Lógica trivalorada e paraconsistência em Peirce

José Renato Salatiel

doi:10.23925/2316-5278.2025v26i1:e57507

Authors

José Renato Salatiel Universidade Federal do Espírito Santo

DOI:

https://doi.org/10.23925/2316-5278.2025v26i1:e57507

Keywords:

Charles S. Peirce, Logic, Non-classical logics, Paraconsistency, Sequents, Three-valued logics

Abstract

Peirce is now recognized as one of the pioneers of mathematical and algebraic logic, but his original work on non-classical logic still receives little attention outside the narrow circle of Peirce scholars. This is particularly evident in his development of the three-valued propositional calculus, recorded in Peirce’s Logic Notebook more than a decade before the rise of many-valued logics. The triadic logic, as named by Peirce, was formalized by Turquette in the late 1960s. Turquette presented an axiomatic interpretation of the three-valued tables in a series of articles that have since become a cornerstone in such studies. Recently, I have proposed a new approach, emphasizing a non-explosive fragment of triadic logic. This paper aims to expand the research in the following points: (i) a critical analysis of Turquette’s works, including the discussion of the Rosser-Turquette method of axiomatization; and (ii) the reconstruction of a fragment of the triadic logic in a system based on Sobociński’s material implication, formalized in sequent calculus. I conclude that Peirce’s three-valued matrix induces a paraconsistent, relevant, and substructural logic with investigative potential for contemporary research in non-classical logics.

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