Modalities in vector logic

Abstract

Vector logic is a mathematical model of logic in which the truth values are mapped on elements of a vector space. The binary logical functions are performed by rectangular matrices operating on the Kronecker product of their vectorial arguments. The binary operators acting on vectors representing ambiguous (fuzzy) truth values, generate many-valued logics. In this article we show that, within the formalism of vector logic, it becomes possible to obtain truth-functional definitions of the modalities “possibility” and “necessity”. These definitions are based on the matrix operators that represent disjunction and conjunction respectively, and each modality emerges by means of an iterative process. The construction of these modal operators was inspired in Tarski’s truth-functional definition of possibility for the 3-valued logic of Łukasiewicz. The classical Aristotelian link between possibility and necessity becomes, in the basic vector logic, a corollary of the De Morgan’s connection between disjunction and conjunction. Finally, we describe extensional versions of the existential and universal quantifiers for vector logics. © 1994 Duke University Press.