괴델의 불완전성 정리가 양진주의의 근거가 될 수 있는가?

최승락,Choi, Seungrak Korean Association for Logic 2017 論理硏究 Vol.20 No.2

양진주의는 참인 모순이 존재한다는 입장이다. 필자는 이 글에서 괴델 정리가 양진주의의 근거라는 프리스트의 논변이 설득력이 없음을 논할 것이다. 이는 괴델 증명이 우리에게 주는 교훈은 임의의 충분히 강한 산수에 관한 이론이 완전하면서 일관적일 수 없다는 것이기 때문이다. 다음으로 필자는 프리스트의 비일관적이고 완전한 산수에서 모순이 도출될 수 있음을 설명할 것이다. 그리고 괴델 문장이 비일관적이고 완전한 산수이론에 적용되어 양진주의에 관한 대안논변을 제시할 수 있음을 소개하고 이 경우에는 순환성의 문제가 있음을 논할 것이다. 요약해서, 필자는 괴델 정리 및 그와 관련된 정리는 완전한 이론들과 일관적인 이론들 간의 관계를 보여줄 뿐임을 주장할 것이다. 괴델 문장의 적용을 통해 도출된 모순이 중간값과 같은 참인 문장의 값을 지닐 수 있는 것 역시 산수에 관한 비일관 모형에서일 뿐이다. 비일관성이나 완전성에 관한 가정을 하지 않는다면, 괴델 문장의 적용이 참인 모순을 이끌어 낼 수 없으며 그렇기에 괴델 정리 및 그와 관련된 정리는 양진주의의 근거가 될 수 없다. Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest's argument for Dialetheism from $G{\ddot{o}}del^{\prime}s$ theorem is unconvincing as the lesson of $G{\ddot{o}}del^{\prime}s$ proof (or Rosser's proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest's inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying $G{\ddot{o}}del$ sentence to the inconsistent and complete theory of arithmetic. We argue, however, that the alternative argument raises a circularity problem. In sum, $G{\ddot{o}}del^{\prime}s$ and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of $G{\ddot{o}}del$ sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of $G{\ddot{o}}del$ sentence. Hence, $G{\ddot{o}}del^{\prime}s$ and its related theorem never can be a ground for Dialetheism.

괴델의 불완전성 정리가 양진주의의 근거가 될 수 있는가?

최승락 ( Seungrak Choi ) 한국논리학회 2017 論理硏究 Vol.20 No.2

Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest`s argument for Dialetheism from Godel`s theorem is unconvincing as the lesson of Godel`s proof (or Rosser`s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest`s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Godel sentence to the inconsistent and complete theory of arithmetic. We argue, however, that the alternative argument raises a circularity problem. In sum, Godel`s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Godel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Godel sentence. Hence, Godel`s and its related theorem never can be a ground for Dialetheism.

테넌트의 자기지시적 역설에 관한 가설과 그에 대한 고전적 반례

최승락 ( Seungrak Choi ) 한국논리학회 2021 論理硏究 Vol.24 No.1

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the Liar paradox which generates a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence.

거짓말쟁이 유형 역설과 직관주의 자연연역체계

최승락 ( Seungrak Choi ) 한국논리학회 2018 論理硏究 Vol.21 No.1

It is often said that in a purely formal perspective, intuitionistic logic has no obvious advantage to deal with the liar-type paradoxes. In this paper, we will argue that the standard intuitionistic natural deduction systems are vulnerable to the liar-type paradoxes in the sense that the acceptance of the liar-type sentences results in inference to absurdity (⊥). The result shows that the restriction of the Double Negation Elimination (DNE) fails to block the inference to ⊥. It is, however, not the problem of the intuitionistic approaches to the liar-type paradoxes but the lack of expressive power of the standard intuitionistic natural deduction system. We introduce a meta-level negation, □s, for a given system S and a meta-level absurdity, □, to the intuitionistic system. We shall show that in the system, the inference to ⊥ is not given without the assumption that the system is complete. Moreover, we consider the Double Meta-Level Negation Elimination rules (DMNE) which implicitly assume the completeness of the system. Then, the restriction of DMNE can rule out the inference to ⊥.

선언적 삼단논법을 지닌 1차 고전 자연연역 체계의 정형화 정리

최승락 ( Seungrak Choi ) 한국논리학회 2021 論理硏究 Vol.24 No.2

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system S_CD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of S_CD, we extend S_CD to the system S_PCD for the first-order classical logic with disjunctive syllogism. It can be shown that SPCD is conservative extension to S_CD. Then, the normalization theorem and the consistency of S_PCD are given.

증명의 동일성과 허용가능한 환원 절차의 기준

최승락 ( Seungrak Choi ) 한국논리학회 2021 論理硏究 Vol.24 No.3

Dag Prawitz (1971) put forward the idea that an admissible reduction process does not affect the identity of proofs represented by derivations in natural deduction. The idea relies on his conjecture that two derivations represent the same proof if and only if they are equivalent in the sense that they are reflexive, transitive and symmetric closure of the immediate reducibility relation. Schroeder-Heister and Tranchini (2017) accept Prawitz’s conjecture and propose the triviality test as the criterion for admissible reductions. In the present paper, we will consider two main troubles of the triviality test. The first is the obscurity of a method of evaluating admissible reductions. The second is the circularity problem that the triviality test already assumes the set of admissible reduction procedures. For the solution of the problems, we will propose the spoiler test which immunes the problems of the triviality test and has the role of the criterion for admissible reductions. At last, we shall cover a plausible problem of the spoiler test that can be caused by Crabbé’s case.