It is commonly known that C. I. Lewis started modern modal logics (S1-S5) because of his dissatisfaction with material implication. But his original intention was to construct a logic of strict implication rather than a logic of modality. Then he had got involved in the latter gradually against his original intention. In this paper, I clarify where his original intention lay and how he was involved in modal logics tracing his writings chronologically. Considering his contemporary logicians' responses, I sum up the argument.

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This work proposes a new semantics named "context semantics", which interprets predicate modal logic in which the modality symbols means logical validity. Although the possible world semantics is the most well-known method for semantics of modal logic, it is not so useful or so essential in studying the predicate modal logic. Especially, the transworld identification always makes serious problem. In order to avoid the problem, we propose the new semantics. Our semantics interprets a formula with finite information. This point is the most essential difference between our semantics and possible world semantics.

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The aim of this paper is to propose anomalies of classical logic in view of relevant logic and to suggest how to treat them in relevant logic. Although a semantics for relevant logic exposed the fact that there are two kinds of anomaly in classical logic, the very fact makes the semantics unnatural (that is, pure) due to its disunited treatment. Thus, in order to obtain a natural (that is, applied) semantics for relevant logic, we should offer some unified and natural explanation for those anomalies.

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論理記号の意味を論理記号導入規則と呼ばれる推論規則に基づいて与えて論理的意味論を展開する研究が論理学者達によっていろいろな観点からなされてきた。例えばGentzen ([4] II.§5) による説明はその最初のものであろう。Dummett, PrawitzおよびMartin-Löf等による最近の一連の論理的意味論の運動もこのような流れの上にある。また, LorenzenおよびLorenzの学派によるoperational logicおよびdialogical logicもこのような意味論の変形と考えられる。Gentzenはまた, そのような意味論を彼の与えた数論の整合性証明の哲学的意義の説明に際して展開した。Dummettの意図はそのような意味論に基づいて直観主義推論の哲学的基礎付けを行うことにあった 。これらの哲学的立場の共通する特徴はFrege-Russellのパラドックスの出現以来生じてきた論理学, 数学基礎論における哲学論争を哲学的意味論の論争として捉え直すという前提に立っている点にある。伝統的な捉え方に従うと, QuineやPutnamらが指摘するように, 数学基礎論における論理主義, 直観主義, 形式主義の論争は中世の存在論哲学の普遍論争における実念論, 概念論, 唯名論の間の論争に対応すると考えられてきた。ここで普遍論争とは普遍観念 (または一般観念) の存在論的位置付けに関する論争のことであり, 実念論, 概念論, 唯名論は各々一般観念がプラトンのイデアのような形で実在するという立場, 我々の精神において構成されるという立場, 単に記号として使用しているだけでありその存在を前提する必要がないとする立場, に対応している。数学や論理学で使用される観念についての論争もこのような存在論的なレベルの論争と解釈されてきたわけである。これに対して先に挙げた人々の共通した前提の一つはこの論争を存在論的枠組の中でではなく, 意味論の問題として, つまり言語哲学的レベルの枠組みで捉えようとする傾向があることである。つまり, 論理主義, 直観主義, 形式主義等の立場の違いを同じ数学言語, 論理言語に対して採用される意味論の違いによって説明しようとするわけである。例えば古典論理の立場では正しいとされる「Aまたは非A」という形の排中律は直観主義論理の立場では一般には正しいと見なされないが, この立場の違いもそれぞれが別な意味論体系を採用していて, この別な意味論体系に従って, 「または」という論理結合詞に対して違った意味解釈を与える, ということによるとみなされる。
以下において我々はこのような枠組みの中で「論理記号導入規則による意味論」と呼ばれる意味論の採用がいかに我々の論理的数学的言語行為に合ったものであるかをPrawitz-Dummettの議論を踏まえながら述べ (§1), 次にこの意味論の中核部分を最も基本的な形で展開する (§2) 。最後にこの意味論による直観主義論理推論の基礎付けの可能性およびそこに含まれる問題点について検討する (§3) 。

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It has been common to formalize counterfactuals (or subjunctive conditionals) in natural language in terms of a certain binary sentential connective, as in Stalnaker [12] and D. Lewis [8]. This paper suggests that another formalization by means of unary multi-modal operators is natural and appropriate for some counterfactuals. To see this naturalness and appropriateness, we observe an instance of transitive inference constituted of three counterfactuals in natural language, and formalize it by using expressive power of multi-modal logic, in particular Hennessy-Milner logic(HML) and Dynamic logic (DL). As a result, the instance of transitive inference turns out to be justified by the multi-modalized version of the most fundamental and familiar rules of modal logic, that is, the necessitation rule (N_Act) and the axiom (K_Act).

