집합론은 메타논리학에 필수불가결한가?
Is Set Theory Indispensable to Metalogic?
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초록
본 논문의 목적은 집합론이 메타논리학에 필수불가결하다는 주장, 즉 필수불가결성 논제에 반대하는 것이다. 만일 집합론이 메타논리학에 필수불가결하다면, 집합론을 포함하게 되는 논리적탐구는 논리학의 가장 근본적인 특성들인 주제중립성과 보편적 적용가능성을 결여하게 되기 때문이다. 논리학의 주제중립성은 논리학의 명제들이 개별 과학과 같은 특정한 지식 분야에 국한되지 않는다는 것을 말하며, 논리학의 보편적 적용가능성은 논리학의 명제들과 추론 규칙들이 모든 과학분야들과 합리적 담론들에서 사용될 수 있다는 것을 말한다. 나아가 주제중립성과 보편적 적용가능성을 지니기 위해서는, 논리학을 기술하는 메타논리적 용어들과 개념들 역시 이러한 특성들을 지녀야만 한다. 하지만 필수불가결성 논제를 받아들이게 되면, 우리는 논리학이 적용되는 모든 분야에서 집합론의 용어들과 집합론적 개념들이 필수불가결하다는 것을 받아들여야만 한다. 그리고 이는 분명 불합리한 일이다. 필수불가결성 논제가 그럴듯하지 않다는 것을 보이기 위해서 나는 집합과 관련된 존재론적 문제를 살펴볼 것이다. 이러한 탐구는 집합이 어떤 식으로 이해되든지 간에 존재론적으로 보수적인 “논리적 존재자” 로 간주되기 어렵다는 것을 보여줄 것이다.
Abstract
The purpose of this paper is to argue against indispensability thesis according to which, set theory is indispensable to metalogic. The reason for this is straight forward. If set theory was indispensable to metalogic, the study of logic, which covers set theory, would lack the most fundamental characters of logic, such as topic-neutrality and universal applicability. The topic neutrality of logic conveys that the propositions of logic are not limited to specific areas of knowledge like individual science. In addition, the universal applicability of logic portrays that propositions of logic and rules of inference can be used in all areas of science (or during a discourse for a rational topic). Furthermore, to have neutrality and universal applicability in propositions of logic, the terms of logic must also possess the characteristics of neutrality and universal applicability. However, once we accept the indispensability thesis, we are recognizing that specific terms and concepts used in a focused field of set theory are indispensible to logic. To state that terms, concepts, and contents derived from facts in concentrated areas of logic are topic-neutral and universally applicable is an irrational judgment. To reject indispensability thesis, I will prove that a set created by a mathematical entity cannot be regarded as an ontologically conservative “logical entity” by concentrating my arguments on ontological relations that are connected to sets.
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