A Note on the signed occurrences of propositional variables

Abstract

This note concerns the positive and negative occurrences of propositional variables. Just like the theory of infectious truth-values provides an algebraic understanding of the position according to which identity of subject-matter between two formulas can approximated syntactically by the identity of propositional variables occurring in these formulas, we develop an algebraic understanding of the similar position which considers signed occurrence (i.e., positive or negative) instead of mere occurrence. We apply our framework to classical logic, yielding this first (to our knowledge) semantic characterisation of the logic called SCL by Hornischer (2020). Moreover, we settle two conjectures by Humberstone (2014) which use signed occurrences to study the equational logic of the power algebra of the two-valued Boolean algebra.