Vectors for Topics: The use of Topical Vector Spaces in Epistemic Logics

Abstract Semantics for logics with Topic Sensitive Intentional Modal operators require a structure, often a mereology, of topics. These structures have varied from mere semi-lattices and sets of partitions of possible worlds to collections of subalgebras. We will argue that a natural representation of topic is as vectors in a vector space. This idea is motivated by the use of vector space models in large language models. There, the relation between vectors corresponds to the semantic relation between the topics of the words being modeled by the vectors. As such, we reconstruct a logic of conditional hyperintensional belief given in Özgün and Berto (2021), replacing their semi-lattice of topics with a vector space of topics. In these vector-based models, the containment relation of topics is given an intuitive reading that (unlike the semi-lattice reading) is straightforwardly compatible with a selection of modifications and extensions that we will discuss. We are offering a new semantics for a logic: the main upshots are a formal framework for topical vector spaces, and that the introduced semantics is philosophically satisfying.