This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of \( {\mathbb{R}} \) called \( the\,hyperreal\,numbers \), which contains infinite and infinitesimal numbers, the two paradoxes are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by \( semantic \,\,indeterminacy \), and the sorites paradox for gradable adjectives is caused by \( epistemic \) \( indiscriminability \). If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language.

本文在经典逻辑的框架内对集体名词和可分级形容词的串联悖论进行了分析。这个悖论是通过区分定性和定量表示来解释的。这种区别通过对每种表示类型使用不同的数学模型来正式表示。定量表示引入阿基米德模型，但定性表示引入非阿基米德模型。通过使用名为\( the\,hyperreal\,numbers \)的非标准模型\( {\mathbb{R}} \)，其中包含无穷大和无穷小数，这两个悖论被证明具有不同的结构。集体名词的连排悖论源于无穷数的使用，而可分级形容词的连排悖论源于无穷小数的使用。每个悖论都可以追溯到不同的模糊来源。集体名词的 sorites 悖论是由\(semantic \,\,indeterminacy \) 引起的，而可分级形容词的 sorites 悖论是由\(epistemic \) \( indiscriminability \)引起的。如果正确，则该分析意味着无穷大和无穷小数在认知上是真实的，并且它们在自然语言的语义解释中起作用。