Paraconsistent logic helps to clarify the concepts of negation and contradiction. On the one hand there are authors for whom contradictions play a quasi-mystical role, used to explain nearly everything in the universe, on the other hand excellent specialists think that contradiction is something unintelligible. Paraconsistent logic not only is useful to demystify contradiction but contributes to calm anyone who is afraid of it.
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Newton da Costa
September 16, 1929 - April 16, 2024
by Jean-Yves Beziau
Central Figure of Paraconsistency
Central Figure of Paraconsistency
Newton da Costa is a central figure of paraconsisent logic, having promoted during more than 50 years this new field of logic. He started to work on the topic in the 1950s, giving talks and publishing papers. In 1958 he publsihed the paper "Nota sobre o conceito de contradição" ("Note on the concept of contradiction" - a review of it was published in the Journal of Symbolic Logic, v.25 (1960), p.60). In this paper he wrote: "Inconsistent mathematical theories are permissible though it is necessary to employ new 'logic rules' in the formalization of such theories". He presented such rules in his thesis Sistemas Formais Inconsistentes (Inconsistent formal systems), part of it was published in the prestigious journal Comptes Rendus de l 'Académie des Sciences de Paris, T257 (1963), pp 3790-3793: "Calculs propositionnels pour les systèmes formels inconsistants" ("Propositional calculi for inconsistent formal systems"). His main system is called "C1", and one of its striking features is that it is possible from the paraconsitent negation of it to define a classical negation.
It is important to point out that da Costa did not develop only one system of paraconsistent logic. Besides the C-systems hierarchy, he worked on three-valued paraconsistent logic with Itala D'Ottaviano in relation with Jaśkowski's project, with Ayda Arruda on some relevant logic systems, with Walter Carnielli on some paraconsistent deontic systems, with Roberto Vernengo he created paraclassical logic, with Andrea Loparic he developed paranormal logic systems (systems which are both paraconsistent and paracomplete), as well with myself. Moreover Newton da Costa has been working not only on formal systems of paraconsistent logic, but also on applications and philosophical aspects of paraconsistency.
Newton da Costa was pivotal for the terminology "Paraconsistent logic", which had a strong impact on the dvelopment of the topic. He was not satisfied with the terminology "Inconsistent formal sytsems" and asked his friend, the Peruvian Philosopher Miró Quesada for a better name. Quesada made several suggestions and they chose "Paraconsistent logic"
"Several years ago, I needed a convenient and meaningful denomination for a logic that did not eliminate contradictions from the outset as being false, i.e., as absolutely unacceptable. Miró Quesada helped me. On the one hand, it should be recalled that, by that time, all logics unavoidably condemned contradictions. The new logic in which I worked faced too much resistance, it was badly divulged, and those that got to know it were in general sceptics. By that time I wrote to Miró Quesada, who saw the new logic with great enthusiasm, requesting a name for it. I remember as it was today that he answered with three proposals: it could be called metaconsistent, ultraconsistent or paraconsistent. After commenting on these possible denominations, he stated that, from his viewpoint, he preferred the latter. The term paraconsistent sounded splendid and I began to use it, suggesting that people interested on this logic did the same. Two or three months later, the miracle took place; the term spread through the world, all the centres directly or indirectly related to logic, from northern to southern hemisphere, began to employ it. I believe that few times in the history of science (definitely in the history of logic) something similar has happened, for not only the word run the whole world, but the very logic called by Miró Quesada “paraconsistent” received a formidable push. It became one of the most discussed theories of logic of our time. (da Costa, “La Filosofia de la Logica de Francisco Miró Quesada Cantuarias,” in Logica, Razon y Humanismo, Lima, 1992, pp. 69–78.)"
See also the special issue Miró Quesada of the South American Journal of Logic edited by Luis Alegre and Carlos Cifuentes.
In 1991, Paraconsistent logic was introduced as a subarea 03B53 of the area 03 Mathematical logic and foundations of the Mathematics Subject Classification.
It is also worth to emphasize that Newton da Costa promoted the work of Nicolai A. Vasiliev (together with Ayda Arruda) and Stanisław Jaśkowski.
Many Activies, Many Interactions, Many Travels
Many Activies, Many Interactions, Many Travels
The work of Newton da Costa is not limited to paracaconsistent logic. He has been working in other non-classical logics, producing some original works, just to mention one: the formalization of the principle of sufficent reason. Moreover, he has been working on the application of logic to physics (in parcticular with with Francisco Doria and Décio Krause).
Newton da Costa had a strong and provocative personality, encouraging young people to do reserach. He has been collaborating and/or exchanging with many people. It would be difficult to give an exhaustive list here. He was the main promoter of logic in Brazil and also in South America jointly with Rolando Chuaqui from Chile.
He travelled a lot since the 1960s: Australia, Poland, Bulgaria, Soviet Union, United States, France, South America, etc. During the The Sixth World Congress of Paraconsistency in Toruń (WCP6) and The Second Stanisław Jaśkowski Memorial Symposium the daugther of Jaśkowski gave a vivid talk recalling the visit of da Costa in Poland to her mother after the death of her father. Da Costa told me about his visit to the picturesque Richard Routley in Australia during which, at every breakfast, Routley was claiming "classical logic is wrong".
