- For the 19th-century British mathematical logician of a similar name, see George Boole. George Boole (buːl ( November 2, 1815 &ndash December 8, 1864) was a British Mathematician and Philosopher.
| George Boolos | |
| Born | September 4, 1940 New York, New York, U.S. |
|---|---|
| Died | May 27, 1996 (aged 55) Cambridge, Massachusetts, U.S. |
George Stephen Boolos (September 4, 1940, New York City – May 27, 1996) was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Events 476 - Romulus Augustus, last emperor of the Western Roman Empire, is deposed when Odoacer proclaims himself Year 1940 ( MCMXL) was a Leap year starting on Monday (link will display the full 1940 calendar of the Gregorian calendar. New York ( is a state in the Mid-Atlantic and Northeastern regions of the United States and is the nation's third most populous New York ( is a state in the Mid-Atlantic and Northeastern regions of the United States and is the nation's third most populous The United States of America —commonly referred to as the Events 927 - Simeon the Great, Tsar of Bulgaria, dies 1120 - Richard III of Capua is anointed Year 1996 ( MCMXCVI) was a Leap year starting on Monday (link will display full 1996 Gregorian calendar) The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England The Commonwealth of Massachusetts ( is a state located in the New England region of the northeastern United States. The United States of America —commonly referred to as the Events 476 - Romulus Augustus, last emperor of the Western Roman Empire, is deposed when Odoacer proclaims himself Year 1940 ( MCMXL) was a Leap year starting on Monday (link will display the full 1940 calendar of the Gregorian calendar. The City of New York Events 927 - Simeon the Great, Tsar of Bulgaria, dies 1120 - Richard III of Capua is anointed Year 1996 ( MCMXCVI) was a Leap year starting on Monday (link will display full 1996 Gregorian calendar) Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic.
Contents |
Life
Boolos graduated from Princeton University in 1961 with an A.B. in mathematics. Princeton University is a private Coeducational research university located in Princeton, New Jersey. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Oxford University awarded him the B. The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the Phil in 1963. In 1966, he obtained the first Ph. D. in philosophy ever awarded by the Massachusetts Institute of Technology, under the direction of Hilary Putnam. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career until his death from cancer. Columbia University is a private University in the United States and a member of the Ivy League. Year 1969 ( MCMLXIX) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar. Cancer (medical term Malignant Neoplasm) is a class of Diseases in which a group of cells display uncontrolled [1]
A charismatic speaker well-known for his clarity and wit, he once delivered a lecture (1994a) giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most At the end of his viva, Hilary Putnam asked him, "And tell us, Mr. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy Boolos, what does the analytical hierarchy have to do with the real world?" Without hesitating Boolos replied, "It's part of it". In Mathematical logic and Descriptive set theory, the analytical hierarchy is a higher type analogue of the Arithmetical hierarchy.
An expert on puzzles of all kinds, in 1993 Boolos reached the London Regional Final of the Times crossword competition. The Times is a daily national Newspaper published in the United Kingdom since 1785 when it was known as The Daily Universal Register. His score was one of the highest ever recorded by an American. He wrote a paper on "the hardest logic puzzle ever" -- one of many puzzles created by Raymond Smullyan. The Hardest Logic Puzzle Ever is a title coined by George Boolos in La Repubblica 1992 under the title L'indovinello più difficile del Raymond Merrill Smullyan (born 1919 is an American Mathematician, Logician, Philosopher, and magician.
Work
Boolos coauthored with Richard Jeffrey the first three editions of the classic university text on mathematical logic, Computability and Logic. Richard C Jeffrey ( 5 August 1926 – 9 November 2002) was an American Philosopher, Logician, and probability Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. The book is now in its fourth edition, the last one updated by John P. Burgess.
