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An Application of Peircean Triadic Logic: Modelling Vagueness

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Abstract

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.

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Notes

  1. Though Peirce connects this definition with philosophical logic, it is applicable to Triadic logic as well.

  2. For more details about axiomatic systems, continuity, infinitesimals, and topology see Shield (1981), Bell (2005), Lane (1999) and Eisele (1976).

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Authors and Affiliations

  1. Department of Computer Science and Software Engineering, International Islamic University, Islamabad, Pakistan

    Asim Raza

  2. Department of Computing and Technology, Abasyn University, Islamabad, Pakistan

    Asim D. Bakhshi

  3. Department of Humanities and Social Sciences, Lahore University of Management Sciences, Lahore, Pakistan

    Basit Koshul

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  1. Asim Raza
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Raza, A., Bakhshi, A.D. & Koshul, B. An Application of Peircean Triadic Logic: Modelling Vagueness. J of Log Lang and Inf 28, 389–426 (2019). https://doi.org/10.1007/s10849-019-09287-2

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