Abstract
Formal models of appearance and reality have proved fruitful for investigating structural properties of perceptual knowledge. This paper applies the same approach to epistemic justification. Our central goal is to give a simple account of The Preface, in which justified belief fails to agglomerate. Following recent work by a number of authors, we understand knowledge in terms of normality. An agent knows p iff p is true throughout all relevant normal worlds. To model The Preface, we appeal to the normality of error. Sometimes, it is more normal for reality and appearance to diverge than to match. We show that this simple idea has dramatic consequences for the theory of knowledge and justification. Among other things, we argue that a proper treatment of The Preface requires a departure from the internalist idea that epistemic justification supervenes on the appearances and the widespread idea that one knows most when free from error.
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Notes
For related models of statistical knowledge, see Dorr et al. (2014) and Goodman and Salow (2018). As in economics, the hope is that these models will provide insight into the target phenomenon, not that they capture exceptionless generalizations. In keeping with this literature, throughout we make at least three significant idealizations. First, as usual in epistemic logic, we assume the relevant agents are logically omniscient. Second, we assume that our agents are cognitively unimpaired and attentive. Finally, we focus primarily on cases of perceptual knowledge, where the agent receives perceptual information about the world and makes deductions on this basis.
In Sect. 4.3, we will entertain (though not endorse) the view that whether one is justified in believing one is subject to error depends on how much error one is subject to. We specify that in fact some source is wrong in order not to prejudge this issue.
There is a second, common sense in which an agent can be described as being in error when she has beliefs which are false. We will we employ ‘error’ exclusively in the former sense, to denote the failure of appearances to match reality.
Littlejohn and Dutant (2020) offer the only extended discussion of this issue that we know of.
Throughout, we follow Williamson (2013) in making the simplifying assumption that the agent knows everything that they are in a position to know. Readers who dislike this aspect of the idealization can treat the model as a model of what we are in a position to know.
Suppose S is justified in believing p, so that MKp holds. Then Consistency says that S is not justified in believing \(\lnot p\), so that \(\lnot MK \lnot p\) holds, which is just KMp.
Suppose that MKp. Then at some v accessible from w, p holds throughout. But by Convergence, every world u accessible from w will see some world in Rv, itself contained in p. So at every world u accessible from w, p is possible. So KMp.
Reduction implies that Justification Agglomeration is equivalent to the requirement that MKp and MKq entail \(MK(p \wedge q)\). Within epistemic logic, this condition is equivalent to the following.
Agg. \(\forall w \forall v \forall u (v, u\in R(w) \supset \exists z \in R(w): R(z) \subseteq R(v) \cap R(u))\)
After all, suppose MKp and MKq are true at w. Then w can access a world v where p is known, and a world u where q is known. Agg implies that w can also access a world z where at least as much is known as at both v and u, and so Agg implies that \(MK(p \wedge q)\) is true at w.
See Goodman and Salow (2018) and Beddor et al. (2019) for alternative conceptions of knowledge in terms of sufficient normality that differ from Williamson (2013) and our own model by validating Knowledge Luminosity. Goodman and Salow (2018)[fn10] observe that their theory can be made to satisfy a margin-for-error constraint, thereby invalidating KK, in a manner very similar to that described in this section. We are grateful to an anonymous referee for pointing this out to us.
An alternative interpretation of the basic model appeals to modal distance instead of normality. Here, an agent knows only those propositions which are the case throughout the set of worlds which fall within a specified modal distance of her location. According to a standard gloss, w is closer to v than it is to z just in case what is the case at w could have been the case at v more easily than it could have been the case at z. Given this gloss, the ideology of modal distance offers a simple way of conceptualizing of a safety-theoretic account of knowledge, on which S knows that p iff p could not easily have been false (Sosa 1999; Williamson 2000, 2011, 2014; Pritchard 2005, 2007).
Unfortunately, it seems that Distance cannot be understood in terms of modal distance. Let the modal distance between (r, a) and \((r',a')\) be equal to the sum of the difference between r and \(r'\) and the difference between a and \(a'\).
$$\begin{aligned} d((r,a), (r', a')) = |r-r'|+|a-a'|. \end{aligned}$$Distance implies that what an agent knows cannot be characterized in terms of d. The set of worlds R-accessible from (r, a) does not correspond to a region of points within some distance of (r, a). Consider the model in Fig. 1. (77.5,75) is d-closer to (82.5,75) than it is to (70,75). But Distance implies that though (77.5,75) can see (70,75), it cannot see (82.5,75). For this reason, we embrace a normality interpretation of Distance instead of an interpretation in terms of modal distance.
See Stalnaker (2006, §7) for a similar proposal.
We have focused on cases where each quantity is measured on a single scale (in our case, temperature) and has the same margin for error. But shouldn’t we also be interested in systems which measure quantities of different kinds or which are subject to different margins? For example, we might want to model a system comprising instruments for measuring temperature, humidity and light levels within a single room, where each instruments differs in its reliability. To do this, we could simple map each quantity to a common scale, renormalised relative to an (arbitrary) margin constant. While this presents no particular issue in theory, to avoid the added complexity, we will continue to focus on systems measuring quantities associated with a single scale and assume they have the same margin.
In some Gettier cases in the literature, appearance and reality coincide. However, these examples are not characterized by appearance/reality models. Gettier cases in these models are identified in Williamson (2013), and do require the agent to be subject to a non-zero level of error.
