# The modal logic of forcing

@article{Hamkins2005TheML, title={The modal logic of forcing}, author={Joel David Hamkins and Benedikt Loewe}, journal={Journal of Immunology}, year={2005} }

A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.

## 64 Citations

Fatal Heyting Algebras and Forcing Persistent Sentences

- MathematicsStud Logica
- 2012

This paper considers the modal companion, the intermediate logic KC, and relates it to the fatal Heyting algebra HZFC of forcing persistent sentences, which is equationally generic for the class of fatal HeyTING algebras.

Structural connections between a forcing class and its modal logic

- Philosophy, Environmental Science
- 2012

Every definable forcing class Γ gives rise to a corresponding forcing modality $${\square _\Gamma }$$ where $${\square _{\Gamma \varphi }}$$ means that ϕ is true in all Γ extensions, and the valid…

Choiceless large cardinals and set-theoretic potentialism

- Philosophy
- 2020

We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the…

The Modal Logic of Generic Multiverses

- Philosophy
- 2017

In this thesis, we investigate the modal logic of forcing and the modal logic of grounds of generic multiverses. Hamkins and Löwe showed that the ZFC-provable modal principles of forcing, as well as…

Moving Up and Down in the Generic Multiverse

- PhilosophyICLA
- 2013

The modal logic of the generic multiverse is investigated which is a bimodal logic with operators corresponding to the relations “ is a forcing extension of’ and “is a ground model of”.

A note on the complexity of S4.2

- PhilosophyJ. Appl. Non Class. Logics
- 2021

It is shown that the -completeness result extends to , the multimodal version of monomodal, and it is proved that the 'classical' proof in the standard Halpern-Moses style persists even if the authors restrict ourselves to fragments with bounded modal depth.

THE MODAL LOGIC OF SET-THEORETIC POTENTIALISM AND THE POTENTIALIST MAXIMALITY PRINCIPLES

- PhilosophyThe Review of Symbolic Logic
- 2019

Abstract We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on…

Naive Infinitism: The Case for an Inconsistency Approach to Infinite Collections

- PhilosophyNotre Dame J. Formal Log.
- 2015

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and…

CHARACTERIZING EXISTENCE OF A MEASURABLE CARDINAL VIA MODAL LOGIC

- Philosophy, MathematicsThe Journal of Symbolic Logic
- 2021

Abstract We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the…

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