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Two Maximality Results for the Lattice of Extensions of \(\vdash _{\mathbf {RM}}\)

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Abstract

We use an algebraic argument to prove that there are exactly two premaximal extensions of \(\mathbf {RM}\)’s consequence. We also show that one of these extensions is the minimal structurally complete extension of the unique maximal paraconsistent extension of \(\mathbf {RM}\). Precisely, we show that there are exactly two covers of the variety of Boolean algebras in the lattice of quasivarieties of Sugihara algebras and that there is a unique minimal paraconsistent quasivariety in that lattice. We also obtain a corollary stating that the set of paraconsistent extensions of \(\mathbf {RM}\) forms a complete sublattice of the lattice of all \(\mathbf {RM}\)’s extensions.

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Acknowledgements

I would like to express my gratitude towards Professor Wiesław Dziobiak for formulating the problem of maximal paraconsistent extensions of \(\mathbf {RM}\), which has been passed to me by Professor Kazimierz Świrydowicz. Also, I wish to thank Professor Kazimierz Świrydowicz for invitation to his three inspiring lectures on relevance logic. Finally, special thanks goes to the anonymous referee whose numerous technical and bibliographical suggestions helped to improve the presentation and expand the author’s awareness on the subject.

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  1. Department of Logic, Nicolaus Copernicus University, Toruń, Poland

    Krzysztof A. Krawczyk

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  1. Krzysztof A. Krawczyk
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Correspondence to Krzysztof A. Krawczyk.

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Krawczyk, K.A. Two Maximality Results for the Lattice of Extensions of \(\vdash _{\mathbf {RM}}\). Stud Logica 110, 1243–1253 (2022). https://doi.org/10.1007/s11225-022-10000-x

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