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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