Kurt Gödel wrote the first paper on provability logic, which applies modal logic — the logic of necessity and possibility — to the theory of mathematical proof, but Gödel never developed the subject to any significant extent. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Provability logic is a Modal logic, in which the box (or "necessity" operator is interpreted as 'it is provable that' A modal logic is any system of formal logic that attempts to deal with modalities. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Boolos was one of its earliest proponents and pioneers, and he produced the first book-length treatment of it, The Unprovability of Consistency, published in 1979. The solution of a major unsolved problem some years later led to a new treatment, The Logic of Provability, published in 1993. The modal-logical treatment of provability helped demonstrate the "intensionality" of Godel's Second Incompletness Theorem, meaning that the theorem's correctness depends on the precise formulation of the provability predicate. These conditions were first identified by David Hilbert and Paul Bernays in their Grundlagen der Arithmetik. The unclear status of the Second Theorem was noted for several decades by logicians such as Georg Kreisel and Leon Henkin, who asked whether the formal sentence expressing "This sentence is provable" (as opposed to the Godel sentence, "This sentence is not provable) was provable and hence true. Martin Lob showed Henkin's conjecture to be true, as well as identifying an important "reflection" principle also neatly codified using the modal logical approach. Some of the key provability results involving the representation of provability predicates had been obtained earlier using very different methods by Solomon Feferman. Solomon Feferman (b December 13, 1928) is an American Philosopher and Mathematician with major works in Mathematical logic
Boolos was an authority on the 19th-century German mathematician and philosopher Gottlob Frege. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Boolos proved a conjecture due to Crispin Wright (and also proved, independently, by others), that the system of Frege's Grundgesetze, long thought vitiated by Russell's paradox, could be freed of inconsistency by replacing one of its axioms, the notorious Basic Law V with Hume's Principle. Crispin Wright (born 1942 is a British Philosopher, who has written on neo- Fregean Philosophy of mathematics, Wittgenstein 's later Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Hume's Principle, or HP —the terms were coined by George Boolos &mdashsays that the number of F s is equal to the number of G s if there is a The resulting system has since been the subject of intense work.
Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no ontological commitment to entities other than those over which the first-order variables range. In Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. In Mathematics and logic, plural quantification is the theory that an individual Variable x may take on In Philosophy, ontology (from the Greek, genitive: of being (part First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science The result is plural quantification. In Mathematics and logic, plural quantification is the theory that an individual Variable x may take on David Lewis employed plural quantification in his Parts of Classes to derive a system in which Zermelo-Fraenkel set theory and the Peano axioms were all theorems. David Kellogg Lewis ( September 28, 1941 &ndash October 14, 2001) is considered to have been one of the leading philosophers of the latter Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural While Boolos is usually credited with plural quantification, Peter Simons (1982) has argued that the essential idea can be found in the work of Stanislaw Lesniewski. In Mathematics and logic, plural quantification is the theory that an individual Variable x may take on Peter Simons may refer to Peter Simons (director, Belgian film director Peter Simons (philosopher, Professor of philosophy Stanisław Leśniewski ( March 30 1886 – May 13 1939) was a Polish Mathematician, Philosopher and Logician
Shortly before his death, Boolos chose 30 of his papers to be published in a book. The result is perhaps his most widely regarded work, his posthumous Logic, Logic, and Logic. This book reprints much of Boolos's work on the rehabilitation of Frege, as well as a number of his papers on set theory, second-order logic and nonfirstorderizability, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem. In Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. In Formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in standard First-order logic. In Mathematics and logic, plural quantification is the theory that an individual Variable x may take on Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most There are also papers on Dedekind, Cantor, and Russell. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian
See also
Books
- 2002 (1974) (with Richard Jeffrey). Provability logic is a Modal logic, in which the box (or "necessity" operator is interpreted as 'it is provable that' The Hardest Logic Puzzle Ever is a title coined by George Boolos in La Repubblica 1992 under the title L'indovinello più difficile del Richard C Jeffrey ( 5 August 1926 – 9 November 2002) was an American Philosopher, Logician, and probability Computability and Logic. Cambridge: Cambridge University Press.
- 1979. The Unprovability of Consistency: An Essay in Modal Logic. A modal logic is any system of formal logic that attempts to deal with modalities. Cambridge University Press.
