Expressive Power of the Positional Operator R: a Case Study in Modal Logic and Modal Philosophy

Abstract

Theories of modal notions belong to subjects of permanent study for logicians as well as all philosophers. We describe some fundamental ideas concerning kinds of modal expressions. Then we deliver a concise introduction to basic concepts of modal logic with connectives M of possibility and L of necessity. We describe typical normal modal logic as sets of theorems as well as by means of relational semantics. In the focal points we present some important outcomes in the field of positional logic containing the operator R, with special emphasis on systems MR and MRQ. We show how relational semantics of typical modal logic may be reconstructed within positional logic MRQ with one binary predicate, and therefore normal modal logics turn out to be algebraically proper parts of positional logic or theories based on it. Some philosophical questions and insights are being raised on the ground of those formal research.