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Boolean negation and non-conservativity I: Relevant modal logics
Logic Journal of the IGPL ( IF 0.6 ) Pub Date : 2020-07-15 , DOI: 10.1093/jigpal/jzaa019
Tore Fjetland Øgaard 1
Affiliation  

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 NRR with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation.

中文翻译:

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NRR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。
更新日期:2020-07-16
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