The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation (e.g., negatively affixed gradable adjectives). We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators and an additional operator for formula negation at their disposal. We establish normalization and subexpression (resp. subformula) property results for the systems. The normalization results allow us to formulate a proof-theoretic semantics for formulae composed of the subatomic operators. We illustrate the systems with applications to reasoning with combinations of sentential negation, predicate term negation (of adjectives, verbs, and common nouns), subject term negation, and antonymy.

一阶逻辑（包括否定）的运算符对整个公式进行运算。这使其不适合作为形式分析以非句式否定形式（例如谓词术语否定）（例如，否定的可分级形容词）进行推理的工具。我们使用否定运算符扩展其语言，该运算符的范围比原子公式更窄。利用亚原子证明理论的考虑对亚原子推论结构的研究的有用性，我们定义了直觉亚原子自然演绎系统，该系统具有多个亚原子算子和一个额外的算式求反算子。我们建立系统的归一化和子表达式（子表达式）属性结果。归一化结果使我们能够为由亚原子算子组成的公式建立证明理论的语义。我们用句子否定，谓词否定（形容词，动词和普通名词），主题词否定和反义词的组合来说明适用于推理的系统。