Newton Carneiro Affonso da Costa (born 16 September 1929 in

Curitiba Curitiba () is the capital and largest city in the state of Paraná in Brazil. The city's population was 1,948,626 , making it the eighth most populous city in Brazil and the largest in Brazil's South Region. The Curitiba Metropolitan area ...

, Brazil) is a Brazilian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History O ...

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, and

philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

. He studied engineering and mathematics at the

Federal University of Paraná The Federal University of Paraná ( pt, Universidade Federal do Paraná, UFPR) is a public university headquartered in Curitiba, Paraná, Brazil. UFPR is considered to be one of the oldest universities in Brazil. UFPR ranks as 37th best universi ...

and the title of his 1961 Ph.D. dissertation was ''Topological spaces and continuous functions''.

Work

Paraconsistency

Da Costa's international recognition came especially through his work on

paraconsistent logic A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syst ...

and its application to various fields such as philosophy,

law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...

, computing, and

artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...

. He is one of the founders of this

non-classical logic Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of ...

. In addition, he constructed the theory of quasi-truth that constitutes a generalization of

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

's theory of truth, and applied it to the foundations of science.

Other fields; foundations of physics

The scope of his research also includes

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the st ...

, generalized

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

, axiomatic foundations of

quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ch ...

and relativity, complexity theory, and abstract logics.http://hps.master.univ-paris-diderot.fr/sites/hps.master.univ-paris-diderot.fr/files/users/fcontami/Paty,M-2000d-QuantClasDom.pdf Da Costa has significantly contributed to the

philosophy of logic Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application ...

, paraconsistent

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities ex ...

, and

philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ulti ...

. He served as the President of the Brazilian Association of Logic and the Director of the Institute of Mathematics at the

University of São Paulo The University of São Paulo ( pt, Universidade de São Paulo, USP) is a public university in the Brazilian state of São Paulo. It is the largest Brazilian public university and the country's most prestigious educational institution, the best ...

. He received many awards and held numerous visiting scholarships at universities and centers of research in all continents. Da Costa and physicist Francisco Antônio Dória axiomatized large portions of

classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...

with the help of

Patrick Suppes Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology ...

' predicates. They used that technique to show that for the axiomatized version of

dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called ' ...

, chaotic properties of those systems are undecidable and Gödel-incomplete, that is, a sentence like ''X is chaotic'' is undecidable within that axiomatics. They later exhibited similar results for systems in other areas, such as mathematical economics. Da Costa believes that the significant progress in the field of logic will give rise to new fundamental developments in computing and technology, especially in connection with non-classical logics and their applications.

Variable-binding term operators

Da Costa is co-discoverer of the truth-set principle and co-creator of the classical logic of variable-binding term operators—both with John Corcoran. He is also co-author with Chris Mortensen of the definitive pre-1980 history of variable-binding term operators in classical

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...

: “Notes on the theory of variable-binding term operators”, History and Philosophy of Logic, vol.4 (1983) 63–72.

P = NP

Together with Francisco Antônio Dória, Da Costa has published two papers with conditional relative proofs of the consistency of

P = NP The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term ''quickly'', used abov ...

with the usual set-theoretic axioms ZFC. The results they obtain are similar to the results of DeMillo and

Lipton Lipton is a British brand of tea, owned by Ekaterra. Lipton was also a supermarket chain in the United Kingdom, later sold to Argyll Foods, after which the company sold only tea. The company is named after its founder, Sir Thomas Lipton, who ...

(consistency of P = NP with fragments of arithmetic) and those of Sazonov and Maté (conditional proofs of the consistency of P = NP with strong systems). Basically da Costa and Doria define a formal sentence = NP which is the same as P = NP in the standard model for arithmetic; however, because = NP by its very definition includes a disjunct that is not refutable in ZFC, = NP is not refutable in ZFC, so ZFC + = NP is

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

(assuming that ZFC is). The paper then continues by an informal proof of the implication : If ZFC + = NP is consistent, then so is ZFC + = NP However, a review by Ralf Schindler points out that this last step is too short and contains a gap. A recently published (2006) clarification by the authors shows that their intent was to exhibit a conditional result that was dependent on what they call a "naïvely plausible condition". The 2003 conditional result can be reformulated, according to da Costa and Doria 2006, as : If ZFC + = NP is ω-consistent, then ZFC + = NPis consistent. So far no formal argument has been constructed to show that ZFC + = NP is ω-consistent. In his reviews for ''

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also p ...

'' of the da Costa/Doria papers on P=NP, logician

Andreas Blass Andreas Raphael Blass (born October 27, 1947) is a mathematician, currently a professor at the University of Michigan. He works in mathematical logic, particularly set theory, and theoretical computer science. Blass graduated from the University ...

states that "the absence of rigor led to numerous errors (and ambiguities)"; he also rejects da Costa's "naïvely plausible condition", as this assumption is "based partly on the possible non-totality of certain functionF and partly on an axiom equivalent to the totality of F".

