Encyclopedia Entry

Testa, R. (2023). Paraconsistency. In J. Mattingly (Ed.), The SAGE encyclopedia of theory in science, technology, engineering, and mathematics (Vol. 1, pp. 629-632). SAGE Publications, Inc., https://dx.doi.org/10.4135/9781071872383.n144 (preprint available here)

About the Encyclopedia

Theories are part and parcel of every human activity that involves knowing about the world and our place in it. In all areas of inquiry from the most commonplace to the most scholarly and esoteric, theorizing plays a fundamental role. The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics focuses on the ways that various STEM disciplines theorize about their subject matter. How is thinking about the subject organized? What methods are used in moving a novice in given field into the position of a competent student of that subject? Within the pages of this landmark work, readers will learn about the complex decisions that are made when framing a theory, what goes into constructing a powerful theory, why some theories change or fail, how STEM theories reflect socio-historical moments in time and how – at their best – they form the foundations for exploring and unlocking the mysteries of the world around us. Featuring more than 200 authoritative articles written by experts in their respective fields, the encyclopedia includes a Reader′s Guide that organizes entries by broad themes; lists of Further Readings and cross-references that conclude each article; and a Resource Guide listing classic books in the field, leading journals, associations, and key websites.


Testa, Rafael R.; Bizio, Lucimar; Moraes, João Antônio de. “Metodologias para o Ensino de Lógica em Libras: Notas sobre o desenvolvimento de uma aula de Lógica para o projeto IFSP FILOLIBRAS”. CLE e-prints, Vol. 20 No. 3 (2022). url: https://www.cle.unicamp.br/eprints/index.php/CLE_e-Prints/article/view/1615

Carnielli, Walter; Testa, Rafael. “Paraconsistent Logics for Knowledge Representation and Reasoning: advances and perspectives”. 18th International Workshop on Nonmonotonic Reasoning (2020). https://philpapers.org/rec/CARPLF-3

Moraes, João A.; Testa, Rafael R. “A sociedade contemporânea à luz da ética informacional”. Acta Scientiarum. Human and Social Sciences 42 3 (2020): e56496. http://dx.doi.org/10.4025/actascihumansoc.v42i3.56496

Testa, Rafael R. “BOOK REVIEW: CARNIELLI, Walter & MALINOWSKI, Jacek (eds.). Contradictions, from Consistency to Inconsistency (Trends in Logic 47, Springer International Publishing, 2018, VI+322 pages)”. Manuscrito 42 1 (2019): 219-228. https://doi.org/10.1590/0100-6045.2019.V42N1.RT

Testa, Rafael; Fermé, Eduardo; Garapa, Marco; Reis, Maurício. “How to construct Remainder Sets for Paraconsistent Revisions: Preliminary Report”. 17th International Workshop on Nonmonotonic Reasoning (NMR) (2018). http://www4.uma.pt/nmr2018/NMR2018Proceedings.pdf#page=125

Testa, R.R.; Coniglio, M.E.; Ribeiro, M.M.. “AGM-like paraconsistent belief change”. Logic Journal of the IGPL 25 4 (2017): 632-672. https://doi.org/10.1093/jigpal/jzx010 (preprint available here).

Testa, R. “The cost of consistency: information economy in Paraconsistent Belief Revision”. South American Journal of Logic 1 2 (2015): 461-480. ISSN: 2446-6719 http://www.sa-logic.org/sajl-v1-i2/09-Testa-SAJL.pdf

Testa, R.; Coniglio, M.E.; Ribeiro, M.M.. “Paraconsistent Belief Revision based on a formal consistency operator”. CLE e-prints 15 8 (2015): https://www.cle.unicamp.br/eprints/index.php/CLE_e-Prints/article/view/992

Testa, R.; Coniglio, M.E.. “Dilemas deônticos e escolha: considerações pragmáticas”. Revista Brasileira de Filosofia 232 (2009): 231-246 (preprint available here).

Testa, R. R.; Coniglio, M.E.. “Solving Normative Conflicts Using Preferences Relations”. 8 (2008): https://www.cle.unicamp.br/eprints/index.php/CLE_e-Prints/article/view/944


Popularization and Education of Logic

Carroll, Lewis. O que a Tartaruga disse a Aquiles. Edited and translated by Rafael Testa. Ebook Kindle ASIN: B09TFTYS1C. (2022).

