George Boolos

Life

Boolos was of Greek-Jewish descent. 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. 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.

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.

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.

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 (Richard Jeffrey and John P. Burgess, eds.). Logic, Logic, and Logic Harvard University Press. ISBN 978-0674537675

2007 (1974) (with Richard Jeffrey 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 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.