Boolos, George
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George Stephen Boolos (; September 4, 1940 – May 27, 1996) was an
American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, p ...
philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
and a
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
ian who taught at the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
.


Life

Boolos was of
Greek Greek may refer to: 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 of all kno ...
-
Jewish Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
descent. He graduated with an A.B. in
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
from
Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
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, Smullyan's first career choice was in stage magic. He ...
.
Oxford University The University of Oxford is a collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the second-oldest continuously operating u ...
awarded him the
B.Phil. Bachelor of Philosophy (BPhil, BPh, or PhB; or or ) is the title of an academic degree in philosophy that usually involves considerable research, either through a thesis or supervised research projects. Unlike many other bachelor's degrees, the ...
in 1963. In 1966, he obtained the first
PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...
in
philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
ever awarded by the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
, under the direction of
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
. After teaching three years at
Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...
, 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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
asked him, "And tell us, Mr. Boolos, what does the
analytical hierarchy Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemica ...
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 Newspaper#Daily, daily Newspaper#National, national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its modern name on 1 January 1788. ''The Times'' and its si ...
''
crossword A crossword (or crossword puzzle) is a word game consisting of a grid of black and white squares, into which solvers enter words or phrases ("entries") crossing each other horizontally ("across") and vertically ("down") according to a set of cl ...
competition. His score was one of the highest ever recorded by an American. He wrote a paper on "
The Hardest Logic Puzzle Ever The Hardest Logic Puzzle Ever is a logic puzzle so called by American philosopher and logician George Boolos and published in '' The Harvard Review of Philosophy'' in 1996. Boolos' article includes multiple ways of solving the problem. A translatio ...
"—one of many puzzles created by
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, Smullyan's first career choice was in stage magic. He ...
. Boolos died of
pancreatic cancer Pancreatic cancer arises when cell (biology), cells in the pancreas, a glandular organ behind the stomach, begin to multiply out of control and form a Neoplasm, mass. These cancerous cells have the malignant, ability to invade other parts of ...
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
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, ''Computability and Logic''. The book is now in its fifth edition, the last two editions updated by John P. Burgess.
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
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 Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
—the logic of necessity and possibility—to the theory of
mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
, 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, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
. 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 philos ...
. 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, ske ...
(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 published 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, given two collections of objects \mathcal F and \mathcal G with properties F and G respectively, the number of objects with property F is equal to the number of objects with property G if and only 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 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 Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
to entities other than those over which the first-order variables range. The result is
plural quantification In mathematics and mathematical logic, logic, plural quantification is the theory that an individual Variable (mathematics), variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, ...
. 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 suc ...
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 nea ...
were all theorems. While Boolos is usually credited with
plural quantification In mathematics and mathematical logic, logic, plural quantification is the theory that an individual Variable (mathematics), variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, ...
, Peter Simons (1982) has argued that the essential idea can be found in the work of
Stanislaw Leśniewski Stanislav and variants may refer to: People *Stanislav (given name), a Slavic given name with many spelling variations (Stanislaus, Stanislas, Stanisław, etc.) Places * Stanislav, Kherson Oblast, a coastal village in Ukraine * Stanislaus County, ...
. 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 Set (mathematics), 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 mathema ...
,
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 ...
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 In mathematics and mathematical logic, logic, plural quantification is the theory that an individual Variable (mathematics), variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, ...
,
proof theory Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
, and three short insightful papers on Gödel's Incompleteness Theorem. There are also papers on
Dedekind Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
,
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. ...
, and Russell.