Carroll, Lewis. Um Paradoxo Lógico. Edited and translated by Rafael Testa. Ebook Kindle ASIN: B09TWWCYXM. (2022).

Edited Proceedings

Bertato, F.; Testa, R. R.. Book of Abstracts: 2nd CLE4Science. CLE e-prints. 2017. https://www.cle.unicamp.br/eprints/index.php/cle4science/article/view/775/649

Bueno-Soler, J.; Carnielli, W.; Testa, Rafael. Trends in Logic XVI: consistency, contradiction, paraconsistency and reasoning, 40 years of CLE – Book of abstracts. 2016. https://www.cle.unicamp.br/trendsxvi/book_of_abstracts_trendsxvi.pdf [Philpapers]


Testa, Rafael. “Revisão de Crenças Paraconsistente baseada em um operador formal de consistência”. PhD, Universidade Estadual de Campinas, 2014. https://doi.org/10.47749/T/UNICAMP.2014.935185 [Unicamp Repository] [PhilPapers]

Testa, Rafael. “Dilemas Deônticos: uma abordagem baseada em relações de preferência”. Master, Universidade Estadual de Campinas, 2008. https://hdl.handle.net/20.500.12733/1608116. [PhilPapers].

Testa, Rafael. “Uma análise de algumas lógicas deônticas para a representação de normas jurídicas”. Bachelor Degree, Universidade Estadual de Campinas, 2006. [PhilPapers].


(with published abstracts)

Testa, Rafael R.; Moraes, João Antonio; Bizio, Lucimar. “Projeto LogLibras – Pedagogia Visual no Ensino de Lógica para Surdos”. I Seminário de Pesquisadores em Acessibilidade na Unicamp (Ciclo Anticapacitista da Unicamp). Unicamp, Brazil. November 23-24, 2023. Youtube video: https://youtu.be/ib46UWiJmUE

Moraes, J.; Testa, R.; Bizio, Lucimar. “Cultura surda: experiência visual no ensino de Filosofia e Lógica”. IX Congresso Internacional sobre Culturas. Universidade Federal de Minas Gerais (UFMG), Brazil. November 6 -10, 2023.

Testa, R. (2022) “Belief Revision in a Nutshell”. 7th World Congress and School on Universal LogicBelief Revision Workshop. April 7. Crete, Greece. Youtube video: https://youtu.be/M83rsUTaEeA

Testa, R. (2022) “Filosofia para quê? Aplicações e desafios da Filosofia na contemporaneidade”. Aula inaugural do Departamento de Filosofia – FAJOPA. February 7th. Marília-SP.

Moraes, J. A.; Testa, R. (2021) “Sociedade da informação, ética informacional e divisão digital”. Café Filosófico – Claretiano. May 14th (online).

Testa, R. R. (2020) “Representação do Conhecimento e Raciocínio: aplicações e motivações às lógicas paraconsistentes”. CLE/GTAL Seminars. September 30th. Unicamp (online) [pdf]

Carnielli, W.; Testa, R. (2020) “Paraconsistent Logics for Knowledge Representation and Reasoning: advances and perspectives”. 18th International Workshop on Nonmonotonic Reasoning. September 12th – 14th – Rhodes, Greece (online) [slides].

Testa, R. R.; Fermé, E., Garapa, M. and Reis, M. (2018) “How to construct remainder sets for paraconsistent revisions”. 17th International Workshop on non-monotonic reasoning (NMR’2018). October 27th – 29th – Tempe-Arizona, USA.

Moraes, J. A.; Testa, R. (2018) “Social networks and epistemic arrogance: An ethical debate.” 10th International meeting on information, knowledge and action (Workshop 1: Information, Big Data and Complexity). São Paulo, Brazil.

Testa, R. R. (2018) “On Paraconsistent Belief Change” First Joint Workshop on Ontologies, Uncertainty, and Inconsistency Handling. Technische Universität Dresden, Institute of Theoretical Computer Science, June 26–29. Dresden, Germany. [pdf]

Testa, R. R. (2018) “Belief Revision in Paraconsistent Logics” Seminars of Cardiff School of Computer Science and Informatics, Cardiff University. Wales, UK. [slides]

Testa, R. R. (2018) “On Theory Change: consistency and the dynamic of belief sets”. Seminars of the Group for Theoretical and Applied Logic (Fapesp thematic group). Unicamp, Brazil.

