• Korean Journal of Logic
• /
• v.14 no.2
• /
• pp.39-76
• /
• 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
• /
• v.16 no.3
• /
• pp.407-436
• /
• 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.

• Honam Mathematical Journal
• /
• v.26 no.3
• /
• pp.309-330
• /
• 2004

In this paper, we apply the concept of intuitionistic fuzzy sets to theory of semigroups. We give some properties of intuitionistic fuzzy ideals and intuitionistic fuzzy bi-ideals, and characterize which is left [right] simple, left [right] duo and a semilattice of left [right] simple semigroups or another type of semigroups in terms of intuitionistic fuzzy ideals and intuitionistic fuzzy bi-ideals.

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

• Journal of the Korean Mathematical Society
• /
• v.54 no.5
• /
• pp.1521-1536
• /
• 2017

This paper introduces a representation style of variable binding using dependent types when formalizing meta-theoretic properties. The style we present is a variation of the Coquand-McKinna-Pollack's locally-named representation. The main characteristic is the use of dependent families in defining expressions such as terms and formulas. In this manner, we can handle many syntactic elements, among which wellformedness, provability, soundness, and completeness are critical, in a compact manner. Another point of our paper is to investigate the roles of free variables and constants. Our idea is that fresh constants can entirely play the role of free variables in formalizing meta-theories of first-order predicate logic. In order to show the feasibility of our idea, we formalized the soundness and completeness of LJT with respect to Kripke semantics using the proof assistant Coq, where LJT is the intuitionistic first-order predicate calculus. The proof assistant Coq supports all the functionalities we need: intentional type theory, dependent types, inductive families, and simultaneous substitution.