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)

We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination theorem isproved (...) in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics. (shrink)

The simple substitution property provides a systematic and easy method for proving a theorem from the additional axioms of intermediate prepositional logics. There have been known only four intermediate logics that have the additional axioms with the property. In this paper, we reformulate the many valued logics S' n defined in Gödel [3] and prove the simple substitution property for them. In our former paper [9], we proved that the sets of axioms composed of one prepositional variable do not have (...) the property except two of them. Here we provide another proof for this theorem. (shrink)

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)

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)

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 as opposed to proper ones. Improper inference rules are more complicated than proper ones and more difficult to understand. In 2022, we provided a sequent system based solely on the application of proper rules. In the present paper, on the basis of our system from 2022, we classify improper inference (...) rules. (shrink)

The idea of interpretability logics arose in Visser [Vis90]. He introduced the logics as extensions of the provability logic GLwith a binary modality. The arithmetic realization of A B in a theory T will be that T plus the realization of B is interpretable in T plus the realization of A. More precisely, there exists a function f on the formulas of the language of T such that T + B C implies T + A f.The interpretability logics were considered (...) in several papers. An arithmetic completeness of the interpretability logic ILM, obtained by adding Montagna ''s axiom to the smallest interpretability logic IL, was proved in Berarducci [Ber90] and Shavrukov [Sha88]. [Vis90] proved that the interpretability logic ILP, an extension of IL, is also complete for another arithmetic interpretation. The completeness with respect to Kripke semantics due to Veltman was, for IL, ILMand ILP, proved in de Jongh and Veltman [JV90]. The fixed point theorem of GLcan be extended to ILand hence ILMand ILP. The unary pendant "T interprets T + A" is much less expressive and was studied in de Rijke [Rij92]. For an overview of interpretability logic, see Visser [Vis97], and Japaridze and de Jongh [JJ98]. (shrink)

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)

敗戦後の歴史的現実において「観念論から唯物論への過渡」を哲学した珠玉の論集。卒論「親鸞に於ける自然法爾の論理」から「民衆に捧げる知識」への若き梅本克己の苦悩と情熱。.

私たちは、暴力的な存在なのか?人間にとって、暴力とは何か?その不可避性と、回避/克服の可能性をともに見すえつつ思考をつむぐ、「暴力の哲学/倫理学」の試み。.

Japan created in 1992 to provide a legal framework for international peacekeeping activities, following its financial involvement in the Gulf War of 1991. This paper argues that the PKO Law imposed certain restrictions which complicated the missions of the Japanese Self Defence Forces (SDF) and civilian personnel in operational fields. Post 11 September (2001), the Japanese government created a new legal framework for counter-terrorism and dispatched its SDF personnel to the United Nations Transitional Administration in East Timor (UNTAET) and the (...) United Nations Mission of Support in East Timor (UNMISET). (shrink)

This article examines the crisis management capabilities of Japan's Self Defense Forces (SDF) in the areas of United Nations (UN) peacekeeping, counter-terrorism, and disaster relief. The three types of overseas operations were all initiated by Japan as a response to international crises. While SDF crisis management capabilities for UN peacekeeping operations have steadily evolved, room for improvement remains. For example, Japan's commitment to logistic and rapid deployment missions could be strengthened. Regarding the second type of operations, counter-terrorism, Japan's crisis management (...) capabilities were enhanced by the passing of a novel Anti-Terrorism Law in October 2001 after the 9/11 attacks. The law legalized a SDF oil-fuelling mission in the Indian Ocean in support of the US-led war on terror in Afghanistan. Compared to the first two types of missions, SDF crisis management capabilities in disaster relief are the least controversial. This is due to the nature of the SDF as politically neutral and their humanitarian and non-military activities. This article highlights and compares the strengths and weaknesses of Japan's capabilities in the three types of SDF overseas operations. It demonstrates that a number of problem areas remain, including the need for legal amendment as well as the enhancement of public support and political consensus. The Japanese government and the SDF should face up to these challenges so that Japan can become better positioned to react quickly to crisis situations that require the dispatch of the SDF. (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)

Modeling legal argumentation is one of the most important research in AI and Law, and a lot of models have been proposed. However, most research has not treated value judgement and debate. In this paper, we introduce a legal reasoning model which covers various aspects of legalreasoning such as making argument, selecting argument and debate.Furthermore, we present how criminal law is described and reasoned inthis model.

Volume 20, Issue 8, August 2020, Page 32-36.