Moraes, J., Testa, R. R. (2018) “Ética Informacional e Arrogância Epistêmica: Iniciando um debate (Information Ethics and Epistemic Arrogance: Starting a Debate”. V Workshop on Contemporary Discussions in Philosophy of Mind and Philosophy of Information. Federal University of Rio de Janeiro.

Testa, R. R. (2018) “Belief Revision in non-classical logics”. Seminars of the KRR group of the Faculty of Exact Sciences and Engineering. University of Madeira, Portugal.

Testa, R. R. (2017) “Paraconsistent Belief Revision”, 3rd Madeira Workshop on Belief Revision, Argumentation, Ontologies and Norms. November 16-20. Madeira, Portugal

Testa, R. R. (2016) Advances and new perspectives in Paraconsistent Belief Revision. 2016. First Campinas Workshop in Contemporary Epistemology.

Testa, R. R., Coniglio, M. E., Ribeiro, M. (2015) “Clarifying some rationality criteria of AGM-like Paraconsistent Belief Revision”, 5th World Congress and School on Universal Logic, June 20-30, Istambul, Turkey (Full Handbook at Academia)

Testa, R. R., Coniglio, M. E., Ribeiro, M. (2013) “External Revision in Belief Sets via Paraconsistency” (2013), 12th Asian Logic Conference, 2011, Wellington – New Zealand. In The Bulletin of Symbolic Logic 19, number 2, pp. 281-282, June 2013.

Testa, R. R. (2013) Revisão de crenças em lógicas paraconsistentes: Novas perspectivas à justificativa coerentista? Seminars of the Group for Theoretical and Applied Logic (Fapesp thematic group). Unicamp, Brazil.

Didatic Videos

Testa, R., Moraes, J. A., Bizio, L., Calo, E. (2021) Projeto IFSP FiloLibras – Aula 7: Lógica. https://youtu.be/mlLac2X2zOs

Science Popularization and Communication

Testa, R. (2021) “Logic for the understanding of Belief Revision”. Series “Logic for…” of LógicaMX. https://youtu.be/yJzuEnm6B8Q

Testa, R. (2021) “O que a IA tem a ver conosco? – carros autônomos e outras questões éticas da robótica e da inteligência artificial”. Semana da Robótica e da Inteligência Artificial. Secretaria Municipal de Educação de Campinas/Departamento Pedagógico/Coordenadoria Setorial de Formação. 28/07/2021.

Carnielli, W.; Testa, R. (2021) “Inteligência Artificial: da Lógica às Humanidades”. Pint of Science Jundiaí. url: https://youtu.be/hiZRjmByIXM

Testa, R. (2021) “Ultron estava certo? Sobre carros autônomos e máquinas morais”. Oficina online – Museu Exploratório de Ciências da Unicamp. url: youtu.be/YW0aIuY69KM

Testa, R. R. (2020) “Inteligência Artificial e seus dilemas lógico-filosóficos (Artificial Intelligence and its logic-philosophical dilemmas)”. Webinar Fajopa. url: youtu.be/aQCku7Id45Q

Testa, R. R. (2012) “Um olhar lógico e filosófico à obra de Alan Turing”. Apresentação de palestra para o Centenário Alan Turing. Instituto Federal de Educação, Ciência e Tecnologia, Muzambinho-MG.

Check Activities for more.

Reception of my work

Fermé, Eduardo & Hansson, Sven Ove (2018). Belief Change: Introduction and Overview. Springer Verlag.

10.4 Paraconsistent Belief Change

Consistency preservation is a central requirement in AGM revision. The reason for this is that the underlying logic is supraclassical and therefore satisfies the explosion principle, namely that anything follows from a contradiction (ex contradictione quodlibet, {p,¬p} ⊢ q). Consequently there is, as we just noted, only one inconsistent belief set, namely the whole language. If we arrive at an inconsistent belief set, then we have lost all distinctions. To avoid this we have to steer clear of contradictions in all operations on belief sets in a supraclassical logic.

