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全文: "Russell's paradox"
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  • Kazuyuki NOMOTO
    Annals of the Japan Association for Philosophy of Science
    2000年 9 巻 5 号 219-230
    発行日: 2000/03/05
    公開日: 2009/03/26
    ジャーナル フリー
  • H. B. CURRY
    Tohoku Mathematical Journal, First Series
    1935年 41 巻 371-401
    発行日: 1935年
    公開日: 2010/03/19
    ジャーナル フリー
  • Turksen I.Burhan
    日本ファジィ学会誌
    1992年 4 巻 6 号 1113-1118
    発行日: 1992/12/15
    公開日: 2018/02/19
    ジャーナル フリー
  • Kazuyuki NOMOTO
    Annals of the Japan Association for Philosophy of Science
    2006年 14 巻 2 号 73-97
    発行日: 2006/03/05
    公開日: 2009/03/26
    ジャーナル フリー
    In this essay I try to give a general outline of the structural and methodological characteristics of Frege's logical-philosophical investigations as a whole.
    Frege's lifelong enterprise is to establish so-called ‘logicism’, i.e., to show ‘that arithmetic is developed logic.’ This project is, philosophically, an attempt to answer the epistemological question concerning what kind of epistemic status an arithmetical proposition has, and Frege's presumptive reply is that arithmetic is a priori and analytic.
    In order to verify this claim, however, it is necessary to solve the logical-mathematical problem, that is, it must be shown that any arithmetical concept is definable by means of logical terms alone and that the axiomatic system of arithmetic is actually deducible without any gap in inference from the fundamental laws of logic alone. Nevertheless, in the second half of the 19c. there was no logical system powerful enough to reduce arithmetic as a whole. So in order to realize his own project, Frege himself had to carry out a revolutionary revision of traditional Aristotelian logic and construct a completely new logic.
    Thus one can divide Frege's logical-philosophical investigations roughly into three parts as follows:
    I. The invention of a new logic, its axiomatic systematization and the development of the logicist philosophy of arithmetic.
    II. Philosophy of logic.
    III. Controversies with his distinguished scholars of his time concerning psychologism, empiricism, physicalism, formalism, etc., though I do not take these up in the present paper.
  • Toshiharu WARAGAI, Keiichi OYAMADA
    Annals of the Japan Association for Philosophy of Science
    2007年 15 巻 2 号 123-149
    発行日: 2007/03/25
    公開日: 2009/03/26
    ジャーナル フリー
    Concepts can be arranged in a taxonomical order by means of a partial ordering relation, and a taxonomical structure of concepts is usually supposed to have a linkage to reality. However, here is a logico-ontological jump of significant theoretical importance: a taxonomical structure of concepts does not include in itself any logical ability to relate concepts to objects. That is to say, a mere arrangement of concepts does not logically require at all of any linkage between concepts and reality. There can logically be no direct jump from concepts to reality.
    The partial ordering relation required to describe a conceptual taxonomy is in close logical relation to the particle known historically as copula in syllogistic systems. By and large the copula corresponds to the relation IS-A that is today widely used in describing a conceptual taxonomy. We use the symbol ‘⊂’ for IS-A relation. Another relation that relates concepts to objects often called today ‘an-instance-of relation’ will be introduced by definition. This relation will be expressed by the symbol ‘ε’. To define this relation we shall use the functor of (non-reflexive) identity to which we assign the symbol ‘=’.
    We aim in this paper to construct a logical system that is appropriate to describing the logical relations between a taxonomy of concepts and reality, which is tantamount to constructing a version of logical Ontology (first constructed by Lesniewski in 1920) on the basis of two primitive functors, i. e. ⊂ and =.
