Chapter 1 Word Vectors and Quantum Logic Experiments with negation and disjunction

A calculus which combined the flexible geometric structure of vector models with the crisp efficiency of Boolean logic would be extremely beneficial for modelling natural language. With this goal in mind, we present a formulation for logical connectives in vector spaces based on standard linear algebra, giving examples of the use of vector negation to discriminate between different senses of ambiguous words. It turns out that the operators developed in this way are precisely the connectives of quantum logic (Birkhoff and von Neumann, 1936), which to our knowledge have not been exploited before in natural language processing. In quantum logic, arbitrary sets are replaced by linear subspaces of a vector space, and set unions, intersections and complements are replaced by vector sum, intersection and orthogonal complements of subspaces. We demonstrate that these logical connectives (particularly the orthogonal complement for negation) are powerful tools for exploring and analysing word meanings and show distinct advantages over Boolean operators in document retrieval experiments. This paper is organised as follows. In Section 1.1 we describe some of th