Foundations without Foundationalism: A Case for Second-order Logic [Paperback]

Stewart Shapiro (Author)

Book Description

Publication Date: May 18, 2000 | ISBN-10: 0198250290 | ISBN-13: 978-0198250296

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today.

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.

Product Details

• Paperback: 304 pages
• Publisher: Oxford University Press, USA (May 18, 2000)
• Language: English
• ISBN-10: 0198250290
• ISBN-13: 978-0198250296
• Product Dimensions: 9 x 6.1 x 0.7 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: #1,666,209 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

4 of 4 people found the following review helpful:

4.0 out of 5 stars Excellent survey of the scope and limits of second-order logic - but don't expect a fully spelled-out argument, February 14, 2009

By 

G.D. (Norway) - See all my reviews
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This review is from: Foundations without Foundationalism: A Case for Second-order Logic (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. The independence of the axiom of choice and continuum hypothesis counts against Shapiro's claim that these notions are `clearly understood' (cardinalities the size of the continuum and larger are easily expressed in higher-order logics). Secondly, I miss a detailed discussion of Boolos' plural quantification approach. And there are indeed some points where I am left nonplussed (e.g. where is the discussion of Parsons really headed?). Furthermore, it is sometimes unclear whether he is advocating a "let a thousand flowers bloom" approach or making an outright attack on first-order logic (though I take the purpose generally to be the former, and Shapiro does at least suggest that his view is that there is no one correct logic, but rather that the appropriateness of using a particular logic is relative to the purpose (and where the purpose of second-order logic is foremostly to codify mathematical practice) - but this view is certainly controversial, even among anti-foundationalists, and could need arguments that are clearer than the ones Shapiro supplies). The historical part is also interesting in itself, but I do not see what it contributes to the overall discussion. But maybe the most important point in Shapiro's favor is his ability to present the often quite technical material in an eminently accessible manner. I do not, then, regret reading this book, and I would recommend it to anyone interested in the issues, although maybe more as a presentation of those issues than a cogent argument for any specific position.

13 of 21 people found the following review helpful:

3.0 out of 5 stars Good, but could be much better, September 3, 2005

By 

J. Porter "Julian Porter" (UK) - See all my reviews
(REAL NAME)   

This review is from: Foundations without Foundationalism: A Case for Second-order Logic (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?