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Might-beliefs and asymmetric disagreement

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Abstract

What we can call asymmetric disagreement occurs when one agent is in disagreement with another, but not vice-versa. In this paper, I give an example of and develop a framework for understanding this phenomenon. One pivotal feature of my example is that one of the agents in the scenario has a belief about what might be the case—a might-belief. I show that a traditional account of might-beliefs and disagreement cannot explain the initially surprising phenomenon of asymmetric disagreement. In order to provide an explanation, I develop a dynamic account of might-beliefs and a corresponding account of disagreement. I close by exploring a choice point for our account—showing that the simple dynamic account has some controversial (though, perhaps, true) consequences. I explore how revisionary notions of validity, inconsistency, and disagreement can allow us to avoid these consequences if we wish.

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Notes

  1. This is based on an example from Ross and Schroeder (2013). Dietz (2008) also uses an example involving might-beliefs but talks about an asymmetry in correct disagreement. I will follow Ross and Schroeder (2013) in thinking that the disagreement, itself, is asymmetric. Dietz discusses the sort of examples that I use as problems for a relativist semantics for epistemic modals (including ‘might’). But, as we will see, asymmetric disagreement is a challenge for more traditional approaches as well.

  2. It may also be correct to say that you don’t know whether Moriarty is the murderer, that you’re not sure whether Moriarty is the murderer, that you suspend judgment on the question of whether Moriarty is the murderer, etc. However, as long as we agree that you also believe that Moriarty might be the murderer and believe that Moriarty might not be the murderer, the central issue in this paper arises. Thanks to anonymous reviewer for discussion.

  3. As will be made explicit later, disagreement is not fundamentally a two-place relation, holding between agents. Instead, it is a relation holding between agents and their beliefs. When we talk simply about two agents disagreeing, we are using this as a shorthand.

  4. In saying these things, Holmes is not evaluating whether your belief that Moriarty might be the murderer is rational given your evidence or whether you were right or wrong to form that belief. Holmes can think you are wrong in your belief (and say, “No”, “You’re wrong”, or “I disagree”), while also thinking you are completely rational to have that belief (and could add, “but I see why you think that, since you don’t have enough evidence about the situation”). Thanks to an anonymous reviewer for discussion. See also 23.

  5. Thanks to John Hawthorne for suggesting this way of setting up the scenario. It is an example of asymmetric synchronic interpersonal disagreement. Asymmetry also can arise in diachronic intrapersonal cases. This is related to the reversibility cases of Ross and Schroeder (2013). Suppose that, in the morning, you believe Moriarty might be the murderer and believe that he might not be the murderer. And suppose that, in the evening, you find out for sure whether or not he is the murderer. In the evening, you are in a position to say that your earlier self was wrong. Nonetheless, in the morning, you don’t disagree with your later self. As Dietz (2008) notes (discussing a related example), it would be inappropriate, for instance, for your earlier self to say, “If I come to think that Moriarty is the murderer or to think that Moriarty is not the murderer, I will be wrong.” You leave it open that you will, at a later time, come to think one of these things. So, it looks like your earlier self does not disagree with your later self, though your later self does disagree with your earlier self. That is, it looks like we have asymmetric disagreement.

  6. I mean to be talking about the state of disagreement, not the act of disagreeing. The act of disagreeing is obviously not symmetric. See Cappelen and Hawthorne (2009, pp. 60–61) and MacFarlane (2014, pp. 119–120) for more on this distinction.

  7. It is natural (especially in light of the first three sections of this paper) to understand this as a claim about beliefs with dynamic contents. As we will see in Sect. 4.2 and Appendix B, if the account presented there is plausible, the class of theories that can explain asymmetric disagreement is slightly broader. It must include, at least, accounts that involve static contents which are, nonetheless, true or false only relative to an information state—provided that such accounts are paired with particular non-orthodox theories of validity, inconsistency, and disagreement.

  8. Given the account of inconsistency that I will put forth, this is not quite right, since \(\hbox {c}_{1}\) or \(\hbox {c}_{2}\) might be a contradiction. This would lead to seemingly incorrect predictions of disagreement. We can avoid this by restricting the contents that can feature in Inconsistentism to those that are not contradictions. I will tacitly assume this throughout.

  9. The propositional view of the contents of beliefs isn’t necessary for this result. It is merely a very popular view on which the result holds. Really, all the result requires is a symmetric account of the relation, inconsistency with, and Inconsistentism.

  10. Finlay (2014, Ch. 8) is engaged in this project for a similar asymmetry involving deontic modals—according to which disagreement results from the robustness of an attitude, where one attitude can be robust enough to underpin disagreement while the other is not.