  • Moto-o TAKAHASHI
    Annals of the Japan Association for Philosophy of Science
    1970年 3 巻 5 号 205-215
    発行日: 1970/03/31
    公開日: 2009/02/16
    ジャーナル フリー
  • Nik WEAVER
    Annals of the Japan Association for Philosophy of Science
    2017年 25 巻 89-100
    発行日: 2017年
    公開日: 2017/09/07
    ジャーナル フリー
  • Walter DEAN
    Annals of the Japan Association for Philosophy of Science
    2017年 25 巻 45-55
    発行日: 2017年
    公開日: 2017/09/07
    ジャーナル フリー
  • 照井 一成
    科学哲学
    2003年 36 巻 2 号 49-64
    発行日: 2003/12/30
    公開日: 2009/05/29
    ジャーナル フリー
    It is observed by Grishin that inconsistency of naive set theory can be avoided by restricting the logical law of contraction, as it is contraction that enables us to derive logical inconsistency from set-theoretic paradoxes such as Russell's paradox.
    In this paper, we examine Grishin's contraction-free naive set theory to better understand Russell's paradox and the naive comprehension principle from a purely formal standpoint. We study both static-propositional and dynamic-procedural aspects of naive comprehension and argue that it could lead to an ideal formalization of (part of) mathematics, where both propositional knowledge (theorems) and procedural knowledge (algorithms) reside in harmony.
  • 三平 正明
    科学哲学
    2003年 36 巻 2 号 33-48
    発行日: 2003/12/30
    公開日: 2009/05/29
    ジャーナル フリー
    It is a well-known story that Russell's discovery of his paradox shook the foundations of Frege's logical system for arithmetic. But there is another route to this paradox. Hilbert pointed out to Frege that he had already found other even more convincing contradictions which he communicated to Zermelo, thereby initiating Zermelo's independent discovery of Russell's paradox. In this paper, we follow this less familiar route and analyze three paradoxes, namely Hilbert's paradox, Zermelo's version of Russell's paradox and Schröder's paradox of 0 and 1. Furthermore, tradition in which these paradoxes were found is reconsidered. We examine Schröder's place in the foundational study and criticize an alleged dichotomy between the algebraic and logistic traditions.
  • Ryo ITO
    Annals of the Japan Association for Philosophy of Science
    2018年 27 巻 27-44
    発行日: 2018年
    公開日: 2018/11/01
    ジャーナル フリー

    Bertrand Russell presented the very first theory of types in the appendices of The Principles of Mathematics. I will argue that he was led to the theory due to Frege's argument against any philosophical account of classes that assimilates a singleton with its sole member. By so doing I will attempt to show that the original theory of types was not meant to be a mere technical solution to the set-theoretic paradox but a philosophical account of what classes are in themselves.

  • TOHRU NOGUCHI
    ENGLISH LINGUISTICS
    2007年 24 巻 1 号 212-234
    発行日: 2007年
    公開日: 2011/06/08
    ジャーナル フリー
  • 岡田 光弘
    科学哲学
    2003年 36 巻 2 号 79-102
    発行日: 2003/12/30
    公開日: 2009/05/29
    ジャーナル フリー
    In this paper we show some logical presumptions for the contradiction-form to really mean contradiction. We first give an introductory note that the same argument-form of Russell paradox could be interpreted to derive a contradiction (as Russell did) and to derive some positive non-contradictory results (such as Gödel's lemma on incompleteness and Cantor's lemma on cardinality), depending on the context. This surprisingly suggests that a logical argument of a contradiction itself is rather independent of interpreting it as contradiction or non-contradiction. In the main section (Section 2) we investigate further in the hidden logical assumptions underlying a usual derivation of contradiction (such as the last step from the Russell argument to conclude a contradiction). We show the logical form of contradiction does not always mean a contradiction in a deep structure level of logic. We use the linear logical analysis for this claim. Linear logic, in the author's opinion, provides fundamental logical structures of the traditional logics (such as classical logic and intuitionistic logic). The each traditional logical connectives split into two different kinds of connective, corresponding to the fundamental distinction, parallel or choice, of the fundamental level of logic; more precisely the connectives related to parallel-assertings and the connectives related to choice-assertions. We claim that (1) the law of contradiction is indisputable for the parallel-connectives, but (2) the law of contradiction is not justified for the choice-connectives. (In fact, the law of contradiction has the same meaning as other Aristotlean laws (the indentity, the excluded middle) from the view point of the duality