The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.

It can be intuitively understood that sets and their elements in mathematics reflect the atomistic way of thinking in physics: Sets correspond to physical properties, and their elements correspond to particles that have these properties. At the same time, quantum statistics and quantum field theory strongly support the view that quantum particles are not individuals. Some of the problems faced in modern physics may be caused by such discrepancy between set theory and physical theory. The question then arises: Is it (...) possible to reconstruct the concept of set as a collection of objects that model quantum particles rather than as a mere collection of individuals? David Deutsch has argued that identical entities can be diverse in their attributes, and that this nature, what he calls fungibility, must lie at the heart of quantum physics. In line with this idea, a set theory with fungible elements is established, and the collection of such sets is shown to be endowed with an ortholattice structure, which is better known as quantum logic. (shrink)

Quantum logic is only applicable to microscopic phenomena while classical logic is exclusively used for everyday reasoning, including mathematics. It is shown that both logics are unified in the framework of modal interpretation. This proposed method deals with classical propositions as latently modalized propositions in the sense that they exhibit manifest modalities to form quantum logic only when interacting with other classical subsystems.

A non-classical subsystem of orthomodular quantum logic is proposed. This system employs two basic operations: the Sasaki hook as implication and the _and-then_ operation as conjunction. These operations successfully satisfy modus ponens and the deduction theorem. In other words, they form an adjunction in terms of category theory. Two types of semantics are presented for this logic: one algebraic and one physical. The algebraic semantics deals with orthomodular lattices, as in traditional quantum logic. The physical semantics is given as a (...) procedure for deriving a final segment of a series of yes-no experiments. (shrink)

This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum $$\lambda $$ -calculus is also introduced for which strong normalization property is established.

The logical structure derived from the algebra of generalised projection operators on a module is investigated. With the assumption of the operators being linear, the associated logic becomes Boolean, while without the assumption, the logic does not admit negation: the concept of linearity of projection operators on a module corresponds to that of negation in Boolean logic. The logic of nonlinear operators is formalised and its soundness and completeness results are proved.

Poor sleep quality negatively affects memory performance, and working memory in particular. We investigated sleep habits related to sleep quality including sleep duration, daytime nap duration, nap frequency, and dream content recall frequency (DCRF). Declarative working memory can be subdivided into verbal working memory (VWM) and visuospatial working memory (VSWM). We hypothesized that sleep habits would have different effects on VWM and VSWM. To our knowledge, our study is the first to investigate differences between VWM and VSWM related to daytime (...) nap duration, nap frequency, and DCRF. Furthermore, we tested the hypothesis that the effects of duration and frequency of daytime naps and DCRF on VWM and VSWM differed according to sex. We assessed 779 healthy right-handed individuals (434 males and 345 females; mean age: 20.7 ± 1.8 years) using a digit span forward and backward VWM task, a forward and backward VSWM task, and sleep habits scales. A correlation analysis was used to test the relationships between VWM capacity (VWMC) and VSWM capacity (VSWMC) scores and sleep duration, nap duration, nap frequency, and DCRF. Furthermore, multiple regression analyses were conducted to identify factors associated with VWMC and VSWMC scores and to identify sex-related differences. We found significant positive correlations between VSWMC and nap duration and DCRF, and between VWMC and sleep duration in all subjects. Furthermore, we found that working memory capacity (WMC) was positively correlated with nap duration in males and with sleep duration in females, and DCRF was positively correlated with VSWMC in females. Our finding of sex-related differences in the effects of sleep habits on WMC has not been reported previously. The associations between WMC and sleep habits differed according to sex because of differences in the underlying neural correlates of VWM and VSWM, and effectiveness of the sleep habits in males and females. (shrink)