Fullness and Decidability in Continuous Propositional Logic

Abstract

In this paper we consider general continuous propositional logics and prove some basic properties about them. First, we characterize full systems of continuous connectives of the form {¬,,f} where f is a unary connective. We also show that, in contrast to the classical propositional logic, a full system of continuous propositional logic cannot contain only one continuous connective. We then construct a closed full system of continuous connectives without any constants. Such a system does not have any tautologies. For the rest of the paper we consider the standard continuous propositional logic as defined by Yaacov, I.B and Usvyatsov, A. We show that Strong Compactness and Craig Interpolation fail for this logic, but approximated versions of Strong Compactness and Craig Interpolation hold true. In the last part of the paper, we introduce various notions of satisfiability, falsifiability, tautology, and fallacy, and show that they are either NP-complete or co-NP-complete.