FOM: second-order logic is a myth

Robert Black Robert.Black at
Tue Mar 9 17:00:22 EST 1999

Charles Silver:

>	To me, the major point that separates (intuitive) quantification
>from set theory is that you can say "for all x..." without implying that
>the x's must be *in* something.  For example, take: "All Canada geese fly
>south for the winter."  I don't think anything in this statement implies
>that in addition to there being some number of geese there is also a *set*
>of geese.  Does anyone know of a theory with much of the power of set
>theory that doesn't imply the existence of some sort of container?

I'm not persuaded that 'container' is a useful metaphor here: does 'Some
critics admire only one another' imply some sort of container?  But there
is of course something with the power not of set theory but of second-order
logic that, oddly, no-one (including me) has mentioned, namely unrestricted
mereology, as used for example by Hartry Field in _Science Without
Numbers_.  I think the reason I haven't mentioned it is that I'm sort of
assuming that resistence to second-order logic comes from what Boolos calls
its 'staggering' undecidability.  And this staggering undecidability will
hold for unrestricted mereology as well.  But of course if the problem is
commitment to sets qua abstract objects (and assuming Boolos is wrong and
second-order logic does commit us to sets), then mereology deserves a
look-in here.

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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