the impermissible ones), none of them are strictly better in all respects.

    All three of these stances are compatible with General Deference. And General Deference will make somewhat different prescriptions depending on which of these stances one adopts. (If one likes, one can think of these different stances about what inputs are ranked inputs, and what their partial orders are, as something that’s part of the normative principle. On this way of thinking about things, each of these different stances yields a different version of General Deference. And one can take the case made in the text for adopting General Deference as an argument for a principle of this general form, an argument that leaves open the further question of which particular version of General Deference one should adopt.)

  20. I.e., believe that S is the same with respect to the inputs of f, or only differs with respect to inputs for which S is higher ranked than you.

  21. In addition to the the ways just described, General Deference is also more general than Deference in another, more subtle, way. Deference only makes prescriptions regarding cases in which you believe it’s permissible for S to believe Xand you believe S actually believes X. General Deference doesn’t require anything like the second clause—in the epistemic case, it doesn’t require you to believe that S actually believes X. So this is yet another way in which General Deference is more general than Deference.

  22. In Sect. 3, we saw Greco and Hedden (2016) argue that considerations regarding deference give us reason to adopt Evidential Uniqueness. In a similar fashion, Greco and Hedden (2016) argue that considerations regarding planning give us reason to adopt Evidential Uniqueness. Their planning argument faces the same worries are their deference argument, so I’ll just briefly rehearse it here. Greco and Hedden assert that the right principle governing rational planning is this (this is the more plausible of the two disambiguations they consider; see their footnote 22):

    • Planning: For any propositions X and E, it’s obligatory to be such that: if you believe that believing X is permissible given evidence E, then you plan to believe X given evidence E.

    Greco and Hedden argue that opponents of Evidential Uniqueness who accept Planning are led to absurd results. But Planning is implausible and/or unsatisfying in many of the same ways as Deference: it employs mismatching deontic operators, it presupposes Evidential Inputs, it’s arbitrarily constrained to the realm of epistemic norms, and so on. And like Deference, Planning is easily replaced with a more general principle without these demerits:

    • General Planning: For any X and Y, it’s permissible [obligatory] to be such that: if you believe that believing X is permissible [obligatory] given normatively relevant factors Y (i.e., believe that given inputs Y, one of f’s outputs [f’s only output] is a doxastic state which believes X), then you plan to believe X given normatively relevant factors Y.

    And General Planning is neutral with respect to the Evidential Uniqueness debate.

  23. One might worry that if your evidence justifies you in believing that Ann has evidence \(E \wedge F\), then you also have evidence F. And this would seem to be in tension with your belief that Ann has strictly more evidence than you. But this worry only arises if we think of F as a de dicto description of Ann’s additional evidence instead of a de re description of Ann’s additional evidence. And we should be thinking about F as a de re description of Ann’s evidence.

  24. This is a reductio of the Ann and Beth-based argument against Deference sketched in the text. One can try to modify the case in various ways to revive the Ann and Beth-style argument. But it’s hard to find a straightforward way to do this. Perhaps the easiest way to get a feel for how hard this is to try one’s hand at constructing concrete examples (with the details filled in) that leads Deference to make the desired prescriptions, and seeing how they fall through. But I’ll briefly describe a few of the potential moves here, and why they don’t work.

    One way to try to revive the Ann and Beth-based argument against Deference, and maintain that believing \(\hbox {Def}_{{\mathrm{Ann}}} \wedge \hbox {Def}_{{\mathrm{Beth}}}\) could be justified after all, is to consider a case in which even though your total evidence E justifies you in believing that Beth permissibly believes \(\lnot X\), you believe Ann’s extra evidence F makes it the case that Ann isn’t justified in believing that Beth permissibly believes \(\lnot X\). In such a case, it might seem that Ann is justified in believing X after all. But it’s not clear this move works. For suppose your evidence does make you justified in believing that Ann’s extra evidence F makes it the case that Ann isn’t justified in believing that Beth permissibly believes \(\lnot X\). Then it seems your evidence would also fail to justify your believing that Beth permissibly believes \(\lnot X\), and again we have a reductio of the claim that you could be justified in believing \(\hbox {Def}_{{\mathrm{Ann}}} \wedge \hbox {Def}_{{\mathrm{Beth}}}\).

