- George Boolos
Infobox Person

name = George Boolos

birth_date = birth date|1940|9|4|mf=y

birth_place =New York ,New York , U.S.

death_date = death date and age|1996|5|27|1940|9|4|mf=y

death_place =Cambridge ,Massachusetts , U.S.**George Stephen Boolos**(September 4 ,1940 ,New York City –May 27 ,1996 ) was aphilosopher and amathematical logic ian who taught at theMassachusetts Institute of Technology .**Life**Boolos graduated from

Princeton University in 1961 with an A.B. inmathematics .Oxford University awarded him the B.Phil in 1963. In 1966, he obtained the first Ph.D. in philosophy ever awarded by theMassachusetts Institute of Technology , under the direction ofHilary Putnam . After teaching three years atColumbia University , he returned to MIT in 1969, where he spent the rest of his career until his death fromcancer . [*[*]*http://web.mit.edu/philos/www/jjt-rlc-jc.html MIT faculty resolution on Boolos' death*]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 theanalytical 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 byRaymond Smullyan .**Work**Boolos coauthored with

Richard Jeffrey the first three editions of the classic university text onmathematical logic , "Computability and Logic". The book is now in its fourth edition, the last one updated byJohn P. Burgess .Kurt Gödel wrote the first paper onprovability logic , which appliesmodal logic —the logic of necessity and possibility—to the theory ofmathematical 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 Godel'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 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 bySolomon Feferman .Boolos was an authority on the 19th-century German mathematician and philosopher

Gottlob Frege . Boolos proved a conjecture due toCrispin Wright (and also proved, independently, by others), that the system of Frege's "Grundgesetze", long thought vitiated byRussell's paradox , could be freed of inconsistency by replacing one of its axioms, the notorious Basic Law V withHume's Principle . 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. The result isplural quantification . David Lewis employed plural quantification in his "Parts of Classes" to derive a system in whichZermelo-Fraenkel set theory and thePeano axioms were all theorems. While Boolos is usually credited withplural quantification ,Peter Simons (1982) has argued that the essential idea can be found in the work ofStanislaw 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 andnonfirstorderizability ,plural quantification ,proof theory , and three short insightful papers onGödel's Incompleteness Theorem . There are also papers on Dedekind, Cantor, and Russell.**Books***2002 (1974) (with

Richard Jeffrey ). "Computability and Logic". Cambridge: Cambridge University Press.

*1979. "The Unprovability of Consistency: An Essay inModal Logic ". Cambridge University Press.

*1990 (editor). "Meaning and Method: Essays in Honor ofHilary Putnam ". Cambridge University Press.

*1993. "The Logic of Provability". Cambridge University Press. Not a revision of Boolos (1979).

*1998 (Richard Jeffrey andJohn P. 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": 497-513.1969, "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 andHilary Putnam , eds.,1984. "Philosophy of Mathematics: Selected Readings", 2nd ed. Cambridge Univ. Press: 486-502. LLL1973, "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 andJohn 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, " [

*http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf 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 ofCharles Parsons ". Cambridge University Press. LLL.**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.**External links*** [

*http://web.mit.edu/philos/www/boolos.html George Boolos Memorial Web Site*]

* [*http://www.hcs.harvard.edu/~hrp/issues/1996/Boolos.pdf George Boolos. The hardest logic puzzle ever. The Harvard Review of Philosophy, 6:62–65, 1996.*]

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**George Boolos**— George Stephen Boolos (4 de septiembre de 1940, Nueva York – 27 de mayo de 1996) fue un filósofo y estudioso de lógica matemática que enseñó en el Massachusetts Institute of Technology. Contenido 1 Vida 2 Trabajo 3 Véase también … Wikipedia Español**George Boolos**— (* 4. September 1940 in New York; † 27. Mai 1996 in Cambridge, Massachusetts) war ein US amerikanischer Philosoph und Logiker. Inhaltsverzeichnis 1 Leben 2 Werke von Boolos 3 Einzelna … Deutsch Wikipedia**George Boole**— Este artículo no trata sobre George Boolos, otro matemático lógico. George Boole [buːl], (2 de noviembre de 1815 8 de diciembre de 1864) fue un matemático y filósofo. Como inventor del álgebra de Boole, la base de la aritmética computacional… … Enciclopedia Universal**The Hardest Logic Puzzle Ever**— is a title coined by George Boolos in La Repubblica 1992 under the title L indovinello più difficile del mondo for the following Raymond Smullyan inspired logic puzzle:Boolos provides the following clarifications:George Boolos, The Hardest Logic… … Wikipedia**El acertijo lógico más dificil**— Saltar a navegación, búsqueda Los tres dioses: Verdad, Falso y, Aleatorio El acertijo lógico más difícil del mundo es un título que acuñó George Boolos en La Repubblica 1992 bajo el título L indovinello più difficile del mondo para el siguien … Wikipedia Español**Proof sketch for Gödel's first incompleteness theorem**— This article gives a sketch of a proof of Gödel s first incompleteness theorem. This theorem applies to any formal theory that satisfies certain technical hypotheses which are discussed as needed during the sketch. We will assume for the… … Wikipedia**Gödel's incompleteness theorems**— In mathematical logic, Gödel s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of… … Wikipedia**Second-order logic**— In logic and mathematics second order logic is an extension of first order logic, which itself is an extension of propositional logic.[1] Second order logic is in turn extended by higher order logic and type theory. First order logic uses only… … Wikipedia**Scott–Potter set theory**— An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician… … Wikipedia**L'Énigme la plus difficile du monde**— (en italien indovinello più difficile del mondo) est à l origine le titre d un article publié par George Boolos (en), philosophe et logicien américain, dans le quotidien La Repubblica. Cette énigme lui a été inspiré par Raymond Smullyan.… … Wikipédia en Français