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Foundations without Foundationalism: A Case for Second-order Logic

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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.

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?