[Phil-logic] Geach Kaplan (Schema N)

[Dean Buckner]

Dennis:
> I'm still wondering where you [Dean & Tom] are trying to get to.

Tom originally posted to FOM, asking for "comments from fom readers about
the significance for the foundations of mathematics of allowing
non-distributive predication in logic."  Here again is the original posting
(http://www.cs.nyu.edu/pipermail/fom/2003-July/007068.html)
which is a summary of chapter 4 (available on his department's website
(http://philosophy.syr.edu/ ). (Click on "Faculty & Staff" and on his name.)

He commented at the end "The question of how much of mathematics can be
grounded in among theory, or, more generally, whether and where among theory
can be a useful tool in place of set theory, needs further exploration."

Which neatly summarises his (and my) interest.  Unfortunately there was not
a huge interest among FOM readership, except for me of course, and William
Tait, but I still feel Tait did not uderstand exactly what was going on , or
its full potential of MckAy's ideas.  For example Tait writes
(http://www.cs.nyu.edu/pipermail/fom/2003-July/007083.html)"I think you want
also an axiom >exists X exists Y not-XAY > to ensure that all pluralities
are non-empty.", which is a clear misunderstanding of what is going on.  The
notion of a "empty plurality" is as mistaken as the idea of a "non-existing
thing" - you wouldn't have an axiom to ensure that every x in your system
was existing, for example.

As to the potential of this - here is one connection.  There are certain
statements involving number, that (famously) can be analysed without
invoking "second order" entities, or without apparently requiring
"quantification over natural numbers".  Thus, for example

Jake said that there were exactly two people at the door

can be analysed as

Jake said that EyEz(~y=z & Fy & Fz & (w)(Fw -> (w=y v w=z)))

Any statement "of this form" can be written so that the number constants are
eliminated, being replaced by exactly that number of quantifiers and
variables.  The difficulty is as implied by my scare quotes around "of this
form".  Suppose I report what Jake said above, but without specifying
exactly how many people Jake said there were.

For some n, Jake said that there were exactly n people at the door

This seems to be true.  Jake said that there were 2 people at the door, so
for some n, namely n=2, Jake did say that there were exactly n people at the
door.  Now the problem should be obvious.  We can't analyse the second
statement without "quantifying over numbers".  We have to invoke mysterious
Platonic entities in order to say what Jake said.  But the paradox is that
no such entities are invoked by the first report of what he said.  Even more
paradoxical, the second report is more general than the first.  It leaves
out a bit of information.  It correctly report Jake's statement that there
were a certain number people at the door, but it omits the information about
exactly how many he said there were.  It appears to contain less.  But then
it also appears to contain more, the mysterious Platonic entities.

Thre's a (possibly) similar problem discussed by Neil Tennant here

http://www.cs.nyu.edu/pipermail/fom/2003-June/006806.html

Tennant proposed (in his 1987 book Anti-Realism and Logic ---- BUY THIS) an
adequacy condition on a theory of number, using what he called Schema N:

Schema N #xF(x) = _n_ iff there are exactly n Fs

On the right-hand side of Schema N, the condition "there are exactly n Fs"
can be spelled out explicitly (for any given choice of n) without any
reference to, or quantification over, natural numbers (unless, of course, F
itself were a numerical predicate, or had at least one numerical argument).
So, one instance of Schema N is

 #xF(x) = ss0 iff EyEz(-y=z & Fy & Fz & (w)(Fw -> (w=y v w=z)))

for any given n, that is.  The difficulty (as far as I can see) is that we
cannot generalise this statement.  Explaining "n" on the r.h. side, when a
variable, appears to require quantification over the same objects as
explcitly invoked by the l.h.

In case I have misunderstood the problem, I have invited Neil to comment.

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Dean Buckner,    : There is no path to infinity
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