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Tore Fjetland Øgaard, Boolean negation and non-conservativity I: Relevant modal logics, Logic Journal of the IGPL, Volume 29, Issue 3, June 2021, Pages 340–362, https://doi.org/10.1093/jigpal/jzaa019
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Abstract
Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the strong relevant logic R with a S5 modal operator, all fail to be conservatively extended by Boolean negation. The proof, therefore, also covers Meyer and Mares’ proof that NR—R with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation.