Questions, topics and restricted closure

Abstract

Single-premise epistemic closure is the principle that: if one is in an evidential position to know that $\varphi$ where $\varphi$ entails $\psi$, then one is in an evidential position to know that $\psi$. In this paper, I defend the viability of opposition to closure. A key task for such an opponent is to precisely formulate a restricted closure principle that remains true to the motivations for abandoning unrestricted closure but does not endorse particularly egregious instances of closure violation. I focus on two brands of epistemic theory (each the object of sustained recent interest in the literature) that naturally incorporate closure restrictions. The first type holds that the truth value of a knowledge ascription is relative to a relevant question. The second holds that the truth value of a knowledge ascription is relative to a relevant topic. For each approach, I offer a formalization of a leading theory from the literature (respectively, that of Jonathan Schaffer and that of Stephen Yablo) and use this formalization to evaluate the theory’s adequacy in terms of a precise set of desiderata. I conclude that neither theory succeeds in meeting these desiderata, casting doubt on the viability of the underlying approaches. Finally, I offer a novel variant of the topic-sensitive approach that fares better.

Appendix: Proof of Theorem 1

The first part is trivial. We prove the second by contraposition: assume that $atoms(\psi ) \nsubseteq atoms(\varphi )$. We construct resolution model ${\mathscr {M}}$ such that ${\mathscr {M}}, w\,\vDash\,K\varphi \wedge \lnot K\psi$. Let $\lambda ^+ = \lambda ^+_\varphi \wedge \lambda ^+_\psi \wedge \lambda ^+_{\varphi , \psi }$ be some conjunction of literals that contains all and only the atoms that occur in one or both of $\varphi$ and $\psi$, and such that $\lambda ^+$ tautologically entails $\varphi$ (and so also entails $\psi$). In particular, let $\lambda ^+_\varphi$ contain all and only the atoms that occur in only $\varphi$; let $\lambda ^+_\psi$ contain all and only the atoms that occur in only $\psi$; let $\lambda ^+_{\varphi , \psi }$ contain all and only the atoms that occur in both $\varphi$ and $\psi$. Further, let $\lambda ^- = \lambda ^-_\varphi \wedge \lambda ^-_\psi \wedge \lambda ^-_{\varphi , \psi }$ be some conjunction of literals that contains all and only the atoms that occur in one or both of $\varphi$ and $\psi$, and such that $\lambda ^-$ tautologically entails $\lnot \psi$ (and so also entails $\lnot \varphi$). In particular, let $\lambda ^-_\varphi$ contain all and only the atoms that occur in only $\varphi$; let $\lambda ^-_\psi$ contain all and only the atoms that occur in only $\psi$; let $\lambda ^-_{\varphi , \psi }$ contain all and only the atoms that occur in both $\varphi$ and $\psi$. Now, let $W = \{w_1, w_2, w_3\}$. Set valuation ${\mathfrak {v}}$ so that $\lambda ^+$ holds at $w_1$ (and so $\varphi$ holds at $w_1$); $\lambda ^-$ holds at $w_2$ (and so both $\lnot \psi$ and $\lnot \varphi$ hold at $w_2$); and $\lambda ^-_\varphi \wedge \lnot \lambda ^-_\psi \wedge \lambda ^-_{\varphi , \psi }$ holds at $w_3$. Note that since $\lambda ^-$ tautologically entails $\lnot \varphi$ and $\lambda ^-_\psi$ contains no atoms from $\varphi$, it must be that $\lambda ^-_\varphi \wedge \lambda ^-_{\varphi , \psi }$ entails $\lnot \varphi$, and so $\lnot \varphi$ holds at $w_3$. Note then that $\{w_2, w_3\}$ is a (and the only) relevant alternative proposition to $\varphi$ in this model (since it is the proposition expressed by $\lambda ^-_\varphi \wedge \lambda ^-_{\varphi , \psi }$) but $\{w_2\}$ is not. Note further that $\{w_2\}$ is a relevant alternative to $\psi$ (expressed by $\lambda ^-_\psi \wedge \lambda ^-_{\varphi , \psi }$). Finally, set $E_{w_1} = \{w_1, w_2\}$ and set $\preceq _{w_1}$ so that $w_3 \preceq _{w_1} w_2$. Thus, the nearest $\lambda ^-_\psi \wedge \lambda ^-_{\varphi , \psi }$ world is compatible with the evidence at $w_1$, while the nearest $\lambda ^-_\varphi \wedge \lambda ^-_{\varphi , \psi }$ is not. In total: ${\mathscr {M}}, w_1\,\vDash\,K\varphi \wedge \lnot K\psi$.