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.

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.

A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.

The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

We introduce sectional projectivity, which appears to be the correct notion of projectivity when working with orthomodularlattices. We prove some positive results for varieties of OMLs satisfying various finiteness conditions, namely that every finite OML in such a variety is sectionally projective. In contrast, we prove that the eight element modular ortholattice, MO 3, is not projective in the variety of modular ortholattices.

Problem 36 of the third edition of Birkhoff's Lattice theory [2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented.

We discuss [2] of the same title and offer an alternative example. This example is a subalgebra of the ortholattice of closed subspaces of separable real Hilbert space.