Selected publications

Articles and lectures

*N.C.A. da Costa, ''Sistemas Formais Inconsistentes''. Curitiba, Brazil: Universidade Federal do Paraná, 1963. *N.C.A. da Costa, ''Review of the article by Corcoran, Hatcher, and Herring on variable-binding term operators'', Zentralblat fur Mathematik, vol. 247, pp. 8–9, 1973. *N.C.A. da Costa, ''On the theory of inconsistent formal systems''. Notre Dame Journal of Formal Logic 1974 ; 15: 497–510. *N.C.A. da Costa (with L. Dubikajtis), ''On Jaskowski's Discussive Logic''. Non-Classical Logics, Model Theory and Computability, North-Holland Publishing Company, Amsterdam, pp. 37–56, 1977. *N.C.A. da Costa (with C. Mortensen), ''Notes on the theory of variable-binding term operators'', History and Philosophy of Logic, vol.4, pp. 63–72, 1983. *N.C.A. da Costa, ''Pragmatic probability''. Erkenntnis 1986; 25: 141–162. *N.C.A. da Costa (with Walter Carnielli), ''Paraconsistent deontic logics''. Philosophia – The Philos. Quarterly of Israel, vol.16, numbers 3 and 4, pp. 293–305, 1988. *N.C.A. da Costa (with V.S. Subrahmanian), ''Paraconsistent logic as a formalism for reasoning about inconsistent knowledge bases''. Artificial Intelligence in Medicine 1989; 1: 167–174. *N.C.A. da Costa (with F.A. Doria), ''Undecidability and incompleteness in classical mechanics'', International J. Theoretical Physics, vol. 30 (1991), 1041–1073. *N.C.A. da Costa, ''Paraconsistent logic''. In Stanisław Jaškowski Memorial Symposium, pp. 29–35. Department of Logic, Nicholas Copernicus University of Toruń. 1998. *N.C.A. da Costa (with O. Bueno and S. French), ''Is there a Zande Logic?'' History and Philosophy of Logic 1998; 19: 41–54. *N.C.A. da Costa (with O. Bueno and A.G. Volkov), ''Outline of a paraconsistent category theory''. In P Weingartner (ed.), Alternative Logics: Do Sciences Need them? Berlin: Springer-Verlag, 2004, pp. 95–114. *N.C.A. da Costa (with F. A. Doria), ''Consequences of an exotic definition for P = NP''. Applied Mathematics and Computation, vol. 145 (2003), 655–665, and ''Addendum to "Consequences of an exotic formulation for P=NP"''. Applied Mathematics and Computation, vol. 172 (2006), 1364–1367. *N.C.A. da Costa (with F. A. Doria), ''Computing the future,'' in Computability, Complexity and Constructivity in Economic Analysis, ed. K. V. Velupillai, Blackwell, 2005. *N.C.A. da Costa (with F. A. Doria), ''Some thoughts on hypercomputation,'' Applied Mathematics and Computation, vol. 178 (2006) 83–92.

Books

*N.C.A. da Costa, ''Lógica Indutiva e Probabilidade''. Hucitec-EdUSP, 2a. ed., São Paulo, 1993. *N.C.A. da Costa, ''Logique Classique et Non-Classique''. Paris, Masson, 1997. *N.C.A. da Costa, ''O conhecimento científico''. São Paulo, Discurso Editorial, 2a. Ed., 1999. *N.C.A. da Costa, J.M. Abe, J.I. da Silva Filho, A.C. Murolo and C.F.S. Leite ''Lógica Paraconsistente Applicada''. São Paulo, Atlas, 1999. *N.C.A. da Costa and S. French, ''Science and Partial Truth: A Unitary Approach to Models and Scientific Reasoning''. (Oxford Studies in Philosophy of Science), Oxford University Press, 2003. * Shyam Wuppuluri, N.C.A. da Costa (Eds.), "Wittgensteinian (adj.) : Looking at the World from the Viewpoint of Wittgenstein's Philosophy" Springer — The Frontiers Collection, 2019.

Essays on N. C. A. da Costa

* Nicola Grana, ''Sulla teoria delle valutazioni di N.C.A. da Costa''. Naples: Liguori Editore, 1990. Pp. 75.

References

External links

Biography at Unicamp (in Portuguese)
*Carnielli, W., Coniglio, M.E., e Marcos, J.

''Handbook of Philosophical Logic'', 2nd edition, volume 14, pages 15–107. Springer-Verlag.
Google scholar profile
{{DEFAULTSORT:Costa, Newton Da Brazilian philosophers 20th-century Brazilian mathematicians 20th-century Brazilian philosophers Brazilian logicians 1929 births Living people People from Curitiba University of São Paulo faculty Paraconsistent logic Federal University of Paraná faculty Federal University of Paraná alumni