- 1990 (editor). Meaning and Method: Essays in Honor of Hilary Putnam. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy Cambridge University Press.
- 1993. The Logic of Provability. Cambridge University Press. Not a revision of Boolos (1979).
- 1998 (Richard Jeffrey and John P. Richard C Jeffrey ( 5 August 1926 – 9 November 2002) was an American Philosopher, Logician, and probability Burgess, eds. ). Logic, Logic, and Logic. Harvard University Press.
Articles by George Boolos
LLL = reprinted in Logic, Logic, and Logic.
FPM = reprinted in Demopoulos, W. , ed. , 1995. Frege's Philosophy of Mathematics. Harvard Univ. Press.
1968 (with Hilary Putnam), "Degrees of unsolvability of constructible sets of integers," Journal of Symbolic Logic 33: 497-513. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy
1969, "Effectiveness and natural languages" in Sidney Hook, ed. Sidney Hook ( December 20 1902 &ndash July 12 1989) was a prominent New York intellectual and Philosopher who championed , Language and Philosophy. New York University Press.
1970, "On the semantics of the constructible levels," ' 16: 139-148.
1970a, "A proof of the Löwenheim-Skolem theorem," Notre Dame Journal of Formal Logic 11: 76-78. In Mathematical logic, the Löwenheim–Skolem theorem states that if a countable first-order theory has an infinite model then for every infinite Cardinal number
1971, "The iterative conception of set," Journal of Philosophy 68: 215-231. Reprinted in Paul Benacerraf and Hilary Putnam, eds. Paul Benacerraf is a philosopher of mathematics who has been teaching at Princeton University since he joined the faculty in 1960 Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy ,1984. Philosophy of Mathematics: Selected Readings, 2nd ed. Cambridge Univ. Press: 486-502. LLL
1973, "A note on Evert Willem Beth's theorem," Bulletin de l'Academie Polonaise des Sciences 2: 1-2. Evert Willem Beth ( July 7, 1908 &ndash April 12, 1964) was a Dutch Philosopher and Logician, whose work principally
1974, "Arithmetical functions and minimization," Zeitschrift für mathematische Logik und Grundlagen der Mathematik 20: 353-354.
1974a, "Reply to Charles Parsons' 'Sets and classes'. Charles Parsons (1933-- is a distinguished figure in the Philosophy of mathematics and son of social scientist Talcott Parsons. " First published in LLL.
1975, "Friedman's 35th problem has an affirmative solution," Notices of the American Mathematical Society 22: A-646. Harvey Friedman (born circa 1948 is a Mathematical logician at Ohio State University in Columbus Ohio.
1975a, "On Kalmar's consistency proof and a generalization of the notion of omega-consistency," Archiv für Mathematische Logik und Grundlagenforschung 17: 3-7.
1975a, "On second-order logic," Journal of Philosophy 72: 509-527. In Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. LLL.
1976, "On deciding the truth of certain statements involving the notion of consistency," Journal of Symbolic Logic 41: 779-781.
1977, "On deciding the provability of certain fixed point statements," Journal of Symbolic Logic 42: 191-193.
1979, "Reflection principles and iterated consistency assertions," Journal of Symbolic Logic 44: 33-35.
1980, "Omega-consistency and the diamond," Studia Logica 39: 237-243.
1980a, "On systems of modal logic with provability interpretations," Theoria 46: 7-18. A modal logic is any system of formal logic that attempts to deal with modalities.
1980b, "Provability in arithmetic and a schema of Grzegorczyk," Fundamenta Mathematicae 106: 41-45.
1980c, "Provability, truth, and modal logic," Journal of Philosophical Logic 9: 1-7. A modal logic is any system of formal logic that attempts to deal with modalities.
1980d, Review of Raymond M. Smullyan, What is the Name of This Book? The Philosophical Review 89: 467-470. Raymond Merrill Smullyan (born 1919 is an American Mathematician, Logician, Philosopher, and magician.
