Abstract
The status of the knowledge iteration principles in the account provided by Lewis in “Elusive Knowledge” is disputed. By distinguishing carefully between what in the account describes the contribution of the attributor’s context and what describes the contribution of the subject’s situation, we can resolve this dispute in favour of Holliday’s (2015) claim that the iteration principles are rendered invalid. However, that is not the end of the story. For Lewis’s account still predicts that counterexamples to the negative iteration principle (\(\lnot Kp\rightarrow K\lnot Kp\)) come out as elusive: such counterexamples can occur only in possibilities which the attributors of knowledge are ignoring. This consequence is more defensible than it might look at first sight.
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Notes
Lewis (1996). Blome-Tillman (2009, 2012, 2014) and Ichikawa (2011a, b, 2013) are recent defenders of (modified) versions of Lewis’s account; commentators that pay close attention to Lewis’s account in particular include Cohen (1998), Vogel (1999), Williams (2001), Williamson (2001), Schaffer (2004), Hawthorne (2004), and Douven (2005).
I actually think that, in addition to it not being obviously false, there are positive reasons to want something like the Lewisian treatment of \(K\lnot K\) to be correct. For, as I hope to show in future work, it allows us to solve hard problems for the (thoroughly non-Lewisian) thesis, defended by Williamson (2000), that one’s evidence consists of all and only the claims that one knows.
This is not quite right as an interpretation of Lewis, since he uses ‘possibilities’ to mean something slightly different from possible worlds (1996, p. 552). To keep the formalization of his account manageable, I ignore that complication here.
To see that it’s transitive, note that from \(xR_Ky\) and \(yR_Kz\) it follows that \(z \in S\) and xEy and yEz. So \(z \in S\) and xEz (since E is transitive), and so \(xR_Kz\). To see that it’s Euclidean, note that if \(xR_Ky\) and \(xR_Kz\), then \(z\in S\) and xEy and xEz. So \(z\in S\) and yEz (since E is euclidean), and hence \(yR_Kz\).
In an unpublished manuscript, Julien Dutant identifies a “rigid interpretation” of Lewis’s semantics, shows how it conflicts with the factivity of knowledge, and then considers a response analogous to this one. He observes that, even once we acknowledge such a response, the interpretation still predicts that the sentence ‘someone could have known something false’ could be true, which is the inspiration for the objection I offer below.
A variant of this is more familiar in modal logic. We could move to ‘model structures’ \(<W, E, S, w>\) which designate world \(w\in W\) as the actual world. Since the actual world is never properly ignored, we would then want to impose the structural requirement that \(w\in S\). When working with model structures instead of frames, it’s also natural to redefine validity as truth at the designated world of every model. The resulting system is very similar to the one discussed in the main text; in particular, it validates S5 for essentially the same reason.
Why? Let us say that v can be reached from w if there are worlds \(u_1,\ldots u_n\) such that \(wR_Ku_1, u_1R_Ku_2, \ldots u_nR_Kv\). Then truth in a model depends only on what happens in worlds that are either in S or can be reached from a world in S. Moreover, the definition of \(R_K\) ensures that all such worlds are themselves in S. Finally, \(R_K\) is an equivalence relation when restricted to S (though not outside it). Together, these facts ensure that we validate an S5 logic.
In the unpublished manuscript mentioned in footnote 5, Dutant argues that “the subject’s actuality” might be construed instead as the (potentially counterfactual) world on which the conversation is focused; this would allow for context alone to determine propriety. I agree that such a reading is just about possible. But since it would leave us with the unsatisfactory account discussed in Sect. 1.1, and the context of the passage strongly suggests that Lewis is trying to rule out this variant account, I think it safe to assume that this is not how Lewis intended these remarks.
