Vector logics: The matrix-vector representation of logical calculus

Abstract

We describe a kind of logical calculus based on matrix operators. This representation is inspired on a neural network model for a context-dependent associative memory. The binary operations of classical propositional calculus (e.g. conjunction, disjunction, implication, Sheffer's connective) are represented by rectangular matrices that operate over the Kronecker product of two vectors that belong to an arbitrary vector space. The basic matrix operators verify some of the classical laws of logical connectives, like De Morgan's Laws and the Law of Contraposition. We show that the matrices that execute the logical operations negation, conjunction and disjunction are also able to recognize the set operations complementation, intersection and union, respectively. In the presence of fuzzy data the basic binary matrix operators generate a many-valued logic.