1981, "For every A there is a B," Linguistic Inquiry 12: 465-466.
1981a, Review of Robert M. Solovay, Provability Interpretations of Modal Logic," Journal of Symbolic Logic 46: 661-662.
1982, "Extremely undecidable sentences," Journal of Symbolic Logic 47: 191-196.
1982a, "On the nonexistence of certain normal forms in the logic of provability," Journal of Symbolic Logic 47: 638-640.
1984, "Don't eliminate cut," Journal of Philosophical Logic 13: 373-378. LLL.
1984a, "The logic of provability," American Mathematical Monthly 91: 470-480.
1984b, "Nonfirstorderizability again," Linguistic Inquiry 15: 343.
1984c, "On 'Syllogistic inference'," Cognition 17: 181-182.
1984d, "To be is to be the value of a variable (or some values of some variables)," Journal of Philosophy 81: 430-450. LLL.
1984e, "Trees and finite satisfiability: Proof of a conjecture of John Burgess," Notre Dame Journal of Formal Logic 25: 193-197. John Burgess may refer to John Burgess (political scientist, American political scientist John Burgess (TV host, Australian television
1984f, "The justification of mathematical induction," PSA 2: 469-475. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that LLL.
1985, "1-consistency and the diamond," Notre Dame Journal of Formal Logic 26: 341-347.
1985a, "Nominalist Platonism," The Philosophical Review 94: 327-344. LLL.
1985b, "Reading the Begriffsschrift," Mind 94: 331-344. Begriffsschrift is the title of a short book on Logic by Gottlob Frege, published in 1879, and is also the name of the Formal system LLL; FPM: 163-81.
1985c (with Giovanni Sambin), "An incomplete system of modal logic," Journal of Philosophical Logic 14: 351-358.
1986, Review of Yuri Manin, A Course in Mathematical Logic, Journal of Symbolic Logic 51: 829-830.
1986-87, "Saving Frege from contradiction," Proceedings of the Aristotelian Society 87: 137-151. LLL; FPM 438-52.
1987, "The consistency of Frege's Foundations of Arithmetic" in J. J. Thomson, ed. , 1987. On Being and Saying: Essays for Richard Cartwright. MIT Press: 3-20. LLL; FPM: 211-233.
1987a, "A curious inference," Journal of Philosophical Logic 16: 1-12. LLL.
1987b, "On notions of provability in provability logic," Abstracts of the 8th International Congress of Logic, Methodology and Philosophy of Science 5: 236-238.
1987c (with Vann McGee), "The degree of the set of sentences of predicate provability logic that are true under every interpretation," Journal of Symbolic Logic 52: 165-171.
1988, "Alphabetical order," Notre Dame Journal of Formal Logic 29: 214-215.
1988a, Review of Craig Smorynski, Self-Reference and Modal Logic, Journal of Symbolic Logic 53: 306-309.
1989, "Iteration again," Philosophical Topics 17: 5-21. LLL.
1989a, "A new proof of the Gödel incompleteness theorem," Notices of the American Mathematical Society 36: 388-390. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most LLL. An afterword appeared under the title "A letter from George Boolos," ibid. , p. 676. LLL.
1990, "On 'seeing' the truth of the Gödel sentence," Behavioral and Brain Sciences 13: 655-656. LLL.
1990a, Review of Jon Barwise and John Etchemendy, Turing's World and Tarski's World, Journal of Symbolic Logic 55: 370-371. Kenneth Jon Barwise ( June 29, 1942 - March 5, 2000) was a US John W Etchemendy (b 1952 in Reno Nevada is Stanford University 's twelfth and current Provost.
1990b, Review of V. A. Uspensky, Gödel's Incompleteness Theorem, Journal of Symbolic Logic 55: 889-891. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most
1990c, "The standard of equality of numbers" in Boolos, G. , ed. , Meaning and Method: Essays in Honor of Hilary Putnam. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy Cambridge Univ. Press: 261-278. LLL; FPM: 234-254.
