• Korean Journal of Logic
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• v.5 no.2
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• pp.1-21
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• 2002

A semantics for logical necessity, based on Carnap's criterion of adequacy, is given with respect to the ontology of logical atomism. A calculus for sentential (propositional) modal logic is described and shown to be complete with respect to this semantics. The semantics is then modified in terms of a restricted notion of 'all possible worlds' in the interpretation of necessity and shown to yield a completeness theorem for the modal logic S5. Such a restricted notion introduces material content into the meaning of necessity so that, in addition to atomic facts, there are "modal facts" that distinguish one world from another.

• Journal of Korean Philosophical Society
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• v.116
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• pp.257-280
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• 2010

I try to formalize the system of modal logic and interpret it in view of constructivism through this study. As to the meaning of a sentence, as we saw, Frege endorsed extensions in view of the fact that they are enough to provide for a compositional account for truth, in particular that (1) the assignment of extensions to expressions is compositional ; (2) the assignment of extensions to sentences coincides with the assignment of truth values. But nobody would be willing to admit that a truth value is what a sentence means and that consequently all true sentences are synonymous. So, if what we are after is meaning in the intuitive sense, then extensions would not do. This consideration has later become the point of departure of modal and intensional semantics. So, it is clear that the language of modal logic do not allow for an extensional interpretation. ${\square}$ is syntactically on a par with ${\vdash}$, hence within the extensional framework it would have to denote a unary truth function. This means that if modal logic is to be interpreted, we need a semantics which is not extensional. The first attempt to build a feasible intensional semantics was presented by Saul Kripke. He came to the conclusion that we must let sentences denote not truth values, but rather subsets of a given set. He called elements of the underlying set possible world. Hence each sentence is taken to denote the set of those possible world in which it is true. This lets us explicate necessity as 'truth in every possible world' and possibility as 'truth in at least one possible world'. But it is clear that the system of modal logic is not only an enlargement of propositional logic, as long as the former contains the new symbols, but that it is of an other nature. In fact, the modal logic is intensional, in that the operators do not determine the functions of truth any more. But this new element is not given a priori, but a posteriori from construction by logicist.

• The Transactions of the Korea Information Processing Society
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• v.5 no.4
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• pp.995-1004
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• 1998

This paper presents an implementation of model checking tool for LTS process specification, which checks deadlock, livelock and reachability for the state and action. The implemented formal checker using modal mu-calculus is able to verify whether properties expressed in modal logic are true on specifications. We prove experimentally that it is powerful to check, safety and liveness for the state and action on LTS. The tool is implemented by $C^{++}$ language and runs on IBM PC under Windows NT.

• Korean Journal of Logic
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• v.16 no.1
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• pp.45-59
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• 2013

In his book Worlds and Individuals: Possible and Otherwise, Takashi Yagisawa Yagisawa argues that his own theory is better than Lewis's theory by showing that his own theory can deal with important objections to modal realism more successfully than Lewis's. In particular, Yagisawa claims that by adopting modal tenses, he can respond to many important objections to modal realism in a uniform way. In this paper, I argue that Lewis can also successfully respond to Peacocke's objection in an exactly parallel way to Yagisawa's by distinguishing existence at the actual world from existence at other possible worlds and that Yagisawa's response to van Inwagen's objection does not succeed. I conclude that Yagisawa fails to show that his own theory is better than Lewis's.

• Journal of KIISE:Computer Systems and Theory
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• v.26 no.12
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• pp.1485-1492
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• 1999