    Another way to try to revive the Ann and Beth-style argument against Deference is to consider a slightly different case, where you believe both Ann and Beth have strictly more evidence than you, but neither has strictly more evidence than each other. (E.g., perhaps you have total evidence E, Ann has total evidence \(E \wedge F\), and Beth has total evidence \(E \wedge G\).) But, again, it’s not clear that your evidence could justify you in believing Ann and Beth have such evidence and permissibly believe X and \(\lnot X\), respectively. For Ann and Beth have all of your evidence, and so know about each other’s beliefs. How this shakes out depends on the details of what this evidence is, but one could plausibly maintain that no matter how you flesh out these details, something will break – either Ann’s evidence will fail to justify her in believing X, or Beth’s evidence will fail to justify her in believing \(\lnot X\), or both.

  25. There is a large literature about such principles, starting with Gaifman (1988); see Pettigrew and Titelbaum (2014) for recent discussion and references. Most of the discussion in this literature focuses on how one should defer to an “expert”. Of special interest, given the kinds of deference principles being discussed here, are discussions which avoid the (sometimes vague) notion of an “expert”, and instead focus directly on the link between what a subject believes to be rational and what is rational for the subject. For discussions of these kinds of principles, see Christensen (2010), Elga (2013), and Levinstein (2015).

  26. Levinstein (2015) formulates WPE as follows (p. 21):

    • Weak Perm Expert: “For some b with total evidence E and for all probability functions \(r, b(\cdot \mid r\in \mathring{R}) = r(\cdot \mid E \wedge r\in \mathring{R})\).”

    (Note that since b is just a function and E a proposition, there needs to be an implicit “if you have credences b and total evidence E, then” clause before the equation in order for the norm to apply to particular subjects.) This statement of the principle leaves the location of the deontic operator that gives it normative force implicit, and there are two natural places one might take it to be. First, one might adopt a narrow scope reading of the principle, and take there to be an implicit obligation operator that applies just to the formula in question. This is to read the principle as saying something like “there’s a b and E such that, for all r, if you have credences b and evidence E, then your credences should satisfy the following equation”. Second, one might adopt a wide scope reading of the principle, and take there to be an implicit obligation operator that holds over the entire conditional. This is to read the principle as saying something like “there’s a b and E such that, for all r, you ought to be such that if you have credences b and evidence E, then your credences satisfy the following equation”. In the text I’ve adopted the narrow scope reading because Levinstein’s argument for Unique Outputs requires it (cf. footnote 27).

  27. Following Levinstein, I’ll leave implicit the assumptions needed to ensure that all of the conditional probabilities mentioned in the formulation of WPE and the argument discussed below are well-defined.

  28. Note that this argument requires a narrow scope understanding of WPE (cf. footnote 25). For a wide scope understanding of WPE won’t entail that at all best worlds \(b(\cdot \mid r \in \mathring{R}) = r(\cdot \mid r \in \mathring{R} \wedge E)\), it’ll just entail that at all best worlds either (a) \(b(\cdot \mid r \in \mathring{R}) = r(\cdot \mid r \in \mathring{R} \wedge E)\)or (b) you don’t have credences b and total evidence E. And this disjunctive requirement won’t allow us to derive \(r(\cdot \mid r,r^\prime \in \mathring{R} \wedge E) = r^\prime (\cdot \mid r,r^\prime \in \mathring{R} \wedge E)\). After all, this disjunctive requirement is compatible with (a) never obtaining.

  29. This is a slightly tweaked version of Levinstein’s argument. The original argument attempted to derive \(r(\cdot \mid r,r^\prime \in \mathring{R} ) = r^\prime (\cdot \mid r,r^\prime \in \mathring{R})\), allowing one to avoid any reference to E. However, this conclusion doesn’t seem to follow from WPE. (In correspondence, Levinstein has suggested that his preferred way of getting around this problem would be to try to modify WPE.) In any case, the differences between these two versions of the argument don’t matter for our purposes.

