A lattice-valued set theory is formulated by introducing the logical implication $\to$ which represents the order relation on the lattice.

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate (...) orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication. (shrink)

This volume investigates how the phenomenology of religion was accepted and developed in different national contexts. It consists of interviews with senior scholars, who are experts on the development of the phenomenology of religion in their countries, along with commentary and analysis.

Does religion bring peace or war? In order to discuss this fundamental question, it is essential to reflect upon religious education that shapes the views of religion among young generations. This book has developed from the special panel on "Religious Education and Peace" for the 19th World Congress of the International Association for the History of Religions, the largest international organization in religious studies, which took place in Tokyo in March 2005. Its international contributors discuss the kinds of religious education (...) used for peace education that is attempted or needed, in their respective societies faced with tensions and conflicts, not only between different religions but also between religion and secularism. This is the first book in the field that includes both Asian and Western writers. It is an innovative attempt to build a bridge between the study of religion/religious education and peace education. This book was previously published as a special issue of _British Journal of Religious Education_. (shrink)

「生命倫理」は研究を不当に妨げている?ヒト由来試料を用いた研究から生じる「倫理的問題」とは何なのか?法学からのアプローチにより、現実的な規制枠組みの確立に向け、基本から問い直す。.

DBS Think Tank IX was held on August 25–27, 2021 in Orlando FL with US based participants largely in person and overseas participants joining by video conferencing technology. The DBS Think Tank was founded in 2012 and provides an open platform where clinicians, engineers and researchers can freely discuss current and emerging deep brain stimulation technologies as well as the logistical and ethical issues facing the field. The consensus among the DBS Think Tank IX speakers was that DBS expanded in (...) its scope and has been applied to multiple brain disorders in an effort to modulate neural circuitry. After collectively sharing our experiences, it was estimated that globally more than 230,000 DBS devices have been implanted for neurological and neuropsychiatric disorders. As such, this year’s meeting was focused on advances in the following areas: neuromodulation in Europe, Asia and Australia; cutting-edge technologies, neuroethics, interventional psychiatry, adaptive DBS, neuromodulation for pain, network neuromodulation for epilepsy and neuromodulation for traumatic brain injury. (shrink)

Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, (...) and allows cut-elimination. A question by Takano regarding the eliminability of the Takeuti-Titani density rule is answered affirmatively. (shrink)

Engagement with fiction often inspires emotional responses. We may pity Sethe while feeling ambivalent about her actions (in Beloved), fear for Ellen Ripley as she battles monstrous creatures (in Alien), get angry at Okonkwo for killing Ikemefuna (in Things Fall Apart), and hope that Kiyoaki and Satoko find love (in Spring Snow). Familiar as they are, these reactions are puzzling. Why do I respond emotionally if I do not believe that these individuals exist or that the events occurred? If (...) I merely imagine that my best friend has betrayed me, I do not become angry with him; if I did, I would be considered irrational. Yet although beliefs seem to be necessary for emotions in other contexts, we respond to fiction without the relevant beliefs. These observations prompt two questions about our emotional responses to fiction (henceforth: fictional emotions). The first is descriptive: Should fictional emotions be classified as the same kind of emotions we experience in other contexts? The second is normative: Are fictional emotions irrational or otherwise inappropriate? I take each of these debates in turn. (shrink)

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are (...) provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1]. (shrink)

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms (...) of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

Strong completeness of S. Titani's system for lattice valued logic is shown by means of Dedekind cuts.

Among improvisers and composers today there is a resurgence of interest in experimental music (EM) practices that welcome contingency; engaging with unforeseen circumstances as an essential component of the music-making process, and a means to sonic discovery. I propose theExperimental Composition Improvisation Continua(ECIC) as a model with which to better understand these experimental musical works. The historical Experimental Music movement of the 1950s and 60s is briefly revisited, and the jazz tradition included as an essential protagonist; both being important historical (...) movements leading to the formulation of ideas around contingent musical practices. The ECIC model is outlined as providing a means to observe the interactions and continua between composition and improvisation on the one hand and more or less experimentally conceived music on the other. This model is applied as an investigative and comparative tool to three distinctive works in order to illuminate the presence or otherwise of various experimental interactions within them. The works are: “Spiral Staircase” – a composition by written by Satoko Fujii in late 2007, John Cage’s4′33″, and a performance of “My Favorite Things” by the John Coltrane Quartet. Further possible applications of the ECIC are suggested in the conclusion. (shrink)

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.

In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and (...) class='Hi'>Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with $\Delta$ , and the Product and Gödel's Logics with $\Delta$ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to $L \Pi$ and $L \Pi \frac{1}{2}$ . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z. (shrink)

We perform a proof-theoretical investigation of two modal predicate logics: global intuitionistic logic GI and global intuitionistic fuzzy logic GIF. These logics were introduced by Takeuti and Titani to formulate an intuitionistic set theory and an intuitionistic fuzzy set theory together with their metatheories. Here we define analytic Gentzen style calculi for GI and GIF. Among other things, these calculi allows one to prove Herbrand’s theorem for suitable fragments of GI and GIF.

The ŁΠ and logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics, including the three mainly investigated ones (Gödel, product and Łukasiewicz logic). The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both logics. Then we will see many other logics contained in the ŁΠ family of logics (namely logics induced by the continuous finitely constructed t-norms and (...) Takeuti and Titani's fuzzy predicate logic). (shrink)

The ŁΠ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics, including the three mainly investigated ones (Gödel, product and Łukasiewicz logic). The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both logics. Then we will see many other logics contained in the ŁΠ family of logics (...) (namely logics induced by the continuous finitely constructed t-norms and Takeuti and Titani's fuzzy predicate logic). (shrink)