1991, "Zooming down the slippery slope," Nous 25: 695-706. LLL.
1991a (with Giovanni Sambin), "Provability: The emergence of a mathematical modality," Studia Logica 50: 1-23.
1993, "The analytical completeness of Dzhaparidze's polymodal logics," Annals of Pure and Applied Logic 61: 95-111.
1993a, "Whence the contradiction?" Aristotelian Society Supplementary Volume 67: 213-233. LLL.
1994, "1879?" in P. Clark and B. Hale, eds. Reading Putnam. Oxford: Blackwell: 31-48. LLL.
1994a, "The advantages of honest toil over theft," in A. George, ed. , Mathematics and Mind. Oxford University Press: 27-44. LLL.
1994a, "Gödel's second incompleteness theorem explained in words of one syllable," Mind 103: 1-3. LLL.
1995, "Frege's theorem and the Peano postulates," Bulletin of Symbolic Logic 1: 317-326. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 LLL.
1995a, "Introductory note to *1951" in Solomon Feferman et al. Solomon Feferman (b December 13, 1928) is an American Philosopher and Mathematician with major works in Mathematical logic , eds. , Kurt Gödel, Collected Works, vol. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher 3. Oxford University Press: 290-304. LLL. *1951 is Gödel’s 1951 Gibbs lecture, "Some basic theorems on the foundations of mathematics and their implications. "
1995b, "Quotational ambiguity" in Leonardi, P. , and Santambrogio, M. , eds. On Quine. Cambridge University Press: 283-296. LLL
1996, "The hardest logical puzzle ever," Harvard Review of Philosophy 6: 62-65. LLL. Italian translation by Massimo Piattelli-Palmarini, "L'indovinello piu difficile del mondo," La Repubblica (16 April 1992): 36-37.
1996a, "On the proof of Frege's theorem" in A. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Morton and S. P. Stich, eds. , Paul Benacerraf and his Critics. Paul Benacerraf is a philosopher of mathematics who has been teaching at Princeton University since he joined the faculty in 1960 Cambridge MA: Blackwell. LLL.
1997, "Constructing Cantorian counterexamples," Journal of Philosophical Logic 26: 237-239. LLL.
1997a, "Is Hume's principle analytic?" In Richard G. David Hume (26 April 1711 25 August 1776 Scottish Philosopher, Economist, and Historian is an important figure in Western philosophy Heck, Jr. , ed. , Language, Thought, and Logic: Essays in Honour of Michael Dummett. Sir Michael Anthony Eardley Dummett FBA DLitt (born 1925 is a leading British Philosopher. Oxford Univ. Press: 245-61. LLL.
1997b (with Richard Heck), "Die Grundlagen der Arithmetik, §§82-83" in Matthias Schirn, ed. , Philosophy of Mathematics Today. Oxford Univ. Press. LLL.
1998, "Gottlob Frege and the Foundations of Arithmetic. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 " First published in LLL. French translation in Mathieu Marion and Alain Voizard eds. , 1998. Frege. Logique et philosophie. Montréal and Paris: L'Harmattan: 17-32.
2000, "Must we believe in set theory?" in Gila Sher and Richard Tieszen, eds. , Between Logic and Intuition: Essays in Honour of Charles Parsons. Charles Parsons may refer to Charles Algernon Parsons (1854&ndash1931 British engineer known for his invention of the steam turbine Charles Cambridge University Press. LLL.
References
- Peter Simons (1982) "On understanding Lesniewski," History and Philosophy of Logic. Peter Simons may refer to Peter Simons (director, Belgian film director Peter Simons (philosopher, Professor of philosophy
- Solomon Feferman (1960) "Arithmetization of metamathematics in a general setting," Fundamentae Mathematica vol. Solomon Feferman (b December 13, 1928) is an American Philosopher and Mathematician with major works in Mathematical logic 49, pp. 35-92.
External links
- George Boolos Memorial Web Site
- George Boolos. The hardest logic puzzle ever. The Harvard Review of Philosophy, 6:62–65, 1996.
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