唯名论与有穷主义（一） (5-2)

We further agreed that for the basic language the requirements of finitism and constructivism should be fulfilled in some sense. We examined various forms of finitism. Quine preferred a very strict form; the number of objects was assumed to be finite and consequently the numbers appearing in arithmetic could not exceed a certain maximum number. Tarski and I preferred a weaker form of finitism, which left it open whether the number of all objects is finite or infinite. . . In order to fulfill the requirement of constructivism I proposed to use certain features of my Language I in my Logical Syntax. We planned to have the basic language serve, in addition, as an elementary syntax language for the formulation of the basic syntactical rules of the total language. The latter language was intended to be comprehensive enough to contain the whole of classical mathematics and physics, represented as syntactical systems. (Carnap 1963, 79)

其中所提到的构建构建唯名论-有穷主义语言所要满足的四个标准是由Tarski提供的，可以总结如下：

• (FN 1) L is first-order.

• (FN 2) All elements of D are “physical things.”

• (FN 3R: restrictive version) D contains a finite number of members; or

• (FN 3L: liberal version) No assumption is made about the cardinality of D.

• (FN 4) L contains only finitely many descriptive predicates.

在后面的讨论中作者称这四个条件为‘finitist- nominalist (FN) conditions’，其中前两个为nominalist，第三个和第四个为finitist。任何满足以上四个条件的语言被称为‘finitist-nominalist language.’

Tarski的构想直接受影响于来自波兰的哲学家，唯名论的先驱Chwistek, reism的代表Kotarbiński，以及有唯名论思想和数学上的“intuitionistic formalism”立场的 Leśniewski。

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