• Korean Journal of Logic
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• v.16 no.3
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• pp.347-380
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• 2013

In this paper I will try to show how Wittgenstein criticized Frege's theory of meaning. Frege's theory of meaning can be compressed as sense-reference theory. Frege distinguishes between sense and reference on all the linguistic expressions. In particular, he regards that a sentence has sense and reference. This distinction was raised from, so to speak, the problem of identity sentences. Wittgenstein's "fundamental thought" of Tractatus Logico-Philosophicus is the key of his direct criticism of Frege's sense-reference theory. That is, it is an attack on Frege's thought that the reference of a sentence is a truth value and truth values are "objects themselves" (in Frege's meaning). According to Wittgenstein, such an object does not exist and according to his picture theory, the function of a name and that of a proposition are fundamentally different. By the way, Frege can reply justly to this criticism that it is insufficient. In short, Frege's 'sense' and 'reference' etc, are the technical terms. Hence Wittgenstein's decisive criticism of Frege's theory consists in accusing his theory of logical flaws. There is an another route to the sense and reference of a sentence which Frege introduces. In discourses of judgement stroke and content stroke in his Begriffsshrift and in those of horizontal stroke since his "Function and Concept", Frege deals with the sense and reference of a sentence. Wittgenstein criticize that the sense of a complex sentence such as ~p would by no means be determined by Frege's determination.

• Journal for History of Mathematics
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• v.27 no.1
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• pp.47-57
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• 2014

This paper aims to evaluate works of Frege and G$\ddot{o}$del, who play the trigger role in development of logic, by Knowledge Change Model. It identifies where their positions are in the model respectively. For this purpose I suggest types of knowledge change and their criteria for the evaluation. Knowledge change are classified into five types according to the degree of its change: improvement, weak glorious revolution, glorious revolution, strong glorious revolution, and total revolution. Criteria to evaluate the change are its contents, influence, pervasive effects, and so forth. The Knowledge Change Model consists of the types and the criteria. I argue that in the model Frege belongs to the total revolution and G$\ddot{o}$del to the weak glorious revolution. If we accept that the revolution in logic initiated by Frege was completed by G$\ddot{o}$del, it is a natural conclusion.

• Korean Journal of Logic
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• v.20 no.1
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• pp.97-112
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• 2017

We discuss Frege's influence on the modern practice of doing mathematical proofs. We start with explaining Frege's notion of variables. We also talk of the variable binding issue and show how successfully his idea on this point has been applied in the field of doing mathematics based on a computer software.

• Korean Journal of Logic
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• v.19 no.1
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• pp.83-106
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• 2016

We compare the approaches to natural numbers and the induction principles in Frege's Grundgesetze and in systems thereafter. We start with an illustration of Frege's approach and then explain the use of induction principles in Zermelo-Fraenkel set theory and in modern type theories such as Calculus of Inductive Constructions. A comparison among the different approaches to induction principles is also given by analyzing them in respect of predicativity and impredicativity.

• Korean Journal of Logic
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• v.18 no.1
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• pp.1-39
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• 2015

Are properties and relations objects in the Tractatus Logico-Philosophicus? In this paper I will discuss essentially important problems concerning that question. That is, I will try to show that in a sense the concept of objects of the Tractatus is closely intertwined with that of Frege, and moreover the former was suggested to overcome Frege's predicament concerning the concept of objects. In the process of our discussions, it must be kept in mind that these discussions have no relations with metaphysical disputes, but proceed only from a logical point view. Futhermore it is Ramsey that made a most decisive contribution on these problems. In this paper I will try to show that in the Tractatus, properties and relations are objects via the discussions of Ramsey who was under the direct influences of Wittgenstein.

