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AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC ${\mathscr {C}}_1$ AND SOME OF ITS PARACONSISTENT EXTENSIONS

Part of: Algebraic logic General logic General

Published online by Cambridge University Press:  04 October 2022

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HUGO ALBUQUERQUE
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INDEPENDENT RESEARCHER E-mail: hugo.albuquerque@ua.pt
CARLOS CALEIRO
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SQIG, INSTITUTO DE TELECOMUNICAÇÕES LISBON, PORTUGAL and DEPARTAMENTO DE MATEMÁTICA, INSTITUTO SUPERIOR TÉCNICO UNIVERSIDADE DE LISBOA LISBON, PORTUGAL E-mail: ccal@math.tecnico.ulisboa.pt
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Abstract

Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$ , with , , and .

Keywords

da Costa’s logic C1paraconsistent logicsprotoalgebraic logicsabstract algebraic logic

MSC classification

Primary: 00-01: Instructional exposition (textbooks, tutorial papers, etc.) 03G27: Abstract algebraic logic 03B53: Paraconsistent logics
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 4 , December 2022 , pp. 477 - 528
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© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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