  11. I won’t consider other alternative theories of the contents of beliefs that might be thought to explain asymmetric disagreement. For instance, I won’t address a relativist suggestion that could be inspired by MacFarlane 2011—though see Ross and Schroeder (2013, §4.3) for an argument against this account.

  12. I don’t assume that a traditional, non-dynamic account can’t explain such data in terms of context-sensitivity and changes in what information is relevant over time.

  13. This may be a non-standard way to use ‘content’. I make only the following assumption about the nature of contents: that contents are whatever play the role of the meanings (in at least one sense of ‘meanings’) of declarative sentences and the objects of attitudes like belief. On the traditional view, propositions play this role, but on the dynamic view, update directions (or, more formally, functions between information states) do.

  14. These symbols are shorthand for the English, “It’s not the case that”, “and”, “or”, “It might be the case that”, and “It must be the case that”, respectively.

  15. Nor are all semantic values objects that determine sets of possible worlds—objects like, for example, structured complexes of objects and properties.

  16. See also Rothschild and Yalcin (2017). All of these authors would characterize the semantics of the system we have developed so far, despite its nontraditional notation, as fundamentally static (or, in Veltman’s words, additive). But, once we add our account of ‘might’, they will characterize our system as dynamic (and non-additive). The addition of ‘might’ allows for a system that meets two of Veltman’s (1996, pp. 222–223) conditions for being dynamic and non-additive; such a system is neither persistent nor idempotent (interestingly, the system developed in Veltman (1996) has a slightly different syntax than mine—‘might’ can only occur with widest logical scope—and, for that reason, only persistence, and not idempotence fails in the system). Similarly, with von Fintel and Gilles (2007, p. 52)—the update function is not distributive in the case of ‘might’-sentences—and Rothschild and Yalcin (2017, p. 35)—the system is neither commutative nor idempotent. Thanks to Daniel Rothschild and Seth Yalcin for discussion.

  17. Equivalent formulations are given in van Bentham (1996, p. 19), Groenendijk et al. (1996, p. 190), Veltman (1996, p. 228), van der Does et al. (1997, p. 367), Beaver (2001, p. 151), Gillies (2004, 601), and von Fintel and Gillies (2007, p. 53). Related, but more nuanced views are given in Yalcin (2012, p. 15) and Willer (2013, p. 57).

  18. Since Grice (1989), there has been a large amount of work on the notion of what is said. Much of this work assumes that the sort of object that serves as what is said is a proposition. Here, I am merely suggesting that we take the same approach while working in the dynamic framework—that the sort of things that fill the theoretical role of what is said are the sort of things that feature as dynamic semantic values. Though there is quite a literature that suggests that the compositional semantic value of a sentence can come apart from what is said [Ninan (2010) and Rabern (2012) in the epistemic modal case and Lewis (1980), Dummett (1973, 1991), Sperber and Wilson (1986), Bach (1994), and Carston (2002) in other cases or in general], this work largely operates with a propositional notion of what is said. In contrast to that background framework, a central characteristic of the dynamic semantic framework is a very close connection between dynamic semantic values and what is said.

  19. This picture is, in the end, probably too weak to be correct. For example, it predicts that you have might-beliefs about many propositions that you’ve never entertained (since they are compatible with your information). For some ways of avoiding this prediction in a similar framework, see Yalcin (2011, 2012) and Willer (2013). Despite this issue, I will proceed with the simple account in the text, as it is sufficient for seeing how the dynamic view allows for asymmetric disagreement.

  20. Thanks to an anonymous reviewer for discussing the advantages of using this notion of validity rather than what Veltman calls validity\(_{1 }\)and van der Does, Groeneveld, and Veltman call ignorant consequence. Letting 0 be the undecided information state, which contains all possible worlds, we can follow Veltman (1996, p. 224) in defining validity\(_{1}\)/ignorant consequence as follows:

    \({\upvarphi }_{1},\ldots , {\upvarphi }_{\mathrm{n}} {\vDash }_{\hbox {IC}}\, {\uppsi }\) if and only if \(\mathbf{0}[{\upvarphi }_{1}]\ldots [{\upvarphi }_{\mathrm{n}}] = \mathbf{0}[{\upvarphi }_{1}]\ldots [{\upvarphi }_{\mathrm{n}}][{\uppsi }]\).

    The key difference is that validity\(_{1}\)/ignorant consequence focuses only on the totally undecided state, whereas validity\(_{2}\)/update-to-test consequence investigates all information states. An account of inconsistency based on the former would predict that every sentence is inconsistent with a ‘can’t’-sentence, including tautologies and even the ‘can’t’-sentence itself. Given the picture that I develop below, this would lead to the prediction that someone who believes that something can’t be the case disagrees with everyone who has a belief in virtue of every belief that they have. Validity\(_{2}\)/update-to-test consequence makes no such predictions.