• Journal for History of Mathematics
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• v.22 no.3
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• pp.1-30
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• 2009

Frege's logicism has been frequently regarded as a development in number theory which succeeded to the so called arithmetization of analysis in the late 19th century. But it is not easy for us to accept this opinion if we carefully examine his actual works on real analysis. So it has been often argued that his logicism was just a philosophical program which had not contact with any contemporary mathematical practices. In this paper I will show that these two opinions are all ill-founded ones which are due to the misunderstanding of the theoretical place of Frege's logicism in the context of contemporary mathematical practices. Firstly, I will carefully examine Cantorian definition of real numbers and Frege's critiques of it. On the basis of this, I will show that Frege's aim was to produce the purely logical definition of ratios of quantities. Secondly, I will consider the mathematical background of Frege's logicism. On the basis of this, I will show that his standpoint in real analysis was much subtler than what we used to expect. On the one hand, unlike Weierstrass and Cantor, Frege wanted to get such real analysis that could be universally applicable. On the other hand, unlike most mathematicians who insisted on the traditional conceptions, he would not depend upon any geometrical considerations in establishing real analysis. Thirdly, I will argue that Frege regarded these two aspects - the independence from geometry and the universal applicability - as those which characterized logic itself and, by logicism, arithmetic itself. And I will show that his conception of real numbers as ratios of quantities stemmed from his methodological maxim according to which the nature of numbers should be explained by the common roles they played in various contexts to which they applied, and that he thought that the universal applicability of numbers could not be adequately explicated without such an explanation.

• Korean Journal of Logic
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• v.8 no.1
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• pp.69-88
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• 2005

Hume's principle says that the number of Fs is the same as the number of Gs iff there are just as many Fs as Gs. Frege seems to suggest at Grundlagen $\S64$ that (i) the content of the two sentences are the same, (ii) the left hand side sentence is a result of 'carving up the content' of the right hand side in a new way, (iii) 'the true order of things' are from the right to left rather than the other way round. We examine here if there is a room for arguing these three theses altogether within Frege's philosophy, and give a positive answer to it.

• Korean Journal of Logic
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• v.19 no.2
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• pp.253-293
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• 2016

In Wittgenstein's Tractatus, the concept of 'identity' gives rise to several puzzles as follows. What is an equation(Gleichung) in the Tractatus? Is an equation identical with so called an identity statement? Frege asserts that identity is not a relation between signs but one between objects or of a thing to itself. Then how does Wittgenstein criticize this Frege's conception? Furthermore Wittgenstein explicitly criticizes about Russell's definition of identity. Then What is the point of such Wittgenstein's critique? In a nutshell, what is early Wittgenstein's idea on the nature of identity? In this paper, I will endeavor to answer these questions.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.663-678
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• 2017

Semiotic studies in mathematical education have been based on Saussure, Peirce, and Frege and many prior researches have explored the concepts in a perspective of semiotics. However, the relationship among semiotical elements and the formation and the evolution of a conception are still ambiguous and veiled in many aspects. This thesis is intended to show how a conception was formed and evolved by expression, which is an element of semiotics. In this process, I sought to partially illuminate the relationship among expressions, concepts, and objects.

• Korean Journal of Logic
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• v.22 no.2
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• pp.309-332
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• 2019

In Kang(2017), Jinho Kang persuasively criticized the attempt of Peter Hanks using his concept of cancelled predication to solve the Frege-Geach problem. According to Kang, Hanks had successfully shown the incoherence of Scott Soames's concept of neutral predication, but if it is true, then Hanks's concept of cancelled predication also falls into the same incoherence. I agree with Kang that Hanks faces the same incoherence, and I think that Hank's answers are unconvincing. As I see, however, it is possible for Hanks to overcome Kang's criticism. In this paper, I will reply to Kang's criticism by using conceptual resources in Hanks's own theory. In particular, the idea is that the final predication is compositionally explained by the type it belongs to without having truth-values because cancelled predication itself gives rise to target-shifting toward the type. By doing so, Hanks can successfully solve the Frege-Geach problem even though he let some remarks about cancelled predication unclear and confusing. In addition, it will be revealed that his notation is misleading as well.