Opaque Updates
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Opaque Updates Michael Cohen1 Received: 14 February 2020 / Accepted: 11 August 2020 / © Springer Nature B.V. 2020
Abstract If updating with E has the same result across all epistemically possible worlds, then the agent has no uncertainty as to the behavior of the update, and we may call it a transparent update. If an agent is uncertain about the behavior of an update, we may call it opaque. In order to model the uncertainty an agent has about the result of an update, the same update must behave differently across different possible worlds. In this paper, I study opaque updates using a simple system of dynamic epistemic logic suitably modified for that purpose. The paper highlights the connection between opaque updates and the dynamic-epistemic principles Perfect-Recall and No-Miracles. I argue that opaque updates are central to contemporary discussions in epistemology, in particular to externalist theories of knowledge and to the related problem of epistemic bootstrapping, or easy knowledge. Opaque updates allow us to explicitly investigate a dynamic (or diachronic) form of uncertainty, using simple and precise logical tools. Keywords Dynamic epistemic logic · Externalism · Basic knowledge · Bootstrapping · Introspection · Perfect-recall · No-miracles · Bayesian update · Opacity
1 Introduction There is a widespread notion of update in formal epistemology that can be semantically summarized in Fig. 1. We have some prior, or initial, model on the left-hand of the figure, containing both p and not-p worlds, representing a situation in which an agent is uncertain whether p is the case. The event E of receiving the information that p results in a new model in which the not-p worlds are eliminated (the model to the right), a model representing a situation in which there is no uncertainty as to p. This simple sketch of Michael Cohen
[email protected] 1
Philosophy Department, Stanford University, Stanford, CA, USA
M. Cohen
Fig. 1 Update as a world insensitive function
updating is at the basis of many systems that represent information change, including the Stalnakerian notion of assertion, Dynamic Semantics, Bayesian updating, and various dynamic epistemic logics. Of course, each implementation of this basic skeleton idea is different, but here I want to point our attention to an assumption that can be detected even at this level of abstraction. The picture in Fig. 1 portrays the event of learning p as a transition from one model to the other, a model transformer. In other words, the event of receiving the information that p is understood as a function (sometimes partial) from epistemic models to epistemic models. Since such a function is assumed to send us from one model to another, it is insensitive to the world of evaluation. In other words, the function behaves the same at every possible world of the prior model—at each world, the function sends us to the same posterior model. Put differently again, we don’t need to know which world is considered actual in order to compute the model which results from an u
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