대규모 시스템 명세의 올바름을 검증하기 위한 유한 상태 LTS에 기반을 둔 CTL논리 적용에 있어 가장 큰 문제점은, 시스템 내부의 병렬 프로세스간의 상호작용으로 인한 상태폭발이다. 그러나 Modal mu-calculus 논리를 시스템 안전성 및 필연성 특성 명세에 사용하면, 행위에 의한 순환적 정의가 가능하므로 상태폭발 문제가 해결 가능하다. 본 논문에서는 LTS로 명세화된 통신 프로토콜 시스템 모델의 안전성 및 필연성 특성을 모형 검사 기법에 의해 검증함에 있어, 시제 논리로 사용된 Modal mu-calculus 안전성 및 필연성 논리식과 CTL 의 안전성 및 필연성 논리식의 극한값이 동일함을 두 논리식을 만족하는 상태 집합이 같다는 것을 보임으로써 증명한다. 증명된 결과는 I/O FSM 모델로 표현된 통신 프로토콜의 안전성 및 필연성 검사를 위해 이론적인 기반으로서, 컴퓨터를 이용한 모형검사 기법에 효과적인 방법으로 응용이 가능하다.Abstract In applying CTL-based model checking approach to correctness verification of large state transition system specifications, the major obstacle is the combinational explosion of the state space arising due to interaction of many loosely coupled parallel processes. If, however, the modal mu-calculus viewed as a CTL-based logic with recursion, is used to specify the safety and liveness property of a given system, it is possible to resolve this problem. In this paper, we discuss the problem of verifying communication protocol system specified in LTS, and prove that a logic expression specifying safety and liveness in modal mu-calculus is semantically identical to the maximum value of the expression in CTL. This relation is verified by the proof that the sets of states satisfying the two logic expressions are equivalent. The proof can be used as a theoretical basis for verifying safety and liveness of communication protocols represented as I/O FSM model.

• Proceedings of the Korea Contents Association Conference
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• 2012.05a
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• pp.287-288
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• 2012

프로그램의 정적 검증을 위한 명세는 1차 술어 논리(First Order Logic)가 주로 사용된다. 하지만 1차 술어 논리가 모든 정보를 표현할 수가 없기에 이를 보완하기위해 양상논리(Modal Logic)를 사용할 수가 있다. 정적 프로그램 검증을 위해 양상 논리를 이용하여 확장된 BML(Bytecode Modeling Language)은 BIRS로 변환 되어야 한다. 본 논문에서는 확장된 BML을 중간 표현 언어인 BIRS(Bytecode Intermediate Representation Specification)로 표현하기 위하여 BIRS 문법을 확장한다.

• Proceedings of the Korea Information Processing Society Conference
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• 2012.04a
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• pp.265-268
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• 2012

검증에 사용하는 명세는 대부분 1차 술어 논리(First Order Logic)로 이루어져 있다. 1차 술어 논리가 자연언어 대부분을 표현하지만 표현하지 못하는 부분도 존재한다. 이를 해결하기위해 양상논리(Modal Logic)를 추가한 명세방법이 존재하지만 간접적인 방법으로만 존재할 뿐 이다. 본 논문에서는, 양상논리를 이용한 명세의 직접적인 표현을 위해 BML(Bytecode Modeling Language)을 확장한다. 이를 통해, 명세정보 표현의 정확성을 향상시킨다.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.279-292
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• 2022

In this paper we adhere to the definition of infra-topological space as it was introduced by Al-Odhari. Namely, we speak about families of subsets which contain ∅ and the whole universe X, being at the same time closed under finite intersections (but not necessarily under arbitrary or even finite unions). This slight modification allows us to distinguish between new classes of subsets (infra-open, ps-infra-open and i-genuine). Analogous notions are discussed in the language of closures. The class of minimal infra-open sets is studied too, as well as the idea of generalized infra-spaces. Finally, we obtain characterization of infra-spaces in terms of modal logic, using some of the notions introduced above.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.803-808
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• 1994

Since L$\ddot{o}$b's announcement of his solution to Henkin's problem (L$\ddot{o}$b (1954, 1955)) there has been successful and fruitful research on provability logic tied up with modal logic. Specially, L$\ddot{o}$b's Theorem is of far-reaching significance in the following meta-mathematical and philosophical sense.

• Journal for History of Mathematics
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• v.23 no.1
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• pp.79-88
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• 2010

G$\ddot{o}$del's proof attempts to establish the existence of God by the definition that God is a being having all positive properties. The proof uses here second order modal logic system $S_5$ with the axiom ${\diamondsuit}{\Box}p{\rightarrow}{\Box}p$. We review the G$\ddot{o}$del's own version and prove his ontological theorems.