• Korean Journal of Logic
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• v.14 no.3
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• pp.137-176
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• 2011

Bob Hale recently proposed a theory of real numbers based on abstraction principles. In his theory, real numbers are regarded as ratios of quantities and the criteria of identities of ratios of quantities are given by an Eudoxan ratio principle. The reason why Hale defines real numbers as ratios of quantities is that he wants to satisfy Frege's requirement that arithmetical concepts should be defined to be adequate for their universal applicability. In this paper I show why Hale's explanation of applications of real numbers fails to satisfy Frege's requirement, and I propose an alternative explanation. At first I show that there are some gaps between his explanation of the concept of quantity and his stipulation of domains of quantities, and that those gaps give rise to some difficulties in his explanation of applications of real numbers. Secondly I introduce a new ratio principle which can be applied to any kinds of quantities, and I argue that it allows us an adequate explanation of the reason why real numbers as ratios of quantities can be universally applicable. Finally I enquire into some procedures of the measurement of quantities, and I propose some principles which we should presuppose in order to successfully apply real numbers to the measurement of quantities.

• Journal of Educational Research in Mathematics
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• v.12 no.2
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• pp.229-246
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• 2002

In this paper, we start with the question "what is algebraic thinking\ulcorner". The problem is that the algebraic thinking is not exactly defined. We consider algebraic thinking from the various perspectives. But in the discussion relating to the definition of algebraic thinking, we verify that there is the algebraic notations in the core of algebraic thinking. So we device algebraic notations into the six categories, and investigate these examples from the school mathematics. In order to investigate this relation of algebraic thinking and algebraic notations, we present 'the algebraic thinking process analysis model' from Frege' idea. In this model, there are three components of algebraic notations which interplays; sense, expression, denotapion. Thus many difficulties of algebraic thinking can be explained by this model. We suppose that the difficulty in the algebraic thinking may be caused by the discord of these three components. And through the transformation of conceptual frame, we can explain the dynamics of algebraic thinking. Also, we present examples which show these difficulties and dynamics of algebraic thinking. As a result of these analysis, we conclude that algebraic thinking can be explained through the semiotic aspects of algebraic notations.

• Language and Information
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• v.16 no.2
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• pp.17-42
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• 2012

The issues of compositionality, materialized ever since Frege (1982), are critically re-examined in language first mainly and then in all other possible representational systems such as thoughts, concept combination, computing, gesture, music, and animal cognition. The notion is regarded as necessary and suggested as neurologically correlated in humans, even if a weakened version is applicable because of non-articulated constituents and contextuality. Compositionality is crucially involved in all linguistically or non-linguistically meaningful expressions, dealing with at-issue content, default content, and even not-at-issue meanings such as implicatures and presuppositions in discourse. It is a constantly guiding principle to show the relation between representation and mind, still posing tantalizing research issues.

• Journal for History of Mathematics
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• v.21 no.1
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• pp.17-28
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• 2008

This Paper introduces a historical background of mathematical logic. Logic and mathematics were not developed dependently until the mid of the nineteenth century, when two streams of logic and mathematics came to form a river so that brought forth synergy effects. Since the mid-nineteenth century mathematization of logic were proceeded while attempts to reduce mathematics to logic were made. Against this background $G{\ddot{o}}del's$ proof shows the limitation of formalism by proving that there are true arithmetical propositions that are not provable.

• Language and Information
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• v.8 no.2
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• pp.113-130
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• 2004

In a prevailing view of meaning and reference (cf. Frege 1892), words pick out entities in the physical world by virtue of meaning. Linguists and philosophers have argued whether the meaning of a word is inside or out-side language users' mind; but, in general, they have taken it for granted that words refer to entities in the physical world. Hilary Putnam (1975), based on his famous twin-earth thought experiment, argued that the meaning of a word could not be inside language users' head. In this paper, I point out that Putnam's argument makes sense only if words refer to entities in the physical world. That is, Putnam did not provide any argument against mentalistic semantics, since he erroneously assumed that meaning, but not reference, was inside our mind in mentalistic semantics. Mentalistic semanticist, however, assume that words pick out their references inside our head (instead of a possible outside world). A number of arguments for the mentalistic position come from psychology: studies on emotion and visual perception provide numerous cases where words cannot pick out entities from the physical world, but inside our head. The mentalistic theory has desirable consequences for the philosophy of language in that some classical puzzles of language (e.g. Russell's (1919) well-known puzzle of excluded middle) are explained well in the proposed theory.

