Uma solução para dois problema abertos: "Logics of Formal Inconsistency Enriched with Replacement: An Algebraic and Modal Account "
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Walter Carnielli
Aug 21, 2022, 7:15:08 PM8/21/22
to Lista acadêmica brasileira dos profissionais e estudantes da área de LOGICA, Juliana Bueno-Soler, David Fuenmayor
Colegas:
Apraz-nos (!) informar que nosso artigo abaixo acaba de ser
publicado pelo The Review of Symbolic Logic:
"Logics of Formal Inconsistency Enriched with Replacement: An
Algebraic and Modal Account " (Walter Carnielli, Marcelo E. Coniglio
and David Fuenmayor)
The Review of Symbolic Logic
https://www.doi.org/10.1017/S1755020321000277.
Published online by Cambridge University Press: 02 July 2021
https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/abs/logics-of-formal-inconsistency-enriched-with-replacement-an-algebraic-and-modal-account/02C1B4CF921AA2998427A258CA35127F
===========================================
Abstract
It is customary to expect from a logical system that it can be
algebraizable, in the sense that an algebraic companion of the
deductive machinery can always be found. Since the inception of da
Costa’s paraconsistent calculi, algebraic equivalents for such systems
have been sought. It is known, however, that these systems are not
self-extensional (i.e., they do not satisfy the replacement property).
More than this, they are not algebraizable in the sense of
Blok–Pigozzi. The same negative results hold for several systems of
the hierarchy of paraconsistent logics known as Logics of Formal
Inconsistency (LFIs). Because of this, several systems belonging to
this class of logics are only characterizable by semantics of a
non-deterministic nature. 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 C_1 , thus obtaining the replacement property (that is, such
LFIs turn out to be self-extensional). Moreover, these logics become
algebraizable in the standard Lindenbaum–Tarski’s sense by a suitable
variety of Boolean algebras extended with additional operations. The
weakest LFI satisfying replacement presented here is called RmbC,
which is obtained from the basic LFI called mbC. Some axiomatic
extensions of RmbC are also studied. In addition, a neighborhood
semantics is defined for such systems. It is shown that RmbC can be
defined within the minimal bimodal non-normal logic E \circplus E
defined by the fusion of the non-normal modal logic E with itself.
Finally, the framework is extended to first-order languages. RQmbC,
the quantified extension of RmbC, is shown to be sound and complete
w.r.t. the proposed algebraic semantics.
======
As primeiras 10 pessoas que tiverem interesse poderão receber
uma cópia oficial (distribuída pelo RSL)
Abraços,
Walter
(em nome de Marcelo e David)
========================
Walter Carnielli
CLE and Department of Philosophy
University of Campinas –UNICAMP, Brazil
https://waltercarnielli.com/
Apraz-nos (!) informar que nosso artigo abaixo acaba de ser
publicado pelo The Review of Symbolic Logic:
"Logics of Formal Inconsistency Enriched with Replacement: An
Algebraic and Modal Account " (Walter Carnielli, Marcelo E. Coniglio
and David Fuenmayor)
The Review of Symbolic Logic
https://www.doi.org/10.1017/S1755020321000277.
Published online by Cambridge University Press: 02 July 2021
https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/abs/logics-of-formal-inconsistency-enriched-with-replacement-an-algebraic-and-modal-account/02C1B4CF921AA2998427A258CA35127F
===========================================
Abstract
It is customary to expect from a logical system that it can be
algebraizable, in the sense that an algebraic companion of the
deductive machinery can always be found. Since the inception of da
Costa’s paraconsistent calculi, algebraic equivalents for such systems
have been sought. It is known, however, that these systems are not
self-extensional (i.e., they do not satisfy the replacement property).
More than this, they are not algebraizable in the sense of
Blok–Pigozzi. The same negative results hold for several systems of
the hierarchy of paraconsistent logics known as Logics of Formal
Inconsistency (LFIs). Because of this, several systems belonging to
this class of logics are only characterizable by semantics of a
non-deterministic nature. 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 C_1 , thus obtaining the replacement property (that is, such
LFIs turn out to be self-extensional). Moreover, these logics become
algebraizable in the standard Lindenbaum–Tarski’s sense by a suitable
variety of Boolean algebras extended with additional operations. The
weakest LFI satisfying replacement presented here is called RmbC,
which is obtained from the basic LFI called mbC. Some axiomatic
extensions of RmbC are also studied. In addition, a neighborhood
semantics is defined for such systems. It is shown that RmbC can be
defined within the minimal bimodal non-normal logic E \circplus E
defined by the fusion of the non-normal modal logic E with itself.
Finally, the framework is extended to first-order languages. RQmbC,
the quantified extension of RmbC, is shown to be sound and complete
w.r.t. the proposed algebraic semantics.
======
As primeiras 10 pessoas que tiverem interesse poderão receber
uma cópia oficial (distribuída pelo RSL)
Abraços,
Walter
(em nome de Marcelo e David)
========================
Walter Carnielli
CLE and Department of Philosophy
University of Campinas –UNICAMP, Brazil
https://waltercarnielli.com/
Joao Marcos
Aug 21, 2022, 8:38:16 PM8/21/22
to Walter Carnielli, Lista acadêmica brasileira dos profissionais e estudantes da área de LOGICA, Juliana Bueno-Soler, David Fuenmayor
Oi, Walter:
Parabéns pelo trabalho!
Talvez valha a pena recordar que o problema particular de definir LFIs que (estendem mbC e) possuem a propriedade de substitutividade (a.k.a. replacement) foi discutido ---e resolvido--- há muitos anos pelas (infinitas) lógicas que reformulam (todas) as lógicas modais normais bem conhecidas numa linguagem contendo a negação paraconsistente definida como não-necessidade, certo? O capítulo 3 da minha tese de doutorado, "Modal Semantics for
Logics of Formal Inconsistency", trata exatamente disto:
Noto que, daí, o artigo mais relevante daquela época é relativamente informal e de fácil leitura, e é citado como [2005b], na bibliografia do seu paper. Claro, o tema também foi retomado em diversos papers posteriores (posso lhes passar várias referências aos nossos trabalhos sobre "modalidades negativas", caso tenham interesse). Entendo, de todo modo, que vocês estudaram agora o caso "mais fraco", não-normal, o que é certamente muito bacana!
***
A conexão entre a propriedade de *substitutividade* de uma dada lógica e a *algebrizabilidade* desta mesma lógica é, quando muito, bastante tênue, não é?
Se me permite a propaganda, gostaria de aproveitar para apontar um trabalho recente que fizemos sobre uma lógica 6-valorada "fully self-extensional yet non-protoalgebraizable" com conectivos de perfeição (consistência + determinação), estendendo simultaneamente as álgebras de De Morgan e a lógica de Dunn-Belnap (de fato, baseada na reformulação da linguagem das álgebras de Stone involutivas, que substitui o misterioso nabla de Monteiro por uma bela bolinha):
Comentários são sempre bem-vindos!
***
Abraços,
Joao Marcos
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