Non-deterministic matrices: theory and applications to algebraic semantics

Abstract

Non-deterministic Matrices (Nmatrizes) are a generalization of the usual concept of multiple-valued matrix. However, in Nmatrizes, the truth value of a given complex formula may be chosen non-deterministically from a certain nonempty set of options, while the usual logical matrices (two-valued or many-valued) the truth value of a complex formula is determined solely by the truth value of its subformulas. An important utility of Nmatrizes is the fact that they serve as finitary semantics, and therefore decidable for many logics that have only one infinite matrix characteristic as semantics. Therefore, this project aims to initially develop a thorough study of usual logical matrices and the Nmatrices, this study will be done from the perspective of comparison to identify unusual properties of matrices that Nmatrizes not have and vice versa. In particular, we will focus our efforts in the study of so-called abstract algebraic logic, inaugurated by W. Blok and D. Pigozzi, aiming to extend the usual techniques involving matrices for the more general context of Nmatrizes. Besides contributing to the formulation of a theory of Nmatrizes extended, this project intends to use the new results related to Nmatrizes, along with those already existing in the literature, to present an algebraic semantics of finite Nmatrizes for some modal logics and some paraconsistent logics such how systems Cn of da Costa that do not have algebraic semantics in the general sense of Blok-Pigozzi, resolving an open problem in the literature. (AU)