  21. We might object to this move on the grounds that our ordinary notion of disagreement is such that only propositions—not dynamic contents—can be the sorts of things that underpin disagreement. Such a worry mischaracterizes our ordinary judgments about disagreement. It is important to keep in mind that ‘proposition’, in the philosophical sense, is a term of art. We are used to talking about disagreeing about propositions, but our pre-theoretic way of talking about disagreement doesn’t involve propositions. Rather, we say things like “Galileo and the Pope disagreed about whether the earth moved around the sun” and “Galileo and the Pope disagreed because Galileo believed that the earth moved around the sun and the Pope did not”. When we say these things, we generally have in mind a conflict or inconsistency between these people’s beliefs. But these ascriptions of disagreement are neutral with respect to what the objects of belief are. Thus, a picture of disagreement where it obtains in virtue of beliefs in dynamic contents is not, as it might first appear, in obvious tension with our ordinary notion of disagreement.

  22. Because Inconsistentism is only a sufficient condition for disagreement, it does not actually predict that you do not disagree with Holmes. It merely fails to predict that you do disagree with him. Simply allowing for the possibility of asymmetric disagreement is already a great improvement over the traditional picture. Furthermore, we could try to strengthen Inconsistentism into necessary and sufficient conditions. I’ve not done so in this paper in order to avoid the argumentative burden, but I think that it is plausible that if there are other types of disagreement which are underpinned in different ways, they are of a quite different sort (for example, disagreement in preferences).

  23. One might worry that there are cases which have the above structure, but which seem to involve the agent with the might-belief disagreeing with the other agent. If such cases are genuine, one could charge that they cannot be explained by Inconsistentism. Consider: Mr. Reckless comes to believe that Moriarty is not the murderer based on a small amount of evidence that does not really support such a belief. Ms. Judicious knows this about Reckless, and, having the same evidence that Reckless has, believes that Moriarty might be the murderer and believes that he might not be the murderer. There is a pull to say that Judicious thinks that Reckless is wrong. If this were a case of Judicious disagreeing with Reckless, it would not be predicted by Inconsistentism. We can respond by distinguishing between two ways that an agent can judge that her interlocutor is wrong: she can judge that he is wrong in thinking some content or she can judge that he is wrong to think that content. Only the former is a way of disagreeing with her interlocutor about that content. The latter is an evaluation of the formation of the thought, but it is not disagreement about the thought. In the purported objection, Judicious does not judge that Reckless is wrong in thinking that Moriarty is the murderer (she is undecided about that matter); she does not disagree with Reckless. But she does think that he is wrong to think that Moriarty is the murderer; she thinks that he has been irresponsible in forming his belief. Thus, though this case might appear to be one where disagreement is symmetric, by carefully reflecting on the different ways of judging another’s thought wrong, we can see that, in fact, it is not.

  24. More generally, though they have different update potentials, [“Moriarty isn’t the murderer”] and [“Moriarty can’t be the murderer”] are accepted by the same information states. That is, for any I:

    I[“Moriarty isn’t the murderer”] = I if and only if I[“Moriarty can’t be the murderer”] = I.

  25. Thanks to an anonymous reviewer for suggesting this informal gloss on my view.

  26. We can view validity\(_{\mathrm{D}}\) as the trivial subcase of validity\(_{\mathrm{IS}}\)—one where the information state is always fixed to 0. Thanks to an anonymous reviewer for discussion of information state-sensitive validity relations.

  27. Thanks to an anonymous reviewer for discussion.

  28. In the introduction to this paper, I bolstered the claim that there was asymmetric disagreement by suggesting that Holmes (who believes that Moriarty isn’t the murderer) could felicitously respond to an utterance of “Moriarty might be the murderer” by saying ‘No’, but you (who believe that Moriarty might be the murderer and believe that Moriarty might not be the murderer) could not felicitously respond to an utterance of “Moriarty isn’t the murderer” by saying ‘No’. In this section we’ve been investigating the nearby scenario where Holmes’s belief and utterance are that Moriarty can’t be the murderer, with the idea that our judgments are the same as in the original case. What semantics for the response particle, ‘No’, could predict this result? We might have thought that saying ‘No’ in response to an utterance of \({\upvarphi } (\hbox {No}_{{\upvarphi } })\) is appropriate just in case your information state doesn’t admit \({\upvarphi }\):

    • \(\hbox {I}[\hbox {No}_{{\upvarphi } }] = \{\hbox {w} \in \hbox {I}: \hbox {I}[{\upvarphi }] = \varnothing \}\).

    But if the view defended in this section is correct, then what is required to felicitously respond to an utterance of \({\upvarphi }\) by saying ‘No’ is that \([{\upvarphi }]\) is inconsistent with\(_{\mathrm{IS}}\) your information state:

    • \(\hbox {I}[\hbox {No}_{{\upvarphi } }] = \{\hbox {w} \in \hbox {I}: \forall \hbox {I}^\prime \subseteq : \hbox {I}^\prime [{\upvarphi }] = \varnothing \}\).

