  Epistemic modals have peculiar logical features that are challenging to
account for in a broadly classical framework. For instance, while a sentence of
the form $p\wedge\Diamond\neg p$ ('$p$, but it might be that not $p$') appears
to be a contradiction, $\Diamond\neg p$ does not entail $\neg p$, which would
follow in classical logic. Likewise, the classical laws of distributivity and
disjunctive syllogism fail for epistemic modals. Existing attempts to account
for these facts generally either under- or over-correct. Some predict that
$p\wedge\Diamond\neg p$, a so-called epistemic contradiction, is a
contradiction only in an etiolated sense, under a notion of entailment that
does not always allow us to replace $p\wedge\Diamond\neg p$ with a
contradiction; these theories underpredict the infelicity of embedded epistemic
contradictions. Other theories savage classical logic, eliminating not just
rules that intuitively fail but also rules like non-contradiction, excluded
middle, De Morgan's laws, and disjunction introduction, which intuitively
remain valid for epistemic modals. In this paper, we aim for a middle ground,
developing a semantics and logic for epistemic modals that makes epistemic
contradictions genuine contradictions and that invalidates distributivity and
disjunctive syllogism but that otherwise preserves classical laws that
intuitively remain valid. We start with an algebraic semantics, based on
ortholattices instead of Boolean algebras, and then propose a more concrete
possibility semantics, based on partial possibilities related by compatibility.
Both semantics yield the same consequence relation, which we axiomatize. We
then show how to lift an arbitrary possible worlds model for a non-modal
language to a possibility model for a language with epistemic modals.