However, this does not seem to be how cognitive agents behave in practice. Real agents can believe in contradictory statements without believing everything and losing all distinctions. In order to model that feature of actual reasoning, we can weaken the consequence relation and make it paraconsistent (which means that the explosion principle does not hold). Relatively little work has been performed on paraconsistent belief revision, but important contributions have been made for instance by Restall and Slaney [280], Priest [273], Mares [248], Tanaka [320], and Testa, Coniglio and Ribeiro [322].

The underlying logic used by Restall and Slaney [280] avoids the explosion principle by demanding a connection between the premises and the conclusion of an inference. In a valid inference the premises have to be relevant to the conclusion.

Mares [248] developed a model in which an agent’s belief state is represented by a pair of sets. One of these is the belief set, and the other consists of the sentences that the agent rejects. A belief state is coherent if and only if the intersection of these two sets is empty, i.e., if and only if there is no statement that the agent both accepts and rejects. In this model, belief revision preserves coherence but does not necessarily preserve consistency.

Priest [273] and Tanaka [320] suggested that in a paraconsistent logic, revision can be performed by just adding sentences without removing anything. In other words, if the logic tolerates inconsistencies, then expansion can serve the function usually assigned to revision. Furthermore, Priest [273] pointed out that in a paraconsistent framework, revision on belief sets can be performed as external revision, i.e., with the reversed Levi identity. In a supraclassical framework, external revision can only be used on belief bases. (See further Section 6.3.) Testa, Coniglio and Ribeiro [322] showed that this holds for semi-revision as well. In a supraclassical system, semi-revision (defined in Section 8.2) can only be used for belief bases, but in a paraconsistent system it can also be used for belief sets. The reason for this difference is that the intermediate inconsistent belief set that arises in external revision and semi-revision extinguishes all distinctions if the underlying logic is supraclassical but not if it is paraconsistent.


[248] Mares, E.D.: A paraconsistent theory of belief revision. Erkenntnis 56(2), 229–246 (2002)

[273] Priest, G.: Paraconsistent belief revision. Theoria 67:3, 214–228 (2001)

[280] Restall, G., Slaney, J.K.: Realistic belief revision. In: Proceedings of the Second World Conference on Foundations of Artificial Intelligence. WOCFAI, vol. 95, pp. 367–378 (1995)

[320] Tanaka, K.: The AGM theory and inconsistent belief change. Logique et Analyse 48(189–192), 113–150 (2005)

[322] Testa, R., Coniglio, M., Ribeiro, M.: Paraconsistent belief revision based on a formal consistency operator. CLE e-prints 15(8), University of Campinas (2015)

Carnielli, W and Coniglio, M. (2016) Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series. (New York: Springer, 2016. ISSN: 2214-9775.)

Chapter 9. Paraconsistency and philosophy of science

The notion of quasi-truth, or partial truth, is therefore, closely connected to the paraconsistent LFI paradigm and constitutes a non-dogmatic overture to the dynamics of theory change in science, tolerant to the flounderings of scientific practice. In [266] and [267] this question is studied under the perspective of the AGM theory of Belief Change based on LFIs (a good general reference for Belief Change is S. Hansson’s book [175]).

Paraconsistent Belief Revision systems apply their tools to elicit the very notion of rationality within a paraconsistent setting. It is possible to explain, in particular, the role the consistency operator, as introduced by LFIs, has to play within a dynamic context. By considering the existence of contradictions as a natural consequence of the dynamics of rational thinking, the strictures set by the Belief Revision systems’ operations within a paraconsistent approach are to be reinterpreted. Thus, such approach could also be taken as a paradigm for scientific reasoning. One innovation of [266, 267] is to understand consistency as an epistemic attitude, thus clearing the way for further inquiries about the epistemological features of paraconsistency, as the ones discussed in Chapter 1 and in the present chapter.


[175] Sven O. Hansson. A Textbook of Belief Dynamics, volume 11 of Applied Logic Series. Kluwer Academic Publishers, Dordrecht, 1999.

[266] Rafael R. Testa. Revisão de Crenças Paraconsistente Baseada em um Operador Formal de Consistência (Paraconsistent Belief Revision Based on a Formal Operator of Consistency, in Portuguese). PhD thesis, IFCH, State University of Campinas, Brazil, 2014.

[267] Rafael R. Testa, Marcelo E. Coniglio, and Márcio M. Ribeiro. Paraconsistent belief revision based on a formal consistency operator. CLE e-Prints, 15(8), 2015.

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