    Thanks to an anonymous reviewer for this suggestion.

  29. This is related to Veltman’s (1996, p. 223) and Gillies’s (2006, p. 130) definitions of persistence—two major differences being that my notion of persistence holds of pairs of sentences and that my notion is not relativized to a consequence relation, as Gillies’s is. See also Gillies (2004) and von Fintel and Gillies (2007).

  30. Thanks to an anonymous reviewer for discussion.

  31. It is interesting that these are the two sorts of systems that predict the inconsistency of epistemic contradictions (Yalcin 2007; Willer 2013), though further discussion of this connection must be left for another time.

  32. Thanks to an anonymous reviewer for encouraging me to include this comparison section.

  33. This takes on the picture from Sect. 4.2, rather than the one from Sect. 3. See note 28 for a more extensive discussion of this point.

  34. We could give a similar treatment of ‘You’re wrong’ as a response particle. The same may not be said of assessments of truth and falsity, especially in so-called eavesdropper cases. Though such cases have been used by theorists like Egan (2007) and MacFarlane (2011) to motivate a relativist account of ‘might’, judgments about them appear to be mixed at best (see, for example, Dowell 2011; Yalcin 2011). Indeed, Knobe and Yalcin (2014) present experimental data that suggest that speakers won’t judge what you say when you utter the might-sentence in the conversation above to be false, even when they know that Moriarty isn’t the murderer. Knobe and Yalcin suggest that this, together with some other data, could incline us toward a dynamic picture. I don’t take asymmetric disagreement to be central to this debate. Nonetheless, the fact that the dynamic picture nicely explains asymmetric disagreement while also being preferred in light of Knobe and Yalcin’s experimental data is a boon for the account.

  35. Kölbel (2004, 2009) and Moltmann (2010) explicitly make this suggestion. See Dietz (2008) and Schaffer (2011) for discussion.

  36. Just as validity\(_{2}\)/update-to-test consequence (or what I have called validity\(_{\mathrm{D}})\) is the trivial subcase of validity\(_{\mathrm{IS}}\)—one where the information state is always fixed to 0 (as I mentioned in note 26)—validity\(_{1}\)/ignorant consequence is the trivial subcase of validity\(_{\mathrm{BP}}\)—again, where the information state is always fixed to 0. See note 20 for a reminder of these notions.

  37. Thanks to an anonymous reviewer for pointing me to these parts of Beaver’s and Pavese’s work.

  38. Some reflection on validity\(_{\mathrm{Y}}\) suggests that we could have implemented the system in Sect. 4.2 somewhat differently. Instead of basing our information-sensitive validity relation on Veltman’s validity\(_{2}\) or update-to-test consequence (see note 20), we could have relied on an information-sensitive version of Veltman’s validity\(_{3}\) or what is often called test-to-test consequence (for example in van Bentham 1996 and van der Does et al. 1997). This is the basis of validity\(_{\mathrm{Y}}\). In the dynamic semantic system, this would look as follows:

    \({\upvarphi } {\vDash }^{\mathrm{TT}}_{\mathrm{I}} \, {\uppsi }\hbox { if and only if }\forall \hbox {I}^\prime \subseteq \hbox {I}:\hbox { if } \hbox {I}^\prime [{\upvarphi }] = \hbox {I}^\prime ,\hbox { then }\hbox {I}^\prime [{\uppsi }] = \hbox {I}^\prime \).

    From this, we could define inconsistency with\(_{\mathrm{TT}}\) and incorporate this into a picture of disagreement like Inconsistentism\(_{\mathrm{IS}}\), which would yield the same predictions as both Inconsistentism\(_{\mathrm{IS}}\) and Inconsistentism\(_{\mathrm{Y}}\). Thanks to an anonymous reviewer for encouraging me to discuss this point.

References

  • Bach, K. (1994). Conversational impliciture. Mind and Language, 9(2), 124–162.

    Article  Google Scholar 

  • Beaver, D. (2001). Presupposition and assertion in dynamic semantics. Stanford: CSLI Publications.

    Google Scholar 

  • Cappelen, H., & Hawthorne, J. (2009). Relativism and monadic truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Carston, R. (2002). Thoughts and utterances: The pragmatics of explicit communication. Oxford: Blackwell Publishers.

    Book  Google Scholar 

  • Dietz, R. (2008). Epistemic modals and correct disagreement. In M. Garcia-Carpintero & M. Kölbel (Eds.), Relative truth. Oxford: Oxford University Press.

    Google Scholar 

  • Dowell, J. (2011). A flexible contextualist account of epistemic modals. Philosophers’ Imprint, 11(14), 1–25.

