Finite-valued Logics for Information Processing

Abstract

We examine the issue of collecting and processing information from various sources, which involves handling incomplete and inconsistent information. Inspired by the framework first proposed by Belnap, we consider structures consisting of information sources which provide information about the values of formulas of classical propositional logic, and a processor which collects that information and extends it by deriving conclusions following from it according to the truth tables of classical logic, applied forward and backward. Our model extends Belnap's in allowing the sources to provide information also about complex formulas. As that framework cannot be captured using finite ordinary logical matrices, if we want to represent each of the relevant logics with a single matrix, we employ Nmatrices for that purpose. In opposition to the approach proposed in our earlier work, we assume that the information sources are reasonable, i.e. that they provide information consistent with certain coherence rules. We provide sound and complete sequent calculi admitting strong cut elimination for the logic of a single information source, and (several variants of) the logic generated by the source and processor structures described above. In doing this, we also provide new characterizations for some known logics. We prove that, in opposition to the variantwith unconstrained information sources considered earlier, the latter logic cannot be generated by structures with any bounded number of sources.