ラッセルの論理主義における非基礎付け主義

Abstract

It is widely believed that logicists, such as Frege and Russell, attempted to secure the certainty of mathematics by reducing it to logic, but only in vain. However, this is not the case with Russell, at least after the discovery of settheoretic and logical paradoxes, when Russell became doubtful even about the certainty of logical principles. Moreover, in his struggle to find a way to avoid these paradoxes, he found himself obliged to admit some axioms into his system that are far from logically true. For these reasons, Russell gave up his intention to found mathematical truth on logic, and took a position quite opposite to his former one. He claimed that mathematics cannot be justified by self-evident principles of logic, but that logical premises of mathematics are justified by the fact that these premises enable us to deduce the whole body of existing mathematics. Then it follows that logical principles are less evident and less certain than mathematical theorems. By reducing mathematics to logic, Russell didn't intend to give mathematics a firm foundation, but to elucidate what is basic to mathematical concepts like sets, relations, or natural numbers, and what premises are essential to mathematical theorems. This may sound like a claim of a mathematical naturalist such as Quine, and what Russell did seems almost the same as what, for example, Zermelo did when he constructed his axiomatic set theory. Although Russell has much in common with them, he is still different from them in some crucial respect. For him, there was further requirement for a system to be logical, and it has much to do with his general philosophy and ontology. In this article, I will try to characterize Russell's logicism, and then show what significance it has, from a philosophical as well as mathematical point of view. The conclusion will be as follows. Russell's logicism can be characterized as an attempt to make clear what the subject matters of mathematics are and what mathematics says about them. He first founded a strictly constructive logical system----called the ramified theory of types----and then showed that, when supplied with some additional axioms, it will become equivalent to set theory.