    Google Scholar 

  • Dummett, M. (1973). Frege: Philosophy of language. London: Gerald Duckworth & Company Limited.

    Google Scholar 

  • Dummett, M. (1991). The logical basis of metaphysics. London: Gerald Duckworth & Company Limited.

    Google Scholar 

  • Egan, A. (2007). Epistemic modals, relativism, and assertion. Philosophical Studies, 133(1), 1–22.

    Article  Google Scholar 

  • Finlay, S. (2014). Confusion of tongues: A theory of normative language. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Gillies, A. (2004). Epistemic conditionals and conditional epistemics. Nous, 38(4), 585–616.

    Article  Google Scholar 

  • Gillies, A. (2006). What might be the case after a change in view. Journal of Philosophical Logic, 35(2), 117–145.

    Article  Google Scholar 

  • Grice, P. (1989). Logic and conversation. In Studies in the ways of words. Cambridge, MA: Harvard UP.

  • Groenendijk, J., Stokhof, M., & Veltman, F. (1996). Coreference and modality. In S. Lappin (Ed.), The handbook of contemporary semantic theory. Oxford: Blackwell Publishers.

    Google Scholar 

  • Heim, I. (1992). Presupposition projection and the semantics of attitude verbs. Journal of Semantics, 9(3), 183–221.

    Article  Google Scholar 

  • Knobe, J., & Yalcin, S. (2014). Epistemic modals and context: Experimental data. Semantics and Pragmatics, 7(10), 1–21.

    Google Scholar 

  • Kölbel, M. (2004). Faultless disagreement. Proceedings of the Aristotelian Society, 104(1), 53–74.

    Article  Google Scholar 

  • Kölbel, M. (2009). The evidence for relativism. Synthese, 166(2), 375–395.

    Article  Google Scholar 

  • Lennertz, B. (2014). Simple contextualism about epistemic modals is incorrect. Thought: A Journal of Philosophy, 3(4), 252–262.

    Google Scholar 

  • Lewis, D. (1980). Index, context, and content. In S. Kanger & S. Ohman (Eds.), Philosophy and grammar. Amsterdam: D. Reidel Publishing Company.

    Google Scholar 

  • MacFarlane, J. (2011). Epistemic modals are assessment sensitive. In A. Egan & B. Weatherson (Eds.), Epistemic modality. Oxford: Oxford University Press.

    Google Scholar 

  • MacFarlane, J. (2014). Assessment sensitivity: Relative truth and its applications. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Moltmann, F. (2010). Relative truth and the first person. Philosophical Studies, 150(2), 187–220.

    Article  Google Scholar 

  • Ninan, D. (2010). Semantics and the objects of assertion. Linguistics and Philosophy, 33, 355–80.

    Article  Google Scholar 

  • Pavese, C. (2017). On the meaning of ‘therefore’. Analysis, 77(1), 88–97.

    Article  Google Scholar 

  • Rabern, B. (2012). Against the identification of assertoric content with compositional value. Synthese, 189, 75–96.

    Article  Google Scholar 

  • Ross, J., & Schroeder, M. (2013). Reversibility or disagreement. Mind, 122(485), 43–84.

    Article  Google Scholar 

  • Rothschild, D., & Yalcin, S. (2017). On the dynamics of conversation. Nous, 51(1), 24–48.

    Article  Google Scholar 

  • Schaffer, J. (2011). Perspective in taste predicates and epistemic modals. In A. Egan & B. Weatherson (Eds.), Epistemic modality. Oxford: Oxford University Press.

    Google Scholar 

  • Sperber, D., & Wilson, D. (1986). Relevance: Communication and cognition. Oxford: Blackwell Publishers.

    Google Scholar 

  • Stephenson, T. (2007). Judge dependence, epistemic modals, and predicates of personal taste. Linguistics and Philosophy, 30(4), 487–545.

    Article  Google Scholar 

  • van Bentham, J. (1996). Exploring logical dynamics. Stanford: CSLI Publications.

    Google Scholar 

  • van der Does, J., Groeneveld, W., & Veltman, F. (1997). An update on ‘might’. Journal of Logic, Language, and Information, 6(4), 361–380.

    Article  Google Scholar 

  • Veltman, F. (1996). Defaults in update semantics. Journal of Philosophical Logic, 25(3), 221–261.

    Article  Google Scholar 

  • von Fintel, K., & Gillies, A. (2007). An opinionated guide to epistemic modality. In T. Szabó Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 2). Oxford: Oxford University Press.

    Google Scholar 

  • von Fintel, K., & Gillies, A. (2010). Must. Stay. Strong!. Natural Language Semantics, 18(4), 351–383.

