Margins for error in context
Résumé
According to the epistemic theory of vagueness defended in particular by Sorensen (2001) and Williamson (1994: 237), vagueness is due to our limited powers of discrimination: looking at a particular shade of red fabric, I may not be able to recognize that it is red, as a result of the specific granularity of my perceptual apparatus, which for instance makes the shade look somewhere between orange and red to me. Conversely, whenever I am confident that a particular shade of color is red, then this means that a slight variation in color should leave intact the fact that the shade is red. In Williamson's account of vagueness, this idea is expressed in terms of what Williamson calls margin for error principles for knowledge: whenever my knowledge is inexact in the sense of being approximative, it requires a sufficient margin for error in order to hold. More abstractly, the margin for error principle says that in order for me to know that some property P holds of an object d, then a slight modification of some relevant parameter in the object d should leave it in the extension of P . Expressed in terms of propositions and contexts, this means that in order to know that some proposition p holds in a context w, then p should still hold in a context w′ that is only slightly different from w. An important consequence of Williamson's margin for error semantics for inexact knowledge concerns the problem of iterations of knowledge. When applied to knowledge itself, the margin for error principle says that for me to know that I know p in a context w, I should know p in all contexts sufficiently similar to w. If knowledge is taken to be positively introspective, namely to be such that I know that I know p whenever I know p, then one can build a soritical argument to the effect that if I know p in a context w, then step by step, I should continue to know p even in contexts that are informationally very remote from w, including contexts in which p is false. For Williamson, the contradiction shows that knowledge, just like other mental states, is not luminous, namely that one can know p without knowing that one knows p. In Dokic & E ́gre ́ (2004), an axiomatic version of Williamson's argument was challenged, resting on the idea that higher-order knowledge and first-order knowledge need not obey the samemarginsoferror.Morerecently,inBonnay&E ́gre ́(2006),weshowedhowWilliamson's margin for error semantics for knowledge can be modified in order to let the margin for error principle and the principle of positive introspection coexist. Despite their common inspiration, there remains a gap between the two approaches. On the one hand, as we shall see below, it can be shown that our semantic approach, like the one of Dokic & E ́gre ́, turns out to deny one of the premises of the syntactic version of Williamson's argument, namely the idea that the agent can have systematic knowledge of her margin of error. On the other hand, although both the syntacticapproachfollowedbyDokic&E ́gre ́andthesemanticapproachputforwardbyBonnay & E ́gre ́ consist in adding a contextualist component to the evaluation of epistemic sentences, this common ingredient needs further articulation. In Bonnay and E ́gre ́ (2006), in particular, we formulate a two-dimensional semantics for knowledge whose contextualist inspiration can be made more explicit if one introduces actuality operators. Moreover, the introduction of actuality operators will also allow us to handle the axiomatic version of Williamson's argument, while remaining closer to its initial premises. In what follows, our aim will therefore be to spell out the details of our contextualist approach to Williamson's paradox. The main point which we will develop concerns the idea that iterations of knowledge should remain anchored to the initial context of epistemic evaluation, in a way that blocks the soritical progression. In this respect, the paper should be seen as an effort to bring the two paradigms of epistemicism and contextualism about vagueness closer together. The paper is structured as follows: in the first section we review the syntactic and semantic versions of Williamson's paradox; in section 2 we present a solution to the semantic version in the framework of Centered Semantics, a two-dimensional semantics for epistemic logic. We discuss in the following section the connection between Centered Semantics and actuality operators, and use this connection for the analysis of the syntactic version of the paradox. We conclude with some comparisons in section 4, in particular with Kamp's contextualist treatment of the sorites.