ETL和NFL的非分配相对逻辑
Non-distributive relatives of ETL and NFL
AI总结：
本文构造了ETL和NFL的非分配相对逻辑，使用非分配格M3作为真值格，提供解析表列系统并证明其正确性与完备性。
摘要：
在本文中，我们构造了Pietz和Riveccio提出的“精确真逻辑”（ETL）及其对偶逻辑——Shramko、Zaitsev和Belikov提出的“非假逻辑”（NFL）的非分配相对逻辑。我们考虑两个预序，它们是ETL和NFL在De Morgan格$\mathbf{4}$上的蕴涵关系的代数对应物。我们推广这些预序，并确定在$\mathbf{4}$上成立的哪些分配性质不被任一预序强制。然后我们构造ETL和NFL的相对逻辑，但这些逻辑缺乏这些分配性质。对于这些逻辑，我们还设计了一种真值表语义，使用非分配格$\mathbf{M3}$作为它们的真值格。我们还提供了处理形如$\phi\vdash\chi$的相继式的解析表列系统。我们还证明了这些证明系统的正确性和完备性结果，并为基于某一族非分配模格的非分配ETL和NFL类逻辑提供了一个简洁的推广。
英文摘要：
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 $ϕ\vdashχ$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
