Unifying Conservation as Translation for General Calculi
Abstract
We present an abstract framework for conservation and translation theorems between logical calculi. Unlike previous approaches, our setting does not require the underlying consequence relations to satisfy structural properties such as cut, allowing in particular for cut-free calculi. Moreover, we study translations between arbitrary calculi rather than only from a stable extension to the original calculus. Translations are formulated at the level of sequents by pairs of functions acting on antecedents and succedents, and the framework is developed for multi-succedent calculi while subsuming the single-succedent case. This yields a uniform treatment of classical results including negative translations, minimality theorems, Orevkov's conservation classes, and a characterisation of the least logic satisfying Kuroda's double negation theorem.
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BibTeX
@misc{fellin2026unifying,
title = {{Unifying Conservation as Translation for General Calculi}},
author = {Giulio Fellin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35727},
eprint = {2609.35727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35727},
}
RIS
TY - UNPB
TI - Unifying Conservation as Translation for General Calculi
AU - Giulio Fellin
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35727
UR - https://arxiv.org/abs/2609.35727
AB - We present an abstract framework for conservation and translation theorems between logical calculi. Unlike previous approaches, our setting does not require the underlying consequence relations to satisfy structural properties such as cut, allowing in particular for cut-free calculi. Moreover, we study translations between arbitrary calculi rather than only from a stable extension to the original calculus. Translations are formulated at the level of sequents by pairs of functions acting on antecedents and succedents, and the framework is developed for multi-succedent calculi while subsuming the single-succedent case. This yields a uniform treatment of classical results including negative translations, minimality theorems, Orevkov's conservation classes, and a characterisation of the least logic satisfying Kuroda's double negation theorem.
ER -