# [FOM] Plural quantification and set theory

Tom McKay tjmckay at syr.edu
Sun Sep 14 14:31:52 EDT 2003

This responds to Dean Buckner's email of September 13.

Buckner writes:
>William Tait writes ("Non-distributive predication" 14 July), concerning Tom
>Mckay's system of plural quantification
>
>>  there is a more or less obvious translation of second-order
>>  predicate logic (where the second order variables range over
>subsets of the domain D of individuals) into [McKay's system]
>>  and conversely---the pluralities corresponding to non-empty
>>  sets.
>
>>But for this reason, how can [McKay's] theory have any foundational
>>significance? One understands your notion of plurality over a
>domain D exactly to the extent that we understand the notion of a
>>  non-empty subset of D.  And any contradiction arising for the one
>>  notion immediately translates into a contradiction for the other.
>
>>So why not, when we are just dealing with a domain and the
>>  subsets/pluralities over it, simply opt for the former, since it is
>>  most familiar and would involve no revision of our customary
>>  mathematical teminology?
>
>I agree with this, if there such a translation, but I have a further point.

I already responded to this on July 14. Maybe you didn't find my
response convincing. Why not?
Here is a little expansion.
In order to generate set-theoretic paradoxes, you need to have a
membership relation where one thing is a member of another thing that
itself can be a member of something. My system doesn't have any such
membership relation.
I have the 'x is one of Y' relation and the 'X are among Y' relation.
(I regard the first as just a special case of the second, where X is
assigned just one individual rather than more than one.) The upper
case 'Y' can be associated (non-distributively) with several
individuals; it is not automatically (or even ordinarily) associated
with just one.

My question was how much of mathematics could be developed within
such a language and without sets. It seems to me that the results
will be similar to those for full second-order arithmetic, but I
wondered (and still wonder) if any readers of the fom list see
reasons to doubt that.

Buckner:
>As well as there being no correlates in plural theory of empty set and
>singleton set, I'm not sure there is any correlate of an infinite set.
>Consider the function f such that
>
>     f(1) = the numbers 0 & 1
>     f(2) = the numbers 0, 1 & 2
>     f(3) = the numbers 0, 1, 2 & 3
>     f(n) = the (consecutive) numbers 0 - n
>
>The function maps natural numbers onto the plural entities defined in
>McKay's system.

I do not have any plural entities in my system. Instead I am revising
the logic to include non-distributive plural predication. A function
like the one you describe would be (defined in terms of) a relation
that is non-distributive in its second argument place. I would not
regard the numbers 0, 1 and 2 as any entity. They are three entities.

Buckner:
>   Now consider
>
>             N = { n: (E s) s = f(n) }
>
>N is the set containing every finite number, and nothing else, since to
>every finite number n there corresponds some consecutive sequence of numbers
>from 0 to n.  But the existence of N is consistent with there being no
>"infinite pluralities", i.e. without there being anything but what is
>referred to by "the numbers from 0 to
>n" for finite n, and so nothing referred to by "all the numbers in N".

I don't have sets, so the relationship of this to my approach is not
entirely clear.
In any case, though, the fact that the existence of N is consistent
with there being no infinite pluralities (does this mean "consistent
with there being only finitely many things"?) does not mean that it
is not also consistent with there being infinitely many things. But I
thought that you were trying to show that my system does not allow
the possibility of there being infinitely many things. (You said,
"I'm not sure there is any correlate of an infinite set.") My plural
logic certainly allows the possibility that there are only finitely
many things, but I don't see that as a problem. It also allows the
possibility of there being infinitely many things. I don't see that
as a problem either.

For the purposes of mathematics, I take the natural numbers as given
(as second-order arithmetic does), so I start out with an infinite
set.

Buckner:
>This results in the apparent paradox that the set N contains every natural
>number, but there are no things corresponding to "the natural numbers".  But
>it's only apparent: it's paradoxical only for those who insist on reading
>
>     {x: x is F}
>
>as "the things that are F" or something like that.  "The natural numbers" is
>not an expression that belongs to set theory.  It's an English definite
>description, and all such descriptions refer to the things that satisfy the
>description, unlike the corresponding sets.

I regard '{x: x is F}' as referring to a set. Accordingly I would not
use that expression in this context.
I agree that "The natural numbers" plurally (and non-distributively)
refers to the natural numbers and not to a set. (They are many in
number and the set containing them is not. It is just one thing, but
with many members.)

Buckner:
>This underscores the point made by Martin Davis, and also William Tait (in
>the same posting further down) that set theory represents a language in its
>own right.  We run into paradox only when we try to interpret or "translate"
>set theory in ordinary language.

I agree with William Tait that the language of set theory is an
important part of ordinary language. My question for FOM is about
what can be done in mathematics without the set-theoretic membership
relation if we allow plural, non-distributive predication.

(I couldn't find Tait making the last claim you attribute to him,

What is the reference for Martin Davis's remarks that you refer to?

Tom McKay
--
Tom McKay
Philosophy Department
Syracuse University
Syracuse  NY  13244
315 443 4501
tjmckay at syr.edu