Moving up and down in the generic multiverse

TL;DR

This paper explores the modal logic framework of the generic multiverse, focusing on the relations of forcing extensions and ground models, and investigates possible combinations of their modal logics.

Contribution

It introduces the modal logic of grounds, a new area of study, and analyzes the possible combinations of modal logics for both forcing and grounds relations.

Findings

Modal logic of grounds is introduced and studied.

Possible combinations of modal logics for the two relations are characterized.

The modal logic of forcing was studied in earlier work.

Abstract

We give a brief account of the modal logic of the generic multiverse, which is a bimodal logic with operators corresponding to the relations "is a forcing extension of" and "is a ground model of". The fragment of the first relation is called the modal logic of forcing and was studied by us in earlier work. The fragment of the second relation is called the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.