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 $(X,\vartriangleleft, R)$.

翻译：经典模态逻辑的非经典推广已在构造性数学和自然语言语义学等背景下得到发展。本文通过代数表示定理，探讨了非经典模态逻辑语义的一般性方法。我们以完备格 $L$ 为出发点，配备一个将 $1$ 映射至 $0$ 的反序运算 $\neg$、一个完全可乘运算 $\Box$ 以及一个完全可加运算 $\Diamond$。此类格扩张可通过一个集合 $X$ 及其上的二元关系 $\vartriangleleft$、$R$ 和 $Q$ 表示，这些关系满足若干一阶条件，分别用于刻画 $(L,\neg)$、$\Box$ 和 $\Diamond$。事实上，任何配备上述 $\neg$、可乘 $\Box$ 和可加 $\Diamond$ 的格 $L$ 均可嵌入到一个框架 $(X,\vartriangleleft,R,Q)$ 的命题格中。基于我们近期对"根本逻辑"的研究，我们重点关注 $\neg$ 满足对偶自伴性（$a\leq \neg b$ 蕴含 $b\leq\neg a$）且 $\Diamond \neg a\leq\neg\Box a$ 的情形。在此条件下，表示可以约束为 $R=Q$，即仅需在 $(X,\vartriangleleft)$ 上增加一个单一关系即可同时表示 $\Box$ 和 $\Diamond$。利用这些结果，我们证明了一个根本模态逻辑系统相对于双关系结构 $(X,\vartriangleleft,R)$ 的一个初等等价类是可靠且完备的。