What is the role of the ‘permissive’ rules, such as the Rules of Reliability, Method, and Conservatism (1996, pp. 558–559)? I have to confess to finding these rather puzzling. As I understand Lewis, any world that isn’t being attended to is automatically ignored, and thus properly ignored if no ‘restrictive’ rule prevents this from happening. But then what role could there be for the permissive rules to play? One hypothesis is that they aren’t rules about the propriety of ignoring at all, but are rather empirical generalizations about what kind of worlds are in fact ignored in ordinary contexts. Another thought, suggested to me by Bob Stalnaker, is that they function as constraints on what ‘restrictive’ rules Lewis would be willing to add to his account: they had better be consistent with it being proper, except in very specific circumstances, to ignore worlds in which our faculties and methods are unreliable.
Or what the agent should believe, but I will set that complication aside.
Given the above statement of the rule of belief, one might worry that this is much too strong: there, Lewis seems to say that a possibility believed to obtain isn’t properly ignored, not that a possibility not believed not to obtain isn’t properly ignored. But Lewis later clarifies that what he really means is that “a possibility may not be properly ignored if the subject gives it [...] a degree of belief that is sufficiently high,” (1996, p. 556) and context makes clear that “sufficiently high” is usually far below .5 (as it has to be, since otherwise almost no reasonable agent will have a “sufficiently high” degree of belief in any single possibility). So ‘the worlds consistent with X’s beliefs’ is a better approximation of Lewis’s rule than ‘the world (if there is one) uniquely consistent with X’s beliefs.’ It is nonetheless merely an approximation of what Lewis was after; one consequence of this choice will be that, contrary to Lewis’s (1996, p. 556) explicit intentions, our formalization will not allow for knowledge without belief in cases like that of the reliable but underconfident examinee.
Thanks to an anonymous referee for extremely helpful discussion on this point.
Though Holliday (2013) considers imposing the constraint corresponding to the rule of belief.
That is, we require that, in all our models, \(wR_Bv\) entails wEv. Cf Holliday (2013).
It’s worth noting that, while the counterexample to KK relies on the intransitivity of C, the counterexample to \(K\lnot K\) does not. For we can simply drop y from the example, rendering C irrelevant; the resulting model will validate KK, but \(K\lnot K\) will still fail at z.
Moreover, Bob Stalnaker tells me that, while Lewis initially thought that his theory should satisfy an S5 logic, he became convinced of the implausibility of the \(K\lnot K\) principle whilst presenting early versions of “Elusive Knowledge”. This change of heart coincided with the introduction of his extended discussion of the Rule of Actuality, and we saw earlier that this is the crucial passage warning us against the iteration-friendly formalization of Sect. 1.1.
Formally: \(\forall x\forall y (xR_Ky\rightarrow \forall z (xR_Bz\leftrightarrow yR_Bz))\). Given Lewis’s account, this claim can be true on every interpretation of ‘knows’ only if a difference in beliefs always makes for a difference in phenomenal state; Smithies (2014) develops a notion of ‘phenomenal state’ designed to have this feature, and argues that one’s justification supervenes on what phenomenal state (in this sense) one is in, so this might be a way of incorporating the introspection assumption into a broadly Lewisian account. It’s also worth noting that, even if we deny that agents in general always know what they believe, it is still interesting and surprising that the Lewisian account predicts our result to hold of those that do.
Recall that the actual world may not be salient to the attributors; the result thus doesn’t entail that the \(K\lnot K\) principle will be true.
A more radically conciliatory response would give up on the thought that worlds that aren’t ignored always need to be eliminated. To preserve any of the Lewisian spirit, we would then have to offer a different account of the role S plays in defining \(A^-\) or A.
This may be a little misleading, since, as I argue later, it’s not very intuitive to think that our standards for knowledge are higher than Soraya’s, which is what the notation suggests.
In saying this, we can be neutral on whether this is the belief expressed by her utterance, as it might not be if her conversational partners do not, in fact, take the same things seriously as she does. See DeRose (2004) for discussion.