How I met da Costa and my Work with him
How I met da Costa and my Work with him
I discovered the work of da Costa in an unexpected way, by chance. In the second half of the 1980s, I was a student in Paris, studying at the same time philosophy, cinema and logic. At some point, I decided to focus on mathematical logic. I was doing a Master in the mathematical department of the University Paris 7 where there was the main group of logicians in France: Krivine, Girard, Lascar, just to give a few names. These people were concerned with the foundations of mathematics and, later on, of computer science. Daniel Andler was giving a class on modal logic and non-monotonic logic. But nobody knew paraconsistent logic.
At this time, I had a girl friend whose father was a psychoanalyst, having been analyzed by Jacques Lacan himself. During one week-end we were at his home near Angers by the river Loire and I read an interview with Newton da Costa in the Lacanian magazine L'âne, about the relations between paraconsistent logic and the inconscious, ending by a quote by Pierre. Curie “Il faut faire de la vie un rêve et faire d'un rêve une réalité” (“We must make life a dream and make a dream a reality.” Back to Paris, I started to read da Costa's papers on paraconsistent systems, and I did my Master research work (1990) on the system C1 with new results, see here.
I met da Costa for the first time when he visited Paris in December 1990. He asked me why I was interested in paraconsistent logic and I told him it was because I was interested on the foundations of logic and was curious to know up to which point the so-called principle of non-contradiction was a fundamental law of logic or not. He was satisfied with my answer and told me that he had been worry because of some fanatics having the idea that contradiction was the essence of everything. He invited me to work with him in São Paulo, Brazil.
Newton da Costa, Otávio Bueno, Jean-Yves Beziau
at the Brazilian Congress of Philosophyy, São Paulo, 1995
I arrived in Brazil at the beginning of August 1991 for a one year stay. We worked together on different topics but mainly on a general approach of logic based on bivaluations (see this paper) an approach he had starting to develop working with Elias Alves on a bivalent semantics for C1.
In October 1992 I went to Poland, where I stayed up to December 1993, coining the expression "Universal Logic". Back to Paris I translated da Costa's main book Essay on the Foundations of Logic in French (with two appendices written by myslef, one on paraconsistent logic, on on the theory of bivaluations). I also at this time gave some private lessons of logic to the President of the International Lacanian Association, Liliane Mahjoub-Trobas, who was a friend of the President o fthe Brazilian Lacanian Association, Jorge Forbes, with whom da Costa did a psychoanalysis.
I came back to São Paulo for a second time in July 1994, this time affiliated to the Institute of Adanced Studies. This is when I coined the expression "paranormal logic". Together with da Costa and Otávio Bueno, who was a student there, we wrote two papers and a book on paraconsistent logic.
After that I came to UCLA for a Fulbright research stay at the Dept of Mathematics, invited by Herbert Enderton. I defended a PhD in Paris in mathematical logic in July 1995 (Daniel Andler was my advisor) and after that came back to Brazil, this time in Rio. I defended a PhD at the USP (University of São Paulo) in philosophical logic with Newton da Costa in 1996.
Arthur Buchsabum, Newton da Costa, Jean-Yves Beziau
at the 2nd World Congress on Paraconsistency, Juquehy, Brazil, 2000
After doing posdocs in Rio and Stanford, I settled down for some years in my country of origin, Switzerland, where I started to develop my main project of Universal Logic, which originated with my work on paraconsistent logic; see my paper "From paraconsistent logic to universal logic".
But I kept continuously working on paraconsistent logic, publishing papers with new systems and new results and organizing events, in particular the 5th edition of the World Congress on Paraconsistency in India in 2014, together with Mihir Chakraborty, including the contest Picturing Contradiction. And more recently in 2024 the 7th edition of the World Congress on Paraconsisteny in Mexico with María del Rosario Martínez-Ordaz and Luis Estrada.
In 2014 I created the Newton da Costa Logic Prize for his 85 birthday, and, following the same model, I created logic pzizes in the world, and launched the 1st World Logic Prizes Contest at the 6th UNILOG (World Congress and School on universal Logic) in Vichy, France.
The last paper I wrote with Newton da Costa is entitled "Is god paraconsistent?", part of the project on logic and religion (LARA) that I initiated with Ricardo Silvestre.
In 2019 for his 90th Birthady, I organized a celebration on September 16 in São Paulo and a big international event in December in Rio de Janeiro: Creativity'2019. At the begiining of 2024, I started the organization of an event for his 95th birthday, a follow up of Creativity'2019, entitled ici-Rio 2024: Imagination, Creativity, Intelligence. Due to his death, the event was dedicated to his memory, and a book will be published.
1st Newton da Costa Prize of Logic, Award Ceremony 2015, Florianópolis, August, 20, 2015
Marcelo Coniglio (president of the Brazilian Logic Society), Itala D'Ottaviano (co-organizer of the prize), Newton da Costa, Rodrigo Freire (winner) , Jean-Yves Beziau (founder and organizer of the prize), Décio Krause (president of the jury)
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