George Boolos - Work

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 Lesniewski.

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.