Publications


Books

*1979. ''The Unprovability of Consistency: An Essay in
Modal Logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
''. Cambridge University Press. *1990 (editor). ''Meaning and Method: Essays in Honor of
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
''. Cambridge University Press. *1993
''The Logic of Provability''
Cambridge University Press. *1998 (
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 ...
and John P. Burgess, eds.). ''Logic, Logic, and Logic'' Harvard University Press. *200
(1974)
(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 ...
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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
), "Degrees of unsolvability of constructible sets of integers," ''Journal of Symbolic Logic 33'': 497–513. * 1969, "Effectiveness and natural languages" in
Sidney Hook Sidney Hook (December 20, 1902 – July 12, 1989) was an American philosopher of pragmatism known for his contributions to the philosophy of history, the philosophy of education, political theory, and ethics. After embracing communism in his youth ...
, 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 In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order ...
," ''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 (; 26 March 1930 – 13 January 2025) was 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 ...
and
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
, 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 to ...
'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 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 ...
," ''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 Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
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 Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
," ''Journal of Philosophical Logic 9'': 1–7. * 1980d, Review of
Raymond M. 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, Smullyan's first career choice was in stage magic. He e ...
, ''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 mathematical proof, 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), \dots  all hold. This is done by first proving a ...
," ''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-writing") 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 notati ...
," ''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 He was born in Indepen ...
and
John Etchemendy John W. Etchemendy (born 1952) is an American logician and philosopher who served as Stanford University's twelfth Provost (education), Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017 ...
, ''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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
''. 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 philos ...
's theorem and the Peano postulates," ''Bulletin of Symbolic Logic 1'': 317–326. LLL. * 1995a, "Introductory note to *1951" in
Solomon Feferman Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
et al., eds., ''
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
, 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 The Hardest Logic Puzzle Ever is a logic puzzle so called by American philosopher and logician George Boolos and published in '' The Harvard Review of Philosophy'' in 1996. Boolos' article includes multiple ways of solving the problem. A translatio ...
," ''
Harvard Review of Philosophy ''The Harvard Review of Philosophy'' is an annual peer-reviewed academic journal of philosophy edited by a student collective at Harvard University.Scanlon, Thomas (2002) "Foreword" ''In'' Upham, S. Phineas, ''Philosophers in Conversation: Intervi ...
'' 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 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 philos ...
's theorem" in A. Morton and S. P. Stich, eds., ''
Paul Benacerraf Paul Joseph Salomon Benacerraf (; 26 March 1930 – 13 January 2025) was 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 ...
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." H ...
''. 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 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 philos ...
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 Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
?" in Gila Sher and Richard Tieszen, eds., ''Between Logic and Intuition: Essays in Honour of Charles Parsons''. Cambridge University Press. LLL.


See also

*
American philosophy American philosophy is the activity, corpus, and tradition of philosophers affiliated with the United States. The ''Internet Encyclopedia of Philosophy'' notes that while it lacks a "core of defining features, American Philosophy can neverthe ...
* Axiomatic set theory S of Boolos (1989) *
General set theory General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. ...
, Boolos's axiomatic set theory just adequate for
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stan ...
and
Robinson arithmetic In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical inducti ...
. *
List of American philosophers American philosophy is the activity, corpus, and tradition of philosophers affiliated with the United States. The ''Internet Encyclopedia of Philosophy'' notes that while it lacks a "core of defining features, American Philosophy can neverthe ...


Notes


References

* Peter Simons (1982) "On understanding Lesniewski," ''History and Philosophy of Logic''. *
Solomon Feferman Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
(1960) "Arithmetization of metamathematics in a general setting," ''Fundamentae Mathematica'' vol. 49, pp. 35–92.


External links


George Boolos Memorial Web Site
* {{DEFAULTSORT:Boolos, George 1940 births 1996 deaths 20th-century American educators 20th-century American essayists 20th-century American historians 20th-century American Jews 20th-century American male writers 20th-century American mathematicians 20th-century American philosophers Alumni of the University of Oxford American historians of mathematics American historians of philosophy American male essayists American male non-fiction writers Analytic philosophers Columbia University faculty Deaths from pancreatic cancer in Massachusetts American game theorists History of logic History of mathematics Jewish American non-fiction writers Jewish American historians Jewish philosophers Mathematical logicians MIT School of Humanities, Arts, and Social Sciences faculty American philosophers of logic American philosophers of mathematics American philosophy writers Princeton University alumni Puzzle designers Set theorists American people of Greek-Jewish descent