标题： 构造性量子逻辑 显示英文标题

标题： Constructive Quantum Logics

摘要： 根据Birkhoff和Von Neumann [Ann. Math. 37 (1936), 23-32]的建议，我们对量子逻辑和直觉主义逻辑进行了联合研究。我们展示了一个线性时间的翻译，该翻译对于每个量子逻辑$Q$和每个超直觉主义逻辑$I$，都能从$Q$和$I$的公理化中得出$Q\cap I$的公理化。该翻译围绕一个特定的公理(Ex)展开，该公理（与连接词的引入和消除规则一起）被证明能公理化正交逻辑和直觉主义逻辑的交集，解决了Holliday [Logics 1 (2023), pp. 36-79]提出的问题。我们证明了所有超Ex逻辑的格与量子逻辑和超直觉主义逻辑在符号$\{\land,\lor,\neg\}$下的乘积格同构。我们证明了存在无限多个扩展Holliday的基本逻辑的子Ex逻辑。 显示英文摘要

摘要： Following a suggestion of Birkhoff and Von Neumann [Ann. Math. 37 (1936), 23-32], we pursue a joint study of quantum logic and intuitionistic logic. We exhibit a linear-time translation which for each quantum logic $Q$ and each superintuitionistic logic $I$ yields an axiomatization of $Q\cap I$ from axiomatizations of $Q$ and $I$. The translation is centered around a certain axiom (Ex) which (together with introduction and elimination rules for connectives) is shown to axiomatize the intersection of orthologic and intuitionistic logic, solving a problem of Holliday [Logics 1 (2023), pp. 36-79]. We prove that the lattice of all super-Ex logics is isomorphic to the product of the lattices of quantum logics and superintuitionistic logics in the signature $\{\land,\lor,\neg\}$. We prove that there are infinitely many sub-Ex logics extending Holliday's fundamental logic.