Suppose the agent justifiedly believes p and justifiedly believes q at s. Then by Supervenience she justifiedly believes p and justifiedly believes q in the corresponding good case \(s_{gc}\). But then by Optimism she knows p and knows q at \(s_{gc}\); so since knowledge agglomerates, she knows \(p \wedge q\) at \(s_{gc}\); so by Optimism again she is justified in believing \(p \wedge q\) at \(s_{gc}\); so by Supervenience again she is justified in believing \(p \wedge q\) at s.
The strongest claim known about quantity i at a global good case is that the real value is within m of its apparent value. Yet the strongest claim an agent is justified in believing at a sequence is the strongest claim known at the corresponding global good case.
We are grateful to an anonymous referee for encouraging us to address the issue of one-dimensional models in our framework.
Here another natural option would appeal to squared distance. We are sympathetic to this alternative, but stick to the simpler definition above throughout.
As a simplifying idealization we assume that this constant is a real number, but we could equally identify it with a real interval instead, to model the case where the total quantity of error is normal as long as it falls within some range.
Justification Agglomeration holds in the basic model because epistemic accessibility has a sphere structure, nested around the good case. By contrast, our model rejects Agg (that \(\forall w \forall v \forall u (v, u\in R(w) \supset \exists z \in R(w): R(z) \subseteq R(v) \cap R(u))\)) in the above case. s can see \( s' \) and \( s'' \). The only world which sees a subset of the points seen by both s and \( s'' \) is the global good case. This is because the global good case is the only case at which the temperature in both locations is known to fall within [70, 80]. Yet s cannot see the global good case. So s cannot see a point which sees only points seen by both \(s'\) and \( s''\).
Proofs of observations in this section are confined to a supplementary appendix, available at https://www.dropbox.com/s/xe3aqh6lom4gan7/error_appendix_8.29.20.pdf?dl=0.
Any world which is accessible from both the global good case and from a world with a normal level of error (where \(\textsf {Error}(s)=\textsf {NormalError}\)) is accessible from every world. Yet for accessibility to be convergent, each world with a normal level of error must be able to see some world which can also be seen from the global good case. Otherwise, at any world which could see both the global good case and that world with a normal level of error, an agent would be justified in believing two pairwise inconsistent propositions. As a result, as long as justification is convergent, we know that there are some worlds in the model which are accessible from every world. Accordingly, accessibility will also be n-wise convergent. Any set of n worlds will be able to see a world in common.
For illustration, consider Fig. 3. Local consistency requires that the blue region has a non-empty overlap with the intersection of the red and yellow regions (i.e., the region accessible from \(s_{gc}\)). Yet the overlap of these three regions will be accessible from any point. So global consistency is also satisfied.
To see why, suppose that the proposition that error is greater than 0 is justified at every world. Then, by Reduction, there must be at least some world at which error is known to be greater than 0. This will be a world which cannot see the global good case (since the total error at the global good case is 0). Since Justification Agglommeration fails, there is something known at the global good case which is not known at any other world (that is, there is no s such that \(\mathbf{R} (s)\subseteq \mathbf{R} (s_{gc}\))). What is known at the global good case is justified at the global good case, but not at any world which cannot see the global good case. So Supervenience fails.
From a logical point of view, Rosenkranz (2018) comes closest to our model. Like Williamson (2013), this theory rejects Knowledge Luminosity while retaining Reduction and Consistency. In addition, Rosenkranz (2018) departs further from Williamson (2013) by denying Justification Agglomeration.
One significant feature of Rosenkranz (2018) is the decision to define justification in terms of being in a position to know, rather than knowledge itself. In addition, Rosenkranz (2018) follows Heylen (2016) (and Rosenkranz 2016) in adopting a non-normal logic of being in a position to know, so that being in a position to know does not agglomerate. Rosenkranz (2018) suggests that an agent can be in a position to know p and be in a position to know \(\lnot Kp\) without being in a position to know the Fitch conjunction \(p \wedge \lnot Kp\).
For example, imagine that the agent is alone in a room with an even number of books on the wall. They could easily count the number of books, but have not. In this case, Heylen (2016) suggests that the agent is in a position to know that the number of books is even. But since they have not counted yet and know this, they know and hence are in a position to know that they don’t know that the number of books is even.
Whether this argument is compelling depends on exactly how we understand the notion of being in a position to know. Schaffer (2007) embraces the normality of being in a position to know because he understands being in a position to know in terms of satisfying the evidential component of knowledge. Notice that in the case above, the agent does not actually possess evidence regarding the number of books, though they easily could have. More precisely, Schaffer (2007) follows Williamson (2000) and Hawthorne (2004) in understanding the notion of being position to know in terms of ‘only needing a belief-based-on-competent-deduction-while-retaining-knowledge-of-the-premises in order to know’ (249). In the case above, the agent needs to perform more than just a competent deduction in order to know that the number of books is even. The agent must also gather new evidence.
In this paper, we are perfectly happy to interpret justification in terms of being in a position to know, rather than knowledge. But our own investigation assumes that knowledge or being in a position to know is a normal modal operator.
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Acknowledgements
We are grateful to Bob Beddor, Julien Dutant, Jeremy Goodman, John Hawthorne, Dan Hoek, Ben Holguin, Nico Kirk-Giannini, Lauren Lyons, and Ezra Rubenstein, along with audiences at Rutgers and the Melbourne Logic Seminar. We are particularly grateful to an anonymous referee at Philosophical Studies.
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Carter, S., Goldstein, S. The normality of error. Philos Stud 178, 2509–2533 (2021). https://doi.org/10.1007/s11098-020-01560-6
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DOI: https://doi.org/10.1007/s11098-020-01560-6