Open AccessPosted Content
Twist-Valued Models for Three-valued Paraconsistent Set Theory
TLDR
It is argued that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 .Abstract:
Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vop\v{e}nka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, L\"{o}we and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by L\"{o}we and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing L\"{o}we and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF).read more
Citations
More filters
Set Theory Boolean Valued Models And Independence Proofs
TL;DR: Thank you very much for reading set theory boolean valued models and independence proofs and maybe you have knowledge that, people have search hundreds of times for their favorite readings, but end up in malicious downloads.
Journal ArticleDOI
First-order swap structures semantics for some logics of formal inconsistency
TL;DR: A novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras using Tarkian structures based on twist structures is considered.
Posted Content
Logics of Formal Inconsistency enriched with replacement: an algebraic and modal account
TL;DR: This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than £1, thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional).
Posted Content
Leibniz's law and its paraconsistent models
TL;DR: In this paper, the importance of Leibniz law to Paraconsistent Set Theories is discussed and discussed in terms of their importance in getting models for set theory models.
Essay on modality across different logics
TL;DR: Modal structures that accommodate communication between logic systems by fixing a common lattice L where different logics build their semantics, where each logic being considered in the modal structure is a sublattice of L.
References
More filters
Stanford Encyclopedia of Philosophy
TL;DR: To understand the central claims of evolutionary psychology the authors require an understanding of some key concepts in evolutionary biology, cognitive psychology, philosophy of science and philosophy of mind.
Book ChapterDOI
Logics of Formal Inconsistency
TL;DR: The Logicas da Inconsistencia Formal (LIFs) as mentioned in this paper form a classe of logicas paraconsistentes particularmente expressivas, in which a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to.
BookDOI
Introduction to Boolean Algebras
Paul R. Halmos,Steven Givant +1 more
TL;DR: The Representation Theorem and Canonical Extensions of Homomorphisms and Isomorphism Theorems have been proposed in this article for the representation of s-algebras.
Book ChapterDOI
A Taxonomy of C-systems
Walter Carnielli,João Marcos +1 more
TL;DR: An enormous variety of paraconsistent logics in the literature is shown to constitute C- System, and a novel notion of consistency is introduced.
Related Papers (5)
Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory
Sourav Tarafder,Sourav Tarafder +1 more