Vector Logic: A Natural Algebraic Representation of the Fundamental Logical Gates

Vector logic is a matrix–vector representation of the logical calculus inspired in neural network models. In this algebraic formalism, the truth values map on orthonormal Q-dimensional vectors, the monadic operations are represented by square matrices, and the dyadic operations produce rectangular matrices that act on the Kronecker product of the vector truth values. In this formalism, the theorems and tautologies of classical logic are demonstrated using the rules of matrix algebra. In the present work, we analyse a three-valued vector logic that adds to the ‘yes’ and ‘no’ vectors, a third ‘uncertain’ vector that represents the truth value corresponding to undecidable propositions. Fuzziness is produced both via linear combinations of ‘yes’ and ‘no’ vectors, and by the supplementary dimension of the logical vector subspace. We describe the basic matrix operators, and we show that for this three-valued vector logic, the modalities ‘possibility’ and ‘necessity’ are simple square matrices instead of infinite recursive processes. Finally, we explore the application of this formalism to represent the complex-valued operator $\sqrt{\mathrm{NOT}}$, and the usefulness of vector logic to understand the powers and limitations of some reversible logical computations.