In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. (...) A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic. (shrink)

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 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. (shrink)

The recently established universal uncertainty principle revealed that two nowhere commuting observables can be measured simultaneously in some state, whereas they have no joint probability distribution in any state. Thus, one measuring apparatus can simultaneously measure two observables that have no simultaneous reality. In order to reconcile this discrepancy, an approach based on quantum logic is proposed to establish the relation between quantum reality and measurement. We provide a language speaking of values of observables independent of measurement based on quantum (...) logic and we construct in this language the state-dependent notions of joint determinateness, value identity, and simultaneous measurability. This naturally provides a contextual interpretation, in which we can safely claim such a statement that one measuring apparatus measures one observable in one context and simultaneously it measures another nowhere commuting observable in another incompatible context. (shrink)

A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: An example of an 1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull completeness theorem (...) and the Loeb measure construction are extended to objects in the generic extension of the internal universe. The reduction theory of the Boolean-valued complex numbers are developed as a prototype of the applications to the topological reduction theory of Boolean sheaves or operator algebras. (shrink)

Quantum set theory and topos quantum theory are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, (...) ‘Q-worlds’. Here, we provide a unifying framework that allows us to better understand the relationship between different Q-worlds, and define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory. (shrink)

An ordered field is said to be Scott complete iff it is complete with respect to its uniform structure. Zakon has asked whether nonstandard real lines are Scott complete. We prove in ZFC that for any complete Boolean algebra B which is not (ω, 2)-distributive there is an ultrafilter U of B such that the Boolean ultrapower of the real line modulo U is not Scott complete. We also show how forcing in set theory gives rise to examples of Boolean (...) ultrapowers of the real line which are not Scott complete. (shrink)

Halvorson and Clifton have given a mathematical reconstruction of Bohr’s reply to Einstein, Podolsky and Rosen (EPR), and argued that this reply is dictated by the two requirements of classicality and objectivity for the description of experimental data, by proving consistency between their objectivity requirement and a contextualized version of the EPR reality criterion which had been introduced by Howard in his earlier analysis of Bohr’s reply. In the present paper, we generalize the above consistency theorem, with a rather elementary (...) proof, to a general formulation of EPR states applicable to both non-relativistic quantum mechanics and algebraic quantum field theory; and we clarify the elements of reality in EPR states in terms of Bohr’s requirements of classicality and objectivity, in a general formulation of algebraic quantum theory. (shrink)

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for (...) bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws. (shrink)

This work contrasts Giovanni Sartor’s view of inferential semantics of legal concepts with a probabilistic model of theory formation. The work further explores possibilities of implementing Kemp’s probabilistic model of theory formation in the context of mapping legal concepts between two individual legal systems. For implementing the legal concept mapping, we propose a cross-categorization approach that combines three mathematical models: the Bayesian Model of Generalization, the probabilistic model of theory formation, i.e., the Infinite Relational Model first introduced by Kemp et (...) al. and its extended model, i.e., the normal-IRM proposed by Herlau et al.. We apply our cross-categorization approach to datasets where legal concepts related to educational systems are respectively defined by the Japanese- and the Danish authorities according to the International Standard Classification of Education. The main contribution of this work is the proposal of a conceptual framework of the cross-categorization approach that, inspired by Sartor, attempts to explain reasoner’s inferential mechanisms. (shrink)

