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Newton Carneiro Affonso da Costa  

""
Newton da Costa at Berkeley in 1973


Born  "Curitiba, "Brazil 
16 September 1929
Known for  "Paraconsistent logic 
Scientific career  
Fields  "Logic, "Mathematics, "Philosophy and "Philosophy of Science 
Doctoral students  "JeanYves Béziau "Marcelo Samuel Berman "Walter Carnielli 
Newton Carneiro Affonso da Costa (born 16 September 1929 in "Curitiba, "Brazil) is a Brazilian "mathematician, "logician, and "philosopher.^{[1]} He studied engineering and mathematics at the "Federal University of Paraná in "Curitiba and the title of his 1961 Ph.D. dissertation was Topological spaces and continuous functions.^{[1]}
Da Costa's international recognition came especially through his work on "paraconsistent logic and its application to various fields such as philosophy, "law, "computing, and "artificial intelligence.^{[2]} He is one of the founders of this "nonclassical logic.^{[3]} In addition, he constructed the theory of quasitruth that constitutes a generalization of "Alfred Tarski's theory of truth, and applied it to the foundations of science.
The scope of his research also includes "model theory, generalized "Galois theory, axiomatic foundations of quantum theory and "relativity, "complexity theory, and abstract logics.^{[4]} Da Costa has significantly contributed to the "philosophy of logic, paraconsistent "modal logics, "ontology, and "philosophy of science. He served as the President of the Brazilian Association of Logic and the Director of the Institute of Mathematics at the "University of Sao Paulo. He received many awards and held numerous visiting scholarships at universities and centers of research in all continents.^{[1]}
Da Costa and physicist "Francisco Antônio Dória axiomatized large portions of "classical physics with the help of "Patrick Suppes' predicates. They used that technique to show that for the axiomatized version of "dynamical systems theory, chaotic properties of those systems are undecidable and Gödelincomplete, 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 nonclassical logics and their applications.
Da Costa is codiscoverer of the truthset principle and cocreator of the classical logic of variablebinding term operators—both with "John Corcoran. He is also coauthor with Chris Mortensen of the definitive pre1980 history of variablebinding term operators in classical first order logic: “Notes on the theory of variablebinding term operators”, History and Philosophy of Logic, vol.4 (1983) 63–72.
Together with "Francisco Antônio Dória, Da Costa has published two papers with conditional relative proofs of the consistency of "P = NP with the usual settheoretic axioms "ZFC. The results they obtain are similar to the results of DeMillo and Lipton (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 [P = NP]' which is the same as P = NP in the standard model for arithmetic; however, because [P = NP]' by its very definition includes a disjunct that is not refutable in ZFC, [P = NP]' is not refutable in ZFC, so ZFC + [P = NP]' is "consistent (assuming that ZFC is). The paper then continues by an informal proof of the implication
However, a review by Ralf Schindler^{[5]} 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 (in press), as
So far no formal argument has been constructed to show that ZFC + [P = NP]' is omegaconsistent.
In his reviews for "Mathematical Reviews of the da Costa/Doria papers on P=NP, logician "Andreas Blass 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 nontotality of [a certain function] F and partly on an axiom equivalent to the totality of F".