# Two variants of noncontingency operator

## TL;DR

This paper introduces two variants of the noncontingency operator with distinct semantics, analyzes their expressiveness over different classes of models, and provides axiomatizations and completeness results for these modal logics.

## Contribution

It defines and compares two variants of the noncontingency operator, explores their expressiveness and frame definability, and develops axiomatizations with completeness proofs.

## Key findings

- $oxdot$-logic is less expressive than $oxplus$-logic on certain model classes.
- Axiomatizations for $oxplus$ and $oxdot$ are provided over various frame classes.
- Completeness is established using $oxdot$-morphisms for serial and symmetric frames.

## Abstract

By slightly adapting two equivalent semantics of noncontingency operator, we obtain two variants, $\boxdot$ and $\boxplus$, with non-equivalent semantics. We show that on the class of models satisfying any of five basic properties (i.e. seriality, reflexivity, transitivity, symmetry, Euclidicity), the logic $\mathcal{L}(\boxdot)$, which has $\boxdot$ as the sole modal primitive, is less expressive than the logic $\mathcal{L}(\boxplus)$, which has $\boxplus$ as the sole modal primitive. We investigate the frame definability of both languages. We then axiomatize $\mathcal{L}(\boxplus)$ and $\mathcal{L}(\boxdot)$ over various classes of bimodal frames. Among other results, a notion of morphisms, called `$\boxdot$-morphisms', are provided to show the completeness of axiomatizations of $\mathcal{L}(\boxdot)$ over serial frames and also over symmetric frames.