AI 中文总结
研究次自反逻辑，通过多种方式构建语义，包括引入海廷和布尔半代数、定义半伴随和单位等，证明其具有可靠且完备语义，能将蕴含解释为稳健后承，还允许句法消去切割，在经典情形下特定指称集语义也完备。
AI 中文摘要
本文表明，没有恒等原则A->A的子结构逻辑（即次自反逻辑）具有合理的可靠且完备语义，并支持多种应用。这种可判定的命题逻辑推广将蕴含自然解释为稳健后承。证明了次自反逻辑允许句法消去切割。引入了海廷和布尔半代数作为海廷和布尔代数的推广，并表明它们能提供完备代数语义而不会意外重新引入自反性。定义了半范畴上的半伴随和（无恒等性的）（余）单位以给出完备半范畴语义。在经典情形下，将蕴含解释为稳健实质蕴含的指称集语义对于次自反逻辑是完备的。
英文摘要
This paper shows that the substructural logic without the identity principle A->A (i.e., subreflexive logic) has principled sound and complete semantics and supports a variety of applications. This decidable generalization of propositional logic naturally interprets implication as robust consequence. Subreflexive logic is proved to admit syntactic cut elimination. Heyting and Boolean semialgebras are introduced as generalizations of Heyting and Boolean algebras and are shown to provide complete algebraic semantics without inadvertently reintroducing reflexivity. Semi-adjunctions on semi-categories and (identity-free) (co-)units are defined to give complete semi-categorical semantics. In the classical case, denotational set semantics that interpret implication as robust material implication are proved complete for subreflexive logic.
