George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

ian who taught at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

Life

Boolos is of

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

Jewish Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...

descent. He graduated with an

A.B. Bachelor of arts (BA or AB; from the Latin ', ', or ') is a bachelor's degree awarded for an undergraduate program in the arts, or, in some cases, other disciplines. A Bachelor of Arts degree course is generally completed in three or four yea ...

in mathematics from

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

after completing a senior thesis, titled "A simple proof of Gödel's first incompleteness theorem", under the supervision of

Raymond Smullyan Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher. Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc from ...

Oxford University Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to th ...

awarded him the B.Phil. in 1963. In 1966, he obtained the first PhD in philosophy ever awarded by the

, under the direction of Hilary Putnam. After teaching three years at

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

, 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 In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers ...

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 ''The Times'' is a British daily national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its current name on 1 January 1788. ''The Times'' and its sister paper '' The Sunday Times'' (f ...

crossword A crossword is a word puzzle that usually takes the form of a square or a rectangular grid of white- and black-shaded squares. The goal is to fill the white squares with letters, forming words or phrases, by solving clues which lead to the ans ...

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

. Boolos died of pancreatic cancer on 27 May 1996.

Work

Boolos coauthored with

Richard Jeffrey Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of ...

the first three editions of the classic university text on

, ''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 Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples ...

, which applies modal logic—the logic of necessity and possibility—to the theory of

mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...

, 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 Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to th ...

. Boolos was an authority on the 19th-century German mathematician and philosopher

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...

. Boolos proved a conjecture due to

Crispin Wright Crispin James Garth Wright (; born 21 December 1942) is a British philosopher, who has written on neo-Fregean (neo-logicist) philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skep ...

(and also proved, independently, by others), that the system of Frege's ''Grundgesetze'', long thought vitiated by

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...

, could be freed of inconsistency by replacing one of its axioms, the notorious Basic Law V with

Hume's Principle Hume's principle or HP says that the number of ''F''s is equal to the number of ''G''s if and only if there is a one-to-one correspondence (a bijection) between the ''F''s and the ''G''s. HP can be stated formally in systems of second-order logic. ...

. 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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...

plurally, then second-order logic can be interpreted as having no

ontological commitment An ontological commitment of a language is one or more objects postulated to exist by that language. The 'existence' referred to need not be 'real', but exist only in a universe of discourse. As an example, legal systems use vocabulary referring t ...

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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...

and the

Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

and

nonfirstorderizability In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic whic ...

, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem. There are also papers on Dedekind,

Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...

, and Russell.

Publications

Books

*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. *1998 (

and John P. Burgess, eds.). ''Logic, Logic, and Logic'' Harvard University Press. *200
(1974)
(with

and John P. Burgess). ''Computability and Logic'', 4th ed. Cambridge University Press.

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 Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He ...

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 Evert Willem Beth (7 July 1908 – 12 April 1964) was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics. He was a member of the Significs Group. Biography Beth was born in Almelo, a small ...

'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

," ''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 Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

," ''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 ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notatio ...

," ''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 Kenneth Jon Barwise (; June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. Education and career Born in Independence, ...

and

John Etchemendy John W. Etchemendy (born 1952 in Reno, Nevada) is an American logician and philosopher who served as Stanford University's twelfth Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017. E ...

, ''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 Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...

's theorem and the Peano postulates," ''Bulletin of Symbolic Logic 1'': 317–326. LLL. * 1995a, "Introductory note to *1951" in

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

's theorem" in A. Morton and S. P. Stich, eds., ''

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 Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He w ...

''. 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, "

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

?" in Gila Sher and Richard Tieszen, eds., ''Between Logic and Intuition: Essays in Honour of Charles Parsons''. Cambridge University Press. LLL.

Notes

References

* Peter Simons (1982) "On understanding Lesniewski," ''History and Philosophy of Logic''. *

(1960) "Arithmetization of metamathematics in a general setting," ''Fundamentae Mathematica'' vol. 49, pp. 35–92.

External links

George Boolos Memorial Web SiteGeorge Boolos. The hardest logic puzzle ever. The Harvard Review of Philosophy, 6:62–65, 1996.
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