  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 $\mathbf{M3}$ as their lattice of
truth values. We also provide analytic tableaux systems that work with sequents
of the form $\phi\vdash\chi$. 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.
