Non-distributive Relatives of ETL and NFL

Abstract

In this paper we devise non-distributive relatives of Exactly true logic (ETL) by Pietz and Riveccio and its dual Non-falsity logic (NFL) by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL’s and NFL’s entailment relations on the de Morgan lattice 4. We generalise these pre-orders and determine which distributive properties that hold on 4 are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics we also devise a truth table semantics which uses non-distributive lattice M3 as their lattice of truth values. We also provide analytic tableaux systems which work with sequents of the form \(\phi \vdash \chi \). We then prove correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices.

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Acknowledgements

The author would like to express his gratitude to his doctoral advisor professor Dmitryi Zaitsev for his fruitful advise and help as well as to professors Heinrich Wansing and Omori Hitoshi for their discussion of the earlier version of these results and kind permission on behalf of prof. Wansing to present the earlier version on a session of the research colloquium at Ruhr-Universität Bochum. The author also wishes to thank two anonymous referees whose detailed comments helped to enhance the quality of the paper. Author’s work was partially supported by RUB Research School scholarship ‘Ph.D.-Exchange’.

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Correspondence to Daniil Kozhemiachenko.

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Kozhemiachenko, D. Non-distributive Relatives of ETL and NFL. Stud Logica 109, 137–165 (2021). https://doi.org/10.1007/s11225-020-09904-3

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Keywords

  • Exactly true logic
  • Non-falsity logic
  • Non-distributive lattices
  • Analytic tableaux
  • ETL-like logic
  • NFL-like logic