# The positivication of coalgebraic logics

## TL;DR

This paper develops a general method to derive positive coalgebraic logics from boolean ones, enabling the transfer of semantics and completeness results to the positive fragment.

## Contribution

It introduces the processes of posetification and positivication to systematically obtain positive coalgebraic logics from boolean counterparts.

## Key findings

- Explicit computation of posetifications and positivications for several modal logics.
- Semantics of boolean coalgebraic logic can be lifted to positive logic.
- Weak completeness transfers from boolean to positive coalgebraic logics.

## Abstract

We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing an endofunctor $T': Pos\to Pos$ from an endofunctor $T: Set\to Set$, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor $L': DL\to DL$ from a syntax-building functor $L: BA\to BA$, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case of several modal logics. We show how the semantics of a boolean coalgebraic logic can be canonically lifted to define a semantics for its positive fragment, and that weak completeness transfers from the boolean case to the positive case.