Boolean negation and non-conservativity I: Relevant modal logics

Logic Journal of the IGPL 29 (3):340-362 (2021)
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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.

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Tore Fjetland Øgaard
University of Bergen

References found in this work

Relevant Logics and Their Rivals.Richard Routley, Val Plumwood, Robert K. Meyer & Ross T. Brady - 1982 - Ridgeview. Edited by Richard Sylvan & Ross Brady.
Principia Mathematica.Morris R. Cohen - 1912 - Philosophical Review 21 (1):87.
The semantics of entailment II.Richard Routley & Robert K. Meyer - 1972 - Journal of Philosophical Logic 1 (1):53 - 73.
Universal Logic.Ross Brady - 2006 - Bulletin of Symbolic Logic 13 (4):544-547.
Universal Logic.Ross Brady - 2007 - Studia Logica 87 (2-3):359-362.

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