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Newton Carneiro Affonso da Costa | |
---|---|
Born | |
Died | 16 April 2024 94) | (aged
Known for | Paraconsistent logic |
Scientific career | |
Fields | Logic, Mathematics, Philosophy and Philosophy of Science |
Doctoral students |
Newton Carneiro Affonso da Costa (16 September 1929 – 16 April 2024) was a Brazilian mathematician, logician, and philosopher. [1] Born in Curitiba, 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 was one of the founders of this non-classical logic. [3] 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.
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 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. [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 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.
Da Costa was 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.
Together with Francisco Antônio Dória, Da Costa 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 [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, 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".
Da Costa died on 16 April 2024, at the age of 94. [6]
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree. The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
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" systems of logic, which reject the principle of explosion.
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas can be effectively determined. A theory in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership can exist for them.
In mathematical logic, independence is the unprovability of a sentence from other sentences.
Originally, fallibilism is the philosophical principle that propositions can be accepted even though they cannot be conclusively proven or justified, or that neither knowledge nor belief is certain. The term was coined in the late nineteenth century by the American philosopher Charles Sanders Peirce, as a response to foundationalism. Theorists, following Austrian-British philosopher Karl Popper, may also refer to fallibilism as the notion that knowledge might turn out to be false. Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision. Fallibilism is often juxtaposed with infallibilism.
Non-classical 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 commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.
Jean-Yves Beziau (French:[bezjo]; born January 15, 1965, in Orléans, France is a Swiss Professor in logic at the University of Brazil, Rio de Janeiro, and Researcher of the Brazilian Research Council. He is permanent member and former president of the Brazilian Academy of Philosophy. Before going to Brazil, he was Professor of the Swiss National Science Foundation at the University of Neuchâtel in Switzerland and researcher at Stanford University working with Patrick Suppes.
Stanisław Jaśkowski was a Polish logician who made important contributions to proof theory and formal semantics. He was a student of Jan Łukasiewicz and a member of the Lwów–Warsaw School of Logic. He is regarded as one of the founders of natural deduction, which he discovered independently of Gerhard Gentzen in the 1930s. He is also known for his research into paraconsistent logic. Upon his death, his name was added to the Genius Wall of Fame. He was the President (rector) of the Nicolaus Copernicus University in Toruń.
Nicolai Alexandrovich Vasiliev, also Vasil'ev, Vassilieff, Wassilieff, was a Russian logician, philosopher, psychologist, poet. He was a forerunner of paraconsistent and multi-valued logics.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Francisco Antônio de Moraes Accioli Dória is a Brazilian mathematician, philosopher, and genealogist. Francisco Antônio Dória received his B.S. in Chemical Engineering from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1968 and then got his doctorate from the Brazilian Center for Research in Physics (CBPF), advised by Leopoldo Nachbin in 1977. Dória worked for a while at the Physics Institute of UFRJ, and then left to become a Professor of the Foundations of Communications at the School of Communications, also at UFRJ. Dória held visiting positions at the University of Rochester (NY), Stanford University (CA), and the University of São Paulo (USP). His most prolific period spawned from his collaboration with Newton da Costa, a Brazilian logician and one of the founders of paraconsistent logic, which began in 1985. He is currently Professor of Communications, Emeritus, at UFRJ and a member of the Brazilian Academy of Philosophy.
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether an arbitrary program eventually halts when run.
Walter Alexandre Carnielli is a Brazilian mathematician, logician, and philosopher, who works as a full professor of Logic at the State University of Campinas. After obtaining his Bachelor and M.Sc. degrees in mathematics at the State University of Campinas, he also obtained his Ph.D. in 1984 there under the supervision of Newton da Costa; subsequently, he worked as a post-doc at the University of California at Berkeley, following an invitation by Leon Henkin.
The following is a list of works by philosopher Graham Priest.
Itala Maria Loffredo D'Ottaviano is a Brazilian mathematical logician who was president of the Brazilian Logic Society. Topics in her work have included non-classical logic, paraconsistent logic, many-valued logic, and the history of logic.