One might worry that this is in tension with our stipulation that Soraya is ignoring the possibility of misleading lighting; for if she is, how could she even articulate what it takes to know\(_{hi}\)? If ‘ignoring’ is understood in terms of presuppositions, the worry is easily dissolved, since Soraya can think about the possibilities of misleading lighting when determining what she knows\(_{hi}\) without taking them seriously; that is, presumably, what most contextualists do when they agree that they know very little by sceptical standards. If ‘ignoring’ is understood in terms of salience, the worry has more bite; but we can still imagine that Soraya reflected earlier about what she would know\(_{hi}\) in various situation, and that those earlier beliefs, which do not feature amongst her conscious thoughts when she is looking at the wall, are sufficient to constitute a belief that she does not know\(_{hi}\) that the wall is red.
The thought that context-sensitive expressions embedded in attitude ascriptions are not simply interpreted relative to the context of utterance is quite familiar; see e.g. Stalnaker (1988) for a classic articulation and defence. It is frequently applied by contextualists to handle embeddings under ‘says that’ or ‘believes that’; see e.g. Cappelen and Hawthorne (2009).
This strategy does face an important challenge with embeddings under factive attitude verbs such as ‘knows’ [cf Weatherson (2008) and Lasersohn (2009, pp. 369–372)]. For it seems to predict that we could say ‘Soraya knows that roller-coasters are fun’, even though we hate them (provided only that we think that Soraya loves them and knows that she does), which is clearly incorrect. We thus need to supplement the simple shifting story with a, perhaps pragmatic, account of why knowledge ascriptions seem to entail the proposition which their complement would have expressed had it not been embedded. But note that simply denying that embedding under ‘knows’ (unlike embedding under ‘believes’) shifts the parameter is also implausible. For we can say ‘Soraya knows roller-coasters are fun’ even if we know that she (falsely) believes that we hate them.
A less optimistic reaction to these problems is to conclude that they sink contextualism about such terms as ‘fun’ or ‘might’, and should push us towards relativism or expressivism instead. But then it seems like we could equally well rehabilitate a broadly Lewisian account of ‘knows’ in a relativist or expressivist framework. Abandoning the contextualist aspect of Lewis’s account for relativism or expressivism seems to preserve all the applications Lewis makes of his contextualism; and it may have independent advantages, as claimed by MacFarlane (2005) for relativism and Chrisman (2007) for expressivism.
If we understand ‘ignoring’ in terms of salience, we cannot handle the cases of hypothetical \(K\lnot K\) failures described above, since (i) a scenario is salient even if it is discussed only hypothetically, and (ii) subject and attributor attend to all the same possibilities in that case.
Cf DeRose (1992).
Perhaps there will still be potential counterexamples in cases where attributors and subject do, intuitively, differ in their standards. Suppose that we are sceptics, refusing to dismiss any possibilities as abnormal. Should we describe ordinary people as failing to know without knowing that they fail? If so, such an ascription will have to be handled via the ‘shifting’ strategy developed in Sect. 2.1. But I actually have rather mixed feelings about this case; it strikes me as fairly natural to say that ordinary people, at least those that have encountered sceptical worries, do know that they don’t really know, while a similar claim sounds absurd to me in the case of Soraya (provided we hold fixed that, in Soraya’s case, the attributors don’t generally take misleading lighting seriously). If that’s right, it suggests that shifting, while perhaps possible, isn’t obligatory, which would make trouble for the hard-nosed response.
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Acknowledgments
I’m grateful to Kevin Dorst, Julien Dutant, Jeremy Goodman, Sophie Horowitz, Brendan de Kenessey, Justin Khoo, Harvey Lederman, Ginger Schultheis, Alex Silk, Declan Smithies, Jack Spencer, Jonathan Vogel, Roger White, Steve Yablo, and one anonymous referee for helpful comments and discussion. I’m especially grateful to Bob Stalnaker and a second anonymous referee, whose critical yet sympathetic comments have improved the following discussion immeasurably, with respect to both numerous specific details (too many to acknowledge individually) and overall structure.
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Salow, B. Lewis on iterated knowledge. Philos Stud 173, 1571–1590 (2016). https://doi.org/10.1007/s11098-015-0568-0
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DOI: https://doi.org/10.1007/s11098-015-0568-0