The previous studies have shown that human infants and domestic dogs follow the gaze of a human agent only when the agent has addressed them ostensively—e.g., by making eye contact, or calling their name. This evidence is interpreted as showing that they expect ostensive signals to precede referential information. The present study tested chimpanzees, one of the closest relatives to humans, in a series of eye-tracking experiments using an experimental design adapted from these previous studies. In the ostension conditions, a (...) human actor made eye contact, called the participant’s name, and then looked at one of two objects. In the control conditions, a salient cue, which differed in each experiment (a colorful object, the actor’s nodding, or an eating action), attracted participants’ attention to the actor’s face, and then the actor looked at the object. Overall, chimpanzees followed the actor’s gaze to the cued object in both ostension and control conditions, and the ostensive signals did not enhance gaze following more than the control attention-getters. However, the ostensive signals enhanced subsequent attention to both target and distractor objects (but not to the actor’s face) more strongly than the control attention-getters—especially in the chimpanzees who had a close relationship with human caregivers. We interpret this as showing that chimpanzees have a simple form of communicative expectations on the basis of ostensive signals, but unlike human infants and dogs, they do not subsequently use the experimenter’s gaze to infer the intended referent. These results may reflect a limitation of non-domesticated species for interpreting humans’ ostensive signals in inter-species communication. (shrink)

The cellular basis of motor learning in the cerebellum has been attributed mostly to long-term depression (LTD) at excitatory parallel fiber (PF)-Purkinje cell (PC) synapses. LTD is induced when PFs are activated in conjunction with a climbing fiber (CF), the other excitatory input to PCs. Recently, by using whole-cell patch-clamp recording from PCs in cerebellar slices, a new form of synaptic plasticity was discovered. Stimulation of excitatory CFs induced a long-lasting (usually longer than 30 min) of 30 sec) and the (...) durations of RP (>30 min) strongly suggests that some intracellular biochemical machinery is involved. Pharmacological evidence suggests that protein kinases are involved in RP of inhibitory synapses and LTD of excitatory PF synapses. Besides the well-described LTD, RP could be a cellular mechanism that plays an important role in motor learning. (shrink)

Ātman (soul) and Nairātmya (no soul) are, for the Brahmanical schools and the Buddhists respectively, equally fundamental tenets which neither side can concede to the other. Among the 16 formulations presented by Uddyotakara, the fifteenth, which is a proof of Ātman and is originally an indirect proof ( avīta/āvīta ), is presented in a prasaṅga -style, and contains double negation ( na nairātmyam ) in the thesis. However, it is perhaps Dharmakīrti who first transformed it into a normal style ( (...) sātmakam ). He is well aware of the law of excluded middle, and insisits that the negation is paryudāsa . On the Nyāya side, Uddyotakara at least seems to be unaware of the law of the logical equivalence of contraposition concerning pervasion ( vyāpti ). After Uddyotakara, however, Vyoman (Vyomaśiva), Bhāsarvajña and Vācaspatimiśra, all seem to be well aware of it. Dharmakīrti, in his conter-argument against the proof of ātman , discusses the negative expressions ‘‘ nairātmya ” and ‘‘ a-nairātmya ” Dharmakīrti here uses two logical arguments skillfully and tactically. As a critic of both the authenticity of the Veda and the existence of ātman , he insists on the theory of dichotomy and the equivalence of anvaya and vyatireka , whereas as an apologist he denies the application of these theories to the relation between the existence of ātman and the concept of nairātmya , because for him as a Buddhist the latter is not a negative but essentially positive state of affairs. (shrink)

夏目漱石を友とし、阿部次郎、小宮豊隆ら多くの人々に影響を与えた、わが国自然科学思想史の開拓者狩野亨吉。一高校長、京大文科大学長を歴任、野に下っては江戸期の思想家志筑忠雄、本多利明、安藤昌益らを発掘する 。その事跡と生涯を、同郷の筆者が膨大な資料を駆使して綴る記念碑的大著。.

Comme son titre l’indique, cet ouvrage collectif renouvelle réflexions et analyses autour des mouvements et figures féministes du Japon dans différents domaines. L’introduction nous donne un aperçu historique et synthétique très utile pour resituer les chapitres qui suivent. Elle souligne l’apparition de figures féministes (Kishida Toshiko, Fukuda Hideko) dès le début de la construction de l’État moderne japonais. Leur combat pour accéder à l’espace politique resta minoritaire alors que le pr...

日本思想史の真髄に迫る論考と座談。天皇制・丸山眞男・日本仏教・慈円の考察から日本文化に潜む外来と固有の対立・融合を読み解く。.