  30. As we saw in Sect. 3.2, given a wide scope principle like Deference, one can try to avoid these implausible consequences by adopting the (question begging) assumption that all rational subjects believe Unique Outputs, and so will only ever think one doxastic state is permissible. But even that (question begging) option isn’t available here, since WPE is a narrow scope principle (cf. footnote 27) that makes prescriptions based on a subject’s actual doxastic state. Thus WPE entails that a (possibly irrational) subject who believes several different credence functions are permissible is obligated to line up their credences with all of them. And WPE has this consequence regardless of whether rational subjects believe Unique Outputs or not.

  31. If WPE employed matching permission operators, then it wouldn’t yield inconsistent prescriptions because instead of requiring\(b(\cdot \mid r,r^\prime \in \mathring{R})\) to equal \(r(\cdot )\) and \(r^\prime (\cdot )\), it would merely allow\(b(\cdot \mid r,r^\prime \in \mathring{R})\) to equal \(r(\cdot )\) or \(r^\prime (\cdot )\). And if WPE employed matching obligation operators, and so applied constraints on b conditional on r being obligatory, then it wouldn’t yield inconsistent prescriptions because there wouldn’t be any probability functions r and \(r^\prime\) which satisfy the conditions the derivation requires; i.e., which assign a value of 1 to both r and \(r^\prime\) being obligatory.

    In correspondence, Levinstein has suggested that his preferred way of getting around the conflicting prescriptions issue would be to modify WPE in certain ways (such as by restricting the probabilistic r-functions WPE makes claims about). These changes alone won’t change the fact that WPE employs mismatching operators, however, so these changes alone still leave us with an implausible principle. (That said, Levinstein has also expressed sympathy for modifying WPE so that its operators match.)

  32. Note that the point being made in these two paragraphs is different from the point being made in footnote 30. Footnote 30 spells out why the matching-operator variants of WPE, unlike WPE, won’t yield conflicting prescriptions. The two paragraphs above spell out why the matching-operator variants of WPE, unlike WPE, won’t yield an argument for Unique Outputs.

  33. We’ve been following Levinstein in avoiding notational clutter by leaving implicit all of the assumptions required to ensure that the relevant conditional probabilities are well-defined (cf. footnote 26). But we can’t make these implicit assumptions in this case, since every probability function will assign a value of 0 to \(r,r^\prime \in \mathring{O}\), making the relevant conditional probabilities undefined.

  34. While this assumption is relatively uncontroversial for epistemically best worlds (and thus for deriving General Deference (Same Inputs) for epistemic permission and obligation), it’s less clear why it’s plausible with respect to, say, prudentially or morally best worlds. I think this assumption is still tenable in those cases given the kind of “subjectivist” approach motivating AAPP and AAPO. Because this approach incorporates features of a subject’s epistemic life into its prescriptions, it naturally takes on features of epistemic norms that would not show up on an “objectivist” approach. In any case, those concerned about this assumption can either take the fact that one can derive General Deference (Same Inputs) for subjects whose beliefs are closed under logical and analytic entailment to merely provide a motivation for (instead of a derivation of) General Deference (Same Inputs), or restrict General Deference (Same Inputs) to subjects whose beliefs are closed under logical and analytic entailment.

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Acknowledgements

I’d like to thank Maya Eddon, Daniel Greco, Brian Hedden, Ben Levinstein, Alejandro Perez-Carballo, and Jonathan Vogel for helpful comments and discussion.

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Authors and Affiliations

  1. Department of Philosophy, E305 South College, University of Massachusetts, Amherst, MA, 01003-9274, USA

    Christopher J. G. Meacham

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Correspondence to Christopher J. G. Meacham.

Appendix

Appendix

Let’s see how we can use the following anti-akratic principles:

  • Anti-Akratic Principle (Permission) (AAPP): For any proposition X, it’s permissible to be such that if you believe X is permissible, then you X.

  • Anti-Akratic Principle (Obligation) (AAPO): For any proposition X, it’s obligatory to be such that if you believe X is obligatory, then you X.

to derive General Deference (Same Inputs), which consists of two conjuncts:

General Deference (Same Inputs, Permission) (GDSIP): For any X and subject S, it’s permissible to be such that: if you believe that it’s permissible for S to X, and believe that the normatively relevant facts are the same for you and S (i.e., believe that the inputs of f are the same for you and S), then you X.

