Twist-Valued Models for Three-Valued Paraconsistent Set Theory

Walter A. Carnielli; Marcelo E. Coniglio

doi:10.12775/LLP.2020.015

Authors

Walter A. Carnielli Institute of Philosophy and the Humanities – IFCH and Centre for Logic, Epistemology and the History of Science – CLE, University of Campinas (Unicamp)
Marcelo E. Coniglio Institute of Philosophy and the Humanities – IFCH and Centre for Logic, Epistemology and the History of Science – CLE, University of Campinas (Unicamp) https://orcid.org/0000-0002-1807-0520

DOI:

https://doi.org/10.12775/LLP.2020.015

Keywords

Paraconsistent set theory, Boolean-valued models, axiomatic set theory, twist structures, logics of formal inconsistency, three-valued paraconsistent logics, Leibniz’s Law

Abstract

We propose in this paper a family of algebraic models of ZFC based on the three-valued paraconsistent logic LPT0, a linguistic variant of da Costa and D’Ottaviano’s logic J3. The semantics is given by twist structures defined over complete Boolean agebras. The Boolean-valued models of ZFC are adapted to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. This allows for inconsistent sets w satisfying ‘not (w = w)’, where ‘not’ stands for the paraconsistent negation. Finally, our framework is adapted to provide a class of twist-valued models generalizing Löwe and Tarafder’s model based on logic (PS 3,∗), showing that they are paraconsistent models of ZFC. The present approach offers more options for investigating independence results in paraconsistent set theory.

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