Quantum Logic of Word Meanings: Concept Lattices in Vector Space Models

This paper systematically develops the logical and algebraic possibilities inherent in vector space models for language, considerably beyond those which are customarily used in semantic applications such as information retrieval and word sense disambiguation. The cornerstone of the approach lies in a simple implementation of the connectives of quantum logic as introduced by Birkhoff and von Neumann (1936), which defines the negation of a concept as the projection onto its orthogonal subspace, and the disjunction and conjunction of two concepts as the vector sum and intersection of their subspaces. This enables us to use the full lattice structure of a vector space, bringing these models much closer to traditional semantic lattice representations such as taxonomic concept hierarchies. We describe selected examples of this process with both negation and disjunction, and summarise experiments which show that the non-local nature of these connectives has clear advantages over their Boolean counterparts in removing the synonyms and neighbours of negated terms in information retrieval, as well as removing the negated terms themselves. Having thus validated the approach, we explore its implications for assigning semantics to some compositional phrases, showing cases where a quantum interpretation is preferable to a traditional Boolean formulation (and vice versa). Finally, we draw attention to the danger that quantum connectives may overgeneralise, and suggest another (also non-Boolean) alternative.