Abstract
As is mentioned in Leigh (Journal of Symbol Logic 80(3):845-865, 2015), it is an open problem whether for several axiomatic theories of truth, including Friedman–Sheard theory \(\mathsf {FS}\) (Friedman and Sheard in Annals of Pure and Applied Logic 33:1–21, 1987) and Kripke–Feferman theory \(\mathsf {KF}\) (Kripke in Journal of Philosophy 72(19):690-716, 1976), there exist cut-elimination arguments that give the upper bounds of their proof-theoretic strengths. In this paper, we give complete cut-elimination results for several well-known axiomatic theories of truth. In particular, we treat the systems \(\mathsf {B,C}\), and \(\mathsf {D}\) \((= \mathsf {FS})\) of Friedman and Sheard’s theories (1987) and \(\mathsf {KF}\).
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Acknowledgements
This work was partially supported by JSPS KAKENHI, Grant Number 20J12361, and by JSPS Core-to-Core Program (A. Advanced Research Networks). This paper is based on my master’s thesis at Hokkaido University. I am deeply indebted to my thesis advisor, Professor Katsuhiko Sano, with whom I had many fruitful discussions. Parts of the results in this paper were presented at the SAML 2018 symposium (Kobe, Japan, 2018) and the FLoV logic seminar (Gothenburg, Sweden, 2019). I would like to thank all participants, especially Bahareh Afshari, Toshiyasu Arai, Mamoru Kaneko, Hidenori Kurokawa, Graham Leigh, Mattias Granberg Olsson, and Yuta Takahashi for their helpful comments and suggestions. I would also like to thank the members of FLoV, especially Graham Leigh, for their hospitality during my visit to University of Gothenburg. Finally, I am grateful to the two anonymous reviewers for their careful reading of the manuscript and their many detailed and important suggestions.
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Hayashi, D. On Cut-Elimination Arguments for Axiomatic Theories of Truth. Stud Logica 110, 785–818 (2022). https://doi.org/10.1007/s11225-021-09978-7
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DOI: https://doi.org/10.1007/s11225-021-09978-7