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 $$\phi \vdash \chi $$ ϕ ⊢ χ . 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.

中文翻译：

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ETL 和 NFL 的非分配亲属

在本文中，我们设计了 Pietz 和 Riveccio 的 Exactly true logic (ETL) 的非分配亲属以及 Shramko、Zaitsev 和 Belikov 的对偶非假逻辑 (NFL)。我们考虑两个前序，它们是 ETL 和 NFL 在 de Morgan 格 4 上的蕴涵关系的代数对应物。我们概括了这些预序，并确定了 4 的哪些分配属性不受任何预序的强制。然后我们构建了 ETL 和 NFL 的亲戚，但缺乏这样的分配特性。对于这些逻辑，我们还设计了一个真值表语义，它使用非分配格 M3 作为它们的真值格。我们还提供了分析表系统，它与 $$\phi \vdash \chi $$ ϕ ⊢ χ 形式的序列一起工作。