In this paper, we explore semantics for comparative epistemic modals that avoid the entailment problems shown by Yalcin (2006, 2009, 2010) to result from Kratzer’s (1991) semantics. In contrast to the alternative semantics presented by Yalcin and Lassiter (2010, 2011) based on finitely additive measures, we introduce semantics based on qualitatively additive measures, as well as semantics based on purely qualitative orderings, including orderings on propositions derived from orderings on worlds in the tradition of Kratzer (1991, 2012). All of these semantics avoid the entailment problems that result from Kratzer’s semantics. Our discussion focuses on methodological issues concerning the choice between different semantics.
Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic.
This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers.
As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study.
In this paper, I argue for a solution to the surprise exam paradox, designated student paradox, and variations theoreof, based on an analysis of the paradoxes using modal logic. The solution to the paradoxes involves distinguishing between two setups, the Inevitable Event and the Promised Event, and between the two-day and n-day cases of the paradoxes. For the Inevitable Event, the problem in the two-day case is the assumption that the student knows the teacher’s announcement; for more days, the student can know the announcement, and the base case of the student’s backward induction is correct, but there is a mistake in the induction step. For the Promised Event, even the base case is questionable. After defending this analysis, I argue that it also leads to a solution to a modified version of the surprise exam paradox, due to Ayer and Williamson, based on the idea of a conditionally expected exam.