In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness--a world makes a disjunction true only if it makes one of the disjuncts true--which classically implies totality--for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

翻译：在经典逻辑及其扩展（如模态逻辑）的传统语义学中，命题被解释为集合的子集（如离散对偶性）或Stone空间的闭开集（如拓扑对偶性）。此类集合中的点可视为“可能世界”，而世界的核心属性是素性——世界仅当使析取项之一为真时才使析取式为真——这在经典逻辑中蕴含了完全性——对于每个命题，世界要么使其为真，要么使其否定为真。本章概述一种更为通用的逻辑语义学方法，即可能性语义学，它用可能不完整的“可能性”替代可能世界。在经典可能性语义学中，命题被解释为偏序集的正则开集（如集合论力迫）或上Vietoris空间的紧致正则开集（如近期“无选择Stone对偶性”理论）。作为可能性看待的这些集合元素可能是不完整的，即在未确定哪个析取项为真的情况下使析取式为真。我们阐释了如何将可能性用于经典逻辑和模态逻辑的语义学，并将其推广至直觉主义逻辑的语义学。其目标在于：克服或深化传统语义学的不完全性结果，避免传统语义学的非构造性，并为新语言的解释提供更丰富的结构。