Modal definability in Kripke's theory of truth

TL;DR

This paper develops a modal language to formalize Kripke's concepts of groundedness and paradoxicality in the theory of truth, proving that some notions like intrinsicality cannot be captured modally.

Contribution

It introduces a modal framework for formalizing Kripke's informal definitions and characterizes the modally definable relations, providing a complete axiomatization.

Findings

Groundedness and paradoxicality are expressible in the modal language.

Intrinsicality is not modally definable.

Complete axiomatization of the modal semantics.

Abstract

In Outline of a Theory of Truth, Kripke introduces some of the central concepts of the logical study of truth and paradox. He informally defines some of these -- such as groundedness and paradoxicality -- using modal locutions. We introduce a modal language for regimenting these informal definitions. Though groundedness and paradoxicality are expressible in the modal language, we prove that intrinsicality -- which Kripke emphasizes but does not define modally -- is not. We characterize the modally definable relations and completely axiomatize the modal semantics.