George Boolos
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- This article is not about George Boole, the 19th century British mathematical logician.
| George Boolos | |
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| Born | September 4, 1940 New York, New York, USA |
| Died | May 27, 1996 Cambridge, Massachusetts, USA |
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.
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[edit] Life
Boolos graduated from Princeton University in 1961 with a B.A. in mathematics. Oxford University awarded him the B.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. After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career.
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. A possibly apocryphal story has it that at the end of his viva, Hilary Putnam asked him, "And tell us, Mr. Boolos, what does the analytical hierarchy have to do with the real world?" Without hesitating Boolos replied, "It's part of it".
An expert on puzzles of all kinds, in 1993 Boolos reached the London Regional Final of the Times crossword competition. 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.
[edit] Work
Boolos coauthored with Richard Jeffrey the first three editions of the classic university text on mathematical logic, Computability and Logic.
Kurt Gödel wrote the first paper on provability logic, modal logic — the logic of necessity and possibility — applied to the theory of mathematical proof. But Boolos took it much further than Gödel ever did, making it the subject of an entire monograph The Unprovability of Consistency. A few years after this book appeared, he discovered major work on this subject of which he had been unaware because it was written in Russian. Translating this Russian work with the help of a dictionary led him to rewrite Boolos (1979); the result is The Logic of Provability.
Boolos was an authority on the 19th-century German mathematician and philosopher Gottlob Frege. Boolos argued 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. Edward Zalta and others have pursued this idea, which has given a new lease on life to (a chastened form of) logicism, the argument that the natural numbers and the operations thereon (what is now called Peano arithmetic) can be seen as theorems of logic; see Anderson and Zalta (2004).
Boolos argued that if one reads the second-order variables in monadic second-order logic as plural terms, this logic can be interpreted as making no ontological commitments to entities other than those over which the first-order variables range. The result is plural quantification. 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. 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.
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. There are also papers on Dedekind, Cantor, and Russell.
[edit] See also
- Hume's principle
- plural quantification
- second order logic
- provability logic
- The hardest logic puzzle ever
[edit] Books
- 2002 (1974) (with Richard Jeffrey). Computability and Logic. Cambridge: Cambridge University Press.
- 1979. The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press.
- 1990 (editor). Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press.
- 1993. The Logic of Provability. Cambridge University Press. A revision of Boolos (1979).
- 1998 (Richard Jeffrey and John Burgess, eds.). Logic, Logic, and Logic. Harvard University Press.
[edit] 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.
1969, "Effectiveness and natural languages" in Sidney Hook, ed., 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.
1971, "The iterative conception of set," Journal of Philosophy 68: 215-231. Reprinted in Paul Benacerraf and Hilary Putnam, eds.,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.
1974, "Arithmetical functions and minimization," Zeitschrift für mathematische Logik und Grundlagen der Mathematik 20: 353-354.
1974a, "Reply to Charles Parsons' 'Sets and classes'." First published in LLL.
1975, "Friedman's 35th problem has an affirmative solution," Notices of the American Mathematical Society 22: A-646.
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. 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.
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.
1980d, Review of Raymond M. Smullyan, What is the Name of This Book? The Philosophical Review 89: 467-470.
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.
1984f, "The justification of mathematical induction," PSA 2: 469-475. 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. 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. 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.
1990b, Review of V. A. Uspensky, Gödel's Incompleteness Theorem, Journal of Symbolic Logic 55: 889-891.
1990c, "The standard of equality of numbers" in Boolos, G., ed., Meaning and Method: Essays in Honor of Hilary Putnam. 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. LLL.
1995a, "Introductory note to *1951" in Solomon Feferman et al., eds., Kurt Gödel, Collected Works, vol. 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. Morton and S. P. Stich, eds., Paul Benacerraf and his Critics. 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. Heck, Jr., ed., Language, Thought, and Logic: Essays in Honour of Michael Dummett. 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." 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. Cambridge University Press. LLL.
[edit] Reference
- Anderson, D., and Edward Zalta (2004) "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1-26.
- Peter Simons (1982) "On understanding Lesniewski," History and Philosophy of Logic.
