The Logic of Provability by George S. Boolos (PDF)

Description

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author’s earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the ‘self-referential’ sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As the other reviewers note, this book is remarkable in that it both offers a reasonably gentle introduction to the subject even for those without too much mathematical background (logically-inclined philosophy students come to mind), and at the same time remains to this day an indispensable reference for the specialist—this for example is the only source where you can find the proofs of Solovay’s “other” modal completeness results for transitive models of set theory. George Boolos was (is) famous for, among other things, his gift of didactically brilliant exposition.There is only one other book on the subject—C.Smory’nski’s “Self-Reference and Modal Logic”, now out-of-print, which provides a nice complement to Boolos with surprisingly small overlap. For an introduction, I would not hesitate to recommend Boolos over Smory’nski.Since 1993 the “baton” has not completely “fallen out of Boolos’ hands”—the interested reader can follow up on this via e.g. a couple of handbook-chapter expositions, easily found by googling around, that point to current research literature. However, no further textbook-level exposition appears to be forthcoming anytime soon.As a comment to galloamericanus’ review, I would note that Boolos does not restrict his monograph exclusively or even primarily to his own results. The reader gets a fair overview of the research field by 1993 as shaped by many contributors.

⭐George Boolos regrettably died too early, at age 56. He was a witty engaging writer, and a brilliant logician. This book,completed not long before his untimely death, is probablythe high point of his career. The key insight, that modal logiccan be interpreted as a sort of metamathematical algebra,is Godel’s from the 1930s. And a key result is due to MartinLob about 50 years ago. But Boolos built a towering edificeon this slim material. I think that the provability interpretationof the modal operator is the most philosophically and mathematically satisfying of all, and I hope someone comesalong to pick up the baton that has fallen from Boolos’s hand.

⭐この本のはじめの方で、多くの本では「これらの議論を形式化すれば、～が出てきて、第二不完全先生定理が帰結する」というようになっている部分を、丁寧に追っている。可証性述語と様相論理の関係というのがこの本の一番のテーマであるから、こうした一部だけを取り上げてレビューするのは適当ではないかもしれないが、この点だけでも貴重な一冊なので挙げてみた。田中一之さんの本でも可導性性条件の証明は多少書いてあるが、やはりある程度丁寧に追った本を探していたので、この本はぴったりだった。特に、不完全性定理の議論を初めから形式化する方法が載っている本は少ない（と思う）のでかなり貴重。このやり方だと、第一不完全性定理を証明するのが通常より大変になるが、その代わり第二不完全性定理が自然に出てくる。ただし、Boolosは（論文でもその傾向があるように思われるが）本の書き方として、それほど分りやすくない点があり、それで四つ星にした。

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