FOM: second-order logic is a myth

Charles Silver csilver at
Mon Mar 1 06:55:20 EST 1999

On Sun, 28 Feb 1999, Stephen G Simpson wrote:

> Today I spent a little more time with Shapiro's book `Foundations
> Without Foundationalism: A Case for Second-order Logic'.  
> Although I still think this book is completely misguided, I have to
> give Shapiro credit for presenting a thoughtful account.  Contrary to
> my earlier impression, I'll now say that this book is *much* better
> than Hersh's.  I now regard Shapiro's `anti-foundationalism' as little
> more than an attempt to be trendy.  I don't assign any serious weight
> to it.

	I have to say that I personally liked Shapiro's book, but I wasn't
sure what to make of many of his arguments.  There's an interesting review
of it by Craig Smorynski in "Modern Logic," vol. 4, no 3, pp. 341-344.
Smorynski makes the following distinctions (p. 341): "Mathematics (M),
Foundations of Mathematics (FM), Mathematical Logic (ML), Philosophical
Logic (PL), Philosophy of Mathematics as practiced by Philosophers (PMP),
Philosophy of Logic (PoL), Foundations of Logic (FL), and Epistemology
(E)."  Smorynski goes on to say: 

   "While there are relations among these subjects (e.g., PMP, E),
   no two of them coincide (except possibly PoL and FL), and what 
   may be inadequate for one may be perfectly adequate for another.
   What I never understood in the book is which one of these 
   perspectives is operational.  Is first-order logic inadequate 
   for M, FM, ML, PL, PMP,... or what?"

	What do people think of Smorynski's criticism of Shapiro's book? 
I imagine that there must be many people on this list who think highly of
the book.  I'd be interested in seeing more discussion of the value of
this work.

Charlie Silver

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