  Non-classical generalizations of classical modal logic have been developed in
the contexts of constructive mathematics and natural language semantics. In
this paper, we discuss a general approach to the semantics of non-classical
modal logics via algebraic representation theorems. We begin with complete
lattices $L$ equipped with an antitone operation $\neg$ sending $1$ to $0$, a
completely multiplicative operation $\Box$, and a completely additive operation
$\Diamond$. Such lattice expansions can be represented by means of a set $X$
together with binary relations $\vartriangleleft$, $R$, and $Q$, satisfying
some first-order conditions, used to represent $(L,\neg)$, $\Box$, and
$\Diamond$, respectively. Indeed, any lattice $L$ equipped with such a $\neg$,
a multiplicative $\Box$, and an additive $\Diamond$ embeds into the lattice of
propositions of a frame $(X,\vartriangleleft,R,Q)$. Building on our recent
study of "fundamental logic", we focus on the case where $\neg$ is dually
self-adjoint ($a\leq \neg b$ implies $b\leq\neg a$) and $\Diamond \neg
a\leq\neg\Box a$. In this case, the representations can be constrained so that
$R=Q$, i.e., we need only add a single relation to $(X,\vartriangleleft)$ to
represent both $\Box$ and $\Diamond$. Using these results, we prove that a
system of fundamental modal logic is sound and complete with respect to an
elementary class of bi-relational structures $(X,\vartriangleleft, R)$.
