Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence (AI). Charles Sanders Peirce (1839–1914) was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic.

中文翻译：

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Peircean 三元逻辑的应用：建模模糊性

决策支持和智能代理系统的开发需要对不确定性和模糊性进行数学描述，以便对模糊性进行建模。本文旨在概述 Peirce 的三元逻辑，作为在人工智能 (AI) 背景下对模糊性进行建模的实用新方法。查尔斯·桑德斯·皮尔斯 (Charles Sanders Peirce，1839-1914 年) 是美国科学家兼哲学家和伟大的逻辑学家，他的三元逻辑是符号学研究和反康托连续性和无穷小模型数学研究的顶峰。在从 AI 的角度介绍 Peircean 符号学之后，给出了 Peircean 三元组的数学公式与经典和模糊集的关系。使用基本逻辑运算符，所有可能的各自蕴涵运算符，双等价运算符，有效的推理规则，通过真值函数方法推导出并验证结合、分配和交换的逻辑属性。为了提出实用的方向，Peirce 三元逻辑的聚合算子已经制定。使用 Peirce 三元关系的医学诊断问题的数学公式和图书馆管理系统的 ER 图显示了所提出的三元组和三元逻辑的进一步应用的潜力。此外，与二进制实现相比，实现了经典 AI 游戏 The Wumpus World 以显示实际功效。除了给出三分集合论的一些初步公式和有限自动机的定义外，