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∧♢¬p ('p, but it might be that
not p') appears to be a contradiction, ♢¬p does not entail ¬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 theories predict that
p∧♢¬p, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of
entailment that does not allow substitution of logical equivalents; these theories underpredict the infelicity
of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that
intuitively fail, like distributivity and disjunctive syllogism, 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. Then we show how to extend our semantics
to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain
what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.