principle in linear logic, and the disputability of the law of contradiction is exactly the same as the disputability of the law of excluded middle, in the linear logic level.) Here, although (1) admits the law of contradiction, the meaning of contradiction is quite diferrent and, in the author opinion, more basic than the traditional sense of contradiction. (2) tells us that the disputability of the law of contradiction for the choice-connectives is equivalent to the disputability of the law of excluded middle. However, this disputability is more basic than the traditional logicist-intuitionist issue on the excluded middle, since admitting the traditional law of excluded middle (from the classical or logicist viewpoint) is compatible with this disputability of the excluded middle (and equivalently the law of contradiction) with respect to the choice-connectives of the linear logic. Then, the traditional logics (classical and intuitionistic logics) are perfectly constituted from this fundamental level of logic by the use of reconstructibity or re-presentation operator, (which is the linear-logical modal operators). With the use of modal operator the originally splitted two groups of logical connectives merge into a single group, which makes the traditional logical connectives. (The use of slightly different modalities results in the difference between the traditional classical logic and the traditional intuitionistic logic.) With the use of modal operator, the contradiction-form becomes to get the traditional sense of contradiction. This situation shows that the traditional sense of contradiction presumes re-presentation or reconstruction of the inference-resources, which is now explicit by the use of linear logical modal operator(s), and which also makes possible the denotational or objectivity interpretation of logical language. The merge of the two different aspects (the parallel-connectives and the choice-connectives) into one, by the presence of the modal operators, also eliminates the original conflict (on the indisputability of the law of contradiction in the parallel-connectives side and the disputability of that in the choice-connectives side.)
  • 向井 国昭
    科学哲学
    2003年 36 巻 2 号 65-77
    発行日: 2003/12/30
    公開日: 2009/05/29
    ジャーナル フリー
    The Cantorian set theory survived the Russell Paradox by means of axiomatizing the theory into the standard theory named ZF. ZF has a set theoretical counterpart named FA (Foundation Axiom) to the vicious-circle principle of Russell's ramified type theory. Despite the principle, circular objects and phenomena are ubiquitous in many applications fields of ZF. For modeling such circular things directly as circular sets, Aczel replaced FA with his Anti-Foundation Axiom (AFA) to allow non-well-founded sets in a strong way. The foundation of the new set theory is explained in details.
  • 中川 大
    科学哲学
    2003年 36 巻 2 号 21-32
    発行日: 2003/12/30
    公開日: 2009/05/29
    ジャーナル フリー
    In this paper we intend to place the early Russell in the context of the refutation of idealism in the school of Meinong. We look into Mally's arguments against idealism, which have recourse to Russell's paradox, and Meinong's critique of them. Then we propose the hypothesis that the early Russell made up his thought in the Meinongian framework. From this point of view, we could point out that the origin of the paradox might be in the early Russell's criticism of Bradley's idealism. And Wittgenstein's resolution which could make Russell's theory of types dispensable might be compared to Mally's method of arguments which Meinong never adopts.
  • 井上 直昭
    科学哲学
    2001年 34 巻 1 号 49-60
    発行日: 2001/05/30
    公開日: 2009/05/29
    ジャーナル フリー
    This paper deals with the so-called Julius Caesar Problem. Crispin Wright has recently shown that it is possible to derive the axioms of second-order arithmetic from a principle which is called Hume's Principle (HP). Depending upon this result, Wright resurrected a version of Fregean logicistic project. But historical Frege suspected HP as not a fundamental law of arithmetic in the face of Caesar Problem in his Die Grundlagen der Arithmetik section 66. He supposed, I think, that this problem was to be solved through axiom V, the basic law in his Grundgesetze der Arithmetik. But this strategy failed because of the inconsistency of axiom V. And this failure must be seen from a point of view of semantic ill-foundedness, which in general would be included in Fregean abstract principle. This difficulty is an important reason for Russell's Paradox, thus makes it impossible to give any answer to Julius Caesar Problem.
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