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This is a Quantitative Study study.

October 19, 2025

Maximality principles in modal logic and the Axiom of Choice

Key Points

Stronger maximality principles are equivalent to the Axiom of Choice, providing valuable insights into logic theories.
We showed that weaker formulations relate to the Boolean Prime Ideal Theorem, highlighting a key distinction.
The analysis covers Fine’s and Esakia’s Maximality Principles, deepening our understanding of modal and intuitionistic logic.
This research calls for further examination of set-theoretic implications in the context of modal logics.

Abstract

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on well-known Fine’s and Esakia’s Maximality Principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

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