American philosopher and mathematical logician
George Stephen Boolos (;
[1] 4 September 1940 – 27 May 1996) was an
American
philosopher and a
mathematical logician who taught at the
Massachusetts Institute of Technology.
[2]
Life
Boolos was of
Greek-
Jewish descent.
[3] He graduated with an
A.B. in
mathematics from
Princeton University after completing a senior thesis, titled "A simple proof of
Gödel's first incompleteness theorem", under the supervision of
Raymond Smullyan.
[4]
Oxford University awarded him the
B.Phil. in 1963. In 1966, he obtained the first
PhD 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 (1994b) giving an account of
Gödel's second incompleteness theorem, employing only words of one syllable. 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.
Boolos died of
pancreatic cancer on 27 May 1996.
[5]
Work
Boolos coauthored with
Richard Jeffrey the first three editions of the classic university text on
mathematical logic, Computability and Logic. The book is now in its fifth edition, the last two editions 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. 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 Gödel's Second Incompleteness 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 Gödel sentence, "This sentence is not provable") was provable and hence true. Martin Löb 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.
Boolos was an authority on the 19th-century German mathematician and philosopher
Gottlob Frege. 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. The resulting system has since been the subject of intense work.[
citation needed]
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. 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 Leśniewski.
Shortly before his death, Boolos chose 30 of his papers to be published in a book. The result is perhaps his most highly 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.
Publications
Books
Articles
- 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.
- 1975b, "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.
- 1994b, "
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 Logic 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.
See also
Notes
References
-
Peter Simons (1982) "On understanding Lesniewski," History and Philosophy of Logic.
-
Solomon Feferman (1960) "Arithmetization of metamathematics in a general setting," Fundamentae Mathematica vol. 49, pp. 35–92.
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