• Korean Journal of Logic
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• v.17 no.1
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• pp.1-32
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• 2014

Both 5.52 and 5.521 of the Tractatus Logico-Philosophicus raise several questions. In this paper I will explicate Wittgenstein's concept of generality by answering such questions. These questions and problems are closely intertwined. I will try to show what follows. It is ${\xi}$-conditions that are most decisive on the concept of generality of the Tractatus. Except Ramsey, commentators such as Anscombe, Glock, Kenny etc. failed in accurately grasping the Wittgenstein's thoughts concerning ${\xi}$-condition and their claims are not fair at all. Futhermore, from a view point of history of logic, 5.52 has very important significances. That is to say, it anticipates for the first time a possibility of infinitary logic and the concept of universe of discourse in model theory.

• Korean Journal of Logic
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• v.23 no.2
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• pp.117-159
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• 2020

In the early Wittgenstein's Tractatus, both philosophy of logic and that of mathematics belong to the most crucial subjects of it. What is the philosophical view of the early Wittgenstein in the Tractatus? Did he, for example, accept Frege and Russell's logicism or reject it? How did he stipulate the relation between logic and mathematics? How should we, for example, interpretate "Mathematics is a method of logic."(6.234) and "The Logic of the world which the proposition of logic show in the tautologies, mathematics shows in equations."(6.22)? Furthermore, How did he grasp the relation between mathematical equations and tautologies? In this paper, I will endeavor to answer these questions.

• Korean Journal of Logic
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• v.5 no.1
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• pp.147-157
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• 2001

In this study I try to show some numerical analogy between Leibniz's binary system anc I-ching's symbolic system of duo rerum principia, imagines quator, octo figurae am 64 hexagrams. But, there is really a formal logical accordance in their symbolic foundations, on which are based especially the Wittgenstein's 16 truth-tables in his Tractatus-logico-philosophicus(5.101) am 16 hexagrams, as long as we interpret with the binary values 0 am 1, i.e. the Bi-Polarity, the logical tradition from J. Boole, G. Frege through B. Russell and AN. Whitehead to R. Wittgenstein. So, I argue that the historical and theoretical root of that tradition goes back to the debate between Bouvet and Leibniz about the mathematical structure of I-ching' symbols and the Leibnizian binary arithmetic. In the letter on 4. 11. 1701 from Peking to Leibniz, Bouvet wrote that the I-Ching's symbolism has an analogous structure with Leibniz's binary arithmetic. Corresponding to his suggestion, but without exact knowledge, in the letter of 2. January 1967 to the duke August in Braunschweig-Lueneburg-Wolfenbuettel had Leibniz shown already an original idea for the creation of the world with imago Dei which comes from binary progression, dark and light on water.

• Korean Journal of Logic
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• v.21 no.1
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• pp.1-37
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• 2018

As is well known, Wittgenstein criticizes Russell's theory of types explicitly in the Tractatus. What, then, is the point of Wittgenstein's criticism of Russell's theory of types? In order to answer this question I will consider the theory of types on its philosophical aspect and its logical aspect. Roughly speaking, in the Tractatus Wittgenstein's logical syntax is the alternative of Russell's theory of types. Logical syntax is the sign rules, in particular, formation rules of notation of the Tractatus. Wittgenstein's distinction of saying-showing is the most fundamental ground of logical syntax. Wittgenstein makes a step forward with his criticism of Russell's theory of types to the view that logical grammar is arbitrary and a priori. His criticism of Russell's theory of types is after all the challenge against Frege-Russell's conception of logic. Logic is not concerned with general truth or features of the world. Tautologies which consist of logic say nothing.

• Korean Journal of Logic
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• v.16 no.3
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• pp.407-436
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• 2013

I examine the notion of truth in the intuitionistic type theory and provide a better explanation of the objective intuitionistic conception of mathematical truth than that of Dag Prawitz. After a brief explanation of the distinction among proposition, type and judgement in comparison with Frege's theory of judgement, I examine the judgements of the form 'A true' in the intuitionistic type theory and explain how the determinacy of the existence of proofs can be understood intuitionistically. I also examine how the existential judgements of the form 'Pf(A) exists' should be understood. In particular, I diagnose the reason why such existential judgements do not have propositional contents. I criticize an understanding of the existential judgements as elliptical judgements. I argue that, at least in two respects, the notion of truth explained in this paper is a more advanced version of the objective intuitionistic conception of mathematical truth than that provided by Prawitz. I briefly consider a subjectivist's objection to the conception of truth explained in this paper and provide an answer to it.