Non-distributive relatives of ETL and NFL

In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic 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 $\mathbf{4}$. We generalise these pre-orders and determine which distributive properties that hold on $\mathbf{4}$ 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 $\mathbf{M}\mathbf{3}$ as their lattice of truth values. We also provide analytic tableaux systems that work with sequents of the form $\phi ⊢\chi$. We also prove the 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.