    Article  Google Scholar 

  • von Fintel, K., & Gillies, A. (2011). Might made right. In A. Egan & B. Weatherson (Eds.), Epistemic modality. Oxford: Oxford University Press.

    Google Scholar 

  • Willer, M. (2013). Dynamics of epistemic modality. The Philosophical Review, 122(1), 45–92.

    Article  Google Scholar 

  • Yalcin, S. (2007). Epistemic modals. Mind, 116(464), 983–1026.

    Article  Google Scholar 

  • Yalcin, S. (2011). Nonfactualism about epistemic modals. In A. Egan & B. Weatherson (Eds.), Epistemic modality. Oxford: Oxford University Press.

    Google Scholar 

  • Yalcin, S. (2012). Context probabilism. In M. Aloni, et al. (Eds.), Amsterdam colloquium 2011, LNCS 7218. Berlin: Springer.

    Google Scholar 

Download references

Acknowledgements

Thanks to Mark Schroeder, Greg Ackermann, John Hawthorne, Caleb Perl, Johannes Schmitt, Shyam Nair, Stephen Finlay, Scott Soames, three anonymous reviewers, and my audience at the 2014 meeting of the Society for Exact Philosophy for comments on earlier versions of this paper and/or discussion of these issues.

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Correspondence to Benjamin Lennertz.

Appendices

Appendix A

Beaver (2001, p. 153) and Pavese (2017) also use information-state sensitive (or context-sensitive) validity relations. Their purpose in doing so is to give a dynamic treatment of conclusion indicators like ‘so’ and ‘I conclude that’ (Beaver) and ‘therefore’ (Pavese). Their notion, what I’ll call validity\(_{{BP}}\), is similar to validity\(_{\mathrm{IS}}\), but differs in one crucial respect. While validity\(_{\mathrm{IS}}\) depends on the result of update at every subset of the relevant information state, validity\(_{\mathrm{BP}}\) only depends on the update of the state itselfFootnote 36:

  • \({\upvarphi } {\vDash }_{\hbox {I}}^{\mathrm{BP}} {\uppsi } \,\hbox {if and only if I}\,[{\upvarphi }] = \hbox {I}[{\upvarphi }][{\uppsi }]\).

Validity\(_{\mathrm{BP}}\) could not serve my purposes in Sect. 4.2, since it leads to the truth of (MCSD). The key fact that allows validity\(_{\mathrm{IS}}\) to avoid the commitment to (MCSD) is that it allows that [can’t \({\upvarphi }\)] is consistent with\(_{\mathrm{IS}}\) a non-defective information state of a person who believes [might \({\upvarphi }\)]. But, this won’t be true if we use a notion inconsistency with based on validity\(_{\mathrm{BP}}\). Take non-null information state, I, for which:

  • \(\hbox {I}[\hbox {might} {\upvarphi }] = \hbox {I}\)

Then:

  • \(\hbox {I}[\hbox {can't}\, {\upvarphi }] = \varnothing \).

So, [can’t \({\upvarphi }\)] is inconsistent with\(_{\mathrm{BP}}\) I. Given a sufficient condition for disagreement quite like Inconsistentism\(_{\mathrm{IS}}\), anyone who believes [might \({\upvarphi }\)] disagrees with anyone who believes [can’t \({\upvarphi }\)]. Thus, (MCSD) is true. Of course, we were undecided about whether we wanted to accept or reject (MCSD). However, if we are going to accept it, there is no reason to move to an information-sensitive validity relation in the first place. We could simply stick with validity\(_{\mathrm{D}}\). So, validity\(_{\mathrm{BP}}\) does not serve my purpose.

Does validity\(_{\mathrm{IS}}\) serves Beaver’s and Pavese’s purpose, at least over the fragment of English we have focused on in this paper? The answer is a bit arcane. Validity\(_{\mathrm{IS}}\) and validity\(_{\mathrm{BP}}\) do make different predictions about conclusion indicators, but the only substantive differences occur in cases involving ‘must \({\upvarphi }\)’ (or ‘can’t \({\upvarphi }\)’) as a premise and where the relevant information, I, is such that:

  • \(\hbox {I}[\hbox {must} \,{\upvarphi }] = \varnothing \, \hbox {but for some I}^\prime \,\subseteq , \hbox {I}, \hbox {I}^\prime [\hbox {must} \,{\upvarphi }] = \hbox {I}^\prime \).

Differences could occur in cases like the following: In a context where we don’t know whether or not Moriarty is the murderer, consider an utterance of “Moriarty must be the murderer; therefore, a contradiction is true.” According to validity\(_{\mathrm{BP}}\), the reasoning expressed here is valid, but according to validity\(_{\mathrm{IS}}\), it is not. To see why let \(\hbox {I} = \{\hbox {w}_{1}, \hbox {w}_{2}\}\) where \(\hbox {w}_{1}\) is a world in which Moriarty is the murderer and \(\hbox {w}_{2}\) is a world in which he isn’t. Then:

  • \(\hbox {I}[\hbox {``Moriarty must be the murderer''}] = \hbox {I}[\hbox {``Moriarty must be the murderer''}][\perp ]= \varnothing \).

