• Korean Journal of Logic
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• v.14 no.3
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• pp.101-137
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• 2011

Based on his analysis of judgement forms in intuitionistic type theory, Martin-L$\ddot{o}$f claims that the usual logical laws and interesting mathematical judgements are synthetic, not analytic. He further claims that the logic of analytic judgements is decidable and complete, while the logic of synthetic judgements is undecidable and incomplete. The aim of this article is to clarify and examine his claims. In section 1, I explain and give some comments on the monomorphic version of intuitionistic type theory. In section 2, after clarifying Martin-L$\ddot{o}$f's distinction between analytic and synthetic judgements, I examine some possible objections to it and evaluate the thesis that the usual logical laws and interesting mathematical judgements are synthetic. In section 3, I clarify and examine the thesis that the logic of analytic judgements is decidable and complete, while the logic of synthetic judgements is undecidable and incomplete.

• Korean Journal of Logic
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• v.14 no.2
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• pp.39-76
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• 2011

We explain some basic elements of Martin-L$\ddot{o}$f's type theory and examine the distinction between propositions and judgments. In section 1, we introduce the problem. In section 2, we explain the concept of proposition in the intuitionistic type theory as a development of the intuitionistic conception of proposition. In section 3, we explain the concept of judgment in the intuitionistic type theory. In section 4, we explain some basic inference rules and examine a particular derivation in the theory. In section 5, we examine one route from the Fregean distinction between propositions and judgments to the distinction between them in the intuitionistic type theory, paying attention to the alleged necessity for introducing different forms of judgments.

• 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.