Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: the first is sound for Takeuti’s quantum set theory, and the second is sound for a variant of Weaver’s quantum logic.

正交模逻辑是伯克霍夫和冯·诺依曼意义上的量子逻辑的弱化。正交模逻辑被证明是一种非线性非交换逻辑。序列被赋予物理动机的语义，该语义与使用否定、合取和蕴涵的命题公式的语义完全一致。特别是，蕴涵必须被解释为佐佐木箭头，它满足该逻辑中的演绎定理。作为一个应用，这个演绎系统被扩展到两个谓词逻辑系统：第一个对于 Takeuti 的量子集合论来说是合理的，第二个对于 Weaver 的量子逻辑的变体来说是合理的。