KF, PKF, and Reinhardt's Program

TL;DR

This paper explores the interpretative framework of the Kripke-Feferman (KF) truth theory, proposing a shift from sentences to inferences to potentially justify nonsignificant sentences within significant ones, and establishes a connection with the Partial KF (PKF) theory.

Contribution

It introduces a novel perspective by focusing on provably true inferences in KF, linking them to PKF, and suggests a positive outlook on Reinhardt's program under this interpretation.

Findings

Provably true inferences of KF match provable sequents of PKF.

Reinterpreting KF in terms of inferences may justify nonsignificant sentences.

The approach offers a new perspective on Reinhardt's instrumentalist interpretation.

Abstract

In 'Some Remarks on Extending an Interpreting Theories with a Partial Truth Predicate' Reinhardt famously proposed an instrumentalist interpretation of the truth theory Kripke-Feferman (KF) in analogy to Hilbert's program. Reinhardt suggested to view KF as a tool for generating 'the significant part of KF', that is, as a tool for deriving sentences of the form $T\ulcorner\varphi\urcorner$. The constitutive question of Reinhardt's program was whether it was possible "to justify the use of nonsignificant sentences entirely within the framework of significant sentences"? This question was answered negatively by Halbach and Horsten (2006) but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from KF-provably true sentences to KF-provably true inferences, that is, we shall identify the significant part of KF with the set of pairs $\langle\Gamma, \Delta\rangle$, such that KF proves that if all members of $\Gamma$ are true, at least one member of $\Delta$ is true. In way of addressing Reinhardt's question we show that the provably true inferences of KF coincide with the provable sequents of the theory Partial Kripke-Feferman (PKF).