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：针对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非经典模态逻辑语义的一般性方法。我们以完备格$L$为起点，该格配备将$1$映射至$0$的反序运算$\neg、完全可乘运算$\Box$以及完全可加运算$\Diamond$。此类格扩张可通过集合$X$及满足若干一阶条件的二元关系$\vartriangleleft$、$R$和$Q$进行表示，分别用于刻画$(L、\neg)$、$\Box$与$\Diamond$。事实上，任何配备此类$\neg$、可乘$\Box$及可加$\Diamond$的格$L$均可嵌入框架$(X，\vartriangleleft,R,Q)$的命题格中。基于我们近期对"基础逻辑"的研究，本文重点考察以下情形：$\neg$具有对偶自伴随性（若$a\leq \neg b$则$b\leq\neg a$）且满足$\Diamond \neg a\leq\neg\Box a$。此时可约束表示形式使$R=Q$，即仅需向$(X，\vartriangleleft)$添加单一关系即可同时表示$\Box$与$\Diamond$。借助这些结果，我们证明基础模态逻辑系统相对于双关系结构$(X，\vartriangleleft,R)$初等类的可靠性与完备性。