Binary Kripke Semantics for a Strong Logic for Naive Truth

TL;DR

This paper introduces a novel Kripke semantics for a strong naive truth logic by modifying existing intuitionistic semantics and provides a simplified natural deduction system for it.

Contribution

It presents a new Kripke semantics for the logic TJK^{d+} by relaxing reflexivity constraints and offers a streamlined natural deduction proof system.

Findings

The semantics relaxes the reflexivity condition on the accessibility relation.

Reflexive worlds are used specifically as counterexamples to logical consequence.

A simplified natural deduction system with restricted conditional proof is developed.

Abstract

I show that the logic $\textsf{TJK}^{d+}$, one of the strongest logics currently known to support the naive theory of truth, is obtained from the Kripke semantics for constant domain intuitionistic logic by (i) dropping the requirement that the accessibility relation is reflexive and (ii) only allowing reflexive worlds to serve as counterexamples to logical consequence. In addition, I provide a simplified natural deduction system for $\textsf{TJK}^{d+}$, in which a restricted form of conditional proof is used to establish conditionals.