So:

  • “Moriarty must be the murderer” \({\vDash }_{\hbox {I}}^{\mathrm{BP}} \perp \).

But letting \(\hbox {I}^\prime = \{\hbox {w}_{1}\} \subseteq \) I:

  • \(\hbox {I}'\)[“Moriarty must be the murderer”] = \(\hbox {I}'\ne \,\hbox {I}'\)[“Moriarty must be the murderer“]\([\perp ]= \varnothing \).

Thus:

  • “Moriarty must be the murderer” \({\vDash }_{\hbox {I}} \perp \).

What does this mean for the utterance of “Moriarty must be the murderer; therefore, a contradiction is true”, relative to a context that is undecided about whether Moriarty is the murderer? Given Beaver’s framework, validity\(_{\mathrm{BP}}\) predicts that there is a true meta-level assertion that Moriarty must be the murderer entails that a contradiction is true in the context (and a false assertion that Moriarty must be the murderer), while validity\(_{\mathrm{IS}}\) predicts that that meta-level assertion is false. Given Pavese’s background framework, validity\(_{\mathrm{BP}}\) predicts that the sentence is false/unsound in the context, while validity\(_{\mathrm{IS}}\) predicts that the sentence is undefined in the context. I don’t have strong judgments about which of these predictions are correct, given that we’re dealing with somewhat novel theoretical notions (like meta-level assertions and undefinedness) that aren’t always easy to have judgments about and are especially hard to think about in this case.Footnote 37

Appendix B

We can generate the same predictions as the system in Sect. 4.2 using a static notion of content together with notions much like validity\(_{\mathrm{IS}}\), inconsistency with\(_{\mathrm{IS}}\), and Inconsistentism\(_{\mathrm{IS}}\). We can use Yalcin’s semantics for modals to do this (2007, p. 994). Yalcin’s system assigns sentences truth-values relative to contexts, worlds, and information states. Though this picture is not dynamic, the semantic values of ‘might’-sentences do depend on information states, and relativity to information states is necessary for our result. Yalcin’s semantics for ‘might’ goes as follows, for a sentence, \({\upvarphi }\), a context, c, an information state, I, and possible worlds, w and \(\hbox {w}'\) (and where we now let [\(\bullet \)] be the semantic value function for Yalcin’s semantics):

  • \([\hbox {might}\,{\upvarphi }]^{\mathrm{c, I, w}}\) is true if and only if \(\exists w^\prime \in \) I: \([{\upvarphi }]^{\mathrm{c, I, w'}}\) is true.

We can assume standard semantic clauses for atomic sentences and connectives. And we can give a Yalcin-inspired account of belief, where our agent, A, has information state I\(_{\mathrm{A}}\):

  • A believes \([{\upvarphi }]^{\mathrm{c}}\) if and only if \(\forall \hbox {w} \in \hbox {I}_{\mathrm{A}}: [{\upvarphi }]^{\mathrm{c, IA, w}}\) is true.

To allow for asymmetric disagreement, we will use a notion of validity that is similar to validity\(_{\mathrm{IS}}\) in that it is information state-sensitive—call it \({ validity}_{Y}\) and represent it, for an information state \(\mathrm{S}, \vDash ^{\mathrm{Y}}_{\mathrm{S}}\) (in the following, we let I be any information state and \({\uppsi }\) any sentence):

\({\upvarphi } {\vDash }^{\mathrm{Y}}_{\mathrm{I}} {\uppsi }\) if and only if \(\forall \hbox {I}^\prime \subseteq \hbox {I}\): if \(\forall \hbox {w} \in \hbox {I}^\prime : [{\upvarphi }]^{\mathrm{c, I', w}}\) is true, then \(\forall \hbox {w} \in \hbox {I}^\prime : [{\uppsi }]^{\mathrm{c, I', w}}\) is true.Footnote 38

We can then define inconsistency as follows (where \(\perp \) is a sentence that is false relative to all context, information state, world triples):

  • \({\upvarphi }\hbox { is }{} \textit{inconsistent with}_{Y}\hbox { I if and only if }{\upvarphi } \, \vDash ^{\mathrm{Y}}_{\mathrm{I}} \perp \).

Simplifying, we get:

  • \({\upvarphi }\) is \(\textit{inconsistent with}_{Y}\) I if and only if \(\lnot \exists \hbox {I}^\prime \subseteq \) I: \(\exists \) w \(\in \,\hbox {I}'\): \([{\upvarphi }]^{\mathrm{c, I', w}}\) is true.

