# [FOM] Non-distributive predication

Tom McKay tjmckay at syr.edu
Mon Jul 14 09:20:56 EDT 2003

```On Sunday, July 13, 2003, William Tait responded with a fine summary
of the formal system that I had sent in for consideration on Friday.
He continues

>As you note, 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 your system, and
>conversely---the pluralities corresponding to non-empty sets.
>
>But for this reason, how can your 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.

The set-theoretic paradoxes generated by naive set theory aren't
generated if we just talk plurally about the things.

If we have only first-level sets (which, if I understand it
correctly, is what we have in second-order arithmetic), then there is
a certain kind of translatabity. However, when we translate to the
set-theoretic language we change the subject (to talk about a single
thing) and change the predicate (to a distributive predicate). If I
say:
Some students are surrounding the building.
They are from many different countries.
How do you translate that into set-theoretic statements while
preserving the same predicates? I think that you can't. These involve
non-distributive predication.

Now that might be a point that has more to do with philosophy of
language than mathematics. But my question about the foundations of
mathematics is what can be done if we add non-distributive
predication (and plural quantification) to logic and do not talk
about sets (as individuals). In other words, where do we really need
the sets in order to do mathematics? It is not that I think that we
don't need them. I would like to have a better sense of where sets
become important. (My mathematics is pretty limited, so I might not
understand the answer. But I thought that I would try.)

>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?

In order to see where the customary terminology becomes important.

>Of course, in set theory, we are often interested in going from D,
>not only to the domain D* of subsets of D,

Of course, relying solely on non-distributive plurals, I wouldn't
even go to the domain D* of subsets of D. When I say that some
students surrounded the building, I am talking about the students,
not about a set. (Sets don't surround buildings.)

>  but to the domain D** of subsets of D*, to D***, etc. (possibly
>transfinite iterations even). This iteration of the operation * of
>taking subsets of is unavailable in the ideology of pluralities,
>since the analogues of D**,  D***, etc., just collapse to the
>analogue of D*. But why shouldn't we then just say: so much for
>pluralities?

Of course we won't get set theory out of the system I was proposing.
That is a part of mathematics that definitely needs sets.

>I should also like to comment on Dean Buckner's remark ( Sat Jul 12,
>2003  9:13:26  AM US/Central) in reply to Tom McKay's posting
>
>>  be warned it's best to avoid words like "ordinary language" in
>>FOM.
>
>There has been a lot of talk on the list about ordinary or natural
>ordinary language in mathematics, and indeed was so in the 1930's,
>when Wittgenstein was treating it as `language on holiday' .
>Whatever the issues are concerning the cumulative hierarchy of sets,
>obtained as we described above by iterating the operation of `set
>of', this latter notion, of passing from D to D*,  itself was well
>established in many areas of mathematics by then.

I absolutely agree with this. Ordinary language also has another way
of talking about several things that is not available within
traditional first-order logic: non-distributive predication. When you
use the more limited (traditional) first-order logic that uses only
distributive predication, sets become very important immediately. I
want to know where they become important if you allow
non-distributive predication.

>Often to make sense of what I have read on the list about `ordinary'
>or  `natural language', I have had to understand by it the
>restricted language of those who have not learned the language of
>mathematics. This is a `natural' language in its own right, which
>(as in the case of any language) one has to learn by learning how
>its terms are used: it does not translate via dictionary definitions
>into non-mathematical language. This is ultimately the reason that
>the demand for formulating mathematics in ordinary or natural
>language (in the narrow sense) cannot be met.
>
>So writing about and in ordinary language on the list is just fine:
>but first learn the relevant (ordinary) language.
>
But I take it that it might be foundationally interesting to know
what difference it would make to expand the expressive resources of
the base logic.

As I suggested, it seems to me that what is expressed within in
second-order arithmetic may be what can be expressed and proved with
my proposed system (with non-distributive predication and no
reference to sets). I wanted to know if people regard this as
obviously true, obviously false, full of potential problems, probably
true but not quite obvious, or what.

system again in a way that might be more readable for some fom