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In this paper, reality-in-itself and the absolute whole (of everything) —two of the most radically metaphysical ideas—are critically examined from a contemporary philosophical viewpoint. The legitimacy of both ideas has long been doubted, particularly since the criticisms of Kant's thing-in-itself (by Fichte and others), and since some paradoxes of set theory (such as Cantor's paradox), respectively. In section 1 of this paper, the truth/justification conditions of the existence of something real-in-itself are presented, and examined to confirm (more explicitly than ever before) that the idea of reality-in-itself cannot be easily maintained. Likewise, in section 2, the truth/justification conditions of the existence of the absolute whole (of everything) are presented, and examined to confirm that this idea cannot easily be maintained either. In section 3, however, the concept of an absolute whole of reality-in-itself (hereafter, |R|) is introduced by combining the above two ideas. Because this concept is formulated by combining two of the most radically metaphysical ideas, |R| can/could be called the “most metaphysical” reality. In view of the results presented in sections 1 and 2, the existence of |R| might be expected to be doubly doubtful. However, the results presented in section 3 are quite the opposite. It is argued that both the truth/ justification conditions are exceptionally satisfied in |R| (hence |R| exists), and thus, both ideas can be exceptionally maintained in |R|.

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It is still an open question whether there exists an absolute truth, i.e., a proposition which is true in every epistemically possible world.
I have proposed an argument for the existence of such a truth, in my paper "'Ontological Argument' for the Existence of Absolute Truth". In this paper, I shall examine this argument in the following two points: whether avoiding selfreference by differentiating language levels can invalidate the argument, and whether this argument is circular (com-mits a petitio principii) or not.

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We define "absolute truth", like "absolutely a priori truth" of H. Putnam, as a proposition (statement) which is true in every epistemically possible world, namely, true no matter how the world turns out (epistemically) to be.
We have yet to know whether there are such absolute truth(s) or not. In this paper, we try to propose, only as a program of course, an argument for the existence of the absolute truth.
Our argument utilizes the logical structure of "the modal ontological argument" for the existence of God. Using this logical structure, we can enable what cannot have been done by any types of so called "refutation of relativism": to deduce the necessity (or actuality) of the existence of the absolute truth from the possibility of the existence of the absolute truth.

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The formalization of human deductive reasoning is a main issue in artificial intelligence. Although classical logic (CL) is one of the most useful ways for the formalization, the implication of CL has some fallacies. For example, in CL, A ⟶ B can be inferred from B for an arbitrary formula A. This inference is not relevant from the viewpoint of the meaning of implication which human has. In human reasoning, when A ⟶ B is inferred, A and B should be related. Relevant logic has been studied for removal of the implication fallacies of CL. For the relevance of A ⟶ B, several principles are introduced. One of the most important principles is Variable-sharing, where, if A ⟶ B is a theorem, then A and B share an atomic proposition. Relevant logical system should satisfy this principle. Another principle is that the truth values of A and B do not decide A ⟶ B. Classification of the fallacies of implication is also introduced. Fallacies are classified into those of relevance, validity, or necessity. Since the fallacies of relevance and validity are strong fallacies, they are removed from almost all relevant logical systems. Relevant logic, however, is a weaker logical system than CL. In this paper, we propose the relevant logic ER. Then we prove that Variable-sharing holds in ER, that fallacies of relevance and validity are removed from ER. We also prove that ER is not weaker than R. Respecially, disjunctive syllogism holds in ER but does not hold in R. In this sense, ER is a natural formalization of human reasoning.

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The formalization of human deductive reasoning is a main issue in artificial intelligence. Although classical logic (CL) is one of the most useful ways for the formalization, the implication of CL has some fallacies. For example, in Cl, A→B can be inferred form B for and arbitrary formula A. This inference is incorrect from the viewpoint of the meaning of implication which human has. In human deductive reasoning, when A⇾B is inferred, A and B should be related. Relevant logic has been studied for removal of implication fallacies in CL. The system R is a typical logical system from which fallacies of relevance and validity are removed. ER is a relevant typical logical system from which fallacies of relevance and validity are removed. ER is a relevant logical system from which these fallacies are removed and this system is not weaker than R. Especially, it is known that disjunctive syllogism holds in ER but does not hold in R. This inference rule is considered to be natural in human reasoning. In this paper, we prove that ER is properly stronger than R. This means that, for the formalization of human deductive reasoning, ER is more suitable than other relevant logical systems. The proof consists of the following steps: First, the natural deduction systems FR and FR' are introduced. FR is a natural deduction system equivalent to R. It is proved that FR' is stronger than FR. Next, we show that he normalization theorem holds in FR' and that there is a proof of ER corresponding to each normal proof of FR'. In addition, We show the fact that there is theorem of ER which can not be inferred in R. It follows that ER is properly stronger than R.

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