  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
non-polynomially 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.
