Newton da Costa
Newton Carneiro Affonso da Costa (born 16 September 1929 in Curitiba, Brazil) is a Brazilian mathematician, logician, and philosopher. 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". Work. Paraconsistency. 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. He is one of the founders of this non-classical logic. In addition, he constructed the theory of quasi-truth that constitutes  a generalization of Alfred Tarski'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, generalized Galois theory, axiomatic foundations of quantum theory and relativity, complexity theory, and abstract logics. 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 São Paulo. 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 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 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: “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 with the usual set-theoretic 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 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 So far no formal argument has been constructed  to show that ZFC + [P = NP]' is ω-consistent. 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 non-totality of [a certain function] F and partly on an axiom equivalent to the totality of F".