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.

中文翻译：

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布尔求反和非保守性I：相关模态逻辑

布尔取反可以保守地扩展许多相关的逻辑。但是，Mares表明，E是一个明显的例外。总体来说，Mares的证明是一个涉及模型理论的证明。本文提出了一个更简单的证明理论证明，它不仅涵盖了E，而且得到了概括，从而还添加了原始模态运算符来涵盖相关的逻辑。结果表明，即使是由弱K ish模态运算符扩充的非常弱的相关逻辑，再到具有S5模态算符的强相关逻辑R，都无法通过布尔求反进行保守扩展。因此，该证明还涵盖了Meyer和Mares关于NR — R的证明。带有原始S4的-模态-也不能通过布尔求反保守地扩展。