Article contents
AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC
${\mathscr {C}}_1$ AND SOME OF ITS PARACONSISTENT EXTENSIONS
Published online by Cambridge University Press: 04 October 2022
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
.
- Type
- Articles
- Information
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2048.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2049.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2050.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2051.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2052.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2053.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2054.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2055.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2056.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2057.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2058.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230115044918234-0911:S1079898622000361:S1079898622000361_inline2059.png?pub-status=live)
- 1
- Cited by