In this paper, we discuss semantical properties of the logic \(\textbf{GL}\) of provability. The logic \(\textbf{GL}\) is a normal modal logic which is axiomatized by the the Löb formula \( \Box (\Box p\supset p)\supset \Box p \), but it is known that \(\textbf{GL}\) can also be axiomatized by an axiom \(\Box p\supset \Box \Box p\) and an \(\omega \) -rule \((\Diamond ^{*})\) which takes countably many premises \(\phi \supset \Diamond ^{n}\top \) \((n\in \omega )\) and returns a conclusion \(\phi \supset (...) \bot \). We show that the class of transitive Kripke frames which validates \((\Diamond ^{*})\) and the class of transitive Kripke frames which strongly validates \((\Diamond ^{*})\) are equal, and that the following three classes of transitive Kripke frames, the class which validates \((\Diamond ^{*})\), the class which weakly validates \((\Diamond ^{*})\), and the class which is defined by the Löb formula, are mutually different, while all of them characterize \(\textbf{GL}\). This gives an example of a proof system _P_ and a class _C_ of Kripke frames such that _P_ is sound and complete with respect to _C_ but the soundness cannot be proved by simple induction on the height of the derivations in _P_. We also show Kripke completeness of the proof system with \((\Diamond ^{*})\) in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations \(\Box x\le \Box \Box x\) and \(\bigwedge _{n\in \omega }\Diamond ^{n}1=0\) is not a variety. (shrink)

「西欧」と「近代」を相対化し、今、エスニックの次元から『日本哲学』を創始すること=「われわれの問題に発して思索する」スタイルを提唱。.

Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to (...) find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models. (shrink)

Examining the interconnections between genes and culture is crucial for a more complete understanding of psychological processes. Genetic predispositions may predict different outcomes depending on one's cultural context, and culture may predict different outcomes depending on genetic predispositions - that is, genes and culture interact. Less is understood, however, about how genes and culture interact, or the psychological mechanisms through which gene–culture interactions occur. In this Element, Joni Y. Sasaki and Heewon Kwon review key findings and theories in gene–culture (...) interaction research. They then go on to discuss current issues and future directions in gene–culture research that may illuminate the path toward an explanatory framework. (shrink)

The volume D'al-Khwārizmī à Descartes is a monumental contribution to the world history of mathematical sciences, showing clearly that Arabic mathematics was an indispensable predecessor of early modern European mathematics. Roshdi Rashed is known, first of all, as an editor of classical mathematical writings in Arabic by such authors as al-Khwārizmī, Thābit ibn Qurra, Ibrāhīm ibn Sinān, Ibn al-Haytham, al-Khayyām, Sharaf al-Dīn al-Ṭūsī, as well as of the Arabic versions of Apollonius' Conics , Diophantus' Arithmetica , and Diocles' Burning Mirrors (...) . As the volume under review shows, he is also a historian of mathematics of the first class who has transformed historiography. This book is, in a sense, a manifesto of Prof. Rashed's entire œuvre. (shrink)

This article looks at the past, present, and future of the study of aesthetics. The early modern period, during which aesthetics came into being, was a great historical turning point for civilization. So, too, is our own day. Looked at this way, aesthetics should show a different face than the one we are used to. Aesthetics is generally considered to be the philosophy of art, yet, with art regarded as an autonomous cultural field, aesthetics commonly gives the impression of being (...) isolated from philosophy in general. This article explores a completely different aesthetics. When it was coming into existence, aesthetics was charged with the real and urgent philosophical problem of its time: how to construct a new world. (shrink)

It is well established that gratitude favours prosocial tendencies in neutral and amicable social interactions. Less is clear, however, about the role of gratitude in threatening situations that br...

The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi [4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki [7].Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is (...) thatL is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki [5]. Also the proof of sufficiency part is improved here. (shrink)

This paper describes the situation of philosophy in Japan with respect to the question of internationalism. For more than a hundred years, Japan took Western culture as its model and European philosophy as the authentic philosophy. This tendency was paradoxically strengthened by the post‐war humanism. It gave us Japanese philosophers a clear discipline to work in but cut us off from our own native culture, history, and philosophical traditions. In wishing to recover this loss, I argue that it is necessary (...) to distinguish between two forms of internationalism: humanistic and merely fashionable. The latter in fact derives from domestic reasons. (shrink)

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, (...) and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\). (shrink)