Finally, we can plug this into a sufficient condition for disagreement like \(\mathbf{Inconsistentism}_{\mathrm{IS}}\):

\(\mathbf{Inconsistentism}_{\mathrm{Y}}\): B disagrees with A in virtue of A’s belief in c, if A believes c, I is B’s information state, and c is inconsistent with\(_{\mathrm{Y}}\) I.

This picture predicts disagreement in exactly the cases that Inconsistentism\(_{\mathrm{IS}}\) does. It is worth noting that though this account of ‘might’ is Yalcin’s, the notions of validity\(_{\mathrm{Y}}\), inconsistency with\(_{\mathrm{Y}}\), and Inconsistentism\(_{\mathrm{Y}}\) are not Yalcin’s. I’ve developed the view in this way in order to compare it to the one presented in Sect. 4.2.

The package, together, makes the following predications: First, a person who believes that Moriarty isn’t or can’t be the murderer disagrees with one who believes that Moriarty might be the murderer. Suppose that Holmes believes that Moriarty isn’t the murderer. So, Holmes has an information state, \(\hbox {I}_{\mathrm{H}}\), which contains only worlds in which Moriarty isn’t the murderer. Thus:

  • \(\lnot \exists \) w \(\in \hbox {I}_{\mathrm{H}}\): \(\hbox {[``Moriarty might be the murderer'']}^{\mathrm{c, IH, w}}\hbox { is true}\).

Furthermore, every subset of \(\hbox {I}_{\mathrm{H}}\) contains only worlds in which Moriarty isn’t the murderer. So:

  • \(\lnot \exists \hbox {I}^\prime \subseteq \hbox {I}_{\mathrm{H}}: \exists \hbox {w} \in \hbox {I}^\prime \): \(\hbox {[``Moriarty might be the murderer'']}^{\mathrm{c, I', w}}\hbox { is true}\).

That means that “Moriarty might be the murderer” is inconsistent with\(_{\mathrm{Y}}\)\(\hbox {I}_{\mathrm{H}}\). Thus, according to \(\mathbf{Inconsistentism}_{\mathrm{Y}}\), Holmes disagrees with you. Furthermore, given the account of belief mentioned above, if Holmes believes that Moriarty isn’t the murderer, he also believes that Moriarty can’t be the murderer. Thus, \(\mathbf{Inconsistentism}_{\mathrm{Y}}\) predicts disagreement in exactly the same way in the case where Holmes believes that Moriarty can’t be the murderer and you believe that he might be.

Our modified version of Yalcin’s theory does not predict disagreement in the other direction. Remember that you believe that Moriarty might be the murderer and believe that he might not be the murderer. So, \(\hbox {I}_{\mathrm{Y}}\) contains at least one world in which Moriarty is the murderer and one world in which he is not. For this reason, both “Moriarty isn’t the murderer” and “Moriarty can’t be the murderer” are consistent with\(_{\mathrm{Y}}\hbox { I}_{\mathrm{Y}}\):

  • \(\exists \hbox {I}^\prime \subseteq \hbox { I}_{\mathrm{Y}}: \exists \hbox {w} \in \hbox {I}': \hbox {[``Moriarty isn't the murderer'']}^{\mathrm{c, I', w}}\) is true

and

  • \(\exists \hbox {I}^\prime \subseteq \hbox { I}_{\mathrm{Y}}: \exists \hbox {w} \in \hbox {I}'\):\( \hbox {[``Moriarty can't be the murderer'']}^{\mathrm{c, I', w}}\) is true.

We can choose for I’ the subset of \(\hbox {I}_{\mathrm{Y}}\) which contains only the worlds of \(\hbox {I}_{\mathrm{Y}}\) in which Moriarty is not the murderer. \(\hbox {I}'\) is nonempty and the following is the case:

  • \(\forall \hbox {w} \in \hbox {I}': \hbox {[``Moriarty isn't the murderer'']}^{\mathrm{c, I', w}}\) is true and \(\hbox {[``Moriarty can't be the}\hbox { murderer'']}^{\mathrm{c, I', w}}\) is true.

So, “Moriarty isn’t the murderer” and “Moriarty can’t be the murderer” are both consistent with\(_{\mathrm{Y}}\hbox { I}_{\mathrm{Y}}\). Therefore, \(\mathbf{Inconsistentism}_{\mathrm{Y}}\), does not predict that you disagree with someone with either of those beliefs. Thus, the modified version of Yalcin’s account allows for asymmetric disagreement. The fallout of this result was addressed in Sect. 4.2.

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Lennertz, B. Might-beliefs and asymmetric disagreement. Synthese 196, 4775–4805 (2019). https://doi.org/10.1007/s11229-018-1688-9

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