$\mathrm {KF}$, $\mathrm {PKF}$, AND REINHARDT'S PROGRAM. Issue 1 (8th March 2023)

• Record Type: Journal Article
• Authors: CASTALDO, LUCA; STERN, JOHANNES
• Is Part Of: Review of symbolic logic. Volume 16:Issue 1(2023)
• Page Start: 33
• Page End: 58
• Publication Date: 2023-03-08
• Subjects: 03B50 -- 03F03 -- 03F05; KF and PKF -- Reinhardt's Program -- classical and non-classical logic; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/S1755020322000089
• Languages: English
• ISSNs: 1755-0203

Abstract

Abstract: In "Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate", Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman ( $\mathrm {KF}$ ) in analogy to Hilbert's program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating "the significant part of $\mathrm {KF}$ ", that is, as a tool for deriving sentences of the form $\mathrm{Tr}\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 & Horsten [10] 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 $\mathrm {KF}$ -provably true sentences to $\mathrm {KF}$ -provably true inferences, that is, we shall identify the significant part of $\mathrm {KF}$ with the set of pairs $\langle {\Gamma, \Delta }\rangle $, such that $\mathrm {KF}$ 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 suitable $\mathrm {KF}$ -like theories coincide with the provable sequents of matching versions of the theory Partial Kripke–Feferman ( $\mathrm {PKF}$ ).