General Deference (Same Inputs, Obligation) (GDSIO): For any X and subject S, it’s obligatory to be such that: if you believe that it’s obligatory for S to X, and believe that the normatively relevant facts are the same for you and S (i.e., believe that the inputs of f are the same for you and S), then you X.

GDSIP and GDSIO might seem to follow trivially from AAPP and AAPO. To some extent that’s true. But while the derivation of the narrow scope variants of GDSIP and GDSIO from narrow scope variants of AAPP and AAPO is straightforward, the derivation of the actual (wide scope) principles GDSIP and GDSIO from the actual (wide scope) principles AAPP and AAPO takes a bit more work.

Two preliminary comments. First, these derivations assume that, at best worlds, one’s beliefs are closed under logical and analytic entailment.Footnote 34 Second, instead of constantly repeating “for any X”, I’ll streamline the discussion by leaving the universal quantifier over X implicit in what follows.

Let’s start by showing how to derive GDSIP from AAPP. Note that (at any best world) if you believe X is permissible for you, then you believe that there is a subject (you) with the same inputs for whom X is permissible. Likewise (at any best world) if you believe that there is a subject S with the same inputs as you for whom X is permissible, then you believe X is permissible for you. Thus the following biconditional holds: (it’s permissible to be such that if you believe you can permissibly X, then you X) iff (it’s permissible to be such that if you believe that there’s a subject S with the same inputs who can permissibly X, then you X). That is, AAPP is true iff:

(A) :

There’s a best world at which the following conditional is true: if you believe there’s a subject S that has the same inputs as you who can permissibly X, then you X.

And (A) entails (B), a reformulation of GDSIP:

(B) :

For any subject S, there’s a best world where the following conditional is true: if you believe S has the same inputs as you and can permissibly X, then you X.

We can see that (A) entails (B) by showing that the falsity of (B) entails the falsity of (A) and applying contraposition. For (B) to be false, there must be an S such that there’s no best world where if you believe S has the same inputs as you and can permissibly X, then you X; i.e., there must be an S such that, at all best worlds, you believe S has the same inputs as you and can permissibly X, and you don’t X. But that entails that at all best worlds, you believe that there’s a subject S with the same inputs who can permissibly X, and you don’t X. And that entails that (A) is false.

Since AAPP is true iff (A), and (A) entails GDSIP, it follows that AAPP entails GDSIP.

Now let’s look at how to derive GDSIO from AAPO. Note that (at any best world) if you believe X is obligatory for you, then you believe that there is a subject (you) with the same inputs for whom X is obligatory. Likewise (at any best world) if you believe that there is a subject S with the same inputs as you for whom X is obligatory, then you believe X is obligatory for you. Given this, the following biconditional holds: (it’s obligatory to be such that if you believe that you’re obligated to X, then you X) iff (it’s obligatory to be such that if you believe that there is a subject S with the same inputs as you who is obligated to X, then you X). That is, AAPO is true iff:

(C) :

At all best worlds the following conditional is true: if you believe there is a subject S with the same inputs as you for whom X is obligatory, then you X.

And (C) entails (and, in fact, is equivalent to) (D), a reformulation of GDSIO:

(D) :

For every subject S, at all best worlds, the following conditional is true: if you believe S has the same inputs as you and that X is obligatory for S, then you X.

We can see that (C) entails (D) by showing that the falsity of (D) entails the falsity of (C) and applying contraposition. For (D) to be false, there must be an S such that, at some best world, you believe S has the same inputs as you and is obligated to X, but you don’t X. This entails that there is some best world at which you believe there is some S with the same inputs as you who is obligated to X, but you don’t X. And that is precisely what needs to be the case in order for (C) to be false.

(To see that (C) and (D) are in fact equivalent, note that the entailment also goes the other way. For (C) to be false, there must be some best world at which you believe there is a subject S with the same inputs as you who is obligated to X, but you don’t X. But that entails that there’s some S such that, at some best world, you believe S has the same inputs as you and is obligated to X, but you don’t X, and thus entails that (D) is false.)

Since AAPO is true iff (C), and (C) entails (and is in fact equivalent to) GDSIO, it follows that AAPO entails GDSIO.

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Meacham, C.J.G. Deference and Uniqueness. Philos Stud 176, 709–732 (2019). https://doi.org/10.1007/s11098-018-1036-4

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