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Foundations without Foundationalism: A Case for Second-order Logic (Oxford Logic Guides) Paperback – May 18, 2000

ISBN-13: 978-0198250296 ISBN-10: 0198250290

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Product Details

  • Series: Oxford Logic Guides
  • Paperback: 304 pages
  • Publisher: Oxford University Press (May 18, 2000)
  • Language: English
  • ISBN-10: 0198250290
  • ISBN-13: 978-0198250296
  • Product Dimensions: 0.6 x 6.1 x 9 inches
  • Shipping Weight: 14.4 ounces (View shipping rates and policies)
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,322,872 in Books (See Top 100 in Books)

Editorial Reviews

Review


"Contains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom." --Journal of Symbolic Logic


"The most comprehensive all-round account and defense of second-order logic as a vehicle for mathematics known to the reviewer. It is also very comprehensively documented with a large bibliography and a wealth of interesting material in the numerous footnotes." --Mathematical Reviews


"An excellent book, covering all of the main results in second-order logic and its applications to mathematical theories. There is a great wealth of material about second-order logic in this book. . . . it is really an excellent book, and . . . the author is to be commended for a job well done." --Notre Dame Journal of Formal Logic


About the Author

Stewart Shapiro is Professor of Philosophy at Ohio State University.

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Most Helpful Customer Reviews

Format: Paperback
There has been some criticism directed at this book based on a perceived difficulty in actually identifying the arguments here. That criticism is to a certain extent justifiable, but I am still happy to have read and studied it. First of all, it gathers together and discusses the ramifications of a huge range of important results in higher-order logics and mathematics (even if one sometimes misses the details, i.e. proofs). But secondly Shapiro also presents several important considerations directed against the opposition to viewing second-order logic as a genuine branch of logic - and if these considerations often take the form of laying out what's at issue and spelling out the various possible considerations in favor of either side rather than genuine arguments in favor of one of them, I cannot really object to that. And by doing that, one realizes that many of the considerations seem rather baseless, insofar as a foundationalist approach is generally eschewed anyway. It is, for example, slightly puzzling why Quine, who was adamantly opposed to foundationalism and argued against drawing sharp boundaries between various branches of science would use the claim that higher-order logics is "set-theory in disguise" as a charge against its logicality (of course, his definition of the ontology of a theory as the range of its bound variables is the hidden agenda here), and Shapiro nicely circumvents the charge that second-order logic entails a staggering ontology (set theory does, of course, and second-order logic is able to express that commitment (and thus might require it in its meta-theory), but this is a different matter).

Still, several questions are left unanswered.
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16 of 26 people found the following review helpful By J. Porter on September 3, 2005
Format: Paperback
I have very mixed feelings about this book. It's quite a good introduction to second order logic and makes a convincing case for why second order logic is not only natural but necessary.

THe down-side, though, is that the author is obsessed with mathematics: he seems to think that the sole function of logic is to provide a solid foundation for mathematics, and thus obsesses over things like set theories, ordinals, etc. The philosophical implications of, say, incompleteness simply pass him by. I would have preferred it very much if the sections on philosophy had not, as they did, covered only mathematical issues, but wider issues in analytic philosophy. For example, Quine's complaint that second order logic involves a 'staggering ontology' and hence cannot be acceptable is merely shrugged off with rather glib (and not entirely correct) phrases, rather than stared straight in the face.

A minor quibble also: throughout the book relies on two concepts of set, a standard ZF-style set and a 'logical' set theory. The preface admits that this distinction is incorrect. In that case why wasn't the book re-written?
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