Without introducing quantifiers, minimal axiomatic systems have already been constructed for Peirce's triadic logics. The present paper constructs a dual pair of axiomatic systems which can be used to introduce quantifiers into Peirce's preferred system of triadic logic. It is assumed (on the basis of textual evidence) that Peirce would prefer a system which rejects the absurd but tolerates the absolutely undecidable. The systems which are introduced are shown to be absolutely consistent, deductively complete, and minimal. These dual axiomatic systems (...) reveal an interesting elegance, independent of their historical motivation. (shrink)

In 1909, Peirce recorded in a few pages of his logic notebook some experiments with matrices for three-valued propositional logic. These notes are today recognized as one of the first attempts to create non-classical formal systems. However, besides the articles published by Turquette in the 1970s and 1980s, very little progress has been made toward a comprehensive understanding of the formal aspects of Peirce's triadic logic (as he called it). This paper aims to propose a new approach to Peirce's matrices (...) for three-valued propositional logic. We suggested that his logical matrices give rise to three different systems, one of them - which we called P3 - is an original and non-explosive logic. Besides that, we will show that the P3 system can easily be transformed into paraconsistent and paracomplete calculi, adding to it, respectively, unary operators of consistency and intuitionistic negation. We conclude with a discussion about philosophical motivations. (shrink)

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. Charles Sanders Peirce 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. (shrink)